Portal:Mathematics/Did you know archive

• Portal:Mathematics/Did_you_know/1: ...that the mathematician Grigori Perelman was offered a Fields Medal in 2006, in part for his proof of the Poincaré conjecture, which he declined?
• Portal:Mathematics/Did_you_know/2: ...that a regular heptagon is the regular polygon with the fewest sides which is not constructible with a compass and straightedge?
• Portal:Mathematics/Did_you_know/3: ...that the regular trigonometric functions and the hyperbolic trigonometric functions can be related without using complex numbers through the Gudermannian function?
• Portal:Mathematics/Did_you_know/4: ...that the Catalan numbers solve a number of problems in combinatorics such as the number of ways to completely parenthesize an algebraic expression with n+1 factors?
• Portal:Mathematics/Did_you_know/5: ...that a ball can be cut up and reassembled into two balls, each the same size as the original (Banach-Tarski paradox)?
• Portal:Mathematics/Did_you_know/6: ...that it is impossible to devise a single formula involving only polynomials and radicals for solving an arbitrary quintic equation?
• Portal:Mathematics/Did_you_know/7: ...that Euler found 59 more amicable numbers while for 2000 years, only 3 pairs had been found before him?
• Portal:Mathematics/Did_you_know/8: ...that you cannot knot strings in 4 dimensions, but you can knot 2-dimensional surfaces, such as spheres?
• Portal:Mathematics/Did_you_know/9: ...that there are 6 unsolved mathematics problems whose solutions will earn you one million US dollars each?
• Portal:Mathematics/Did_you_know/10: ...that there are different sizes of infinite sets in set theory? More precisely, not all infinite cardinal numbers are equal?
• Portal:Mathematics/Did_you_know/11: ...that every natural number can be written as the sum of four squares?
• Portal:Mathematics/Did_you_know/12: ...that the largest known prime number is nearly 41 million digits long?
• Portal:Mathematics/Did_you_know/13: ...that the set of rational numbers is equal in size to the set of integers; that is, they can be put in one-to-one correspondence?
• Portal:Mathematics/Did_you_know/14: ...that there are precisely six convex regular polytopes in four dimensions? These are analogs of the five Platonic solids known to the ancient Greeks.
• Portal:Mathematics/Did_you_know/15: ...that it is unknown whether π and e are algebraically independent?
• Portal:Mathematics/Did_you_know/16: ...that a nonconvex polygon with three convex vertices is called a pseudotriangle?
• Portal:Mathematics/Did_you_know/17: ...that it is possible for a three-dimensional figure to have a finite volume but infinite surface area, such as Gabriel's Horn?
• Portal:Mathematics/Did_you_know/18: ... that as the dimension of a hypersphere tends to infinity, its "volume" (content) tends to 0?
• Portal:Mathematics/Did_you_know/19: ...that the primality of a number can be determined using only a single division using Wilson's Theorem?
• Portal:Mathematics/Did_you_know/20: ...that the line separating the numerator and denominator of a fraction is called a solidus if written as a diagonal line or a vinculum if written as a horizontal line?
• Portal:Mathematics/Did_you_know/21: ...that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type the complete works of William Shakespeare?
• Portal:Mathematics/Did_you_know/22: ... that there are 115,200 solutions to the ménage problem of permuting six female-male couples at a twelve-person table so that men and women alternate and are seated away from their partners?
• Portal:Mathematics/Did_you_know/23: ... that mathematician Paul Erdős called the Hadwiger conjecture, a still-open generalization of the four-color problem, "one of the deepest unsolved problems in graph theory"?
• Portal:Mathematics/Did_you_know/24: ...that the six permutations of the vector (1,2,3) form a regular hexagon in 3d space, the 24 permutations of (1,2,3,4) form a truncated octahedron in four dimensions, and both are examples of permutohedra?
• Portal:Mathematics/Did_you_know/25: ...that Ostomachion is a mathematical treatise attributed to Archimedes on a 14-piece tiling puzzle similar to tangram?
• Portal:Mathematics/Did_you_know/26: ...that some functions can be written as an infinite sum of trigonometric polynomials and that this sum is called the Fourier series of that function?
• Portal:Mathematics/Did_you_know/27: ...that the identity elements for arithmetic operations make use of the only two whole numbers that are neither composites nor prime numbers, 0 and 1?
• Portal:Mathematics/Did_you_know/28: ...that as of April 2010 only 35 even numbers have been found that are not the sum of two primes which are each in a Twin Primes pair? ref
• Portal:Mathematics/Did_you_know/29: ...the Piphilology record (memorizing digits of Pi) is 70000 as of Mar 2015?
• Portal:Mathematics/Did_you_know/30: ...that people are significantly slower to identify the parity of zero than other whole numbers, regardless of age, language spoken, or whether the symbol or word for zero is used?
• Portal:Mathematics/Did_you_know/31: ...that Auction theory was successfully used in 1994 to sell FCC airwave spectrum, in a financial application of game theory?
• Portal:Mathematics/Did_you_know/32: ...properties of Pascal's triangle have application in many fields of mathematics including combinatorics, algebra, calculus and geometry?
• Portal:Mathematics/Did_you_know/33: ...work in artificial intelligence makes use of swarm intelligence, which has foundations in the behavioral examples found in nature of ants, birds, bees, and fish among others?
• Portal:Mathematics/Did_you_know/34: ...that statistical properties dictated by Benford's Law are used in auditing of financial accounts as one means of detecting fraud?
• Portal:Mathematics/Did_you_know/35: ...that modular arithmetic has application in at least ten different fields of study, including the arts, computer science, and chemistry in addition to mathematics?
• Portal:Mathematics/Did_you_know/36: ... that according to Kawasaki's theorem, an origami crease pattern with one vertex may be folded flat if and only if the sum of every other angle between consecutive creases is 180º?
• Portal:Mathematics/Did_you_know/37: ... that, in the Rule 90 cellular automaton, any finite pattern eventually fills the whole array of cells with copies of itself?
• Portal:Mathematics/Did_you_know/38: ... that, while the criss-cross algorithm visits all eight corners of the Klee–Minty cube when started at a worst corner, it visits only three more corners on average when started at a random corner?
• Portal:Mathematics/Did_you_know/39: ...that in senary, all prime numbers other than 2 and 3 end in 1 or a 5?
• Portal:Mathematics/Did_you_know/40: ...that, for all prime numbers p, the pth Perrin number is divisible by p?
• Portal:Mathematics/Did_you_know/41: ...that it is impossible to trisect a general angle using only a ruler and a compass?
• Portal:Mathematics/Did_you_know/42: ...that in a group of 23 people, there is a more than 50% chance that two people share a birthday?
• Portal:Mathematics/Did_you_know/43: ...that the 1966 publication disproving Euler's sum of powers conjecture, proposed nearly 200 years earlier, consisted of only two sentences?
• Portal:Mathematics/Did_you_know/44: ...the hyperbolic trigonometric functions of the natural logarithm can be represented by rational algebraic fractions?
• Portal:Mathematics/Did_you_know/45: ... that economists blame market failures on non-convexity?
• Portal:Mathematics/Did_you_know/46: ... that, according to the pizza theorem, a circular pizza that is sliced off-center into eight equal-angled wedges can still be divided equally between two people?
• Portal:Mathematics/Did_you_know/47: ... that the clique problem of programming a computer to find complete subgraphs in an undirected graph was first studied as a way to find groups of people who all know each other in social networks?
• Portal:Mathematics/Did_you_know/48: ... that the Herschel graph is the smallest possible polyhedral graph that does not have a Hamiltonian cycle?
• Portal:Mathematics/Did_you_know/49: ... that the Life without Death cellular automaton, a mathematical model of pattern formation, is a variant of Conway's Game of Life in which cells, once brought to life, never die?
• Portal:Mathematics/Did_you_know/50: ... that one can list every positive rational number without repetition by breadth-first traversal of the Calkin–Wilf tree?
• Portal:Mathematics/Did_you_know/51: ... that the Hadwiger conjecture implies that the external surface of any three-dimensional convex body can be illuminated by only eight light sources, but the best proven bound is that 16 lights are sufficient?
• Portal:Mathematics/Did_you_know/52: ... that an equitable coloring of a graph, in which the numbers of vertices of each color are as nearly equal as possible, may require far more colors than a graph coloring without this constraint?
• Portal:Mathematics/Did_you_know/53: ... that no matter how biased a coin one uses, flipping a coin to determine whether each edge is present or absent in a countably infinite graph will always produce the same graph, the Rado graph?
• Portal:Mathematics/Did_you_know/54: ...that it is possible to stack identical dominoes off the edge of a table to create an arbitrarily large overhang?
• Portal:Mathematics/Did_you_know/55: ...that in Floyd's algorithm for cycle detection, the tortoise and hare move at very different speeds, but always finish at the same spot?
• Portal:Mathematics/Did_you_know/56: ...that in graph theory, a pseudoforest can contain trees and pseudotrees, but cannot contain any butterflies, diamonds, handcuffs, or bicycles?
• Portal:Mathematics/Did_you_know/57: ...that it is not possible to configure two mutually inscribed quadrilaterals in the Euclidean plane, but the Möbius–Kantor graph describes a solution in the complex projective plane?
• Portal:Mathematics/Did_you_know/58: ...that the six permutations of the vector (1,2,3) form a hexagon in 3D space, the 24 permutations of (1,2,3,4) form a truncated octahedron in four dimensions, and both are examples of permutohedra?
• Portal:Mathematics/Did_you_know/59: ...that the Rule 184 cellular automaton can simultaneously model the behavior of cars moving in traffic, the accumulation of particles on a surface, and particle-antiparticle annihilation reactions?
• Portal:Mathematics/Did_you_know/60: ...that a cyclic cellular automaton is a system of simple mathematical rules that can generate complex patterns mixing random chaos, blocks of color, and spirals?
• Portal:Mathematics/Did_you_know/61: ...that a nonconvex polygon with three convex vertices is called a pseudotriangle?
• Portal:Mathematics/Did_you_know/62: ...that the axiom of choice is logically independent of the other axioms of Zermelo–Fraenkel set theory?
• Portal:Mathematics/Did_you_know/63: ...that the Pythagorean Theorem generalizes to any three similar shapes on the three sides of a right-angled triangle?
• Portal:Mathematics/Did_you_know/64: ...that the orthocenter, circumcenter, centroid and the centre of the nine-point circle all lie on one line, the Euler line?
• Portal:Mathematics/Did_you_know/65: ...that an arbitrary quadrilateral will tessellate?
• Portal:Mathematics/Did_you_know/66: ...that it has not been proven whether or not every even integer greater than two can be expressed as the sum of two primes?
• Portal:Mathematics/Did_you_know/67: ...that the sum of the first n odd numbers divided by the sum of the next n odd numbers is always equal to one third?
• Portal:Mathematics/Did_you_know/68: ...that i to the power of i, where i is the square root of -1, is a real number?
• Portal:Mathematics/Did_you_know/69: ...an infinite, nonrepeating decimal can be represented using only the number 1 using continued fractions?
• Portal:Mathematics/Did_you_know/70: ...that 253931039382791 and the following 18 prime numbers all end in the digit 1?
• Portal:Mathematics/Did_you_know/71: ...that the Electronic Frontier Foundation funds awards for the discovery of prime numbers beyond certain sizes?
• Portal:Mathematics/Did_you_know/72: ...that pi can be computed using only the number 2 by the work of Viète?
• Portal:Mathematics/Did_you_know/73: … that the Riemann Hypothesis, one of the Millennium Problems, depends on the asymptotic growth of the Mertens Function?
• Portal:Mathematics/Did_you_know/74: … that every positive integer can be written as the sum of three palindromic numbers in every number system with base 5 or greater?
• Portal:Mathematics/Did_you_know/75: … that the best known lower bound for the length of the smallest superpermutation was first posted anonymously to the internet imageboard 4chan?
• Portal:Mathematics/Did_you_know/76