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Star Polyhedra
In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality. There are two general kinds of star polyhedron: *Polyhedra which self-intersect in a repetitive way. *Concave polyhedra of a particular kind which alternate convex and concave or saddle vertices in a repetitive way. Mathematically these figures are examples of star domains. Mathematical studies of star polyhedra are usually concerned with regular, uniform polyhedra, or the duals of the uniform polyhedra. All these stars are of the self-intersecting kind. Self-intersecting star polyhedra Regular star polyhedra The regular star polyhedra are self-intersecting polyhedra. They may either have self-intersecting faces, or self-intersecting vertex figures. There are four regular star polyhedra, known as the Kepler–Poinsot polyhedra. The Schläfli symbol implies faces with ''p'' sides, and vertex figures with ''q'' sides. Two of them have p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prismatic Uniform Polyhedron
In geometry, a prismatic uniform polyhedron is a uniform polyhedron with dihedral symmetry. They exist in two infinite families, the uniform prisms and the uniform antiprisms. All have their vertices in parallel planes and are therefore prismatoids. Vertex configuration and symmetry groups Because they are isogonal (vertex-transitive), their vertex arrangement uniquely corresponds to a symmetry group. The difference between the prismatic and antiprismatic symmetry groups is that D''p''h has the vertices lined up in both planes, which gives it a reflection plane perpendicular to its ''p''-fold axis (parallel to the polygon); while D''p''d has the vertices twisted relative to the other plane, which gives it a rotatory reflection. Each has ''p'' reflection planes which contain the ''p''-fold axis. The D''p''h symmetry group contains inversion if and only if ''p'' is even, while D''p''d contains inversion symmetry if and only if ''p'' is odd. Enumeration There are: * pr ... [...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'' has Schläfli symbol and can also be constructed as a truncated equilateral triangle, t, which alternates two types of edges. A regular hexagon is defined as a hexagon that is both equilateral and equiangular. It 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). All internal angles are 120 degrees. A regular hexagon has six rotational symmetries (''rotational symmetry of order six'') and six reflection symmetries (''six lines of symmetry''), making up the dihedral group D6. The longest diagonals of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Dodecicosahedron
In geometry, the great dodecicosahedron (or great dodekicosahedron) is a nonconvex uniform polyhedron, indexed as U63. It has 32 faces (20 hexagons and 12 decagrams), 120 edges, and 60 vertices. Its vertex figure is a crossed quadrilateral. It has a composite Wythoff symbol, 3 ( ) , , requiring two different Schwarz triangles to generate it: (3 ) and (3 ). (3 , represents the ''great dodecicosahedron'' with an extra 12 pentagons, and 3 , represents it with an extra 20 triangles.) pp. 9–10. Its vertex figure 6... is also ambiguous, having two clockwise and two counterclockwise faces around each vertex. Related polyhedra It shares its vertex arrangement with the truncated dodecahedron. It additionally shares its edge arrangement with the great icosicosidodecahedron (having the hexagonal faces in common) and the great ditrigonal dodecicosidodecahedron (having the decagrammic faces in common). Gallery See also * List of uniform polyhedra In geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Dodecicosahedron
In geometry, the great dodecicosahedron (or great dodekicosahedron) is a nonconvex uniform polyhedron, indexed as U63. It has 32 faces (20 hexagons and 12 decagrams), 120 edges, and 60 vertices. Its vertex figure is a crossed quadrilateral. It has a composite Wythoff symbol, 3 ( ) , , requiring two different Schwarz triangles to generate it: (3 ) and (3 ). (3 , represents the ''great dodecicosahedron'' with an extra 12 pentagons, and 3 , represents it with an extra 20 triangles.) pp. 9–10. Its vertex figure 6... is also ambiguous, having two clockwise and two counterclockwise faces around each vertex. Related polyhedra It shares its vertex arrangement with the truncated dodecahedron. It additionally shares its edge arrangement with the great icosicosidodecahedron (having the hexagonal faces in common) and the great ditrigonal dodecicosidodecahedron (having the decagrammic faces in common). Gallery See also * List of uniform polyhedra In geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isosceles Triangle
In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter version thus including the equilateral triangle as a special case. Examples of isosceles triangles include the isosceles right triangle, the golden triangle, and the faces of bipyramids and certain Catalan solids. The mathematical study of isosceles triangles dates back to ancient Egyptian mathematics and Babylonian mathematics. Isosceles triangles have been used as decoration from even earlier times, and appear frequently in architecture and design, for instance in the pediments and gables of buildings. The two equal sides are called the legs and the third side is called the base of the triangle. The other dimensions of the triangle, such as its height, area, and perimeter, can be calculated by simple formulas from the lengths of the legs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Face-transitive
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent but must be ''transitive'', i.e. must lie within the same ''symmetry orbit''. In other words, for any two faces and , there must be a symmetry of the ''entire'' figure by translations, rotations, and/or reflections that maps onto . For this reason, convex isohedral polyhedra are the shapes that will make fair dice. Isohedral polyhedra are called isohedra. They can be described by their face configuration. An isohedron has an even number of faces. The dual of an isohedral polyhedron is vertex-transitive, i.e. isogonal. The Catalan solids, the bipyramids, and the trapezohedra are all isohedral. They are the duals of the (isogonal) Archimedean solids, prisms, and antiprisms, respectively. The Platonic solids, which are either se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagrammic Dipyramid
In geometry, the pentagrammic prism is one of an infinite set of nonconvex prisms formed by square sides and two regular star polygon caps, in this case two pentagrams. It is a special case of a right prism with a pentagram as base, which in general has rectangular non-base faces. Topologically it is the same as a convex pentagonal prism. It is the 78th model in the list of uniform polyhedra, as the first representative of uniform star prisms, along with the pentagrammic antiprism, which is the 79th model. Geometry It has 7 faces, 15 edges and 10 vertices. This polyhedron is identified with the indexed name U78 as a uniform polyhedron. The triangle face has an ambiguous interior because it is self-intersecting. The central pentagon region can be considered a angel or exterior depending on how the interior is defined. One definition of the interior is the set of points that have a ray that crosses the boundary an odd number of times to escape the diameter Gallery Pentagr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagram Dipyramid
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square (geometry)
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ''ABCD'' would be denoted . Characterizations A convex quadrilateral is a square if and only if it is any one of the following: * A rectangle with two adjacent equal sides * A rhombus with a right vertex angle * A rhombus with all angles equal * A parallelogram with one right vertex angle and two adjacent equal sides * A quadrilateral with four equal sides and four right angles * A quadrilateral where the diagonals are equal, and are the perpendicular bisectors of each other (i.e., a rhombus with equal diagonals) * A convex quadrilateral wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagrammic Prism
In geometry, the pentagrammic prism is one of an infinite set of nonconvex prisms formed by square sides and two regular star polygon caps, in this case two pentagrams. It is a special case of a right prism with a pentagram as base, which in general has rectangular non-base faces. Topologically it is the same as a convex pentagonal prism. It is the 78th model in the list of uniform polyhedra, as the first representative of uniform star prisms, along with the pentagrammic antiprism, which is the 79th model. Geometry It has 7 faces, 15 edges and 10 vertices. This polyhedron is identified with the indexed name U78 as a uniform polyhedron. The triangle face has an ambiguous interior because it is self-intersecting. The central pentagon region can be considered a angel or exterior depending on how the interior is defined. One definition of the interior is the set of points that have a ray that crosses the boundary an odd number of times to escape the diameter Gallery Pentagra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.stl/1200px-Pentagrammic_bipyramid_(dual_uniform).stl.png "Click for more on -> Pentagrammic Prism")