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Stellation
In geometry, stellation is the process of extending a polygon in two dimensions, 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 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 great stellated dodecahedron. He also stellated the regular octahedron to o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stella Octangula
The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depicted in Pacioli's ''De Divina Proportione,'' 1509. It is the simplest of five regular polyhedral compounds, and the only regular compound of two tetrahedra. It is also the least dense of the regular polyhedral compounds, having a density of 2. It can be seen as a 3D extension of the hexagram: the hexagram is a two-dimensional shape formed from two overlapping equilateral triangles, centrally symmetric to each other, and in the same way the stellated octahedron can be formed from two centrally symmetric overlapping tetrahedra. This can be generalized to any desired amount of higher dimensions; the four-dimensional equivalent construction is the compound of two 5-cells. It can also be seen as one of the stages in the construction of a 3D Koch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on the same plane. Cubes and pyramids are examples of convex polyhedra. A polyhedron is a 3-dimensional example of a polytope, a more general concept in any number of dimensions. Definition Convex polyhedra are well-defined, with several equivalent standard definitions. However, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of "polyhedron" have been given within particular contexts,. some more rigorous than others, and there is not universal agreement over which of these to choose. Some of these definitions exclude shapes that have often been counted as polyhedra (such as the self-crossing polyhedra) or include shapes that are often not considered as valid polyhedr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Stellated Dodecahedron
In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex. It shares the same vertex arrangement as the convex regular icosahedron. It also shares the same edge arrangement with the great icosahedron, with which it forms a degenerate uniform compound figure. It is the second of four stellations of the dodecahedron (including the original dodecahedron itself). The small stellated dodecahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the edges (1-faces) of the core polytope until a point is reached where they intersect. Topology If the pentagrammic faces are considered as 5 triangular faces, it shares the same surface topology as the pentakis dodecahedron, but with much taller isosceles triangle faces, with the heigh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Stellated Dodecahedron
In geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedron, with Schläfli symbol . It is one of four nonconvex regular polyhedra. It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each vertex. It shares its vertex arrangement, although not its vertex figure or vertex configuration, with the regular dodecahedron, as well as being a stellation of a (smaller) dodecahedron. It is the only dodecahedral stellation with this property, apart from the dodecahedron itself. Its dual, the great icosahedron, is related in a similar fashion to the icosahedron. Shaving the triangular pyramids off results in an icosahedron. If the pentagrammic faces are broken into triangles, it is topologically related to the triakis icosahedron, with the same face connectivity, but much taller isosceles triangle faces. If the triangles are instead made to invert themselves and excavate the central icosahedron, the result is a great dodecahedron. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagram Green
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" refers only to the five-pointed star, not the surrounding circle of a pentacle. Pentagrams were used symbolically in ancient Greece and Babylonia. Christians once commonly used the pentagram to represent the five wounds of Jesus. Today the symbol is widely used by the Wiccans, witches, and pagans. The pentagram has magical associations. Many people who practice neopaganism wear jewelry incorporating the symbol. The word ''pentagram'' comes from the Greek word πεντάγραμμον (''pentagrammon''), from πέντε (''pente''), "five" + γραμμή (''gr ... [...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] |
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] |
Octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. A regular octahedron is the dual polyhedron of a cube. It is a rectified tetrahedron. It is a square bipyramid in any of three orthogonal orientations. It is also a triangular antiprism in any of four orientations. An octahedron is the three-dimensional case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan () metric. Regular octahedron Dimensions If the edge length of a regular octahedron is ''a'', the radius of a circumscribed sphere (one that touches the octahedron at all vertices) is :r_u = \frac a \approx 0.707 \cdot a and the radius of an inscribed sphere (tangent to each of the octahedron's faces) is :r_i = \frac a \approx 0.408\cdot a while the midradius, which ... [...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" refers only to the five-pointed star, not the surrounding circle of a pentacle. Pentagrams were used symbolically in ancient Greece and Babylonia. Christians once commonly used the pentagram to represent the Five Holy Wounds, five wounds of Jesus. Today the symbol is widely used by the Wiccans, witches, and pagans. The pentagram has Magic (supernatural), magical associations. Many people who practice neopaganism wear jewelry incorporating the symbol. The word ''pentagram'' comes from the Greek language, Greek word πεντάγραμμον (''pentagrammon''), fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dodecahedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120. Some dodecahedra have the same combinatorial structure as the regular dodecahedron (in terms of the graph formed by its vertices and edges), but their pentagonal faces are not regular: The pyritohedron, a common crystal form in pyrite, has pyritohedral symmetry, while the tetartoid has tetrahedral symmetry. The rhombic dodecahedron can be seen as a limiting case of the pyritohedron, and it has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra, are space-filling. There ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enneagon Stellations
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 than "nonagon". 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). Construction Although a regular nonagon is not constructible with compa ... [...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 than "nonagon". 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). Construction Although a regular nonagon is not constructible with compa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
