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Compound Polygon
In geometry, stellation is the process of extending a polygon in two dimensions, a polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specific elements such as its edges or face planes, usually in a symmetrical way, until they meet each other again to form the closed boundary of a new figure. The new figure is a stellation of the original. The word ''stellation'' comes from the Latin ''stellātus'', "starred", which in turn comes from the Latin ''stella'', "star". Stellation is the reciprocal or dual process to ''faceting''. Kepler's definition In 1619 Kepler defined stellation for polygons and polyhedra as the process of extending edges or faces until they meet to form a new polygon or polyhedron. He stellated the regular dodecahedron to obtain two regular star polyhedra, the small stellated dodecahedron and the great stellated dodecahedron. He also stellated the regular octah ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coprime
In number theory, 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 When the integers and are coprime, the standard way of expressing this fact in mathematical notation is to indicate that their greatest common divisor is one, by the formula or . In their 1989 textbook '' Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed an alte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Obtuse Heptagram
Obtuse may refer to: * Obtuse angle, an angle of between 90 and 180 degrees * Obtuse triangle, a triangle with an internal angle of between 90 and 180 degrees * Obtuse leaf shape * Obtuse tepal shape * Obtuse barracuda, a ray-finned fish * Obtuse, a neighborhood in Brookfield, Connecticut {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enneagram (geometry)
In geometry, an enneagram (🟙 U+1F7D9) is a nine-pointed plane figure. It is sometimes called a nonagram, nonangle, or enneagon. The word 'enneagram' combines the numeral prefix ''wikt:ennea-, ennea-'' with the Greek language, Greek suffix ''wikt:-gram, -gram''. The ''gram'' suffix derives from ''γραμμῆ'' (''grammē'') meaning a line. Regular enneagram A regular enneagram is a 9-sided star polygon. It is constructed using the same points as the regular enneagon, but the points are connected in fixed steps. Two forms of regular enneagram exist: *One form connects every second point and is represented by the Schläfli symbol . *The other form connects every fourth point and is represented by the Schläfli symbol . There is also a star figure, or 3, made from the regular enneagon points but connected as a compound of three equilateral triangles. (If the triangles are alternately interlaced, this results in a Brunnian link.) This star figure is sometimes known as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enneagon
In geometry, a nonagon () or enneagon () 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. Regular nonagon A '' regular nonagon'' is represented by Schläfli symbol and has internal angles of 140°. The area of a regular nonagon of side length ''a'' is given by :A = \fraca^2\cot\frac=(9/2)ar = 9r^2\tan(\pi/9) :::= (9/2)R^2\sin(2\pi/9)\simeq6.18182\,a^2, where the radius ''r'' of the inscribed circle of the regular nonagon is :r=(a/2)\cot(\pi/9) and where ''R'' is the radius of its circumscribed circle: :R = \sqrt=r\sec(\pi/9)=(a/2)\csc(\pi/9). Construction Although a regular nonagon is not constructible with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A regular hexagon is defined as a hexagon that is both equilateral and equiangular. In other words, a hexagon is said to be regular if the edges are all equal in length, and each of its internal angle is equal to 120°. The Schläfli symbol denotes this polygon as \ . However, the regular hexagon can also be considered as the cutting off the vertices of an equilateral triangle, which can also be denoted as \mathrm\ . A regular hexagon is bicentric, meaning that it is both cyclic (has a circumscribed circle) and tangential (has an inscribed circle). The common length of the sides equals the radius of the circumscribed circle or circumcircle, which equals \tfrac times the apothem (radius of the inscribed circle). Measurement The longest diagonals of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Star Figure 2(3,1)
Regular may refer to: Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses * Regular character, a main character who appears more frequently and/or prominently than a recurring character * Regular division of the plane, a series of drawings by the Dutch artist M. C. Escher which began in 1936 Language * Regular inflection, the formation of derived forms such as plurals in ways that are typical for the language ** Regular verb * Regular script, the newest of the Chinese script styles Mathematics Algebra and number theory * Regular category, a kind of category that has similarities to both Abelian categories and to the category of sets * Regular chains in computer algebra * Regular element (other), certain kinds of elements of an algebraic structure * Regular extension of fields * Regular ideal (multiple definit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagon
In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ''regular pentagon'' (or ''star polygon, star pentagon'') is called a pentagram. Regular pentagons A ''regular polygon, regular pentagon'' has Schläfli symbol and interior angles of 108°. A ''regular polygon, regular pentagon'' has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex polygon, convex regular pentagon are in the golden ratio to its sides. Given its side length t, its height H (distance from one side to the opposite vertex), width W (distance between two farthest separated points, which equals the diagonal length D) and circumradius R are given by: :\begin H &= \frac~t \approx 1.539~t, \\ W= D &= \frac~t\approx 1.618~t, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagram
A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle around the five points creates a similar symbol referred to as the pentacle, which is used widely by Wiccans and in paganism, or as a sign of life and connections. The word ''pentagram'' comes from the Greek language, Greek word πεντάγραμμον (''pentagrammon''), from πέντε (''pente''), "five" + γραμμή (''grammē''), "line". The word pentagram refers to just the star and the word pentacle refers to the star within a circle, although there is some overlap in usage. The word ''pentalpha'' is a 17th-century revival of a post-classical Greek name of the shape. History Early history Early pentagrams have been found on Sumerian pottery from Ur c. 3500 Common Era, BCE, and the five-pointed star was at various times the symbol of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digon
In geometry, a bigon, digon, or a ''2''-gon, is a polygon with two sides (edge (geometry), edges) and two Vertex (geometry), vertices. Its construction is Degeneracy (mathematics), degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visualised in elliptic space. It may also be viewed as a representation of a graph theory, graph with two vertices, see "Generalized polygon". A regular digon has both angles equal and both sides equal and is represented by Schläfli symbol . It may be constructed on a spherical geometry, sphere as a pair of 180 degree arcs connecting antipodal points, when it forms a spherical lune, lune. The digon is the simplest abstract polytope of rank 2. A truncation (geometry), truncated ''digon'', t is a square, . An Alternation (geometry), alternated digon, h is a monogon, . In different fields In Euclidean geometry The digon can have one of two visual representat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
