|
Megagon
A megagon or 1,000,000-gon is a polygon with one million sides (mega-, from the Greek μέγας, meaning "great", being a unit prefix denoting a factor of one million). Regular megagon A regular megagon is represented by the Schläfli symbol and can be constructed as a truncated 500,000-gon, t, a twice-truncated 250,000-gon, tt, a thrice-truncated 125,000-gon, ttt, or a four-fold-truncated 62,500-gon, tttt, a five-fold-truncated 31,250-gon, , or a six-fold-truncated 15,625-gon, . A regular megagon has an interior angle of 179°59'58.704" 3.14158637 rad.Darling, David J., The Universal Book of Mathematics: from Abracadabra to Zeno's Paradoxes', John Wiley & Sons, 2004. Page 249. . The area of a regular megagon with sides of length ''a'' is given by :A = 250,000a^2 \cot \frac. The perimeter of a regular megagon inscribed in the unit circle is: :2,000,000 \sin\frac, which is very close to 2π. In fact, for a circle the size of the Earth's equator, with a circumference of 4 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Myriagon
In geometry, a myriagon or 10000-gon is a polygon with 10,000 sides. Several philosophers have used the regular myriagon to illustrate issues regarding thought. Meditation VI by Descartes (English translation). Regular myriagon A regular myriagon is represented by Schläfli symbol and can be constructed as a truncated 5000-gon, t, or a twice-truncated 2500-gon, tt, or a thrice-truncated 1250-gon, ttt{1250), or a four-fold-truncated 625-gon, tttt{625}. The measure of each internal angle in a regular myriagon is 179.964°. The area of a regular myriagon with sides of length ''a'' is given by :A = 2500a^2 \cot \frac{\pi}{10000} The result differs from the area of its circumscribed circle by up to 40 parts per billion. Because 10,000 = 24 × 54, the number of sides is neither a product of distinct Fermat primes nor a power of two. Thus the regular myriagon is not a constructible polygon. Indeed, it is not even constructible with the use of an angle trisector, as the number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chiliagon
In geometry, a chiliagon () or 1000-gon is a polygon with 1,000 sides. Philosophers commonly refer to chiliagons to illustrate ideas about the nature and workings of thought, meaning, and mental representation. Regular chiliagon A '' regular chiliagon'' is represented by Schläfli symbol and can be constructed as a truncated 500-gon, t, or a twice-truncated 250-gon, tt, or a thrice-truncated 125-gon, ttt. The measure of each internal angle in a regular chiliagon is 179°38'24"/\fracrad. The area of a regular chiliagon with sides of length ''a'' is given by :A = 250a^2 \cot \frac \simeq 79577.2\,a^2 This result differs from the area of its circumscribed circle by less than 4 parts per million. Because 1,000 = 23 × 53, the number of sides is neither a product of distinct Fermat primes nor a power of two. Thus the regular chiliagon is not a constructible polygon. Indeed, it is not even constructible with the use of an angle trisector, as the number of sides is neither a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chiliagon
In geometry, a chiliagon () or 1000-gon is a polygon with 1,000 sides. Philosophers commonly refer to chiliagons to illustrate ideas about the nature and workings of thought, meaning, and mental representation. Regular chiliagon A '' regular chiliagon'' is represented by Schläfli symbol and can be constructed as a truncated 500-gon, t, or a twice-truncated 250-gon, tt, or a thrice-truncated 125-gon, ttt. The measure of each internal angle in a regular chiliagon is 179°38'24"/\fracrad. The area of a regular chiliagon with sides of length ''a'' is given by :A = 250a^2 \cot \frac \simeq 79577.2\,a^2 This result differs from the area of its circumscribed circle by less than 4 parts per million. Because 1,000 = 23 × 53, the number of sides is neither a product of distinct Fermat primes nor a power of two. Thus the regular chiliagon is not a constructible polygon. Indeed, it is not even constructible with the use of an angle trisector, as the number of sides is neither a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierpont Prime
In number theory, a Pierpont prime is a prime number of the form 2^u\cdot 3^v + 1\, for some nonnegative integers and . That is, they are the prime numbers for which is 3-smooth. They are named after the mathematician James Pierpont, who used them to characterize the regular polygons that can be constructed using conic sections. The same characterization applies to polygons that can be constructed using ruler, compass, and angle trisector, or using paper folding. Except for 2 and the Fermat primes, every Pierpont prime must be 1 modulo 6. The first few Pierpont primes are: It has been conjectured that there are infinitely many Pierpont primes, but this remains unproven. Distribution A Pierpont prime with is of the form 2^u+1, and is therefore a Fermat prime (unless ). If is positive then must also be positive (because 3^v+1 would be an even number greater than 2 and therefore not prime), and therefore the non-Fermat Piermont primes all have the form , when is a posi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star Figure
In geometry, a generalized polygon can be called a polygram, and named specifically by its number of sides. All polygons are polygrams, but can also include disconnected sets of edges, called a compound polygon. For example, a regular pentagram, , has 5 sides, and the regular hexagram, or 2, has 6 sides divided into two triangles. A regular polygram can either be in a set of regular star polygons (for gcd(''p'',''q'') = 1, ''q'' > 1) or in a set of regular polygon compounds (if gcd(''p'',''q'') > 1). Etymology The polygram names combine a numeral prefix, such as ''penta-'', with the Greek suffix '' -gram'' (in this case generating the word ''pentagram''). The prefix is normally a Greek cardinal, but synonyms using other prefixes exist. The ''-gram'' suffix derives from ''γραμμῆς'' (''grammos'') meaning a line. Generalized regular polygons A regular polygram, as a general regular polygon, is denoted by its Schläfli symbol , where ''p'' and ''q'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also '' is prime to '' or '' is coprime with ''. The numbers 8 and 9 are coprime, despite the fact that neither considered individually is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing Standard notations for relatively prime integers and are: and . In their 1989 textbook ''Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed that the notation a\perp b be used to indicate that and are relatively prime and that the term "prime" be used instead of coprime (as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star Polygon
In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations on regular simple and star polygons. Branko Grünbaum identified two primary definitions used by Johannes Kepler, one being the regular star polygons with intersecting edges that don't generate new vertices, and the second being simple isotoxal concave polygons. The first usage is included in polygrams which includes polygons like the pentagram but also compound figures like the hexagram. One definition of a ''star polygon'', used in turtle graphics, is a polygon having 2 or more turns (turning number and density), like in spirolaterals.Abelson, Harold, diSessa, Andera, 1980, ''Turtle Geometry'', MIT Press, p.24 Etymology Star polygon names combine a numeral prefix, such as ''penta-'', with the Greek suffix '' -gram'' (in this cas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Directed Edge
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pair where * ''V'' is a set whose elements are called '' vertices'', ''nodes'', or ''points''; * ''A'' is a set of ordered pairs of vertices, called ''arcs'', ''directed edges'' (sometimes simply ''edges'' with the corresponding set named ''E'' instead of ''A''), ''arrows'', or ''directed lines''. It differs from an ordinary or undirected graph, in that the latter is defined in terms of unordered pairs of vertices, which are usually called ''edges'', ''links'' or ''lines''. The aforementioned definition does not allow a directed graph to have multiple arrows with the same source and target nodes, but some authors consider a broader definition that allows directed graphs to have such multiple arcs (namely, they allow the arc set to be a m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches of recreational mathematics, most notably the invention of the cellular automaton called the Game of Life. Born and raised in Liverpool, Conway spent the first half of his career at the University of Cambridge before moving to the United States, where he held the John von Neumann Professorship at Princeton University for the rest of his career. On 11 April 2020, at age 82, he died of complications from COVID-19. Early life and education Conway was born on 26 December 1937 in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at a very early age. By the time he was 11, his ambition was to become a mathematician. After leaving sixth form, he studied mathematics at Gonville and Caius College, Camb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element ''g'' such that every other element of the group may be obtained by repeatedly applying the group operation to ''g'' or its inverse. Each element can be written as an integer power of ''g'' in multiplicative notation, or as an integer multiple of ''g'' in additive notation. This element ''g'' is called a ''generator'' of the group. Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order ''n'' is isomorphic to the additive group of Z/''n''Z, the integers modulo ''n''. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dihedral Symmetry
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry. The notation for the dihedral group differs in geometry and abstract algebra. In geometry, or refers to the symmetries of the -gon, a group of order . In abstract algebra, refers to this same dihedral group. This article uses the geometric convention, . Definition Elements A regular polygon with n sides has 2n different symmetries: n rotational symmetries and n reflection symmetries. Usually, we take n \ge 3 here. The associated rotations and reflections make up the dihedral group \mathrm_n. If n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is even, there are n/2 axes of symmetry connecting the midpoints of opposite sides and n/2 axes of symmetry connecting oppo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathematics was central to his method of inquiry, and he connected the previously separate fields of geometry and algebra into analytic geometry. Descartes spent much of his working life in the Dutch Republic, initially serving the Dutch States Army, later becoming a central intellectual of the Dutch Golden Age. Although he served a Protestant state and was later counted as a deist by critics, Descartes considered himself a devout Catholic. Many elements of Descartes' philosophy have precedents in late Aristotelianism, the revived Stoicism of the 16th century, or in earlier philosophers like Augustine. In his natural philosophy, he differed from the schools on two major points: first, he rejected the splitting of corporeal substance into mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
