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The connection between mathematics and art goes back thousands of years. Mathematics has been used in the design of Gothic cathedrals, Rose windows, oriental rugs, mosaics and tilings. Geometric forms were fundamental to the cubists and many abstract expressionists, and award-winning sculptors have used topology as the basis for their pieces. Dutch artist M.C. Escher represented infinity, Möbius bands, tessellations, deformations, reflections, Platonic solids, spirals, symmetry, and the hyperbolic plane in his works.
Mathematicians and artists continue to create stunning works in all media and to explore the visualization of mathematics--origami, computer-generated landscapes, tesselations, fractals, anamorphic art, and more.
Jump to one of the galleries
- 2011 Mathematical Art Exhibition
- Edmund Harriss:: Shapes and Tilings
- 2010 Mathematical Art Exhibition
- Simulated Snowflakes
- Crocheted Lorenz Manifolds
- 2009 Mathematical Art Exhibition
- Jean-Francois Colonna :: A Gateway Between Art and Science
- Gwen L. Fisher :: Woven Beads
- Dejenie A. Lakew :: Hyper Symmetries
- Nathan Selikoff :: Algorithmic Artwork
- Chaim Goodman-Strauss :: Symmetries
- Robert J. Lang :: Origami
- Carlo Séquin :: Mathematical Images
- Anne M. Burns :: Gallery of "Mathscapes"
- George Hart :: Geometric Sculptures
- Fractal Art :: Beauty and Mathematics
- Seifert Surfaces
- Robert Straight :: Toroids and Plaids
- Quilts
- Mike Field :: Realizations
- Bradford Hansen-Smith :: Wholemovement
- Knots
- 3D-XplorMath
- Thomas Hull :: The mathematics of origami
- Notices of the American Mathematical Society :: Cover Art
Mathematics Genealogy Project:
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Source file for Index to OEIS
The alphabetical sections of the Index are created "on the fly" from the following plain text file.
This file is useful if you want to search the whole Index, or suggest additions or changes.
- jeep problem | abelian numbers | absolute primes | abundant numbers | Ackermann function | addition chains | additive bases | Aho-Sloane paper | Airey's converging factor | Aitken's array | Alcuin's sequence | Alekseyev's problem | algebras, Jordan | aliquot trajectories for certain initial values | almost primes | almost-natural numbers | alphabetical order, numbers in | alternating bit sets | alternating bit sum | alternating permutations | alternating sign matrices | amicable numbers | Andrews-Mills-Robbins-Rumsey numbers | Apery numbers | Apery's number zeta(3) | Apocalyptic powers | Apollonian ball packings | Apollonian circle packings | ApSimon mints problem | arborescences | arc-cotangent reducible numbers | arccos(x) | arccosec(x) | arccosech(x) | arccosh(x) | arcos and arccos are both used in the OEIS! | arcosh and arccosh are both used in the OEIS! | arcsec(x) | arcsech(x) | arcsin(x) | arcsinh(x) | arctangent numbers, triangle of | ARIBAS | arithmetic means | arithmetic numbers | arithmetic progressions | Armstrong numbers | Aronson's sequence | Artin's conjecture | Artin's constants
- B-trees | Baby Monster simple group | backgammon | Baker-Campbell-Hausdorff expansion | balanced numbers | Balancing weights | ballot numbers | balls into boxes | balls on the lawn | Barker sequences (or Barker codes) | Barnes-Wall lattices | barriers for omega(n) | baseball | Batcher parallel sort | Baxter permutations | Beans-Don't-Talk | Beanstalk | Beatty sequences | Beethoven | beginning with t | Bell numbers | Bell's formula | Benford numbers | Benny, Jack | Berlekamp's switching game | Bernoulli numbers | Bernoulli polynomials | Bernoulli twin numbers | Bernstein squares | Berstel sequence | Bertrand's Postulate | Bessel function | Bessel numbers | Bessel polynomial | Bessel triangle | Board of Directors Problem | Bode's law | Bohr radius | Bokmal | Bond percolation | Boolean functions | Boolean lattices | Braille | Bravais lattices | Brazilian Portuguese | Brun's constant | Buffon's needle | Burnside's problem in group theory | Busy Beaver problem
- C(n,2) | C(n,3) | C(n,4) | C(n,k) | cab-taxi numbers | Cahen's constant | Cald's sequence | Cantor set | Cantor's sigma function | Carmichael numbers | Carmichael's lambda function | Catalan constant | Catalan numbers | Catalan triangle | Catalan's conjecture | Catalan | Cauchy numbers | Cayley's mistake | Cebychev is spelled | Chacon sequences | Chains | Champernowne constant | Champernowne sequence | Chebycheff is spelled | Chebychev is spelled | Chebyshev function theta(n) | Chebyshev polynomials | Chernoff sequence | Chinese | Chowla sequence | Chowla's function | Chvatal conjecture | Class numbers | Clifford group, | Cluster series | Clusters | coconut problem | | Coefficients | Collatz problem | Colombian numbers | coloring a cube | coloring a map | coloring a triangle | colorings | Congruence property | Congruences | congruent numbers | Connell sequence | Consecutive | Consistent | Convolutions | Conway group Con.0 | Conway sequences | Conway-Guy sequence | Coprime sequences | cos(nx) | cos(x) | cosec(x), Taylor series for | cosecant numbers | cosh x / cos x, Taylor series for | cosh(x) | Costas arrays | cot(x), Taylor series for | cotangent numbers | Cotes numbers | Cotesian numbers | cototient(n) | Coxeter-Todd 12-dimensional lattice | Critical exponents | Croatian | Crystal classes | crystallographic groups | cube numbers, centered | Cube roots | Cubes | Cubic curves | Cubic fields | cuboctahedral numbers | Cullen numbers | Cullen primes | Cycles | cyclic group | cyclic numbers | Cyclic | Czech
- DAGs | Danish | dartboard | Davenport-Schinzel numbers | David, J.-P. | Dawson's chess | Decimal equivalent | Decompositions | Dedekind psi function | Dedekind's function | Dedekind's problem | Deficit | Definite integrals | Degree sequences | Delannoy numbers, central | Deleham's operator DELTA | DELTA operator | Demlo numbers | Determinants | Diagonal length function | diagonal sequences | Diagrams | Diameters | Diamond lattice | Difference equations | Differences periodic | Differential equations | differential structures | digital root | digraphs | digraphs, Eulerian | Dimensions | Diophantine equations | Dirac delta function | Directed graphs | Diregular | Dirichlet divisor problem | Dirichlet series | Discordant | discriminants | Disjunctive | dismal arithmetic | Dissections | Distribution problem | divisor chains | dodecahedral numbers | dodecahedron | Doehlert-Klee designs | dominoes | Don's sequence | Max Alekseyev's problem | Dowling numbers | dragon-curve sequences | Dress's sequence | Duffinian numbers | dumbbells | dungeons | Dutch | Dyck paths | dying rabbits | Dynamic storage
- e, continued cotangent for | e, continued fraction for | e, convergents to | e, decimal expansion of | e-divisors of n | e-perfect numbers | E-trees | Earliest sequences | Early Bird numbers | eban numbers | economical numbers | Eddington's estimate of protons in universe | Egyptian fractions | EHS numbers | Eisenstein integers | Eisenstein series | Eisenstein-Jacobi primes | EKG sequence | elevator buttons | Elliptic functions | ELN | emirps | Enantiomorphs | Endomorphism patterns | Energy functions | Engel expansions | English words for the numbers, dependent on | English | Entringer numbers | Epstein's Put or Take a Square game | Eratosthenes | Erdos-Woods numbers | error function | Esperanto | eta(x), Dedekind's function | ethylene derivatives | Euclid numbers | | Euclid-Mullin sequence | Euclidean fields | Euler characteristics | Euler graphs | Euler numbers | | Euler PHI function | Euler polynomials | Euler transforms | Euler's constant gamma | Euler-Mascheroni constant | Euler's idoneal numbers, or numeri idonei (or numerus idoneus) | Euler's product | Euler-Bernoulli numbers | Euler-Jacobi pseudoprimes | Euler-Mascheroni constant | Eulerian circuits | Eulerian numbers | Eulerian polynomials | even numbers | Even sequences | evil numbers | excess of n | exp(1 - e^x) | exp(Pi*sqrt(163)) | exponential divisors | exponential numbers | Exponentiation | exponents in factorization of n | Expressions | Expulsion array | extremal theta series | extremal weight enumerators | EYPHEKA! num = DELTA + DELTA + DELTA | E_6 Eisenstein series
- fabrics | factorial numbers | factorials | FactorInteger (Mma) | Factorization (MAGMA) | factorization patterns | factorization problems | factorizations of important sequences | falling factorials | fanout-free functions | Farey series or tree | Feigenbaum constants | Fermat coefficients | Fermat numbers, 2^(2^n) + 1 | Fermat numbers | Fermat primes | Fermat quotients | Fermat remainders | Fermat's last theorem | Fermionic string states | Feynman diagrams | Fibbinary numbers | | Fielder sequences | Fifteen Puzzle | Fifth roots | Figurate numbers | figure 8's | filaments | final digits | Fine's sequence | fine-structure constant | Finite automata | Finnish | first factors | First moment | First occurrence of | Flavius's sieve | flexagons | flimsy numbers | folding a map | Ford and Johnson sorting | forests | fortnightly intervals | Fortunate numbers | Foster census | fountains of coins | four 4's problem | four-color theorem | fourth powers | fractal sequences | fractional base | fractions | francais | Franel numbers | free energy series | free subsets | French language, sequences involving | French | Friedman's sequence | friendly numbers | friendly pairs | full sets | Fullerenes | functional cube roots | functional determinants | functional square roots | functors | fundamental discriminants | fundamental units | fusc
- G.C.D. | g.c.d. | Gaelic | Galego | gamma (Euler-Mascheroni constant) | gamma function | gaps | gates | Gauss-Kuzmin-Wirsing constant | Gaussian binomial coefficients | Gaussian integers and primes | Gaussian primes | GCD(x,y) | GCD, greedy sequence | GCD | generated by substitutions | Genocchi medians | Genocchi numbers | geometrical configurations | geometries | Germain primes | GF(2)[X]-polynomials, Matula-Goebel-tree | GF(2)[X]-polynomials, Moebius-analogue | Gijswijt's sequence | Gilbreath's conjecture | Giuga numbers | Glaisher numbers, | Glaisher's chi numbers | Glaisher's G numbers | Glaisher's H numbers | Glaisher's H' numbers | Glaisher's I numbers | Glaisher's J numbers | Glaisher's T numbers | Gleason's theorem | gluons | glycols | Go games | Goedel, Escher, Bach, | Goellnitz's theorem | Golay codes | Golay-Rudin-Shapiro sequence | Goldbach conjecture | | golden sieve | Golomb rulers | Golomb's sequence | golygons | good numbers | Goodstein sequences | gossip problem | Gould's sequence | Graham, Ron's sequence | Gram points | grandchildren of a binary vector | graph reconstruction problem | graphical partitions | Graphs, de Bruijn | Graphs, Euler | Graphs, Eulerian | graphs, Hamiltonian | Graphs, independence number of | Graphs, independent sets in | graphs, inseparable | Graphs, interval | graphs, interval | graphs, irreducible | graphs, K_4-free | graphs, least number of edges in | graphs, line | graphs, mating | graphs, Moore | graphs, nonseparable (or blocks) | Graphs, of maximal intersecting sets | Graphs, oriented | Graphs, partition | Graphs, path | graphs, perfect | Graphs, planar | graphs, plane | graphs, pointed | Graphs, polygonal | Graphs, polyhedral | Graphs, radius of | graphs, regular connected, of degree k | Graphs, regular | graphs, rooted | Graphs, self-complementary | Graphs, self-converse | Graphs, self-dual | graphs, series-parallel | Graphs, series-reduced | Graphs, signed | Graphs, spectra of | Graphs, splittance of | Graphs, square-free | Graphs, stable | Graphs, star | Graphs, Steinhaus | Graphs, tensor sum of | Graphs, transitive | graphs, triangle of numbers of, connected, unlabeled | Graphs, triangle-free | graphs, triangle | Graphs, triangles in | graphs, triangulated | Graphs, trivalent | Graphs, unicyclic | Graphs, valence of | Graphs, vertex-degree sequences of | Graphs, vertex-transitive | graphs, with loops, triangle of | Graphs, with no isolated vertices | Graphs, without endpoints | Graphs, without points of degree 2 | graphs | grasshopper sequence | Gray codes | greatest common divisor | | Green's function | greengrocer's numbers | Greg trees | Grids | Grossman's constant | group | groupoids | groups | Groups, Lorentzian | groups, of Rubik cubes | Grundy's game | Gudermannian
- h.c.p. | Hadamard matrices | half-totient function | Halving | Hamilton numbers | Hamiltonian cycles | Hamiltonian graphs | Hamiltonian paths | Hamiltonian polyhedra | Hamming sequence | Hankel function | Hannah Rollman's numbers | happy factorizations | happy numbers | Harary and Palmer | hard-hexagon model | harmonic coefficients | harmonic means | harmonic numbers | harmonic series | harmonic triangle of Leibniz | Harshad numbers | Harvey Friedman's sequence | Hauptmodul series | Havender tableaux | HCF | hcp | Heawood conjecture | Hebrew | Heegner numbers | hendecagon is spelled 11-gon in the OEIS | heptagonal numbers | heptagonal pyramidal numbers | Hermite constant | Hermite polynomials | Hertzsprung's problem | hex numbers | hexaflexagons | hexaflexagrams | hexagonal close packing | hexagonal numbers | Hexagonal prism | hexagonal pyramidal numbers | Hexanacci numbers | hierarchies | highest common factor (hcf) | highly abundant numbers | highly composite numbers | highly powerful numbers | Hilbert series | Hindi | Hofstadter Q-sequence | Hofstadter-Conway $10,000 challenge sequence | Hofstadter-type sequences | Hoggatt sequences | Holes | home primes | Honeycomb | humble numbers | Hungarian cube | Hurwitz numbers | Hurwitz-Radon numbers | hydrocarbons | hyperbinomial transform | hyperfactorials | hypergraphs | hyperperfect numbers | hyperplane arrangements | Hypertournaments | hypothenusal numbers | hypotheses
- icosahedral numbers | icosahedron | idempotents | idoneal numbers | ifactor (Maple) | Imaginary parts | Impedances | Impossible values | impractical numbers | Incidence matrices | Indefinitely growing | Independence number | Independent sets | Infinitary perfect | infinitesimal generators | initial digits | integers | Integral points | Integrals | interprimes | intersections of diagonals | interval graphs | interval orders | interval schemes | Invariants | INVERT transform | Irish Gaelic | Irish | irreducible polynomials | irregular primes | isogons | isthmuses | Italian | Iterated exponentials | Iterates of number-theoretic functions | i^i
- J-function | J2 simple group | Jack Benny | Jacobi elliptic function sn | Jacobi nome | Jacobi symbol | Jacobi theta series | Jacobi triple product identity | Jacobsthal sequence | Jacobsthal-Lucas numbers | Japanese | join points around circle | joke numbers | Jordan algebras | Josephus problem | Josephus's sieve | Joyce, James, "Ulysses" | | jumping champions | jumping problem | junction numbers | Justified arrays
- k-arcs | K-free sequences | Kagome' lattice | Kaprekar numbers | Kayles | Kempner tableaux | Kempner-Smarandache numbers | Kendall-Mann numbers | Kepler's tree of fractions | keys | Khintchine's constant | Kimberling puzzle | kings problem | kissing numbers | Klarner-Rado sequences | knight's tours | Knopfmacher expansions | knots | Knuth's sequence (or Knuth numbers) | Kobon triangles | Kolakoski sequence | Kolakowski sequence | Kotzig factorizations | Kronecker (-1,n) | Kubelski sequence | Kummer's conjecture
- L-series | L.C.M. | labeled partitions | lacing a shoe | Laguerre polynomials, | Lah numbers | LambertW function | laminated lattices, numbers of | Landau approximation | Landau's function g(n) | Langford-Skolem problem of arranging 11223344...nn | language, words in a certain | largest factors of various numbers | largest prime dividing n | last digits | last occurrence | Latin (the language) | Latin cubes | Latin rectangles | Latin squares | Latin | Lazy Caterer sequence | LCM(x,y) | least common multiple | Least number of powers to represent n | least significant bit (lsb) | Leech lattice | Leech triangle | Leech's path-labeling problem | left factorials | Legendre polynomials | Legendre's conjecture | LEGO blocks | Lehmer's constant | Lehmer's polynomial | Leibniz's triangle | lemniscate function | Lemoine's conjecture | Leonardo logarithms | Les Marvin sequence | Levenshtein distance | Levine's sequence | Levy's conjecture | Li(n) | Lie algebras | light, speed of | light-bulb game of Berlekamp | light-bulbs | linear codes | linear extensions | linear forms | linear inequalities | linear orders | Linear spaces | Linus sequence | Liouville's constant | Liouville's function L(n) | Liouville's function lambda(n) | Liouville's number | list manipulation functions of Lisp | ln | locks | Loeschian numbers | log 10, decimal expansion of | log 2, decimal expansion of | log 3, decimal expansion of | log 5, decimal expansion of | log 7, decimal expansion of | log(3/2) | log(cos(x)) | log(n)*2^n/n | log(n) | log(sin(x)/x) | logarithmic integral | logarithmic numbers | log_10 e, decimal expansion of | log_2 3 | log_2(n) | long but finite sequences | longest common subsequences, most | longest common substrings, most | look and say sequences | loops, Moufang | Losanitsch's triangle | loxodromic sequence of spheres | Loxton-van der Poorten sequence | Loyd's 15-Puzzle | LS ("Look and Say") | lsb = least significant bit | Lucas numbers | Lucas polynomials | Lucas pseudoprimes | Lucas, Theorie des Nombres | Lucas-Carmichael numbers | Lucasian primes | lucky numbers | Lukasiewicz words | Lyndon words | L_infinity norms in lattices
- M-sequences | M-trees | magic numbers | magic series | magic squares | magnetization coefficients | Mahler's number | Mahonian numbers | Mancala solitaire (generalized) | Mandelbrot set | Mangoldt function | Manhattan lattice | Manhattan subways | map, 3x+1 | map, coloring | maps, folding | Maps, Hamiltonian | Maris-McGwire numbers | Markoff numbers | Markov numbers | Matchings | mathematical symbols in OEIS | matrices, (+1,-1) | matrices, (0,1) | matrices, alternating sign | matrices, anti-Hadamard | matrices, binary, n X n | matrices, Hadamard | Matrices, Hilbert | Matrices, Pascal | Matrices, Schur | Matrices, stochastic | matrices, ternary, n X n | matroids , | Maundy cake | Max Alekseyev's problem | max(x,y) | McKay-Thompson sequences for Monster simple group | McKay-Thompson series for Monster simple group (start) | meanders | Meeussen sequences | menage numbers | mens-room permutations | merge sort | Mertens's conjecture | Mertens's function | Mian-Chowla sequences | Miller-Rabin primality test | Mills primes | Mills's constant | min(x,y) | Minkowski's question mark function | mobiles | Mobius | Mock theta numbers | mod(x,y) | models (in statistics) | modest numbers | Moebius (or Mobius) function mu(n) | Moebius function | Moebius transforms | Molecular species | Molien series | money | monoids | Monster simple group, McKay-Thompson series for | months | Montreal solitaire | Moon (1987), "Some enumerative results on series-parallel networks" | Moran numbers | mosaic numbers | Moser-de Bruijn sequence | most significant bit (msb) | motifs | Motzkin numbers | Motzkin triangle | mousetrap game | Mozart | Mrs Perkins's quilt | msb = most significant bit | Mu Torere | mu(n) | MU-numbers | mult | multigraphs | Multinomial coefficients | Multiplication-cost | Multiplicative encodings | multiplicative order of 2 mod n, ord(2,n) | multiplicative order of x mod y, ord(x,y), sequences related to | multiplicatively perfect numbers | multiply-perfect numbers | musical scales | mutinous numbers
- n -> 3n - sigma(n) | n appears n times | n divides concatenation of all numbers up through n | n reversed, R(n) | n!+1 | n!-1 | n!/2 | n! | n# (1st definition of primorial numbers) | N-free graphs | n-phi-torial | Narayana triangle (also called Catalan triangle) | Narayana-Zidek-Capell numbers | narcissistic numbers | natural numbers, in base 2 | natural numbers, in bases 1 through 10 | natural numbers, rearrangement of (conjectured or otherwise)(1) | near-rings | necklaces | necklaces, aperiodic | necklaces, asymmetric | necklaces | negative numbers | Negative pseudo-squares | neofields | Nets | Networks | next prime after terms of various sequences | nexus numbers | Niemeier lattices | nilpotent numbers | Nim-addition | Nim-multiplication | Nim-multiplication in Maple | Nim-products | Nim-sums | Nim-sums in Maple | Nim | Niven numbers | Niven's constant | no-three-in-line problem | Noergaard, Per | nome | non-collinear points in cube | Non-Hamiltonian | non-mathematical sequences | Non-separable | noncubes | nonpositive integers | nonprimes | nonrepetitive sequences | nonsquares | nontotients | Norgard, Per | Norwegian | not always integral | notation in OEIS | noughts and crosses | NSW numbers | number of positive integers | number of syllables to represent n | number of ways the set (1^k, 2^k, ..., n^k) can be partitioned into two sets of equal sums | number of words to represent n | number theory, unsolved problems in | numbers n such that 2^k + n is prime for all k | numbers n such that n*2^k + 1 is composite for all k | numbers n such that n*2^k + 1 is prime for all k | numbers of form k_1 k_2 .. k_n (1/k_1 + .. + 1/k_n), k_i >= 1 | numbers, automorphic | numbers, Bernoulli | numbers, Euler | numbers, feral | numbers, octal | numbers, tri-perfect | numbers, triperfect | numbers, triply perfect | numbers, wild | numeri idonei | numerus idoneus | n^(n+1) | n^(n-1) | n^(n-2) | n^(n-3) | n^2-n+41 is prime | n^n
- O'Nan group | octagonal numbers | octahedral numbers | octahedron, truncated | octal numbers | octupi | odd numbers, fake | odious numbers | omega(n), number of distinct primes dividing n | Omega(n), total number of primes dividing n | one local maximum, arrays with | one odd, two even, etc. | one puddle | ones-counting sequence | operational recurrences | optimal rulers | OR(x,y) | orchard problem | ord(x,y) | order, binary | order, multiplicative order of 2 mod n | ordered factorizations | ordered partitions | orders | ordinals | Ore numbers | orthogonal arrays, number of | orthogonal groups | out-points
- p-adic square roots in PARI | p-adic valuations | packing squares | Packings | Padovan sequence | Pair-coverings | | panarithmic numbers | pancake numbers | paper-folding sequences | paradoxical sequences | parentheses, ways to arrange | parenthesized in 2 ways | PARI code | parity sequence | partially ordered sets | partition function for lattices | partition numbers, prime | partitions, binary | partitions, graphical | partitions, into even number of parts | partitions, into odd number of parts | partitions, into odd parts | partitions, into parts 5k+-1 | partitions, into parts 5k+-2 | partitions, into primes | partitions, into relatively prime parts | partitions, into triangular numbers | partitions, m-ary | partitions, odd | partitions, of a polygon | partitions, of n into 4 squares | partitions, of n into 4th powers | partitions, perfect | Partitions, planar | partitions, protruded | partitions, square | partitions, total number of parts | partitions, total | partitions, wide | Pascal triangle | Pascal's rhombus | Pascal's square | Paths in plane | paths in square grid | Paths through arrays | patience | Patterns | pay-phones | Peaks | Peanuts cartoon sequences | pedal triangles | peeling rinds | Pell numbers | Pell's equation | Pellian equation x^2 - D*y^2 = 1 | Pellian equation | Pennies | | percolation series (1) | perfect lattices | perfect numbers | perfect partitions | perfect powers | perfect rulers | perfect years | period of reciprocal of n | period of reciprocal of nth prime | Periodic differences | Periodic sequences | periodic table | Periods | Permanents | Permutation arrays | permutation groups | Permutations, alternating | permutations, asymmetric | Permutations, Baxter | permutations, bitriangular | Permutations, by cycles | Permutations, by descents | Permutations, by distance | Permutations, by inversions | Permutations, by length of runs | Permutations, by number of peaks | Permutations, by number of runs | Permutations, by number of sequences | Permutations, by order | Permutations, by rises | Permutations, by spread | Permutations, by subsequences | permutations, connected | Permutations, cycles in | Permutations, descending subsequences of | Permutations, discordant | permutations, even | permutations, graceful | permutations, indecomposable | Permutations, inequivalent | Permutations, inversions in | Permutations, isolated reformed | Permutations, key | Permutations, necklace | Permutations, odd | permutations, of n letters | permutations, of order 2 | permutations, of order dividing k, for k=2,3,4,5,... (1) | permutations, of the positive (or nonnegative) integers | permutations, of the integers, induced by Catalan rerankings, each paired with its inverse | permutations, of the integers, signature-permutations induced by Catalan automorphisms, | Perrin sequence | persistence (additive) | persistence (multiplicative) | Phi(n) | phi(n) | pi(10^n) | pi(x), number of primes | Pi, continued cotangent for | Pi, continued fraction for | Pi, convergents to | Pi, decimal expansion of | Pi, strings of digits in | piano keyboard | picture-perfect numbers | pieces | Pierce expansions | Pierpont primes | Pisano numbers | Pisano periods | Pisot sequences, definition | Pisot sequences, warning about recurrences for | Pisot sequences | planar graphs | planar vs plane | Planck's constant | plus perfect numbers | Poincare conjecture | Poincare series | pointed objects | Points on surface of various polyhedra | poker | Polish | Polya's conjecture | polyaboloes | polyares | polycubes | polygamma function | polygonal numbers | polygons (1) | polygons constructible with ruler and compass | polygons inscribed in a circle | polygons with all diagonals drawn | polyhedra (1) | polyhedra, allomorphic | polyhedra, regular | polyhedra | polyhes | polyhexes | polyiamonds | polyknights | polynomial identities | Polynomials, Bell | polynomials, Bernoulli | Polynomials, Bessel | Polynomials, by height | Polynomials, characteristic | Polynomials, Chebyshev | Polynomials, cyclotomic | polynomials, cyclotomic, inverse, Phi(1..9) | polynomials, cyclotomic | Polynomials, discriminants of | polynomials, divisors of x^n-1 | polynomials, Euler | Polynomials, Gandhi | Polynomials, Hammersley | polynomials, Hermite | Polynomials, hit | polynomials, irreducible over a finite field | polynomials, irreducible over GF(q), q >= 4 | polynomials, irreducible, binary, degree divides n | polynomials, irreducible, binary, degree n | polynomials, irreducible, over GF(3), degree divides n | polynomials, irreducible, over GF(3), degree n | polynomials, irreducible, over GF(4), degree divides n | polynomials, irreducible, over GF(4), degree n | polynomials, irreducible, over GF(5) | polynomials, irreducible, over GF(7) | polynomials, irreducible | polynomials, Laguerre | polynomials, Legendre | polynomials, monic irreducible over finite fields | Polynomials, m\'{e}nage | Polynomials, orthogonal | Polynomials, period | polynomials, primitive | Polynomials, relatively prime | Polynomials, rook | polynomials, Shapiro | polyominoes | polyominoes, 3-dimensional | polyominoes, hexagonal | polyominoes, indecomposable | polyominoes, rotationally symmetric | polyominoes, triangular | polytans | polytopes, regular | polytopes | Poonen-Rubinstein paper | Popes | porisms | Portuguese | posets | posets | posets, chains in | posets, connected | posets, forbidden | posets, graded | posets, increasing | posets, irreducible | posets, N-free | posets, reduced | posets, series-parallel | positive integers | Post functions | postage stamp problem | Potts model | power set, chains in | power train | power-sum numbers | powerful numbers | powers (1) | powers of 10 | powers of 11 | powers of 12 | powers of 13 | powers of 14 | powers of 15 | powers of 16 | powers of 17 | powers of 18 | powers of 19 | powers of 20 | powers of 21 | powers of 22 | powers of 23 | powers of 24 | powers of 25 | powers of 26 | powers of 27 | powers of 28 | powers of 29 | powers of 2 | powers of 30 | powers of 31 | powers of 32 | powers of 33 | powers of 34 | powers of 35 | powers of 36 | powers of 37 | powers of 38 | powers of 39 | powers of 3 | powers of 40 | powers of 41 | powers of 42 | powers of 43 | powers of 44 | powers of 45 | powers of 46 | powers of 47 | powers of 48 | powers of 4 | powers of 5 | powers of 6 | powers of 7 | powers of 8 | powers of 9 | powers of a prime but not prime | powers of e rounded up | powers of Pi rounded upwards | powers that be | powers, not the difference of two | powers, perfect | powers, the difference of two | powertrain function | practical numbers | Prague clock sequence | preferential arrangements | preorders | previous prime and next prime etc. for terms of various sequences | previous prime, next prime etc. NP...PP...NPMPP...nMPP...NPMn...MIN | prime factorizations of important sequences | prime factors of n, number of A001222 | prime plus twice a square | prime powers | prime pyramid | Prime quadruplets | Prime races | prime signature | prime triplets | prime(2^n) | prime(n) == +/-k (mod n) | prime, largest dividing n | prime, smallest whose product of digits is (something) | PRIMEGAME | Prime Pi(x), number of primes | primes | primes involving quasi-repdigits D(R)nE | primes involving repunits | primes of the form binomial(k*n, n) ? 1, k=2..6 | primes p such that x^k = 2 has a solution mod p, | primes such that the sum of the predecessor and successor primes is divisible by k | primes with X as smallest positive primitive root | primes, absolute | primes, additive | primes, almost | primes, approximations to | primes, automorphic | primes, balanced | primes, Bertrand | Primes, by class number | primes, by Erdos-Selfridge class n+ | primes, by Erdos-Selfridge class n- | Primes, by number of digits | primes, by order | Primes, by period length | primes, by primitive root | Primes, chains of | primes, characteristic function of | Primes, compressed | Primes, consecutive | primes, cuban | primes, Cullen | primes, deceptive | Primes, decompositions into | primes, differences between | primes, dihedral calculator | primes, dividing n | primes, doubled | primes, duodecimal | primes, Euclid-Pocklington | primes, Euclidean | primes, even | primes, Fermat | primes, final digits of | primes, fortunate | primes, from Euclid's proof | primes, gaps between | primes, Germain | primes, good | primes, half-quartan | primes, happy | primes, Higgs | primes, home | primes, Honaker | primes, iccanobiF | primes, in arithmetic progressions | primes, in decimal expansion of Pi | Primes, in intervals | Primes, in number fields | Primes, in residue classes | Primes, in sequences | Primes, in ternary | primes, in various ranges | Primes, inert | primes, irregular | primes, isolated | Primes, largest | primes, left-truncatable | primes, lonely | primes, long period | primes, Lucas numbers | primes, Lucasian | primes, Mersenne | primes, Mills's | primes, minus a constant | primes, multiplicative and additive | primes, multiplicative | primes, next | primes, number of less than n*10^k | primes, octavan | primes, of a particular form, number that are less than or equal to 10^n | primes, of form n! +- 1 | primes, of form x^2 + kxy + y^2 | primes, of form x^2+27y^2 | primes, of form x^2+y^2 | primes, order of | primes, Pierpont | Primes, primitive roots of | primes, products of | primes, pseudo | primes, quadratic form, discriminant | Primes, quadratic partitions of | Primes, quadratic residues of | primes, quartan | primes, quintan | primes, reciprocals of, periods | primes, regular | Primes, represented by quadratic forms | primes, repunit | primes, right-truncatable | primes, safe | primes, sextan | primes, short period | Primes, single | primes, Sophie Germain | Primes, special sequences of | Primes, square roots of | primes, Stern | primes, strobogrammatic | primes, strong | primes, subsets in range 2^n,2^(n+1), | primes, sum of the first k^n primes, k=2,3,5,6,7,10 | Primes, sums of digits of | Primes, supersingular | primes, that divide sum of all primes | Primes, to odd powers only | primes, transformed by cellular automata | Primes, transforms of | primes, truncatable | primes, truncated | primes, twin primes conjecture | primes, twin | primes, undulating | primes, various subsets in range 2^n,2^(n+1) | primes, weak | primes, which are average of their neighbors | Primes, whose reversal is a square | primes, Wilson | Primes, with consecutive digits | primes, with first digit 1 (or 2, 3, etc.) | Primes, with large least nonresidues | Primes, with prime subscripts | primes, Woodall | primeth recurrence | primitive (1) | Primitive factors | primitive polynomials | primitive roots | primorial numbers | primorial primes | primorials | principal character | prism numbers | probability difference equation | probability orderings | problimes | product_{k >= 1} (1-x^k)^m (1) | profiles | projective planes of order n | projective planes, maps on | projective planes, permanent of | projective planes | promic numbers | pronic numbers | Proth numbers | proton mass | proton-to-electron mass ratio | Prufer codes for trees | pseudo Q-numbers | pseudo-bricks | pseudo-Galois numbers | pseudo-lines | pseudo-powers to base 3 | pseudo-random numbers | pseudo-random numbers | pseudo-Smarandache numbers | pseudo-squares | pseudo-wild numbers | pseudoperfect numbers | pseudoprimes | pseudoprimes, Carmichael numbers | pseudoprimes, Euler-Jacobi | pseudoprimes, Lucas | pseudoprimes, Miller-Rabin primality test | psi function | puzzle sequences (01) | pyramidal numbers (1) | Pythagoras' theorem | Pythagorean triples |
- q-factorials | Q-graphs | quadrangulations | quadratic character, sequences with prescribed | quadratic fields, | quadratic fields, Euclidean | quadratic fields, genera of | quadratic fields, imaginary, by class numbers | quadratic fields, real, by class numbers | quadratic fields, real, discriminants | quadratic fields, simple | quadratic fields, totally real of degree n | quadratic fields, unique factorization domains | quadratic forms, binary | quadratic forms, extreme | quadratic forms, genera of | quadratic forms, minimal norm of | quadratic forms, one class per genus | quadratic forms, perfect | quadratic forms, populations of | quadratic forms, ternary | Quadratic invariants | quadratic nonresidues, consecutive | quadratic residues, consecutive | Quadrilaterals | quadrinomial coefficients | quadruple factorial numbers | quarter-squares | quasi-amicable numbers | quasi-orders | quasigroups, asymmetric | quasigroups, by idempotent | quasigroups, commutative | quasigroups, self-converse | quasigroups, with identity | quasigroups | quaternions, Hurwitz, prime | Quebbemann 32-dimensional lattice | queens problem | Quet transform | quilt, Mrs. Perkins's
- R(n), or n reversed | rabbits, dying | rad(n) | RADD sequences (Reverse then add something) | Radon function | Radon theorem | Ramanujan approximation | Ramanujan numbers tau(n) | Ramanujan's Lost Notebook | Ramsey numbers | random numbers | random sequences | Raney numbers | rapidly growing sequences | rational numbers | Rational points on curves | rationals, enumerating | RATS | Raymond strings | reachable configurations on circles | read n backwards | rebasing notation b[n]q | Recaman's sequence | Recaman's sequence, addition steps | Recaman's sequence, condensed version | Recaman's sequence, heights | Recaman's sequence, quotients and remainders | Recaman's sequence, records for a(n)/n | Recaman's sequence, segments in | Recaman's sequence, simplified version | Recaman's sequence, steps to hit n | Recaman's sequence, subtraction steps | Recaman's sequence, transforms based on | Recaman's sequence, two-dimensional versions | Recaman's sequence, variations on | reciprocal of n, decimal expansion of | reciprocals of primes | rectilinear crossing number of complete graph | recurrences over rings | recurrences, of the form a(n) = k*a(n - 1) +/- a(n - 2), | recurrences, of the form a(0) = 2; a(1) = k; a(n) = k*a(n - 1) + a(n - 2) | reduced residue system | reduced totient function psi | refactorable numbers | reflection coefficients | regions formed by lines in plane | regions formed by spheres in space | regions in regular polygon | regular n-gon with all diagonals drawn | regular primes | regular sequences | relations on n points | relations, antisymmetric transitive | relations, antisymmetric | Relations, connected | Relations, dissimilarity | Relations, reflexive | relations, self-converse | relations, symmetric reflexive | relations, symmetric | relations, transitive | Relations, ultradissimilarity | Relations, vacuously transitive | relations, without symmetry | relations | relatively prime pairs of numbers | relatively prime triples of numbers | remove 2's from n | remove squares! | rencontres numbers, triangle of | repdigit (repeated digit) numbers | repeating substrings | repfigit or Keith numbers | Representation degeneracies | Representations as sums of increasing powers | Representations as sums of Lucas numbers | Representations as sums of squares | Representations as sums of triangular numbers | Representations of 0 | Representations of 1 | Representations of symmetric group | repunit primes | repunits | residues, cubic | Residues | resistances | Reverse Add Then Sort | Reverse and Add! | reversion of series | REVERT transform | rhombic dodecahedral numbers | rhyme schemes | Riemann hypothesis, | Riemann hypothesis, sequences equivalent to | Riemann hypothesis, sequences related to | Riemann zeta function | Riesel numbers | Riesel problem | riffle shuffling | rings (in graph-theoretic sense) | rings, commutative | rings, enumeration of | rings, nonassociative | rings, nonisomorphic and nonantiisomorphic | rings, self-converse | rings, semisimple | rings, with unit | rings | Riordan numbers | Robbins constant | Robbins numbers | Robbins triangle | Robinson's constant | Rogers-Ramanjuan continued fraction | Rogers-Ramanujan identities | Roman numerals for n | Ron's sequence | rook tours | rook walks | Rooks | rooted trees , | rooted trees, achiral | rooted trees, asymmetric (1) | rooted trees, AVL | rooted trees, B-trees | rooted trees, binary (1) | rooted trees, boron | rooted trees, by generators | rooted trees, by internal nodes | rooted trees, carbon | rooted trees, chiral planted | rooted trees, codes for | rooted trees, composite binary | rooted trees, constant | rooted trees, directed | rooted trees, evolutionary | rooted trees, fixed points in | rooted trees, game | rooted trees, genealogical | rooted trees, Greg | rooted trees, Husimi | rooted trees, hybrid binary | rooted trees, identity | rooted trees, increasing | rooted trees, involution | rooted trees, lableling (Goebel) | rooted trees, leaves | rooted trees, M-type | rooted trees, matched | rooted trees, nodes | rooted trees, noncrossing | rooted trees, normalized total height | rooted trees, of subsets | rooted trees, oriented | rooted trees, partially labeled | rooted trees, permutation | rooted trees, phylogenetic | rooted trees, planar | rooted trees, planar, planted | rooted trees, plane, encodings of | rooted trees, plane, encoded as totally balanced binary strings | rooted trees, planted | rooted trees, pointed | rooted trees, powers of enumerator | rooted trees, prime binary | rooted trees, projective plane | rooted trees, quartic planted | rooted trees, red-black | rooted trees, search | rooted trees, series-reduced planted (1) | rooted trees, series-reduced | rooted trees, spanning | rooted trees, steric planted | rooted trees, symmetries in planted | rooted trees, ternary | rooted trees, triangle | rooted trees, trimmed | rooted trees, unary-binary | rooted trees, with a forbidden limb | rooted trees, with(out) a primary branch | rooted trees | rotation distance | royal paths in a lattice | Rubik cube | Rudin-Shapiro word | ruler and compass | ruler function | rulers, complete | rulers, Golomb | rulers, optimal | rulers, perfect | runs in binary expansion | runs, lengths of | Russian | Ruth-Aaron numbers
- safe primes | SAGE program | Salie numbers | Sally sequence | Sam Loyd's 15-Puzzle | same upside down | Sarrus numbers | scheduling | Scheme (programming language) | Schroder is spelled Schroeder in the OEIS | Schroeder numbers | Schroeder's 1st problem | Schroeder's 2nd problem | Schroeder's 3rd problem | Schroeder's 4th problem | Schroeder's problems | Scott, Dana, sequences | Scrabble | sec(x), Taylor series for | secant numbers | secant-tangent numbers | Second moment | segmented numbers | Self numbers | Self-contained numbers | self-describing numbers | Self-dual | Self-generating sequences | semigroups | semigroups, asymmetric | semigroups, by idempotents | semigroups, commutative | semigroups, idempotent | semigroups, inverse | semigroups, non-commutative | semigroups, numerical | semigroups, regular | semigroups, relation | semigroups, self-converse | semigroups, with identity | semiorders | semiperfect numbers | semiprimes (or semi-primes) | semiprimes | separating families | sequence and first differences include all numbers | sequences by number of increases | sequences depending on A-numbers in OEIS | Sequences of prescribed quadratic character | sequences offering a monetary reward | sequences that contain every finite sequence of nonnegative integers | sequences that need extending | sequences which agree for a long time but are different | sequences whose extension requires factoring large numbers | sequences with a gap (some later term is known) (1) | sequences with a large but finite number of terms | Serbian | series-parallel networks | series-parallel numbers | set partitions | sets of lists | sets | sexy prime pairs | shadow of constants | Shannon switching function | Shell sort | shift registers, enumeration of output sequences | shift registers, enumeration of | shift registers, periods | Shifts left when transformed | shoe lacing | shoelaces | shogi (Japanese chess) | shuffle groups | shuffling (1) | | Sierpinsky | sieve, | sieve, badly sieved numbers | sieve, binary | sieve, Eratosthenes | sieve, even | sieve, Flavius Josephus | sieve, generated by a | sieve, golden | sieve, Madore's | sieve, multilevel | sieve, Smarandache | sieve, spiral | sieve, square | sieve, ternary | sigma(n) = sum of divisors of n | sigma_0(n) | sigma_1(n) | sigma_k(n), the sum of the k-th powers of the divisors of n | signature sequences (1) | signature | silver mean, 1+sqrt(2) | silver number | Silverman's sequence | simple cubic lattice, theta series of | simple groups | simplex, barycentric subdivision of | simplices in a cube | Simplices | simplicial arrangements of lines | simplicial polyhedra | sin(x) | sinh(x) | siteswaps (or site swaps) | skat | skeins | skip 1, take 2, etc | slicing a cake | slicing a torus | slicing space | sliding block puzzles | Slowly converging series | smallest number not a product of earlier terms | Smarandache numbers | Smarandache-Wagstaff function | Smarandache-Wellin numbers and primes | Smith numbers | sn | snake-in-box problem | Snoopy cartoon sequences | sociable numbers, infinitary | sociable numbers, unitary | sodalite | Soddy | solid partitions | solitary numbers | Solovay-Strassen primality test | solutions of x^k = 1 in symmetric group, for k=2,3,4,... | solutions to x+y=z | solvable numbers | Somos sequences | songs, popular, sequences from | sopf(n), sum of primes dividing n (without repetition) | sopfr(n), sum of primes dividing n (with repetition) | Sophie Germain primes | sorting, Batcher parallel sort | sorting, bridge hands | sorting, by block moves | sorting, by list merging | sorting, by prefix reversal | sorting, merge sort | sorting, networks | sorting, Shell sort | sorting | spaces, linear | spaces | Spanish | special numbers | specific heat | spectral arrays | spectrum of a number | speed of light | spelling and notation | spelling | Sperner families | Sperner's theorem | sphere, surface area of n-dimensional | sphere, vector fields on | sphere, volume of n-dimensional | spherical designs | Spheroidal harmonics | spirals, enumeration of | spirals, sequences from | split numbers | sqrt(2), continued cotangent for | sqrt(2), continued fraction convergents to | sqrt(2), decimal expansion of | sqrt(3), decimal expansion of | sqrt(n), length of period of continued fraction for | sqrt(n), nearest integer to, etc. | sqrt(p), length of period of continued fraction for | SQS | square arrays, indexing | square lattice (1) | square lattice, polygons on | square lattice, sublattices of | square lattice, theta series of | square lattice, walks on | square numbers | square pyramidal numbers | square root of pi | square roots of integers ( 1) | square roots, functional | square roots, of numbers | square roots, of primes | square, truncated | square-free graphs | square-free numbers, gaps between | square-free numbers | square-free sequences | square-free words | square-full numbers | squared rectangles and squared squares | squared squares | squarefree | squarefull numbers | squares, magic | squares, packing | squares, undulating | Squares | stacking boxes | stacks | stamp-folding | Standard deviation | Stanley | stapled intervals | stapled sequences | star numbers | stars in sky | statistical models | Stechkin's function | Steiner systems, quadruple (SQS's) | Steiner triple systems (STS's) | stella octangula numbers | Stern's sequences | Stern's diatomic sequence | Stern's sequence | Stern's and Stern-Brocot sequences | Stern-Brocot tree | Stirling numbers, associated | Stirling numbers, generalized | Stirling numbers, of 1st kind | Stirling numbers, of 2nd kind | Stirling transform | Stirling's formula | Stolarsky array | Stopping times | Storage systems | Stormer numbers | Strobogrammatic numbers | strongly multiplicative means that a(m*n) = a(m)*a(n) for all m and n >= 1 | structure constants | structures, differential | STS | subfactorial numbers | sublattices | Subsequences of [1 ... n] | subset sums | subtract if you can else add | subtract-a-prime | subtract-a-square | subway stops | subwords | Such sequence | suitable numbers | sum of digits in powers of m | sum of digits, n times | sum of digits | sum of first n squares equals a triangular number | sum-free subsets | sum-full subsets | summarize previous term | sums of squares and sums of cubes , | sums of 16 squares, number of ways of writing as | sums of 2 cubes | sums of 2 squares | sums of 3 cubes | sums of 3 squares | sums of 3 squares | sums of 4 cubes | sums of 4 squares | sums of 4th powers needed to represent n | sums of 5 cubes | sums of 6 cubes | sums of 7 cubes | sums of 8 cubes | sums of 8 or 9 cubes | sums of consecutives squares give squares | sums of distinct cubes | sums of distinct squares | Sums of divisors | Sums of powers | sums of squares needed to represent n | sums of squares, sequences related to (01) | sums of tetrahedral numbers | sums of two distinct prime cubes | Sum_{k = 0..n} f(k) is standard OEIS notation (rather than sum_k^n, sum for k from 0 to n, etc.) | super-abundant numbers | superabundant numbers | superfactorials | superior highly composite numbers | Superpositions of cycles | superqueens | supersingular primes | | supertough | Surfaces | susceptibility (1) | Swedish | switchboard problem | switching classes | switching networks | Sylvester's sequence | Sym, game of | symbols in OEIS | symmetric functions of noncommuting variables | symmetric functions | symmetric group S_n, character table | symmetric group S_n, order of | symmetric group S_n | symmetric numbers | S_n
- t is the first... | T-coordinates for arrays | t-designs, spherical | table (or triangle) of (1) | take 1, skip 2, etc. | take-a-factorial | take-a-prime | take-a-square | take-a-triangle | tan(x), Taylor series for | tan(x) | tangent numbers, generalized | tangent numbers, triangle of | tangent numbers | tanh(x), Taylor series for | tanh(x) | tatami mats | tau(n), number of divisors | tau | tau_k or d_k numbers, number of ordered n-factorizations of n | taxi-cab numbers | taxicab numbers | Tchebycheff is spelled | Tchebychev is spelled | Tchoukaillon (or Mancala) solitaire | telephone country codes | tennis ball problem (1) | tensors | terminology in OEIS | ternary continued fractions | ternary quadratic forms | ternary representation | ternary words | tetrahedral lattice | tetrahedral numbers, sums of | tetrahedral numbers | tetrahedron, coloring a | tetrahedron, truncated | tetrahedron | | theta series of A_3 lattice | theta series of b.c.c. lattice | theta series of Coxeter-Todd lattice | theta series of cubic lattice | theta series of diamond lattice | theta series of D_3 lattice | theta series of f.c.c. lattice | theta series of h.c.p. | theta series of hexagonal lattice | theta series of hexagonal net | theta series of laminated lattices | theta series of Leech lattice | theta series of P_{10b} packing | theta series of square lattice | theta series of Z lattice | theta series | theta(n) | theta_2(q) | theta_3(q) | theta_4(q) | Third One Lucky game | three-way splitting of integers | threshold functions | threshold graphs | Thue-Morse sequence | Thue-Morse ternary sequences (closed under a->abc, b->ac, c->b) | tic-tac-toe | tie, tying a | tiered orders | topologies | topologies, connected | torus, cubes in | torus, slicing a | total height of rooted trees | total orders | total partitions | totative | totient function phi(n) | totient function phi(n), does not take these values | totient function phi(n), half-totient function | totient function phi(n), inverse to | totient function phi(n), iterating | totient function phi(n), values of | tough polyhedra | tournaments, automorphism groups of | tournaments, number of different tournament graphs | tournaments, outcomes of | tournaments, rigid | tournaments, round-robin | tournaments, score sequences in | tournaments, tournament sequences | tournaments | tours, rook | Tower of Hanoi | Towers of Hanoi | traffic light problem | trailing zeros in n! | trails | transform, hyperbinomial | transformations on unit interval | translation planes | transpositions | tree, unary-binary, definition | trees | trees, 2-colored | trees, achiral | trees, alternating | trees, asymmetric | trees, automorphism group of | trees, average height of rooted labeled | trees, balanced ordered | trees, balanced | trees, bicentered | trees, binary rooted | trees, binary | trees, bisectable | trees, boron | trees, by diameter | trees, by height | trees, by stability index | trees, by valency | trees, carbon | trees, centered | trees, chiral | trees, codes for | trees, complexity | trees, constant | trees, coordination sequence | trees, diameter 3 | trees, diameter 4 | trees, diameter 5 | trees, diameter 6 | trees, diameter 7 | trees, diameter 8 | trees, diameter of | trees, directed | trees, E-type | trees, endpoints | trees, evolutionary | trees, exchange | trees, exponentiation of e.g.f. | trees, fat | trees, fixed points in | trees, free | trees, graceful | trees, Greg | trees, heap ordered | trees, hexagon | trees, homeomorphically irreducible | trees, Husimi | trees, identity | trees, in wheel | trees, indecomposable | trees, intransitive | trees, labeled | trees, leaves | trees, Leech labeling problem | trees, leftist | trees, log of e.g.f. | trees, M-type | trees, matched | trees, nodes | trees, of subsets | trees, oriented | trees, partially labeled | trees, path length | trees, permutation | trees, phylogenetic | trees, planar | trees, plane | trees, planted | trees, powers of g.f. | trees, projective plane | trees, quartic | trees, Ramsey theorem for | trees, reversion of g.f. | trees, rooted | trees, rotation distance between | trees, Schroeder | trees, series-reduced | trees, shapes of | trees, signed | trees, spanning (1) | trees, spectra of | trees, squares of | trees, stability index of | trees, stable | trees, Steiner | trees, steric | trees, symmetries in | trees, ternary | trees, triangle of | trees, trimmed | trees, two-colored | trees, Weiner index | trees, with a forbidden limb | trees, with bicentroid | trees, with centroid | trees | tri-perfect numbers | triangle of x+y, etc. | triangle, coloring a | Triangles | triangles, by perimeter | triangular arrays of integers, enumeration of special | triangular arrays, indexing | triangular lattice | triangular numbers, partitions into | triangular numbers, sums of three | triangular numbers, sums of two | triangular numbers | triangular square numbers | triangular triples | triangulations (1) | tribulations game, remoteness numbers | tribulations | tricapped prism | trigonometric functions which either increase or decrease monotonically | trinomials over GF(2) , | triperfect numbers | triple factorial numbers | triply perfect numbers | truncatable primes | truncated cube | truncated octahedron | truncated square numbers | truncated tetrahedron | Turing machines which halt | Turing machines | Turkish | twin primes conjecture | twin primes constant | twin primes | two consecutive residues | two-way infinite sequences (01) | two-way splittings of integers | twopins positions
- U-coordinates for arrays | ugly numbers | Ulam numbers | Ulam-type sequences | unary representations | unary-binary tree, definition | undecagon is spelled 11-gon in the OEIS | undulating numbers | undulating squares | unexplained differences between partition generating functions | unhappy numbers | unidecagon is spelled 11-gon in the OEIS | Unions and sums | unique factorization domains | unit fractions | unit interval graphs | unitary amicable numbers | unitary divisors of n, number of | unitary divisors of n, sum of | unitary divisors | unitary perfect numbers | unitary phi (or unitary totient) function uphi | unpredictable sequence | unsolved problems in number theory | untouchable numbers | up-down permutations | urinals | urns | usigma(n)
- vampire numbers | Van der Pol numbers | Van der Waerden numbers | Van Lier sequences | Vassiliev invariants | vector fields | vertex diagrams | vertex operator algebras (1) | very rapidly growing sequences | Von Staudt-Clausen representation | voting schemes
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- years, perfect | years | yoke-chains | Young tableaux
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- 2 is a 4th power residue mod p | 2 is a cubic residue mod p | 2 is a square mod p | 2 is an mth power residue mod p | 2-adic valuation | 2-graphs | 2-plexes | 20-gonal numbers | 21-gonal numbers | 22-gonal numbers | 23-gonal numbers | 23-smooth numbers | 2^2^n | 2^n + 1, primes dividing | 2^n + 2 is divisible by n | 2^n - 1 | 2^n/n | 2^n | 2^p - 1
- 3-adic valuation | 3-almost primes | 3-colored | 3-connected | 3-gonal numbers | 3-plexes | 3-polyhedra | 3-smooth numbers | 3-trees | 3n - sigma(n) | 3x+1 problem , | 3x+1 problem, (01) | 3x+1 problem, J. C. Lagarias's | 3^n
- 4-gonal numbers | 4^n
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- 6-gonal numbers | 6^n
- 7-gonal numbers | 7-smooth numbers | 7^n
- 8-gonal numbers | 8^n
- 9-gonal numbers | 9-ish numbers | 9^n
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Millennium Problems
In order to celebrate mathematics in the new millennium, The Clay Mathematics Institute of Cambridge, Massachusetts (CMI) has named seven Prize Problems. The Scientific Advisory Board of CMI selected these problems, focusing on important classic questions that have resisted solution over the years. The Board of Directors of CMI designated a $7 million prize fund for the solution to these problems, with $1 million allocated to each. During the Millennium Meeting held on May 24, 2000 at the Collège de France, Timothy Gowers presented a lecture entitled "The Importance of Mathematics", aimed for the general public, while John Tate and Michael Atiyah spoke on the problems. The CMI invited specialists to formulate each problem.
One hundred years earlier, on August 8, 1900, David Hilbert delivered his famous lecture about open mathematical problems at the second International Congress of Mathematicians in Paris. This influenced our decision to announce the millennium problems as the central theme of a Paris meeting.
The rules for the award of the prize have the endorsement of the CMI Scientific Advisory Board and the approval of the Directors. The members of these boards have the responsibility to preserve the nature, the integrity, and the spirit of this prize.
Paris, May 24, 2000
- Birch and Swinnerton-Dyer Conjecture
- Hodge Conjecture
- Navier-Stokes Equations
- P vs NP
- Poincaré Conjecture
- Riemann Hypothesis
- Yang-Mills Theory
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Cake cutting | Calendar Magic | Calisson | Candy's Theorem | Canonical Equation of Conics | Canter | Cantor Function | Cantor set | Cantor set of positive measure | C0 - C0 = [-1, 1] | Cantor's lines | Cantor's point | Cantor's theorem | Cantor-Bernstein-Schroeder theorem | Cardinal | Cardinal Numbers | Cardioid | Hearty Munching on Cardioids | Carnot's Theorem | Carnot's Theorem | Carnot's Theorem (Generalization of Wallace's Theorem on simsons) | Carpets Theorem | Cartesian coordinates | Cartesian coordinate system | Cassini's identity | Cauchy's inequality | Cauchy's limit | Cauchy sequence | Cayley's Lines | Cayley Table | Cayley's Theorem | Ceiling Function | Cellular Automaton | Center Circles | Center of Inversion | Centipede | Central Angle | Central Similarity | Central Limit Theorem | Centroid | Centroid, a Characteristic Property Of | Circle of Similitude | Ceva's Theorem | The Menelaus theorem | Trigonometric Form of Ceva's Theorem | Ceva is equivalent to Menelaus I | Ceva is equivalent to Menelaus II | Cevian | Cevian Nest | Cevian Triangle | Cevian Triangle | Changing Colors | | Characteristic Constants of the n-Line | Characteristic function | Chasles' Theorem | Chebyshev polynomials | Checker - jumping | Chevalier de Méré's Problem | Chinese Remainder Theorem | Chord (in circle) | Chronology of updates | Chvatal's Art Gallery Theorem | Circle | Circle of Apollonius | Circle of Concern | Circle of Influence | Circle-Stacking Theorem | Circuit, same as cycle | Circulant Matrix | Circulant Matrix: example | Circular Coordinates | Circular Poggendorff Illusion | Circumference | Circum-orthic Triangle | Circumcenter | Existence of circumcenter | Circumcevian Triangle | Circumcircle | Circumpedal Triangle | Circumradius | Circumscriptible Polygon | City-block distance | Classification of Triangles | Cleavance center | Cleaver | Clifford's Chain | Clifford's Lemma | Clinant | Closed interval | Closed set | Closed walk | Closure | Coalition | Coaxal circles | Coaxal Circles Theorem | Coincidence | Collatz Conjecture | Collinear points | Collage Theorem | Column vector | Combination | Combinatorial Proofs | Common Notions of Euclid | Commutative group | Commutativity | Compact sets | Compactification | Comparing complex numbers. | Complementary angles | Complementary event | Complementary sequences | Complete graph | Complete graph | Complete order | Complete Quadrangle | Complete Quadrilateral | Complete Quadrilateral, the Theorem of | Diagonals of the complete quadrilateral | Completeness | Complex numbers | Complex product of Complex Numbers | Complex Projective Line | Composite number | Composition | Compound Interest | Concatenation | Concentric circles | Concentric polygons | Concurrent lines | Concyclic Points | Conditional Probability | Cone | Conformal | Congruent sets | Congruence Modulo a Number | Conic Sections | Conjugate complex numbers | Conjugate coordinates | Conjugate families of circles | Conjugate points | Conjunction | Connected component of a graph | Connected graph | Connected set | Consecutive Isosceles Decomposition | Consistency | Constant Chord | Constant Random Variable | Constant width, shapes of | Constructible Numbers | Construction problem | Construction with the Compass Only | Rusty Compass Construction of Equilateral Triangle | Contact Triangle | Content | Continuity | Continued fraction | Continued Fractions on the Stern-Brocot Tree | Continuous Line Illusion | Continuum | Continuum Hypothesis | Contraction Mapping Theorem | Convergent | Convergent Series | Converse theorem | Convex hull | Convex shape | Convolution | Convolution | Conway's Circle | Coolidge theorem | Coordinate | Copernicus' Theorem | Proof of Copernicus' Theorem | Coprime | Cosine Rule | Another derivation | In a vector space | Countable sets | Counting: The Beginning of Mathematics | Let Them Count | Counting numbers | Covering A Chessboard With Domino | Critical Player | Cross product | Crosscap | Crossing Diagonal Method | Crossing Number of a Graph | Crossover | Cross-Ratio | Crossed-Lines Construction of Shapes of Constant Width | Curve | CTK Exchange, the | Curry's Paradox | Curvilinear triangle | Cutting pipes | Cycle (a permutation) | Cycle (on a graph), same as circuit | Cyclic quadrilaterals | Brahmagupta's Theorem | Four Concurrent Lines in a Cyclic Quadrilateral | Incenters in Cyclic Quadrilateral | Remarkable Line in Cyclic Quadrilateral | Cycloids | Cyclotomic Equation | Cylinder
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Galois Extensions | Game of Hex | Gauss-Bodenmiller theorem | Gaussian coordinates | Gaussian Integers | Gelfond's theorem, Gelgond-Schneider theorem | General Equation of a Straight Line | General position, lines in | Generalization in Mathematics | Generating Function | | Geoboard, Toying with the | Geometric Constructions | Geometric mean | Arithmetic-Geometric Mean Inequality | Geometric series | Sum of the geometric series | Gerbino's Illusion | Gergonne and Van Obel Theorems | Gergonne Line | Gergonne Point | Gergonne Triangle | Gergonne's Magic Trick | Gibbs, Josiah Willard | Gion Shrine Problem | Girth of a Graph | GLaD Construction | Glide Reflection | Glider | Gnomon | Golden rectangle | Golden section | | Golden triangle | Gordius, King of Phrygia | Gradient | Graph | Graph Theory | Gray codes | Greatest Common Divisor (gcd) | Grebe's theorem | Griffiths' theorem | Group | Groupoid | Growth | Grundy number | Grundy's Game | Guest book
Hadwiger-Finsler Inequality | Half-axis | Half-turn | Hamilton Cycle | Hamilton Path | Hamilton's Method of Apportionment | Hamilton's theorem | Hamming Distance | Handshake Lemma | Hanoi, Tower of | Happy 8 puzzle | Harmonic bundle | Harmonic conjugates | Harmonic Mean | Harmonic mean | Harmonic Ratio | Harmonic Series | Harmonic Triangle | Hart Circle | Hart's Inversor | Hart's Theorem | Haruki's Lemma | Haruki's Theorem | Hausdorff Distance | Heaviside Step Function | Height in a triangle, formula for | Hering Illusion | Heptadecagon | Heptagon | Heredity | Hering Illusion | Hermann Grid Illusion | Heron's Formula | Derivation | Heron's Problem | Hex Can't End in a Draw | Hex numbers | Hilbert's Plane Filling Curve | Hjelmslev Theorem | Hofstadter Triangles and Points | Hole | Homogeneous Coordinates | Homogeneous Functions | Homologous | Homothety | Homothety | Hooked Langford Sequence | Hooper's Paradox | Hopf Fibration | Horner's Method | Hour glass timing | How to define a permutation? | Human calculators | Hummer's Mind Reader | Huntington-Hill's Method of Apportionment | Huzita's axioms of paper folding | Hyperbinary expansion | Hypercube | Hypocycloids
Ian McGee's Observation | Ideal elements | Ideal subrings | Idempotent law | Identical Twins Puzzle | Identical Transformation | Identity | Identity matrix | iff | Illusory contour | Imaginary numbers | Imaginary Part of a Complex Number | Impartial game | A second look | Implication | Implicit Equation of a Straight Line | Impossible Fork | Impossible page | Improper Isometry | Incidence matrix | Incident points and lines | Incenter | Incenters in Cyclic Quadrilateral | Incircle | Inclusion-Exclusion Principle | Marriage Theorem | Incompatible Events | Indegree | Independent Events | Index | Index Lemma, the | Induction | Mathematical Induction: examples | Infinite Products | Convergent | Divergent | Convergent to 0 | Divergent to 0 | Infinite Sums | Infinitesimal | Infinitude of Primes | Infinity | Infinity and Probability | What Is Infinity? | Inflectin point | Injective function | Inner product | Inner Soddy circle | Inradius | Inscribed Angles | Inscriptable (inscriptible) Polygons | Integer | Integer Iterations on a Circle | Integral Domain | Integration in Spherical Coordinates | Intercept | Intercepted Arc | Interest Calculations | Interior Angles formed by a secant and parallel lines | Intermediate Value Theorem | Internal point | Internal Tangent to Two Circles | Internet News sites | Intersecting Chords (Secants) Theorem | Intersecting Chords Theorem | Intersection of sets | Intouch Triangle | Invariant Set | Inventor's Paradox | Inverse Element | Inversion (combinatorial) | Inversion (geometric) | Inverse image | Inversion | Inversion Tool | Inversion with Negative Power | Invert the Divisor and Multiply | Involution | Irrational numbers | ISBN Encoding | Isogonal lines | Isogonal circle | Isogonal conjugates | Isogonal Conjugality | Isogonal Concurrency | Isogonic Centers | Isolated point | Isometry | Plane Isometries | Isosceles trapezoid | Isosceles triangle | Isotomic lines | Isotomic conjugates | Isoperimetric Inequality | Isoperimetric Theorem | Isoperimetric Theorem, a variation on | Isoperimetric Theorem, a different one | Iterative process
Jam | Japanese Theorem I | Japanese Theorem II | Jefferson's Method of Apportionment | Joachimsthal's Notations | John Sharp's Paradox | Johnson-Trotter Method | Jordan Curve Theorem | Josephus Flavius, a legend and a problem | Judd Illusion | Julia sets | Jump
Kanizsa Triangle | Kantor's Circle | Kaprekar's constant | Kaprekar's process | Kayles | Grundy function for Kayles | Kiepert's Theorem | Kiepert's Perspector | Kirkman's Theorem | Pascal Lines: Steiner and Kirkman Theorems II | Kitaoka's waves | Kite | Klein Bottle | Knots | Koch's snowflake | Koffka's Illusion | Königsberg | Königsberg Bridges | Kronecker's symbol | Kurschak's Tile and Theorem
La Hire's Theorem | La Hire's Theorem, a Variant | Joachimsthal's Notations | Labeling of a graph | Lagrange's identity | Lagrange's polynomial | Lambert's Theorem | Langford Sequence | Langman's Paradox | Latin Squares | Lattice | Lattice | Lattice Multiplication | Latus Rectum | Law of Cosines | Law of excluded middle | Law of Sines | La Hire's Theorem | Leaf | Leap Year | Least Common Multiple (lcm) | | Left-handedness | Leibniz Triangle | Legendre's Lemma | Lemoine antiparallel | Lemoine circles | Lemoine hexagon | Lemoine line | Lemoine point | Léon Anne's theorem | Leonardo's Claws | The Lepidoptera of the Circles | The Lepidoptera of the Quadrilateral | Level Curve | Levenshtein Distance | Lewis Carroll's problem | Lewis Carroll's Game of Logic | Lexicographic order | Complex numbers | Permutations | Pigeonhole | Lights Out | Lindemann's Theorem | Line at infinity | Line at Infinity | Line at infinity | Linear combination | Linear function | Linear order | Linear Programming | Linear Space | Linear Transformation | Lines in general position | Liouville's Theorem | Listing all permutations | Locus | Logarithmic spiral | Logistic equation | Logistic model | Logo of mine | Lone Divider Method | Long Multiplication | Long Runs of Composite Numbers | Looking your best | Loop | Looping Chips | Lowest Terms, fraction in | Loyd's Puzzles, Sam | Can you better Sam Loyd? | Farmer and Wife To Catch Rooster and Hen | Sam Loyd's fifteen | Sam Loyd's Geometric Puzzle | Plato's Geometric Numbers: can you better Sam Loyd? | Puzzlist vs. Mathematician: A Practical Arithmetic Problem | The Gordian Knot | Lubell-Yamamoto-Meshalkin inequality | Lucas' Theorem | Lucas' Theorem | Lucky 7 puzzle. | Lune | LYM inequality
Maclaurin series | Magic in Square | Magic Squares game | Mahavira's theorem | Major circle of ellipse | Malfatti's Problem | Maltitude | Mandelbrot and Julia sets | Iterations and the Mandelbrot Set | Manifesto | Markers Method of Fair Division | Marriage Theorem | Mascheroni constructions | MasterMind, Invitation to | Mathematical Expectation | Matrix | A Matter of Age | Maximal Chain | Maxwell's Theorem | Mean Proportional | Mean Value Theorem | Rolle's and Mean Value Theorems | Mechanical Proof of the Pythagorean Theorem | Medial triangle | Median triangle | Median | Median Point | Medians in a Quadrilateral | Mediant fraction | Mediant fraction | Dots and Fractions | Mendelsohn triple | Menelaus Theorem | Menelaus is equivalent to Ceva I | Menelaus is equivalent to Ceva II | Merlin's Magic Squares | Metric Completeness | Metric Function | Metric Space | Mex rule | Mex rule | Midline (in a triangle) | Millipede | Mini Lights Out | Minimal Lattice | Minimal-change Property | Minimax Principle | Minimax Theorem | Minimum element in a poset | Minimum Spanning Tree | Minimum Spanning Tree | Minkowski addition | Minkowski subtraction | Miquel's Point | Miquel's Point of a 4-line | Miquel's Point of a 4-line Via Spiral Similarity | Mirror property of the orthic triangle | Mirror property of the altitudes | Mixed Product of Three Vectors | Mixtilinear Circles and Concurrence | Radius and Construction of a Mixtilinear Circle | Mobile Integer | Model | Modulo Arithmetic | Modus ponens | Modus tollens | Möbius Strip | Möbius Transformations | Möbius Transformations | Mohr-Mascheroni construction | Monge Point | Monge's Circle Theorem | Monge via Desargues | Monge via Desargues II | Monic polynomial | Monomino | Monty Hall Dilemma | Moore's Plane Filling Curve | Morley's Miracle | Doodling and Miracles | Most likely number of successes in Bernoulli trials | Müller-Lyer Illusion | Most elementary theorem of Euclidean geometry | Multiplicand | Multiplication | Multiplicative Inverse | Multiset | Mutation | Mutually Exclusive Events | Mutually prime integers | My logo
9-Point Center | 9-Point Circle | 9-point Circle as a Locus of Concurrency | Nagel Line | Nagel Point | Nagel Point | Nagel's Theorem | Napier Bones | Near | Necker Cube | Negative Powers | Neighborhood | Nephroid | Neuberg's Theorem | Neuberg's Theorem, Synthetic proof | Neusis Constructions | Neusis | New-States Paradox | Newton-Gauss line | Newton's line | Newton's Theorem | Nim, the game of | Nimble | Nimber | Nim-heap | Nim-sum | Nine-point Circle | Nine dots problem | n-Line | Characteristic Constants of the n-Line | Nobbs' points | Non-degenerate conic | Non-Eclidean Geometries | Nonseparable Graph | Norm (of a vector) | Normal play | Normal topological space | Northcott's game | Nothing is something | Nowhere dense set | N-position | Number Line | Number Mosaics | Number System With Base 36 | Numbers, types of | Numbers, Conway's definition | Numeral
1-1 correspondence | Obtuse angle | Obtuse angle | Obtuse triangle | Odd permutation | Odd vertex | Oddball Problem | Old Japanese Theorem | Olof Hanner's Jigsaw Puzzle | One Pile | One-sided surfaces | Open interval | Open sets | Open walk | Operations on Fractions | Operator | Oppositely Congruent Figures | Optical Illusion | Optical Property of Ellipse | Orbison Illusion | Orbits | Ordinary Generating Function | Ordinate | Orientation | Origami | Orthic Axis | Orthic triangle | Mirror property of the orthic triangle | Orthocenter | Orthocentric system | Orthodiagonal Quadrilateral | Orthodiagonal Quadrilateral | Orthogonal latin squares | Orthogonality | Orthologic Triangles | Orthopole | Other math sites | Outdegree | Outer Soddy circle
Pace | Packing shapes in a plane | Paper Folding | Paper strip activities | Pappus' Generalization of Euclid I.47 | Pappus' Theorem | Dual to Pappus' Theorem | Parabola | Parabola as locus | Parabolic Mirror | Parabola as Envelope | Paradox | Paradox of Tristram Shandy | Parahexagon | Parallel Lines | Parallel Postulate | Parallelogram | Parallelogram Law | Parallelogram law | Parallelogram rule | Parametric Equation of a Straight Line | Parity | Parity of a node on a graph | Parpolygon | Parrondo Paradox | Partial order | Partial order in a lattice | Partition of point sets in a plane | Partitioning a Circle | Partizan game | A second look | Pascal's line | Pascal Lines: Steiner and Kirkman Theorems | Pascal's theorem | Pascal's theorem | Pascal's Theorem, Homogeneous Coordinates | Pascal in a Cyclic Quadrilateral | Pascal in Ellipse | | Pascal Triangle and the Tower of Hanoi | | Peasant Multiplication | Peaucellier Linkage | Pedal circle | Pedal curve | Pedal point | Pedal triangle | Pedal Construction of Parabola | Pedal Curve | Pedoe's Theorem | Peg Solitaire | Peg Solitaire and Group Theory | Pencil of circles | Pencil of lines | Munching on Inscribed Angles | Penosculant | | Percent | Perfect Langford Sequence | Perfect number | Perimeter Splitter | Periodic cycle | Periodic patterns in sin(N) | Permutation | How to define a permutation | Listing all permutations | What, how, and the Web | Perpendicular bisector | Perspective triangles | Triangles perspective from a point | Triangles perspective from a line | Petersen Graph | p | Determination of p | Pick's Theorem | Pigeonhole principle | Pigeonhole Principle and Extensions | Pitot's Theorem | Pivot Theorem | Pivotal Player | Place Value | Plainim | Plainim Misère | Planar graph | Planar polygon | Plane Filling Curves | Plato's Geometric Numbers: can you better Sam Loyd? | Plane Isometry | Plücker's Lines | Plus or Minus | Poggendorff Illusion | Poggendorff Illusion II | Circular Poggendorff Illusion | Point | Point at infinity | Point at Infinity | Point at Infinity | Point at infinity | Point Circle | Point in a Square | Point inside a triangle | Point-Slope Equation of a Straight Line | Polar Circle | Polar coordinates | Polar Equation of conics | Pole and Polar with respect to an angle | Pole and Polar with respect to a circle | Pole and Polar with respect to a triangle | Pole | Poles and Polars | Polygon | Polygonal number | Polyomino | Pompeiu's Triangle and Theorem | Poncelet's Porism | Poncelet's Theorem | Ponzo Illusion | Ponzo Illusion II | Population Paradox | Porism | Poset | Positional Number System | Positive (Counterclockwise) Direction | Power of a point with respect to a circle | Power of a point with respect to a circle | Power of Inversion | P-position | | Prime | | Primorial | Principal | Principle of Conservation of Angular Momentum | Principle of Duality | Principle of Proportionality | Prisoner's Dilemma | Probability | Probability density function | Probability distribution function | Probability paradoxes | Product of Rotations | Sequential Rule | Proizvolov's Identity | Projection | Projective Geometry | Projective Plane | Pronk | Proofs Without Words | Proof | Proper Isometry | Property of an | Protractor Postulate | Ptolemy's Inequality | Ptolemy's Theorem | Ptolemy's Second Theorem | Ptolemy's Theorem implies Pythagoras' Theorem | Sine, Cosine, and Ptolemy's Theorem | Euler's Generalization of Ptolemy's Theorem | Pure existence proof | Purely Imaginary Complex Numbers | puz(G), the graph of a puzzle | Puzzles on graphs | Pythagoras' Theorem | Mechanical Proof of the Pythagorean Theorem | "Extra-geometric" proofs of the Pythagorean Theorem | Pythagorean Theorem, Generalizations of | Pythagorean triples | Proof of the formula | Norm of Gaussian integers | Gaussian integers | Divisibilty in Pythagorean Triples | Trinary Tree(s) underlying Primitive Pythagorean Triples
Quadratic Average | Quadratic Field | Quadratic Formula | Completing a Square and Quadratic Formula | Quadratic Function | Quadrilateral | Quarter-turn | Quasigroup | Quasigroups and block designs from triangular grid | Quota Rule | Math Quotations, a Web poll
Rabinowitz's Point | Radical axis of two circles | Radical center of three circles | Radius of Inversion | Radius-vector | Radix | Ramsey Number | Ramsey Theorem | Ramsey's Theorem is Sharp | Random Dot Stereograms | Random Event | Random Number Generator | Random Variable | Rational approximation | Rational integers | Rational numbers | Rational Root Theorem | Ray | Real numbers | Real Part of a Complex Number | Real Product of two Complex Numbers | Real Product of Complex Numbers | Rectangle | Recurrence relations | Reflection | Reflection in Line | Reflection in Point | Reflection Law in Ellipse | Reflective Properties of Parabola | Reflex angle | Reflexivity | Reflexivity | Regular polygon | Regular polyhedron | Relatively prime integers | Removable singularity | Repelling point | Rep-tile | Residue classes | Retinend | Reuleaux's triangle | Reverse Solitaire | Reversible move | Revolving Circles Illusion | Rhombus | Riemann Sphere | Right angle | Right triangle | Right-handedness | Rigid motions | Rigid motion | Ring | Rolle's Theorem, the | Rolle's and Mean Value Theorems | Rotocenter | Rottary gallop | Rounding numbers | Rouse Ball's Fallacy | Routh's Theorem | Row vector | Russell's Paradox | Russian Peasant Multiplication | Rusty Compass Construction of Equilateral Triangle
7 Coins Puzzle | Sam Loyd's Puzzles | Can you better Sam Loyd? | Farmer and Wife To Catch Rooster and Hen | Sam Loyd's fifteen | Sam Loyd's Geometric Puzzle | Plato's Geometric Numbers: can you better Sam Loyd? | Puzzlist vs. Mathematician: A Practical Arithmetic Problem | The Gordian Knot | Saddle Point | Salinon: From Archimedes' Book of Lemmas | Salmon's points | Sam Loyd's fifteen | Sam Loyd's Geometric Puzzle | Sam Loyd's Son's Dissection | Sangaku | SaRD Construction | SAS | Sawayama's Lemma | Scalar | Scalar product | Scalene triangle | Schlafli formula | Scintillating Grid Illusion | Scoring | Scoring Misère | Scrub Tile Puzzle | Sealed Bids Method | Secant | Secant | Secant Angles | Sector (in circle) | See-Saw Lemma | Segment (in circle) | Self-conjugate triangle | Self-descriptive Strings | Self-documenting sentences | Self-dual configuration | Self-reference | Semigroup | Sending scouts into the desert | Separating point sets with a circle | Sequential Coalition | Sequential compactness | Sequential Rule | Set | Seven Circles Theorem | Seven Terms Periodic Sequence | Shaggy Dog Theorem | Shape Property | Shapes in a lattice | Shapes of constant width | Star Construction of Shapes of Constant Width | Shapley-Shubik Index | Shearing Transform | Shepard's Parallelogram Illusion | Shift | Shimmer Illusion | Ship-Dock Theorem | Shredding the torus | Shuttle Puzzle | Sibling node | | Sieve of Eratosthenes | s-Algebra | Signed segment | Signum function, sign(t) | Silver Dollar Game | Similar orderings | Similar Triangles, theorem of | Similarity | Similarity dimension | Simple continued fraction | Simple graph | Simple polygon | Simplex | Simson Line | Simson Line From Isogonal Perspective | Simson-Wallace Theorem | Simpson's paradox | Simultaneous Contrast Illusion | Simultaneous Contrast Illusion II | Sine Law | Single formula to express any integer with digits 0-9 each used just once | Six Circles Theorem (Elkies) | Six Circles Theorem (Evelyn) | Skolem Sequence | Slider Puzzles | Sliders | Slope | Slope-Intercept Equation of a Straight Line | Social Choice | Soddy circles | Soddy Line | Soddy points | Solitaire in the Circle | Somos-4 Sequence | Soriteses | Soroban | | Spanning Tree | Sperner's Labeling | Sperner's Lemma | Sperner's Theorem | Sphere | Spieker Center | Spira Mirabilis | Spiral Similarity | Miquel's Point of a 4-line Via Spiral Similarity | Splitter | Splitting piles | Sprague-Grundy function | Square | Square numbers | Squaring a circle | Squaring Parabola | Squares and Circles | Squares, Circles, and Triangles | Squinting Eyes Theorem | SSS | Stability | Staircase Illusion | Star Polygons | Star-shaped region | Stationary point | Steiner Porism | Steiner Porism | Steiner's Circle | Steiner's Curve | Steiner's Theorem | Steiner's and Kirkman's Theorems | Pascal Lines: Steiner and Kirkman Theorems II | Steinhaus' Problem | Stereographic projection | Stereographic Projection and Radical Axes | Stern-Brocot Tree | Binary Encoding with the Stern-Brocot tree | Stern's diatomic series | Stewart's theorem | Straight angle | Straight Edge Only Construction of Polar w.r.t. an Angle | Straight Edge Only Construction of Polar w.r.t. a Circle | Straight line | Straight Tromino | Strategy stealing in the game of Hex | Stretching | Suan pan | Subfactorial | Subgraph | Subgroup | Subjective contour | Subtraction Game | Successor | Sum Rule | Sums and Products | Superfactorial | Supergraph | Superposition Principle | Supplementary angles | Supporting lines of convex figures | Surjective function | Surreal numbers | Surreal Numbers | Surreal numbers and games | Syllogism, Modes Ponens | Syllogism, more general | Symmedian | Symmedian point | Symmetric difference | Symmetric group | Symmetric relation | Symmetry in circle | Synthetic division | System of Distinct Representatives
3-4-5 triangle | 3 glass puzzle | 3 glass puzzle, second, more general tool | T-Junction | Take-Away Games | Tangent Pair | Tangent to two circles | Circle of Similitude | Tangential triangle | Tarski-Banach Decompositions | Taylor circle | Taylor series | Telescoping Series | Ternary number system | Tessellation | Tesseract | Thales' Theorem Thales' Theorem | The only rational triangle | The Two Men of Tibet Problem | Thébault's Problem I | Thébault's Problem II | Thébault's Problem III | Theorem of Complete Quadrilateral | Theorem of Theaetetus | Thick set | Thin set | Third Proportional | Three pancakes problem | Three 3s | Three 4s | Three 5s | Three Circles and Common Tangents | Three circles problem | Three Tangents Theorem | Three Tangents Theorem | Toads And Frogs Puzzle | Toothpick Construction of a Square | Topological equivalence | Topological Space | Topological transformation | Topology | Torque | Torricelli point"> | Torus | Totality (of a binary relation) | Totally ordered set | Tower of Hanoi | | Bicolor Towers of Hanoi | Toys and Tots problem | Trachoids | Train wheel quandary | Transcendental numbers | Transfinite Cardinality | Transitive groups | Transitivity in Action | Transivity | Translation Transform | Transpose of a vector | Transposition | What, how, and the Web | Transversal | Transverse gallop | Trapezoid | Tree | Trefoil knot | Trema | Triangle Classification | Triangle Inequality, the | Axioms of distance function | Absolute Value of Complex Number | Triangular Billiard | Triangular numbers | There exist triangular numbers that are also square | Triangulation | Diagonal Triangulation | Trichotomy, the law of | Trigonometric Form of Ceva's Theorem | Trigonometric Form of Complex Numbers | Trilateral Diagrams | Trilinear Coordinates | Trivial Ramsey Number | Tromino | Straight Tromino | Trot | Tucker Circles | Turn | Turning Turtles | Tweedledum & Tweedledee Principle | Twin Circles of Archimedes | Twisted Cord Illusion | Two Butterflies Theorem | Two Butterflies Theorem II | Two coins puzzle | Two Envelopes Paradox | Two Numbers Guessing Game | Two-squares identity
Uniform convergence | Uniform probability distribution | Union of sets | Unique Factorization Theorem | Uniqueness of number 30 | Unit element | Unit Fraction | Unity | UPC-A encoding | Urquhart's Theorem | Urquhart's Theorem
Value | Van Obel Theorem, Determinants, and Barycentric Coordinates | Gergonne and Van Obel Theorems | Van Aubel's theorem | Van Schooten's theorem | Van Schooten's and Pompeiu's Theorems | Varignon Parallelogram | Vecten's Collinearity | Vecten configuration | Vecten Points | Vector | Vector product | Vector Space | Verging | Versine | Vertex | Vertical Angles | Viète's Formulas | Viewing a Statue | Viviani's Theorem
Walk on a graph | Wallace-Bolyai-Gerwien Theorem | Wallace's Theorem | Wasan | Web poll | Webster's Method of Apportionment | Weierstrass-Bolzano property | Weierstrass Product Inequality | Weighted graph | Weird curves bound normal areas | Weitzenböck Inequality | Well-ordered set | What do you call a centipede? | White's Illusion | Whole number | Whole part function, [x] | Wilson's Theorem | Wittenbauer's Parallelogram | Word Ladders | Word Problems | Wunderlich's Plane Filling Curves | Wundt Illusion | Wundt Block Illusion | Wythoff's Nim | Wythoff's Nim II
XOR | X-intercept
Y Can't End in a Draw | Y-intercept | Y. Sawayama's Lemma | Yosan
0!=1 | Zaslavsky's theorem | Zeckendorf's theorem | Zenodorus Theorem | Zero | Zipf's Law | Zöllner's Illusion
- Hyperspace Star Polytope Slicer Java 1.1
- Counter-Rotating Spirals Illusion Java 1.0
- Animated Necker Cube Java 1.0
- Waterman Polyhedra Java 1.1
- Stereoscopic Animated Hyperspace Objects Java 1.0
- (These are all based on the same Java applet.)
- HyperCube
- 24-Cell Object
- Cross-Polytope
- Simplex
- The java source code for these. (This was my first Java applet.)
- My Critique of My Code.
- Stellations of the Dodecahedron Java 1.1
- Fechner Color Illusion Java 1.1
- #1 (Linear motion) Plus explanations.
- #2 (Rotating)
- Guestbook (what other people saw)
- Morphing Polyhedron Java 1.1
- Pulfrich Illusion Java 1.0
- HyperSlice (Hyperspace Polytope Slicer) Java 1.2.
Word Combinations
This script reads a series of alphanumeric characters, and calculates how many unique ways those characters can be rearranged. If desired, it will display all of the combinations.
Jeff Miller Web Pages
- Images of Mathematicians on Postage Stamps
- Earliest Known Uses of Some of the Words of Mathematics
- Earliest Uses of Various Mathematical Symbols
- Ambiguously Defined Mathematical Terms at the High School Level
- A Collection of Word Oddities and Trivia
- History of Beckley, W. Va.
- History of American Broadcasting
- History of Broadcasting in West Virginia
- Some History of West Virginia
- Some Tampa-St. Petersburg Radio History
- History of Pasco County, Florida
- Gulf High School
- Woodrow Wilson High School Alumni
- Remembering WDBS at Duke University
Earliest Known Uses of Some of the Words of Mathematics
These pages show the names of the individuals who first used various common mathematical symbols, and the dates the symbols first appeared. The most important written source is the definitive A History of Mathematical Notations by Florian Cajori.
Zu jedem der folgenden Punkte gibt es eine eigen Seite mit ausführlichen Erklärungen:
- Symbols of operation, including +, -, X, division, exponents, radical symbol, dot and vector product
- Grouping symbols, including (), [], {}, vinculum
- Symbols of relation, including =, >, <
- Fractions, including decimals
- Symbols for various constants, such as pi, i, e, 0
- Symbols for variables
- Symbols to represent various functions, such as log, ln, absolute value; also the f(x) notation
- Symbols used in geometry
- Symbols used in trigonometry; also symbols for hyperbolic functions
- Symbols used in calculus
- Symbols for matrices and vectors
- Set notation and logic
- Symbols used in number theory
- Symbols used in probability and statistics
- Written sources for these pages
Ambiguously Defined Mathematical Terms at the High School Level
Krohn-Rhodes Complexity | Krohn-Rhodes Theorem | Krohn-Rhodes Theory
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Important : Le "Χ" à la fin du mot LaTeX n'est pas un x majuscule mais la lettre grecque "Chi" (dont le code HTML est Χ). LaTeX se prononce donc "lateck" et non "latecs".
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Inhaltsverzeichnis
- 1 Allgemeines 3
- 1.1 The Name of the Game . . . . . . . . . . . . . . . . . . . . . . . 3
- 1.1.1 TEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
- 1.1.2 LATEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
- 1.1.3 LATEX2" . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
- 1.2 Grundkonzept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
- 1.2.1 Autor, Designer und Setzer . . . . . . . . . . . . . . . . . 3
- 1.2.2 Layout-Design . . . . . . . . . . . . . . . . . . . . . . . . 4
- 1.2.3 Vor- und Nachteile . . . . . . . . . . . . . . . . . . . . . . 4
- 1.2.4 Der Arbeitsablauf . . . . . . . . . . . . . . . . . . . . . . 5
- 2 Eingabefile 5
- 2.1 Leerstellen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
- 2.2 LATEX-Befehle und Gruppen . . . . . . . . . . . . . . . . . . . . . 6
- 2.3 Kommentare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
- 2.4 Aufbau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
- 2.5 Dokumentklassen . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
- 2.6 Pakete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
- 2.7 Eingabezeichensatz . . . . . . . . . . . . . . . . . . . . . . . . . . 9
- 3 Setzen von Text 12
- 3.1 Deutschsprachige Texte . . . . . . . . . . . . . . . . . . . . . . . 12
- 3.2 Zeilen- und Seiten-Umbruch . . . . . . . . . . . . . . . . . . . . . 12
- 3.2.1 Blocksatz . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
- 3.2.2 Silbentrennung . . . . . . . . . . . . . . . . . . . . . . . . 13
- 3.3 Wortabstand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
- 3.4 Spezielle Zeichen . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
- 3.4.1 Anf¨uhrungszeichen . . . . . . . . . . . . . . . . . . . . . . 15
- 3.4.2 Binde- und Gedankenstriche . . . . . . . . . . . . . . . . . 15
- 3.4.3 Punkte . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
- 3.4.4 Ligaturen . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
- 3.4.5 Symbole, Akzente und besondere Buchstaben . . . . . . . 16
- 3.5 Kapitel und ¨Uberschriften . . . . . . . . . . . . . . . . . . . . . . 18
- 3.6 Fußnoten . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
- 3.7 Hervorgehobener Text . . . . . . . . . . . . . . . . . . . . . . . . 19
- 3.8 Hochgestellter Text . . . . . . . . . . . . . . . . . . . . . . . . . . 19
- 3.9 Umgebungen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
- 3.9.1 Zitate (quote, quotation, verse) . . . . . . . . . . . . . . . 19
- 3.9.2 Listen (itemize, enumerate, description) . . . . . . . . . . 20
- 3.9.3 Flattersatz (flushleft, flushright, center) . . . . . . . . . . 22
- 3.9.4 Direkte Ausgabe (verbatim, verb) . . . . . . . . . . . . . 22
- 3.9.5 Tabulatoren (tabbing) . . . . . . . . . . . . . . . . . . . . 22
- 3.9.6 Tabellen (tabular) . . . . . . . . . . . . . . . . . . . . . . 23
- 4 Setzen von mathematischen Formeln 24
- 4.1 Allgemeines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
- 4.2 Elemente in mathematischen Formeln . . . . . . . . . . . . . . . 25
- 4.3 Nebeneinander Setzen . . . . . . . . . . . . . . . . . . . . . . . . 28
- 4.4 ¨Ubereinander Setzen . . . . . . . . . . . . . . . . . . . . . . . . . 29
- 4.5 Liste der mathematischen Symbole . . . . . . . . . . . . . . . . . 30
- 5 Setzen von Bildern 34
- 6 Seitenaufbau 34
- 6.1 Kopf- und Fußzeilen . . . . . . . . . . . . . . . . . . . . . . . . . 34
- 6.2 Gleitobjekte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
- 6.2.1 Abbildungen (figure) . . . . . . . . . . . . . . . . . . . . . 35
- 6.2.2 Tabellen (table) . . . . . . . . . . . . . . . . . . . . . . . 36
- 7 Schriften 37
- 7.1 Schriftgr¨oßen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
- 7.2 Schriftstil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
- 7.3 Andere Schriftfamilien . . . . . . . . . . . . . . . . . . . . . . . . 39
- 7.4 Die ”europ¨aischen“ Zeichens¨atze . . . . . . . . . . . . . . . . . . 39
- 8 Spezialit¨aten 41
- 8.1 Abst¨ande . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
- 8.1.1 Zeilenabstand . . . . . . . . . . . . . . . . . . . . . . . . . 41
- 8.1.2 Spezielle horizontale Abst¨ande . . . . . . . . . . . . . . . 41
- 8.1.3 Spezielle vertikale Abst¨ande . . . . . . . . . . . . . . . . . 42
- 8.2 Briefe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
- 8.3 Literaturangaben . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
- 8.4 Zerbrechliche Befehle . . . . . . . . . . . . . . . . . . . . . . . . . 44
".LATEX", LaTeX Source Document (LaTeX3 Project)
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Der LATEX-Index
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Befehlsübersicht
| abovedisplayskip | abovedisplayshortskip | abstract | acute | addcontentsline | address | addtocontents | addtocounter | addtolength | addvspace | Alph | alph | and | appendix | arabic | array | arraycolsep | arrayrulewidth | arraystretch | author | bar | baselineskip | baselinestretch | begin | belowdisplayskip | belowdisplayshortskip | bf | bibitem | bigskip | bigskipamount | boldmath | bottomfraction | bottomnumber | breve | cal | caption | cc | center | centering | centerline | chapter | check | circle | cite | cleardoublepage | clearpage | cline | closing | columnsep | columnseprule | dashbox | date | dblfloatpagefraction | dblfloatsep | dbltextfloatsep | dbltopfraction | dbltopnumber | ddot | description | discretionary | displaymath | displaystyle | document | documentstyle | dot | dotfill | doublerulesep | em | encl | end | enumerate | eqnarray | equation | evensidemargin | extracolsep | fbox | fboxrule | fboxsep | figure | fill | floatpagefraction | floatsep | flushbottom | flushleft | flushright | fnsymbol | footheight | footnote | footnotemark | footnotesep | footnotesize | footnotetext | footskip | frac | frame | framebox | frenchspacing | fussy | fussypar | grave | hat | headheight | headsep | hfill | hline | hoffset | hrulefill | hspace | Huge | huge | imath | include | includeonly | indent | input | intextsep | it | item | itemize | itemsep | jmath | jot | kill | label | LARGE | Large | large | LaTeX | letter | line | linebreak | linethickness | listoffigures | listoftables | makebox | maketitle | marginpar | marginparpush | marginparsep | marginparwidth | markboth | markright | math | mathindent | mbox | medskip | medskipamount | minipage | multicolumn | multiput | name | newcommand | newcounter | newenvironment | newfont | newlength | newline | newpage | newsavebox | newtheorem | noindent | nolinebreak | nonfrenchspacing | nonumber | nopagebreak | normalmarginpar | normalsize | numberline | oddsidemargin | onecolumn | opening | oval | overbrace | overline | pagebreak | pagenumbering | pageref | pagestyle | par | paragraph | parbox | parindent | parsep | parskip | part | picture | poptabs | protect | ps | pushtabs | put | quotation | quote | raggedbottom | raggedleft | raggedright | raisebox | ref | refstepcounter | renewcommand | renewenvironment | reversemarginpar | rm | Roman | roman | rule | samepage (Befehl) | samepage (Umgebung) | savebox | sbox | sc | scriptscriptstyle | scriptsize | scriptstyle | section | setcounter | setlanguage | setlength | settowidth | sf | shortstack | signature | sl | sloppy | sloppypar | small | smallskip | smallskipamount | sqrt | stackrel | stepcounter | subparagraph | subsection | subsubsection | symbol | tabbing | tabbingsep | tabcolsep | table | tableofcontents | tabular | TeX | textfraction | textfloatsep | textheight | textstyle | textwidth | thanks | thebibliography | thicklines | thinlines | thispagestyle | tilde | tiny | title | titlepage | today | topfraction | topmargin | topnumber | topsep | topskip | totalnumber | tt | twocolumn | typein | typeout | unboldmath | underbrace | underline | unitlength | usebox | value | vec | vector | verb | verbatim | verse | vfill | vline | voffset | vspace | widehat | widetilde | Zähler

Select Subject: Basic Math | Everyday Math | Pre-Algebra | Algebra | Geometry | Trigonometry | Statistics | Calculus | Advanced Topics | Others | Math Tutoring
- A: absolute value | acute angle | acute triangle | additive identity | additive inverse | adjacent angles | angle | arc | area | associative property of addition | associative property of multiplication | average | axis of symmetry
- B: base | bisect | box and whisker plot
- C: Cartesian coordinates | central angle | chord | circle | circumference | coefficient | collinear | combination | common factor | common multiple | commutative property of addition | commutative property of multiplication | complementary angles | composite number | cone | congruent | constant | coordinate plane | coplanar | counting numbers | counting principle | cross product | cube | cylinder
- D: data | decimal number | degree | denominator | dependent events | diagonal | diameter | difference | digit | distributive property | dividend | divisor
- E: ellipse | equation | equilateral triangle | equivalent equations | equivalent fractions | error of measurement | evaluate | even number | event | exponent | equivalent fractions
- F: factor | factorial | face | formula | fraction | frequency | frequency table
- G: graph | greatest common factor (GCF)
- H: horizontal | hypotenuse
- I: identity property of addition | identity property of multiplication | improper fraction | independent events | inequality | infinity | inscribed angle | inscribed polygon | integers | intercept | intercepted arc | interpolation | intersecting lines | inverse | inverse operations | irrational number | isosceles triangle
- L: least common denominator | least common multiple | like fractions | line | line of symmetry | line segment | locus | logic | lowest terms
- M: mean | median | midpoint | mixed number | mode | multiple | multiplicative identity | multiplicative inverse | mutually exclusive events
- N: natural numbers | negative number | normal | number line | numerator
- O: obtuse angle | obtuse triangle | octagon | odd number | operation | opposites | ordered pair | origin | outcome
- P: parallel | parallelogram | percent | perimeter | permutation | perpendicular | pi | plane | point | polygon | polyhedron | positive number | power | probability | product | proper fraction | proportion | protractor | pyramid | Pythagorean Theorem
- Q: quadrant | quadrilateral | quotient
- R: radius | range | rate | ratio | rational number | ray | real numbers | reciprocal | rectangle | reflection | regular polygon | repeating decimal | rhombus | right angle | right triangle | root | rotation
- S: sample space | scale drawing | scalene triangle | scattergram | scientific notation | set | simplifying | skew lines | slope | solution | sphere | square | square root | statistics | stem and leaf plot | straight angle | supplementary angles | surface area
- T: terminating decimal | translation | transformation | transversal | trapezoid | tree diagram | triangle
- U: unit price
- V: variable | vertex | vertical angles | volume
- W: whole numbers
- X: x-axis | x-intercept
- Y: y-axis | y-intercept
- Z: zero | zero property of multiplication
Number Theory | Combinatorics | Geometry | Algebra | Calculus & Diff Eqs | Probability & Statistics | Set Theory & Foundations | Reflections on Relativity | History | Physics | Music | Animated Illustrations | Combined List of Articles | Quotations
Number Theory | Symmetric Pseudoprimes | Is e Normal? | Identities for Linear Recurring Sequences | The Half-Totient Tree | Limit Cycles of xy (mod x+y) | A Knot of Congruences | Rounding Up to PI | Fermat's Last Theorem for Cubes | | Self-Similar Reverse-Sum Sequences | The Sum of the Prime Factors of N | Sequence Partitionable Into Powers of 2 or 3 | Iterated Sum-of-Digits Function | If ab+1, ac+1, bc+1 are squares,... | Zeisel Numbers | The Jewel of Arithmetic: Quadratic Reciprocity | Highly Wilsonian Primes | Quadratic Congruences | Squares in Arithmetic Progression (mod p) | Numeri Idonei | Sequences With No Arithmetic Progressions | A Primality Criterion | Can n Divide !n ? | Diophantine Walk-a-thon | Barlow's Observation | Divisors of an n-term Geometric Series | More Results on the Form xy (mod x+y) | Sums of Three Cubes | Congruences Involving the Totient Function | Infinitely Many Rhondas | Why z Is Not a Prime Power in z^p = x^p + y^p | Summations and Recurrences | Evidence for Goldbach | Miscellaneous Diophantine Equations | No Four Squares In Arithmetic Progression | Minimum Difference Function | Gauss' Lemma Without Divisibility Arguments | Product Divisible By Sum of Squares | Lucas and Perrin Pseudoprimes | Pseudoprimes For x^2 - 4x - 9 | Primitive Roots and Exponential Iterations | Sums of Consecutive Nth Powers Equal to an Nth Power | On Case 1 of Fermat's Last Theorem | A Mersenne Coincidence | Coprime Partitions | The 450 Pound Problem (x^3 + y^3 = 6z^3) | Is This a Prime? | Reverse Greed for Unit Fractions | The Amanda Arrangement | Sum of Divisors Equals a Power of 2 | Sublime Numbers | On x^2 + y^3 = z^6 | A Method Of Factoring Based On 1/N | The Distribution of Perfection | Fermat's Last Theorem for Quadratic Integers | Square Triangular Numbers | Differences Between Powers | Recurrences and Pell Equations | In Defense of Base-Related Problems | On x^3 - x + y^3 - y = z^3 - z | On the Density of Some Exceptional Primes | Double Equations from Triangles in Squares | Differently Perfect | A Special Property of 151 | Simple Complex Quadratic Fields | | Casting Out Nines | Detecting Squares | Numbers Expressible As (a^2 - 1)(b^2 - 1) | Concordant Forms | Solving Magic Squares | | Cyclic Divisibility | Waring's Problem | The 2 Ohm Problem | Reflective and Cyclic Sets of Primes | Unit Fraction Partitions | N = (x^2 + y^2)/(1+xy) Is A Square | Perrin's Sequence | Proof of Generalized Little Theorem of Fermat | Minimizing the Denominators in Unit Fraction Expansions | | Pythagorean Graphs | Series Within Parallel Resistor Circuits | Sequences Based On Continued Fraction For PI | Smallest Quadrilateral With Distinct Integer Sides | Mock-Rational Numbers | Coherent Arrays of Squares | Anti-Carmichael Pairs | Magic Square of Squares | Orthomagic Square of Squares | Automedian Triangles and Magic Squares | Continued Fractions and Characteristic Recurrences | One In The Chamber | | Average of Sigma(n)/n | Lucas's Primality Test With Factored N-1 | Highly Heronian Ellipses | How Leibniz Might Have Anticipated Euler | Odd Greedy Unit Fraction Expansions | Factoring Zeta | Accidental Melodies | Some Properties of the Lucas Sequence | Four Squares From Three Numbers | Meandering Convergence of a Dirichlet Series | Geodesic Diophantine Boxes | Least Significant Non-Zero Digit of n! | Discordance Impedes Square Magic | No Progression of Four Rectangles On A Conic? | Whole Permutation Fractions | A Unique Intersection Pattern for Plane Curves
Combinatorics | Partitions into Distinct Parts | Dedekind's Problem | On Eulerian Numbers | Permutation Loops | The Four Color Theorem | Polyomino Enumerations | Powers of Primes Dividing Factorials | The Dartboard Sequence | Meeting Combinations and Interval Graphs | Binary Games | Additive and Multiplicative Partitions | 414298141056 Quarto Draws Suffice! | Coloring The Edges of an Icosahedron | Uniform Squares | Cumulative Permutation Sequences | Center of Gravity With Integer Coordinates | The Factorial Number System | Sums and Differences of Discrete Functions | The Statistics of Counterintelligence | Partition Transform Cycles | Clouds, Shy Squares, and Diophantus | Computing the Partitions of n | Solving Magic Squares | Generating the Monotone Boolean Functions | Meeting Probabilities | The Determinants of 4x4 Magic Squares | Discrete Circular Distributions | Balls In Bins With Limited Capacities | The Importance of Being Urnest (B Balls, K Colors, N Urns) | Sequences of Characters in Random Strings | Complete Bit-Cubes | A Procrustean Protocol | Partitions into Consecutive Integers | Tables of 1 to N with Equal Columns | Regular Polygonal Arrangements | The Pigeon Hole Principle | Two Proofs of Impossibility for Five-Point Star | Golf Tournaments, Polyhedra, and Latin Squares | Generating Functions for Point Set Distances | Derangements and Generalizations | Round-Robin Ties | The Coin in Three Fountains | Impossible Strings for Cellular Automata | Hamilton Cycles on McCauley Graphs | Permutation Puzzles | Collecting k Complete Sets of Coupons | Christmas Ornaments | Pythagorean Graphs | Ringing of the Changes (The Shape of 4!) | f(f(x))=exp(x) Generates the Multinomials | Progress on Dedekind's Problem | Electoral Draws | Generating Functions and Recurrence Relations
Geometry | The Five Squarable Lunes | Constructing the Heptadecagon | Generalizing Pythagoras and Carnot | Iterative Isoscelizing | Parabola Through Four Points | Euclid's Proposition III,20 and its Converse | Sphere Packing In Curved Space | Pascal's Mystic Hexagram | Min-Energy Configurations of Electrons On A Sphere | Generating Functions for Point Set Distances | Curvature of Linear Interpolation | Gravity Sextant | An Optical Illusion | Platonic Solids and Plato's Theory of Everything | Euclid's Plan and Proposition 6 | What Mirrors Do | Volume of n-Spheres and the Gamma Function | Ptolemy's Orbit | | Apollonius' Tangency Problem | Loci of Equi-angular Points | Heron's Formula and Brahmagupta's Generalization | Heron's Formula for Tetrahedrons | The Orbit of Triangles | Perpendicular Regression Of A Line | Rolling Spheres and Cones | Cramer's Paradox | The Lemniscate and Spacetime | Factoring Convex Figures | Smallest Quadrilateral With Distinct Integer Sides | Mean Distance from Vertex to Interior of Plane Figures | The Mystery of the Grazing Goat | Old Notes on Curvature | Did Archimedes Know Gauss-Bonnet? | Leaning Ladders | Pent Up Ratios | Thinking Outside the Triangle | On the Ellipse | The Shape of Coincidence | Double Equations from Triangles in Squares | Main Diagonal Passing Through Interior | Distances In Bounded Regions | Embedding Non-Euclidean Within Euclidean Geometry | On de Bruijn Grids and Tilings | Archimedes on Spheres and Cylinders | Trees on a Complex Plain | Pappus' Theorem | Acute Problem | Quintisecting an Angle | Quadrilateral In a Circle | Volume Under a Triangle | Constructible Points and Coverable Points | A Quarky Formula for Volume of Parallelpiped | Piero della Francesca's Tetrahedron Formula | Automedian Triangles | Piano Keys | On Ptolmey's Theorem | N-Fold Polygonal Spirals | Equal Bisectors and Isosceles Triangles | Equation of a Triangle | Highly Heronian Ellipses | No Equilateral Triangles on a Chess Board | Net Area and Green's Theorem | Ringing of the Changes (The Shape of 4!) | The Resultant and Bezout's Theorem | Stochastic Rhombic Tiling | Diophantine Geodesic Boxes | Angular Angst | Random Tiling of a Sphere | Zeno's Mice and the Logarithmic Spiral | Path Lengths and Coordinates | The Cube Unfolded | Revisiting the Four-Point Parabola | Morley's Trisection Theorem | Isospectral Point Sets in Higher Dimensions | Exponential Spiral Tilings | Harmonia Mensurarum | Constant Headings and Rhumb Lines | Enveloping Circular Arcs | Normals From A Point To An Ellipse | | The Tetratorus and Other Multi-Layered Polyhedra | The Curvatures of Hypersurfaces | Menelaus and Ceva | Carnot, Organizer of Transversals | Non-Periodic Tilings With N-fold Symmetry | Aperiodic Tilings | Tilings Based on Perrin's Sequence | Routh's Theorem by Cross Products
Algebra | Multiple Linear Regression | Kummer's Objection | Irreducibility Criteria | Proving Algebraic Inequalities | Bit-String Orbits Under Rotate-XOR | Piano Keys | The Crystallographic Restriction | Polynomials For Sums of Square Roots | Leaning Ladders | On Ptolmey's Theorem | Machin's Merit | The Twelve-Step Cycle of 1/(sin(x)cos(x)) | Finite Subgroups of the Mobius Group | The Fundamental Theorem of Algebra | String Algebra | Apollonius' Tangency Problem | Asymptotic Approach to 2D Arrays | Iterated Logarithmic Functions | Eigenvalue Problems and Matrix Invariants | Iterated Means | | Square Roots by Pencil and Paper | A Quasi-Periodic Sequence | Sums of Powers in Terms of Symmetric Functions | Self-Similar Reverse-Sum Sequences | Generalized Mediant | Tangents, Exponentials, and PI | Recurrences For Harmonic Sums | Bernoulli Numbers and Harmonic Series | Quasi-Groups | Laminar Transformations | Ancient Square Roots | Pythagoras On Dot and Cross Products | Infinite Products and a Tangent Fan | Exponential Partitions | Geometric Dot Products and Digit Reversals | Radical Expression For cos(2pi/7) | On f(x^2+y^2) = f(x)^2 + f(y)^2 | Series Solution of Non-Linear Equation | Compressor Stalls and Mobius Transformations | Convoluting Arrays | Harmonic Sums of Integers with k Binary 1's | | Gauss' Lemma Without Divisibility Arguments | Mnemonic Coins and Pattern Avoidance | The Shortest Path To Trigonometric Identities | Polynomials From Pascal's Triangle | Bras, Kets, and Matrices | Decimal Representations as Power Series | Reflecting on the Geometric Mean | Sums of Powers | Trees on a Complex Plain | Inverse Square Weighted Interpolation | | Reducing Quartics to Cubics | Platonions | Annuities With Inflation | On Eulerian Numbers | Intersections of Polynomials | Angular Angst | Quarky Vector Triple-Product | From the Geometric Series to Stirling Numbers | Orthomagic Square of Squares | Automedian Triangles and Eigen Vectors | Continued Fractions and Characteristic Recurrences | On the Solution of the Cubic | Phased Summations | Arranging the Solutions of f(x+y+z)=xyz | Tangent To PI | Cross-Linear Interpolation | The Super-Symmetric Mean | Rational Sines of Rational Multiples of PI | Cyclic Redundancy Checks | Permutation Loops | Loxodromic Aliasing | Determining the Galois Group of a Polynomial | Formal-Numeric Series | | Continuous From Discrete Transfer Functions | Expulsion Sets | Bi-Rational Substitutions Giving Squares | Cyclic Divisibility | The Tetratorus and Other Multi-Layered Polyhedra | Maxwell and Second-Degree Commutation | Inverse-Square Forces and Orthogonal Polynomials | Harmonia Mensurarum | Iterative Isoscelizing | Galois Groups | Hermitian Matrices | An Algebraic Duality | Linear Fractional Transformations | The Resultant and Bezout's Theorem | Isospectral Point Sets in Higher Dimensions | Rotation Matrices | Iterative Solutions of Homogeneous Linear Systems | Binary Reverse-Sum Automata | The Cayley-Hamilton Theorem
Calculus and Differential Equations | The Laplace Equation and Harmonic Functions | Fractional Calculus | Analytic Functions, The Magnus Effect, and Wings | | Propagation of Pressure and Waves | The Virial Theorem | Causality and the Wave Equation | Integrating the Bell Curve | Compressor Stalls and Mobius Transformations | Dual Failures with General Densities | | Series Solutions of the Wave Equation | The Limit Paradox | Proof That PI is Irrational | Simple Proof that e is Irrational | The Filter Of Observation | Eigenvalue Problems and Matrix Invariants | Root-Matched Recurrences For DiffEQs | Why Calculus? | High Order Integration Schemes | Do We Really Need Eigen Values? | Markov Models with Aging Components | Leibniz's Rule | A Removable Singularity in Lead-Lag Coefficients | Convergence of Series (How NOT to Prove PI is Irrational) | Sum of n^2 / (n^3 + 1), n=1 to inf | Tilting Pencils | Continuous From Discrete Transfer Functions | Ptolemy's Orbit | Distances In Bounded Regions | Rolling Spheres and Cones | Integrating Factors | Series Solution of Non-Linear Equation | Volume of n-Spheres and the Gamma Function | Choice Without Context? | Infinite Products and a Tangent Fan | Cylinders with Cylindrical Holes | Markov Models with Variable Transition Rates | Age Distributions in Continuous Markov Models | Analytic Continuation | Ratio Populations | Newton's Proposition LXXI | Harmonic Sums of Integers with k Binary 1's | An Infinite Wall | Perpendicular Regression Of A Line | The Prismoidal Formula | On Random Chords | Potential Flow and d'Alembert's Paradox | Curvature of Linear Interpolation | Round-Robin Ties | Irrationality of Quadratic Sums | Time for a Rocking Chair | Net Area and Green's Theorem | How Leibniz Might Have Anticipated Euler | Meandering Convergence of a Dirichlet Series | Mean Distance from Vertex to Interior of Plane Figures | Aliasing and Uncertainty | Frequency Response of First-Order Lag | Mean Partial Sums of Non-Convergent Series | Constant Headings and Rhumb Lines | Inverse-Square Forces and Orthogonal Polynomials | Did Archimedes Know Gauss-Bonnet? | Enveloping Circular Arcs | Laplace Transforms | The Wave Equation and Permutation of Rays | Huygens' Principle | Recurrence Relations for Ordinary Differential Equations | The Curvatures of Hypersurfaces | Poisson Processes and Queues | Lagrangian and Hamiltonian Mechanics | Lead-Lag Frequency Response | Differential Operators and the Divergence Theorem | Precession in a Circle | Higher-Order Wave Equations and Matter Waves | Complete Solutions of Linear Systems | Noether's Theorem | | Series Solution of Relativistic Orbits | Geodesics by Differentiation | Inverse Functions | The Euler-Maclaurin Formula | Change of Variables in Multiple Integrals
Probability and Statistics | Evaluating Probabilities of Boolean Events | The Gambler's Ruin | Integrating the Bell Curve | Poisson Processes and Queues | Ratio Populations | Failure Rates, MTBFs, and All That | Boolean Expansion as Linear Interpolation | At Least One Girl | The Collector's Problem | Periodic and Continuous Repair Models | Collecting k Complete Sets of Coupons | Generalized Birthday Problem (N Items in M Bins) | Independence and Negation | Probabilities With Renaming | Dice Rolling A Given Sum | Redundant Systems With A Common Threat | Meeting Probabilities | A Better Lottery? | Bayesian Balls | Random High-Low Searching | Hierarchial Repair and Quantum State Reduction | Mnemonic Coins and Pattern Avoidance | Reliability with Mixed Periodic Repairs | Balls In Bins | Choice Without Context? | Combining Probabilities | A Barrel of Colors and Null Outcomes | Open-Loop and Closed-Loop Markov Models | The e in Petersburg | Weibull Analysis | Two Properties of Markov Models | Hail Encounters | Dual Failures with General Densities | Optimizing Your Wife | Complete Markov Models and Reliability | Probabilities and Velocities | Two In the Shop | Differences Between Normal Samples | Benford's Law | Perpendicular Regression Of A Line | On 13 Bridges | Random Triangles | The Complex Domain of Probability | Infinite Parallel Redundancy | N Points On Sphere Fall In One Hemisphere | Bernoulli Trials | Two-Dimensional Runs in Bernoulli Trials | Reliability of Series-Parallel Systems | Sequences of Characters in Random Strings | Botches, Failures, and Successes | Undetected Failures of Threat Protection | A Random Walk Through PI | The Shape of Coincidence | Markov Models with Boolean Transition Logic | The Expectation of Posterior Beliefs | Markov Models of Dual-Redundant Systems | Then There Were None | Generating Random Numbers In Order | Biased and Anti-Biased Variance Estimates | Markov's Marbles | The Y10K Problem, Pascal's Wager, and Petersberg | Expected Area of Random Polygon in a Circle | World's Oldest Man Dies | Asymptotic Rate of an Open Loop Markov Model | Probability of Intersecting Intervals | Timed Markov Models
Set Theory and Foundations | Representing Sets of Pure Order | | Global Reversibility of Cellular Automata | Guessing Faster Than Light | Hierarchial Repair and Quantum State Reduction | Spiritualism and Electromagnetism | The Necessity of the Quantum | Maxwell's Paradox of Attraction | Fuzzy Logic | Continuous Turing Machines | Fractal Logic | On Olber's Paradox | Hawking Radiation and Super-Acoustic Diffusion | Generalized Lorentz Transformations and Relative Rotation | Zeno and Uncertainty | Is Arithmetic Consistent? | Reconstructing Brouwer | The 3m Halo | Euclid's Plan and Proposition 6 | Constructible Points and Coverable Points | Round Trips and One-Way Speeds | A Mirror To Physics | Cantor's Diagonal Proof | Conditions for Relativity | Expansion as Gravity and Many Worlds | Function 34776 | An Infinite Wall | Omni-Directional Flux | Interfering With PI | Choice Without Context? | The Shortest Path To Trigonometric Identities | Reflections on The Mirror Question | Certainty In Math and Physics | Dialogue On Mathematical and Physical Knowledge | Is Mathematical Knowledge Cumulative? | Are All Triangles Isosceles? | Higher Order Dynamics | Dialogue on the Foundations of String Theory | Controversies Over the Equivalence Principle | On the Origins of Quantum Mechanics | Lorentz Boosts and the Electromagnetic Field | Did Poincare Anticipate Godel? | Attractive Forces From Quantum Exchanges | Weighing the Evidence | Something Like Quantum Mechanics | Marginalia | Timely Confessions | What is an Inertial Coordinate System? | If the Sun Were Suddenly to Explode | Is the World Provably Indeterminate? | Gamaliel's Principle | Preserving the Wave | This Business of Interpretation | Unreasonable Effectiveness | MathPages Wanted List | Newtonian Gravity In Closed or Curved Space | The Y10K Problem, Pascal's Wager, and Petersberg | Infinite Descent versus Induction | Newtonian and Relativistic Conservation Laws | Expulsion Sets | Interleaving Ad Infinitum | White Ravens and Black Swans | Spacetime Coordinates and Frames | Continuity and the Void | Potential Energy, Inertia, and Quantum Coherence | Cellular Automata | Harmonia Praestabilita | Central Forces from Retarded Action | Pitch and Color Recognition | General Relativity and the Principle of Inertia | Efficient and Final Causes | Feynman's Ants | Another Derivation of Mass-Energy Equivalence | Hawking Radiation as Delayed Choice | Null Cones in Cylindrical Dimensions | On Relativistic Precession | The Spacetime Hole and the EPR Arguments | Benevolent Design and the Anthropic Principle | Causality and the Wave Equation | Dirac's Belt | On The Many Principles of Equivalence | Curved Coordinate Systems and Fictitious Forces | Eigen Duality and Quasi-Eigen Systems
Reflections on Relativity | | Preface 1 | | 1. First Principles | 1.1 Experience and Spacetime 3 | 1.2 Systems of Reference 9 | 1.3 Inertia and Relativity 16 | 1.4 The Relativity of Light 23 | 1.5 Corresponding States 33 | 1.6 A More Practical Arrangement 44 | 1.7 Staircase Wit 58 | 1.8 Another Symmetry 65 | 1.9 Null Coordinates 72 | | 2. A Complex of Phenomena | 2.1 The Spacetime Interval 81 | 2.2 Force Laws and Maxwell's Equations 88 | 2.3 The Inertia of Energy 98 | 2.4 Doppler Shift for Sound and Light 111 | 2.5 Stellar Aberration 118 | 2.6 Mobius Transformations of the Night Sky 131 | 2.7 The Sagnac Effect 140 | | 2.9 Accelerated Travels 159 | 2.10 The Starry Messenger 173 | 2.11 Thomas Precession 183 | | 3. Several Valuable Suggestions | 3.1 Postulates and Principles 192 | 3.2 Natural and Violent Motions 202 | 3.3 De Mora Luminis 209 | 3.4 Stationary Paths 218 | 3.5 A Quintessence of So Subtle a Nature 224 | 3.6 The End of My Latin 230 | 3.7 Zeno and the Paradox of Motion 238 | 3.8 A Very Beautiful Day 245 | 3.9 Constructing the Principles 251 | | 4. Weighty Arguments | 4.1 Immovable Spacetime 256 | 4.2 Inertial and Gravitational Separations 265 | 4.3 Free-Fall Equations 270 | 4.4 Force, Curvature, and Uncertainty 274 | 4.5 Conventional Wisdom 280 | 4.6 The Field of All Fields 292 | 4.7 The Inertia of Twins 297 | 4.8 The Breakdown of Simultaneity 301 | | 5. Extending the Principle | 5.1 Vis Inertiae 308 | 5.2 Tensors, Contravariant and Covariant 313 | 5.3 Curvature, Intrinsic and Extrinsic 323 | 5.4 Relatively Straight 340 | 5.5 Schwarzschild Metric from Kepler's 3rd Law 349 | 5.6 The Equivalence Principle 354 | 5.7 Riemannian Geometry 359 | 5.8 The Field Equations 370 | | 6. Ist Das Wirklich So? | 6.1 An Exact Solution 382 | 6.2 Anomalous Precession 389 | 6.3 Bending Light 400 | 6.4 Radial Paths in a Spherically Symmetrical Field 408 | 6.5 Intersecting Orbits 413 | 6.6 Ideal Clocks in Arbitrary Motion 419 | | 6.8 Sources in Motion 431 | | 7. Cosmology | 7.1 Is the Universe Closed? 438 | 7.2 The Formation and Growth of Black Holes 449 | 7.3 Falling Into and Hovering Near A Black Hole 458 | 7.4 Curled-Up Dimensions 468 | 7.5 Packing Universes In Spacetime 472 | 7.6 Cosmological Coherence 478 | 7.7 Boundaries and Symmetries 486 | 7.8 Global Interpretations of Local Experience 493 | | 8. The Secret Confidence of Nature | 8.1 Kepler, Napier, and the Third Law 505 | 8.2 Newton's Cosmological Queries 510 | 8.3 The Helen of Geometers 518 | | 8.5 Scholium 531 | 8.6 On Gauss' Mountains 536 | 8.7 Strange Meeting 541 | 8.8 Who Invented Relativity? 548 | 8.9 Paths Not Taken 559 | | 9. The Relativistic Topology | 9.1 In The Neighborhood 567 | 9.2 Up To Diffeomorphism 576 | 9.3 Higher-Order Metrics 579 | 9.4 Spin and Polarization 584 | 9.5 Entangled Events 587 | 9.6 Von Neumann's Postulate and Bell's Freedom 593 | 9.7 The Gestalt of Determinism 599 | 9.8 Quaedam Tertia Natura Abscondita 602 | 9.9 Locality and Temporal Asymmetry 606 | 9.10 Spacetime Mediation of Quantum Interactions 611 | | Conclusion 618 | | Appendix: Mathematical Miscellany 620 | | Bibliography 633
History | Mercurius in Sole Visus | The Thought of a Thought - Edgar Allan Poe | The Prismoidal Formula | Gerber's Gravity | Harmonia Mensurarum | Carnot, Organizer of Transversals | Roemer's Hypothesis | Magnetism and Earnshaw's Theorem | Nicolas Fatio and the Cause of Gravity | Dialogues | Stokes' Mistake | Timely Confessions | Conquering the Perihelion | Wonder En Is Gheen Wonder | The Quantity of Motion | Historical Assessments of the Fatio-Lesage Theory | Archimedes and the Square Root of 3 | Bertelsen's Number | Euclid's Plan and Proposition 6 | Mayan Numeration | Saturnday, Sunday, Moonday | The Dullness of 1729 | Berkeley and the Infidel | Newton's Birth Date and the Anni Mirabiles | Whittaker and the Aether | Franklin's Magic Squares | Fatio, Lesage, and the Camisards | Ancient Square Roots | Fermat's Infinite Descent | Negative Numbers | Hold Your Horses | Did Archimedes Know Gauss-Bonnet? | Herbert Dingle and the Twins | Egyptian Unit Fractions | The Metamorphosis of Judas Iscariot | Catch of the Day (153 Fishes) | The Guest Star | From Euclid to Gregory | What Happened to Dingle? | Guillaume, Guillaume, and Einstein | Eccentrics, Deferents, Epicycles, and Equants | Why Was Michelson Surprised? | The Most Teachable of Mortals | Zulu Hitler | Hipparchus on Coumpound Statements | Piero della Francesca's Tetrahedron Formula | Omar Khayyam On Cubics | The Ten Means of Ancient Greece | Heron's Formula and Brahmagupta's Generalization | Gravitational and Inertial Mass in Newton's Principia | Continuity and the Void | | Galois's Analysis of Analysis | How Leibniz Might Have Anticipated Euler | Fermat's Fallibility | Adam's Gold | Constructing the Heptadecagon | The Moszkowski Affair | Prisca Sapientia | Harmonia Praestabilita | Einstein on the Inertia of Energy | Accidental Melodies | The Portraits of Emilie | Poincare Contemplates Copernicus | The Five Squarable Lunes | Reflections on Relativity | Archimedes on Spheres and Cylinders | Newton's Proposition LXXI | Platonic Solids and Plato's Theory of Everything | Lesage's Shadows | Remembering Socrates | The Bulging Earth | Translating Aristotle | The Longitude of Lewis and Clark | An Anomaly In Translation | Galileo's Law of Planetary Motion | Maxwell's Displacement and Einstein's Trace | Zur Farbenlehre | Kinetic Pressure and Tetrode's Star | Did Einstein Misunderstand Aberration? | The Doctrine of Chance
Animated (Java) Illustrations
Here are a few animated illustrations that are discussed in one or more notes on this site.
The Stars In The Sky
This applet shows the positions of all 297 stars with magnitudes less than 3.5, projected onto a sphere like a planetarium, and identifies all those of the first magnitude.
Kinematics of a Hypocycloidal Engine
This applet illustrates the workings of an unconventional internal combustion engine.
24 Electrons on a Sphere
This shows the minimum energy configuration of 24 charged particles on the surface of a sphere. The colors of the edges denote their lengths. Notice that the surface of this solid consists of 24/4 perfect squares, 24/3 perfect triangles, 24/2 additional "bi-gons", and of course 24/1 vertices, all arranged so that each vertex is identical. In addition, the vertices have a "handedness", so there are really two distinct versions of this solid.
Bell-Ringing Solid
This shape is what's called a "truncated octahedron", one of the 13 Archimedian solids. It has six square faces and eight hexagonal faces, making a total of 14 faces. The number of edges is 36, as required by Euler's formula F + V - E = 2. There is an interesting connection between this solid and the patterns of bell ringing.
Kepler's Rhombic Dodecahedron
This is one of the most fascinating of the simple solids, with an interesting correspondence to "the shape of coincidences".
The above illustrations are discussed in the notes
- The Stars In The Sky
- Hypocycloidal Engine
- Min-Energy Configurations of Electrons on a Sphere
- Ringing of the Changes (The Shape of 4!)
- Meeting Probabilities and The Shape of Coincidence
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The Divisions of Mathematics
In order to find one's way around the collection of mathematical ideas, it is useful to organize them and classify them in some way into parts.
...
How many parts of mathematics -- Two? Eight? Sixty-three?
...
The image at right shows a "map" of the subfields of mathematics. These are the major classification groupings used at this site and by most research mathematics projects. The sizes and positions of the "bubbles" are computed to reflect the sizes and relatedness of the various disciplines. On our tour, we'll highlight some of the main groupings of these areas (the different color groups).
...
Core branches of mathematics:Applied and related areas:
- Foundations
- (Abstract) Algebra and Number Theory
- Geometry and Topology
- Mathematical Analysis including
- Calculus and Real Analysis
- Complex variable Analysis
- Differential Equations and related topics
- Functional Analysis and related topics
- Numerical Analysis and Optimization
Top-level subject areas in the MSC
- Computer and Information Sciences
- Probability and Statistics
- Physical Sciences and Engineering
- History, Education, Other
- 00: General material including elementary mathematics
- 01: History and biography
- 03: Mathematical logic and foundations
- 05: Combinatorics and graph theory
- 06: Order, lattices, ordered algebraic structures
- 08: General algebraic systems
- 11: Number theory
- 12: Field theory and polynomials
- 13: Commutative rings and algebras
- 14: Algebraic geometry
- 15: Linear and multilinear algebra; matrix theory
- 16: Associative rings and algebras
- 17: Nonassociative rings and algebras
- 18: Category theory, homological algebra
- 19: K-theory
- 20: Group theory and generalizations
- 22: Topological groups, Lie groups
- 26: Real functions and elementary calculus
- 28: Measure and integration
- 30: Functions of a complex variable
- 31: Potential theory
- 32: Several complex variables and analytic spaces
- 33: Special functions including trigonometric functions
- 34: Ordinary differential equations
- 35: Partial differential equations
- 37: Dynamical systems and ergodic theory
- 39: Difference and functional equations
- 40: Sequences, series, summability
- 41: Approximations and expansions
- 42: Fourier analysis
- 43: Abstract harmonic analysis
- 44: Integral transforms, operational calculus
- 45: Integral equations
- 46: Functional analysis
- 47: Operator theory
- 49: Calculus of variations and optimal control; optimization
- 51: Geometry, including classic Euclidean geometry
- 52: Convex and discrete geometry
- 53: Differential geometry
- 54: General topology
- 55: Algebraic topology
- 57: Manifolds and cell complexes
- 58: Global analysis, analysis on manifolds
- 60: Probability theory and stochastic processes
- 62: Statistics
- 65: Numerical analysis
- 68: Computer science
- 70: Mechanics of particles and systems
- 74: Mechanics of deformable solids
- 76: Fluid mechanics
- 78: Optics, electromagnetic theory
- 80: Classical thermodynamics, heat transfer
- 81: Quantum Theory
- 82: Statistical mechanics, structure of matter
- 83: Relativity and gravitational theory
- 85: Astronomy and astrophysics
- 86: Geophysics
- 90: Operations research, mathematical programming
- 92: Biology and other natural sciences
- 93: Systems theory; control
- 94: Information and communication, circuits
- 97: Mathematics education
Index using "Encyclopedic Dictionary" System
...
The twenty-one groupings in this system areArticle headings are shown with their article number:
- Logic and Foundations
- Sets, General Topology, and Categories
- Algebra
- Group Theory
- Number Theory
- Euclidean and Projective Geometry
- Differential Geometry
- Algebraic Geometry
- Topology
- Analysis
- Complex Analysis
- Functional Analysis
- Differential, Integral, and Functional Equations
- Special Functions
- Numerical Analysis
- Computer Science and Combinatorics
- Probability Theory
- Statistics
- Mathematical Programming and Operations Research
- Mechanics and Theoretical Physics
- History of Mathematics
Logic and FoundationsSets, General Topology, and Categories
- Foundations of Mathematics [156]
- Axiom Systems [035]
- Paradoxes [319]
- Symbolic Logic [411]
- Axiomatic Set Theory [033]
- Model Theory [276]
- Nonstandard Analysis [293]
- Recursive Functions [356]
- Decision Problem [097]
- Constructive Ordinal Numbers [081]
- Analytic Sets [022]
Algebra
- Sets [381]
- Relations [358]
- Equivalence Relations [135]
- Functions [165]
- Axiom of Choice and Equivalents [034]
- Cardinal Numbers [049]
- Structures [409]
- Permutations and Combinations [330]
- Numbers [294]
- Real Numbers [355]
- Complex Numbers [074]
- Ordering [311]
- Ordinal Numbers [312]
- Lattices [243]
- Boolean Algebras [042]
- Topological Spaces [425]
- Metric Spaces [273]
- Plane Domains [333]
- Convergence [087]
- Connectedness [079]
- Dimension Theory [117]
- Uniform Spaces [436]
- Uniform Convergence [435]
- Categories and Functors [052]
- Inductive Limits and Projective Limits [210]
- Sheaves [383]
Group Theory
- Algebra [008]
- Matrices [269]
- Determinants [103]
- Polynomials [337]
- Algebraic Equations [010]
- Fields [149]
- Galois Theory [172]
- Linear Spaces [256]
- Rings [368]
- Associative Algebras [029]
- Commutative Rings [067]
- Noetherian Rings [284]
- Rings of Polynomials [369]
- Rings of Power Series [370]
- Quadratic Forms [348]
- Clifford Algebras [061]
- Differential Rings [113]
- Witt Vectors [449]
- Valuations [439]
- Adèles and Idèles [006]
- Cayley Algebras [054]
- Jordan Algebras [231]
- Modules [277]
- Homological Algebra [200]
- Hopf Algebras [203]
Number Theory
- Groups [190]
- Abelian Groups [002]
- Free Groups [161]
- Finite Groups [151]
- Classical Groups [060]
- Topological Groups [423]
- Topological Abelian Groups [422]
- Compact Groups [069]
- Lie Groups [249]
- Lie Algebras [248]
- Algebraics Groups [013]
- Homogeneous Spaces [199]
- Symmetric Riemannian Spaces and Real Forms [412]
- Discontinuous Groups [122]
- Crystallographic Groups [092]
- Representations [362]
- Unitary Representations [437]
- Invariants and Covariants [226]
Euclidean and Projective Geometry
- Number Theory [296]
- Number Theory, Elementary [297]
- Continued Fractions [083]
- Number-Theoretic Functions [295]
- Additive Number Theory [004]
- Partitions of Numbers [328]
- Distributions of Prime Numbers [123]
- Lattice-Point Problems [242]
- Diophantine Equations [118]
- Geometry of Numbers [182]
- Transcendental Numbers [430]
- Quadratic Fields [347]
- Algebraic Number Fields [014]
- Class Field Theory [059]
- Complex Multiplication [073]
- Fermat's Problem [145]
- Local Fields [257]
- Arithmetic of Associative Algebras [027]
- Zeta Functions [450]
Differential Geometry
- Geometry [181]
- Foundations of Geometry [155]
- Euclidean Geometry [139]
- Euclidean Spaces [140]
- Geometric Constructions [179]
- Regular Polyhedra [357]
- Pi [332]
- Trigonometry [432]
- Conic Sections [078]
- Quadric Surfaces [350]
- Convex Sets [089]
- Vectors [442]
- Coordinates [090]
- Projective Geometry [343]
- Affine Geometry [007]
- Non-Euclidean Geometry [285]
- Conformal Geometry [076]
- Erlangen Program [137]
- Continuous Geometry [085]
- Curves [093]
- Surfaces [410]
- Four-Color Problem [157]
Algebraic Geometry
- Differential Geometry [109]
- Differentiable Manifolds [105]
- Riemannian Manifolds [364]
- Connections [080]
- Tensor Calculus [417]
- Geodesics [178]
- Symmetric Spaces [413]
- G-structures [191]
- Complex Manifolds [072]
- Kähler Manifolds [232]
- Harmonic Integrals [194]
- Differential Geometry of Curves and Surfaces [111]
- Riemannian Submanifolds [365]
- Minimal Submanifolds [275]
- Harmonic Mappings [195]
- Morse Theory [279]
- Differential Geometry in Specific Spaces [110]
- Finsler Spaces [152]
- Integral Geometry [218]
- Siegel Domains [384]
- Pseudoconformal Geometry [344]
- Spectral Geometry [391]
- Global Analysis [183]
Topology
- Algebraic Geometry [012]
- Algebraic Curves [009]
- Algebraic Surfaces [015]
- Algebraic Varieties [016]
- Abelian Varieties [003]
- Riemann-Roch Theorems [366]
Analysis
- Topology [426]
- Fundamental Groups [170]
- Covering Spaces [091]
- Degree of Mapping [099]
- Complexes [070]
- Homology Theory [201]
- Fixed-Point Theorems [153]
- Cohomology Operations [064]
- Homotopy Theory [202]
- Fiber Spaces [148]
- Obstructions [305]
- Topology of Lie Groups and Homogeneous Spaces [427]
- Fiber Bundles [147]
- Characteristic Classes [056]
- K-Theory [237]
- Knot Theory [235]
- Combinatorial Manifolds [065]
- Differential Topology [114]
- Transformation Groups [431]
- Theory of Singularities [418]
- Foliations [154]
- Dynamical Systems [126]
- Shape Theory [382]
- Catastrophe Theory [051]
Complex Analysis
- Analysis [020]
- Continuous Functions [084]
- Inequalities [211]
- Convex Analysis [088]
- Functions of Bounded Variations [166]
- Differential Calculus [106]
- Implicit Functions [208]
- Elementary Functions [131]
- C^\infty-Functions and Quasi-Analytic Functions [058]
- Integral Calculus [216]
- Curvilinear Integrals and Surface Integrals [094]
- Measure Theory [270]
- Integration Theory [221]
- Invariant Measures [225]
- Set Functions [380]
- Length and Area [246]
- Denjoy Integrals [100]
- Series [379]
- Asymptotic Series [030]
- Polynomial Approximation [336]
- Orthogonal Functions [317]
- Fourier Series [159]
- Harmonic Analysis [192]
- Almost Periodic Functions [018]
- Laplace Transform [240]
- Integral Transforms [220]
- Potential Theory [338]
- Harmonic Functions and Subharmonic Functions [193]
- Dirichlet Problem [120]
- Capacity [048]
- Calculus of Variations [046]
- Plateau's Problem [334]
- Isoperimetric Problems [228]
- Variational Inequalities [440]
Functional Analysis
- Holomorphic Functions [198]
- Power Series [339]
- Dirichlet Series [121]
- Bounded Functions [043]
- Univalent and Multivalent Functions [438]
- Transcendental Entire Functions [429]
- Meromorphic Functions [272]
- Distributions of Values of Functions of a Complex Variable [124]
- Cluster Sets [062]
- Algebraic Functions [011]
- Algebroidal Functions [017]
- Riemann Surfaces [367]
- Ideal Boundaries [207]
- Conformal Mappings [077]
- Quasiconformal Mappings [352]
- Teichmüller Spaces [416]
- Kleinian Groups [234]
- Extremal Length [143]
- Function-Theoretic Null Sets [169]
- Analytic Functions of Several Complex Variables [021]
- Analytic Spaces [023]
- Automorphic Functions [032]
Differential, Integral, and Functional Equations
- Functional Analysis [162]
- Hilbert Spaces [197]
- Banach Spaces [037]
- Ordered Linear Spaces [310]
- Topological Linear Spaces [424]
- Function Spaces [168]
- Distributions and Hyperfunctions [125]
- Vector-Valued Integrals [443]
- Linear Operators [251]
- Compact and Nuclear Operators [068]
- Interpolation of Operators [224]
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- Semigroups of Operators and Evolution Equations [378]
- Differential Operators [112]
- Microlocal Analysis [274]
- Banach Algebras [036]
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- Operational Calculus [306]
- Nonlinear Functional Analysis [286]
Special Functions
- Differential Equations [107]
- Ordinary Differential Equations [313]
- Ordinary Differential Equations (Initial Value Problems) [316]
- Ordinary Differential Equations (Boundary Value Problems) [315]
- Ordinary Differential Equations (Asymptotic Behaviour of Solutions) [315]
- Linear Ordinary Differential Equations [252]
- Linear Ordinary Differential Equations (Local Theory) [254]
- Linear Ordinary Differential Equations (Global Theory) [253]
- Nonlinear Ordinary Differential Equations (Local Theory) [289]
- Nonlinear Ordinary Differential Equations (Global Theory) [288]
- Nonlinear Oscillation [290]
- Nonlinear Problems [291]
- Stability [394]
- Integral Invariants [219]
- Difference Equations [104]
- Functional-Differential Equations [163]
- Total Differential Equations [428]
- Contact Transformations [082]
- Partial Differential Equations [320]
- Partial Differential Equations (Methods of Integrations) [322]
- Partial Differential Equations (Initial Value Problems) [321]
- Partial Differential Equations of First Order [324]
- Monge-Ampère Equations [278]
- Partial Differential Equations of Elliptic Type [323]
- Partial Differential Equations of Hyperbolic Type [325]
- Partial Differential Equations of Parabolic Type [327]
- Partial Differential Equations of Mixed Type [326]
- Green's Functions [188]
- Green's Operators [189]
- Integral Equations [217]
- Integrodifferential Equations [222]
- Special Functional Equations [388]
- Pseudodifferential Operators [345]
Numerical Analysis
- Special Functions [389]
- Generating Functions [177]
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- Gamma Function [174]
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- Spherical Functions [393]
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Computer Science and Combinatorics
- Numerical Methods [300]
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Probability Theory
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Statistics
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- Stationary Processes [395]
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- Ergodic Theory [136]
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Mathematical Programming and Operations Research
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Mechanics and Theoretical Physics
- Mathematical Programming [264]
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- Information Theory [213]
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History of Mathematics
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- Dimensional Analysis [116]
- Variational Principles [441]
- Mechanics [271]
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- Abel, Niels Henrik [001]
- Artin, Emil [028]
- Bernoulli Family [038]
- Cantor, Georg [047]
- Cartan, Élie [050]
- Cauchy, Augustin Louis [053]
- Dedekind, Julius Wilhelm Richard [098]
- Descartes, René [101]
- Dirichlet, Peter Gustav Lejeune [119]
- Einstein, Albert [129]
- Euler, Leonhard [141]
- Fermat, Pierre de [144]
- Fourier, Jean Baptiste Joseph [158]
- Galois, Évariste [171]
- Gauss, Carl Friedrich [175]
- Hilbert, David [196]
- Jacobi, Carl Gustav Jacob [229]
- Klein, Felix [233]
- Kronecker, Leopold [236]
- Lagrange, Joseph Louis [238]
- Laplace, Pierre Simon [239]
- Lebesgue, Henri Leon [244]
- Leibniz, Gottfried Wilhelm [245]
- Lie, Marius Sophus [247]
- Newton, Isaac [283]
- Pascal, Blaise [329]
- Poincaré, Henri [335]
- Riemann, Georg Friedrich Bernhard [363]
- Takagi, Teiji [415]
- Viète, François [444]
- Von Neumann, John [445]
- Weierstrass, Karl [447]
- Weyl, Hermann [448]
A positive integer n is called a "base-b Rhonda number" if the product of the base-b digits of n is equal to b times the sum of n's prime factors. These numbers were named by "K. S. Brown" after an acquaintance of his whose residence number 25662 satisfies this property. The etymology of the term is therefore similar to the Smith numbers.
25662 is a Rhonda number to base-10 since its prime factorization is
25662 = 2*3*7*13*47
and the product of its base-10 digits satisfies
2*5*6*6*2 = 720 = 10*(2+3+7+13+47)
The Rhonda numbers to base 10 are 1568, 2835, 4752, 5265, 5439, 5664, 5824, 5832, 8526, 12985, ... (Sloane's A099542). The corresponding sums of prime factors are 24, 24, 28, 30, 54, 72, 32, 24, 48, 72, ... (Sloane's A099543).
Rhonda numbers exist only for bases that are composite since there is no way for the product of integers less than a prime b to have b as a factor.
The first few Rhonda numbers for small composite bases b are summarized in the following table.
...
Many thanks to the artists who gave permission for their work to be displayed here. You can see more of their work by clicking on their names below:
JsMath is a means of including mathematical notation in web pages. It interprets TeX code within your document and uses JavaScript, CSS and unicode fonts to typeset the resulting formulas. It works across multiple browsers and platforms, without MathML
\left(\, \sum_{k=1}^n a_k b_k \right)^2 \le\left(\, \sum_{k=1}^n a_k^2 \right) \left(\, \sum_{k=1}^n b_k^2 \right)\warning{ jsMath is not working! }
TEX-Dateien und -Tools
Dieses Verzeichnis listet nützliche kleine TeX-Dateien auf, deren Makros man bei dem ein oder anderen mathematischen Dokument, wie z.B. einem Vorlesungsskrip oder einem Versuchsprotokoll (in Physik) nicht jedesmal von neu in seinen Editor hämmern möchte.
First published Thu Jul 11, 2002
Set Theory is the mathematical science of the infinite. It studies properties of sets, abstract objects that pervade the whole of modern mathematics. The language of set theory, in its simplicity, is sufficiently universal to formalize all mathematical concepts and thus set theory, along with Predicate Calculus, constitutes the true Foundations of Mathematics. As a mathematical theory, Set Theory possesses a rich internal structure, and its methods serve as a powerful tool for applications in many other fields of Mathematics. Set Theory, with its emphasis on consistency and independence proofs, provides a gauge for measuring the consistency strength of various mathematical statements. There are four main directions of current research in set theory, all intertwined and all aiming at the ultimate goal of the theory: to describe the structure of the mathematical universe. They are: inner models, independence proofs, large cardinals, and descriptive set theory. See the relevant sections in what follows....
- 1. The Essence of Set Theory
- 2. Origins of Set Theory
- 3. The Continuum Hypothesis
- 4. Axiomatic Set Theory
- 5. The Axiom of Choice
- 6. Inner Models
- 7. Independence Proofs
- 8. Large Cardinals
- 9. Descriptive Set Theory
- Bibliography
- Other Internet Resources
- Related Entries
Did a state legislature once pass a law saying pi equals 3?
February 22, 1991
Dear Cecil:
In Science magazine a while back an article about the latest attempts to calculate pi to the umpteen zillionth decimal place made a passing reference to a curious Oklahoma law. It said Oklahoma legislators had passed a law making pi equal to 3.0. I also remember Robert Heinlein in one of his novels mentioning that Tennessee had passed a similar law. Did either of these states ever pass such a law? Are they still on the books? What are the penalties if I proclaim that pi equals 3.14159...?
— Wulf Losee, Andover, Connecticut
Dear Wulf:
Cecil had heard this story too, only the state in question was Kansas, leading him to believe the whole thing was made up by big-city sharpies having a little fun at the expense of the rustics. However, with the help of Joseph Madachy, editor of the Journal of Recreational Mathematics, I've learned the story does have a germ of truth to it.
...
Plotting & GraphicsDiscrete Mathematics
- plot a function plot x^3 - 6x^2 + 4x + 12
- plot a region satisfying multiple inequalities plot x^2+y^2<1 and y>x
Applied Mathematics
- compute a possible formula and continuation for a sequence 5, 14, 23, 32, 41, ...
- analyze a graph specified by adjacency rules 1->2, 2->3, 3->1, 3->4, 4->1
- solve a recurrence f(n)=f(n-1)+f(n-2), f(1)=1, f(2)=2
Mathematical Functions
- minimize or maximize a function maximize x(1-x)e^x
- numerically integrate functions that cannot be integrated symbolically integrate sin(cos x) from x=0 to 1
Advanced Mathematics
- do computations with special functions d/dx Si(x)^2
- do computations with number-theoretic functions phi(110)
- find representations for a function gamma(x) integral representation
Mathematical Definitions
- compute properties of a function of a complex variable (use the variable z) e^z
- do vector analysis computations div (x^3 y, y^3 z, z^3 x)
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Famous Math Problems
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- get historical information about a theorem Fermat's little theorem
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MathWorldTM is the web's most complete mathematical resource, assembled over more than a decade by internet encyclopedist Eric W. Weisstein with assistance from the mathematics and internet communities.
MathWorld is a comprehensive and interactive mathematics encyclopedia intended for students, educators, math enthusiasts, and researchers. Like the vibrant and constantly evolving discipline of mathematics, this site is continuously updated to include new material and incorporate new discoveries.
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Kontorovich-Lebedev Transform