HeptadecagonIn geometry, a heptadecagon, septadecagon or 17-gon is a seventeen-sided polygon. A regular heptadecagon is represented by the Schläfli symbol {17}. As 17 is a Fermat prime, the regular heptadecagon is a constructible polygon (that is, one that can be constructed using a compass and unmarked straightedge): this was shown by Carl Friedrich Gauss in 1796 at the age of 19. This proof represented the first progress in regular polygon construction in over 2000 years.
Mathematics of paper foldingThe discipline of origami or paper folding has received a considerable amount of mathematical study. Fields of interest include a given paper model's flat-foldability (whether the model can be flattened without damaging it), and the use of paper folds to solve up-to cubic mathematical equations. Computational origami is a recent branch of computer science that is concerned with studying algorithms that solve paper-folding problems.
HeptagonIn geometry, a heptagon or septagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of septua-, a Latin-derived numerical prefix, rather than hepta-, a Greek-derived numerical prefix; both are cognate) together with the Greek suffix "-agon" meaning angle. A regular heptagon, in which all sides and all angles are equal, has internal angles of 5π/7 radians (128 degrees). Its Schläfli symbol is {7}.
Neusis constructionIn geometry, the neusis (νεῦσις; ; plural: neuseis) is a geometric construction method that was used in antiquity by Greek mathematicians. The neusis construction consists of fitting a line element of given length (a) in between two given lines (l and m), in such a way that the line element, or its extension, passes through a given point P. That is, one end of the line element has to lie on l, the other end on m, while the line element is "inclined" towards P.
NonagonIn geometry, a nonagon (ˈnɒnəgɒn) or enneagon (ˈɛniəɡɒn) is a nine-sided polygon or 9-gon. The name nonagon is a prefix hybrid formation, from Latin (nonus, "ninth" + gonon), used equivalently, attested already in the 16th century in French nonogone and in English from the 17th century. The name enneagon comes from Greek enneagonon (εννεα, "nine" + γωνον (from γωνία = "corner")), and is arguably more correct, though less common than "nonagon". A regular nonagon is represented by Schläfli symbol {9} and has internal angles of 140°.
IcosagonIn geometry, an icosagon or 20-gon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees. The regular icosagon has Schläfli symbol , and can also be constructed as a truncated decagon, t, or a twice-truncated pentagon, tt. One interior angle in a regular icosagon is 162°, meaning that one exterior angle would be 18°. The area of a regular icosagon with edge length t is In terms of the radius R of its circumcircle, the area is since the area of the circle is the regular icosagon fills approximately 98.
Gaussian periodIn mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis (discrete Fourier transform). They are basic in the classical theory called cyclotomy. Closely related is the Gauss sum, a type of exponential sum which is a linear combination of periods. As the name suggests, the periods were introduced by Gauss and were the basis for his theory of compass and straightedge construction.
Fermat numberIn mathematics, a Fermat number, named after Pierre de Fermat, the first known to have studied them, is a positive integer of the form where n is a non-negative integer. The first few Fermat numbers are: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, ... . If 2k + 1 is prime and k > 0, then k itself must be a power of 2, so 2k + 1 is a Fermat number; such primes are called Fermat primes. , the only known Fermat primes are F0 = 3, F1 = 5, F2 = 17, F3 = 257, and F4 = 65537 ; heuristics suggest that there are no more.
Cyclotomic polynomialIn mathematics, the nth cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial with integer coefficients that is a divisor of and is not a divisor of for any k < n. Its roots are all nth primitive roots of unity where k runs over the positive integers not greater than n and coprime to n (and i is the imaginary unit). In other words, the nth cyclotomic polynomial is equal to It may also be defined as the monic polynomial with integer coefficients that is the minimal polynomial over the field of the rational numbers of any primitive nth-root of unity ( is an example of such a root).
Regular polygonIn Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon (effectively a straight line), if the edge length is fixed. These properties apply to all regular polygons, whether convex or star.
