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In polyhedral geometry a vertex configuration is a short-hand notation for representing a vertex as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron. Geometry (Greek γεωμετÏία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. ...
A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ...
A vertex configuration is given as a sequence of numbers representing the number of sides of the faces going around the vertex. A a.b.c means a vertex has 3 faces around it, with a, b, and c sides.
For example 3.5.3.5 means a vertex has 4 faces, alternating triangles and pentagons. This vertex configuration defines the vertex-uniform icosidodecahedron polyhedron. A triangle is one of the basic shapes of geometry: a two-dimensional figure with three vertices and three sides which are straight line segments. ...
In geometry, a pentagon is any five-sided polygon. ...
An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. ...
Vertex Figures
A vertex configuration can also be represented graphically as vertex figure showing the faces around the vertex. This vertex figure has a 3-dimensional structure since the faces are not in the same plane for polyhedra, but for vertex-uniform polyhedra all the neighboring vertices are in the same plane and so this plane projection can be used to visually represent the vertex configuration. Image File history File links Icosidodecahedron_vertfig. ...
Image File history File links Icosidodecahedron_vertfig. ...
An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. ...
A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ...
This article is about technical drawings. ...
See image category: Category:Polyhedra-vf image
Variations and uses Different notations are used, sometimes with a comma (,), and sometimes a period (.) separator. The period operator is useful because it looks like a product and an exponent notation can be used. For example 3.5.3.5 is sometimes written as (3.5)^2 or (3.5)2.
The order is important and so 3.3.5.5 is different than 3.5.3.5. The first has two triangles followed by two pentagons.
The notation can also be considered an expansive form of the simple Schläfli symbol for regular polyhedra. {p,q} means q p-agons around each vertex. So this can be written as p.p.p... (q times). For example an icosahedron is {3,5} = 3.3.3.3.3 = 3^ 5= 35. In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ...
In solid geometry and some ancient physical theories, a Platonic solid is a convex polyhedron with all its faces being congruent regular polygons, and the same number of faces meeting at each of its vertices. ...
The notation is cyclic and therefore is equivalent with different starting points. So 3.5.3.5 is the same as 5.3.5.3. To be unique, usually the smallest face (or sequence of smallest faces) are listed first. So 3.3.4.3.4 represents the snub cube starting with 2 triangles. The snub cube, or snub cuboctahedron, is an Archimedean solid. ...
This notation applies to polygon tiles as well as polyhedra. A planar vertex configuration can imply a uniform tiling just like a nonplanar vertex configuration can imply a uniform polyhedron.
The notation is ambiguous for chiral forms. For example, the snub cube has a clockwise and counterclockwise form which are identical across mirror images. Both have a 3.3.3.3.4 vertex configuration. In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone. ...
The snub cube, or snub cuboctahedron, is an Archimedean solid. ...
Star polygons
 Vertex figure for Great dirhombicosidodecahedron The notation also applies for nonconvex regular faces, the star polygons. For example a pentagram has 5/2 sides meaning 5 vertex going around the vertex twice. For example, the nonconvex regular polyhedron small stellated dodecahedron has a vertex configuration of Schläfli symbol of {5/2,5} which expands to an explicit vertex configuration as 5/2.5/2.5/2.5/2.5/2. Image File history File links Great_dirhombicosidodecahedron_vertfig. ...
Image File history File links Great_dirhombicosidodecahedron_vertfig. ...
In geometry, a star polygon is a complex, regular polygon, so named for its starlike appearance, created by extending lines in a regular pattern from one vertex of a simple, regular, n-sided polygon to another, non-adjacent vertex and continuing the process until the original vertex is reached again. ...
A pentagram, pentacle, pentalpha, or pentangle A pentagram is a five-pointed star drawn with five straight strokes. ...
In geometry, the small stellated dodecahedron is a Kepler-Poinsot solid. ...
In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ...
The last, U75, nonconvex uniform polyhedron great dirhombicosidodecahedron has a vertex figure of (4.5/3.4.3.4.5/2.4.3/2)/2. This complex vertex figure has 8 faces that pass around the vertex twice. In geometry, the great dirhombicosidodecahedron is a concave uniform polyhedron, indexed last as U75. ...
Inverted polygons Faces on a vertex figure are considered to progress in one direction. Some uniform polyhedra have vertex figures with inversions where the faces progress retrograde. A vertex figure represents this in the Star polygon notation of sides p/q as an improper fraction (greater than one), where p is the number of sides and q the number of turns around a circle. For example 3/2 means a triangle that has vertices that go around twice, which is the same as backwards once. Similarly 5/3 is a backwards pentagram 5/2. In geometry, a star polygon is a complex, regular polygon, so named for its starlike appearance, created by extending lines in a regular pattern from one vertex of a simple, regular, n-sided polygon to another, non-adjacent vertex and continuing the process until the original vertex is reached again. ...
Face Configuration for duals The dual polyhedron are can also be listed by this notation, but prefixed by a V. See face configuration. In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ...
In geometry, a face configuration is notational description of a face-uniform polyhedron. ...
The faces of semiregular polyhedral duals are not regular polygons, but edges vary in length in relation regular polygons in the dual. For example, you can tell a face configuration of V3.4.3.4 represents a rhombus face since every edge is a V3-V4 type, and V3.4.5.4 will be a kite with two types of edges: V3-V4 and V4-V5. This shape is a rhombus In geometry, a rhombus (also known as a rhomb) is a quadrilateral in which all of the sides are of equal length. ...
A separate article is about kite flying. ...
Notation used in articles In solid geometry and some ancient physical theories, a Platonic solid is a convex polyhedron with all its faces being congruent regular polygons, and the same number of faces meeting at each of its vertices. ...
In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ...
In geometry, a prism is a polyhedron made of two parallel copies of some polygonal base joined by faces that are rectangles or parallelograms. ...
An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ...
This table contains an indexed list of the Uniform and stellated polyhedra from the book Polyhedron Models, by Magnus J. Wenninger. ...
The following list contains all 75 nonprismatic uniform polyhedra, 11 uniform tessellations in the plane, and a samplings of the infinite set of prisms and antiprisms. ...
This table shows the 11 uniform tilings of the plane, and their dual tilings. ...
In geometry, a face configuration is notational description of a face-uniform polyhedron. ...
A rhombic dodecahedron In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. ...
A bipyramid is a polyhedron formed by joining two identical pyramids base-to-base. ...
The trapezohedron is the dual polyhedron of the corresponding antiprism. ...
Reference - Williams, Robert (1979). The Geometric Foundation of Natural Structure: A source book of Design, Dover Publications, Inc. ISBN 0-486-23729-X.
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