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In geometry, a polyhedron (pl.: polyhedra or polyhedrons; from Greek πολύ (poly-)  'many' and ἕδρον (-hedron)  'base, seat') is a three-dimensional figure...

the same number of faces meet at each vertex. There are only five such polyhedra: a regular tetrahedron (four triangular faces), a cube (six square faces)...

of selected geodesic polyhedra and Goldberg polyhedra, two infinite classes of polyhedra. Geodesic polyhedra and Goldberg polyhedra are duals of each other...

uniform polyhedra with degenerate vertex figures which have overlapping edges (not counted by Coxeter); The uniform tilings (infinite polyhedra) 11 Euclidean...

authors exclude uniform polyhedra (in which all vertices are symmetric to each other) from the definition; uniform polyhedra include Platonic and Archimedean...

convex regular polyhedra (the Platonic solids), and four regular star polyhedra (the Kepler–Poinsot polyhedra), making nine regular polyhedra in all. In addition...

of the uniform polyhedra are also star polyhedra. There are two infinite classes of uniform polyhedra, together with 75 other polyhedra. They are 2 infinite...

in any general number of dimensions. The following list of polygons, polyhedra and polytopes gives the names of various classes of polytopes and lists...

In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra. They may be obtained by stellating and faceting the regular convex dodecahedron...

figures remain combinatorial or abstract polyhedra, but not all can also be constructed as geometric polyhedra. Starting with any given polyhedron, the...

The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygons and are vertex-transitive,[citation needed] although they...

Catalan solids are the dual polyhedra of Archimedean solids. The Archimedean solids are thirteen highly-symmetric polyhedra with regular faces and symmetric...

Polyhedra is a family of relational database management systems offered by ENEA AB, a Swedish company. The original version of Polyhedra (now referred...

self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon...

Polyhedra Games is a publisher of tabletop role-playing games and video games established by Clipper Arnold in 2019. Polyhedra's titles include Justice...

ways together with prisms and antiprisms; 35 formed by modifying uniform polyhedra, by augmenting, diminishing, or gyrating with primitives; and 9 which...

equiprojective. Hasan and his colleagues later found more equiprojective polyhedra by truncating equally the tetrahedron and three other Johnson solids....

Polyhedra is a book on polyhedra, by Peter R. Cromwell. It was published by in 1997 by the Cambridge University Press, with an unrevised paperback edition...

geodesic polyhedra, and some pollen grains are based on geodesic polyhedra. Fullerene molecules have the shape of Goldberg polyhedra. Geodesic polyhedra are...

lower-case letter chi). The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification...

on Polyhedra: A Study of the "De solidorum elementis" is a book in the history of mathematics, concerning the work of René Descartes on polyhedra. Central...

In geometry, the great dodecahedron is one of four Kepler–Poinsot polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagons),...

greater. Notable examples include the Császár and Szilassi polyhedra. Toroidal polyhedra are defined as collections of polygons that meet at their edges...

five convex regular polyhedra are called the Platonic solids. The vertex figure is given with each vertex count. All these polyhedra have an Euler characteristic...

convex uniform polyhedra o3o5x – doe". Editable printable net of a dodecahedron with interactive 3D view The Uniform Polyhedra Origami Polyhedra – Models made...

equilateral (but not in general equiangular) faces. Simple examples of Goldberg polyhedra include the dodecahedron and truncated icosahedron. Other forms can be...

process of extending edges until they meet to form a new polygon. For polyhedra he distinguished two types of stellation: an echinus (Latin for hedgehog...

the first to be solved. The problem asks the following: Given any two polyhedra of equal volume, is it always possible to cut the first into finitely...

distinguished polyhedra in three dimensions contain fourteen faces or vertices as facets: The cuboctahedron, one of two quasiregular polyhedra, has 14 faces...

the tetrahedron with 4 triangular faces, 6 edges, and 4 vertices. Named polyhedra primarily come from the families of platonic solids, Archimedean solids...

geometry, the great icosahedron is one of four Kepler–Poinsot polyhedra (nonconvex regular polyhedra), with Schläfli symbol {3,5⁄2} and Coxeter-Dynkin diagram...

are a useful aid to the study of polyhedra and solid geometry in general, as they allow for physical models of polyhedra to be constructed from material...

are two general kinds of star polyhedron: Polyhedra which self-intersect in a repetitive way. Concave polyhedra of a particular kind which alternate convex...

analogs of spherical polyhedra – tessellations of the sphere – and toroidal polyhedra – tessellations of the toroids. Projective polyhedra are also referred...

tetrahedron to form the tetrahedral-octahedral honeycomb. The following polyhedra are combinatorially equivalent to the regular octahedron. They all have...

John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations. Conway...

sides (faces). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number...

relationships among the uniform polyhedra. The Wythoff construction is able to construct almost all of the uniform polyhedra from the acute and obtuse Schwarz...

constructed. As well as cubes, all of the regular polyhedra and many of the semi-regular and stellated polyhedra have been made. A cuboid is a rectilinear polyhedron...

dual action to faceting a polyhedron. Originating from studies of star polyhedra in 14th century Europe, a proper mathematical account of polyhedral stellations...

orthogonal polyhedra that can be decomposed into identical cubes and are three-dimensional analogs of planar polyominoes. Orthogonal polyhedra can be either...

and mathematical objects such as polyhedra and the Möbius strip. Magnus Wenninger creates colourful stellated polyhedra, originally as models for teaching...

defined. Regular polyhedra are isohedral (face-transitive), isogonal (vertex-transitive), and isotoxal (edge-transitive). Quasiregular polyhedra, like the cuboctahedron...

as a hosohedron. Some "improper" polyhedra, such as hosohedra and their duals, dihedra, exist as spherical polyhedra, but their flat-faced analogs are...

regular-faced polyhedra when sliced by a plane.[dubious – discuss] Repeated slicing of this type until it cannot produce more such polyhedra again is called...

Alexandrov's theorem on polyhedra is a rigidity theorem in mathematics, describing three-dimensional convex polyhedra in terms of the distances between...

undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the 3-vertex-connected planar graphs. That is, every...

of freedom, and it represents a fixed geometric, just like the regular polyhedra. In general all single ringed uniform polytopes have a uniform truncation...

polyhedra x3o5x - srid". Editable printable net of a Rhombicosidodecahedron with interactive 3D view The Uniform Polyhedra Virtual Reality Polyhedra The...

culture and music theory. It can be the core of polyhedra construction, and it can tile with different polyhedra to create a honeycomb. The vertices and edges...

polygons whose vertices are not all in the same plane, and extended it to polyhedra. While apeirohedra are typically required to tile the 2-dimensional plane...

dimensional features, and counting how many facets it passes through. For polyhedra for which this count does not depend on the choice of the ray, and for...

geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal...

Goldberg polyhedron because it is the initial polyhedron to construct new polyhedra by the process of chamfering. It has a relation with other Platonic solids...

The study of integer points in convex polyhedra is motivated by questions such as "how many nonnegative integer-valued solutions does a system of linear...

a special case of the hypercube graph. The cube is the core of many polyhedra's construction or other geometrical shapes. For example, truncating a cube's...

In geometry, the rhombihexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. It has Schläfli symbol of rr{8,6}. The dual tiling, called...

In geometry, the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one decagon, and one dodecagon...

icosahedral graph represents the skeleton of a regular icosahedron. Many polyhedra and other related figures are constructed from the regular icosahedron...

In geometry, the truncated order-5 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{4,5}. John H. Conway, Heidi...

the Penrose tiling. Kites also form the faces of several face-symmetric polyhedra and tessellations, and have been studied in connection with outer billiards...

Gosset's 1900 definition of the more general semiregular polytope). These polyhedra include: The thirteen Archimedean solids. The elongated square gyrobicupola...

Excluding the infinite sets, there are 75 uniform polyhedra in the third dimension, which incorporate star polyhedra as well. Inclusive of 7 families of prisms...

and with Schläfli symbol {5⁄2, 5}. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting...

Impressions: Goldberg Polyhedra". Simons Science News. Deza, A.; Deza, M.; Grishukhin, V. (1998). "Fullerenes and coordination polyhedra versus half-cube embeddings"...

used these 15 great circles, along with 10 and 6 others in two other polyhedra, to define his 31 great circles of the spherical icosahedron. Being a...

uniform polyhedra x3x3o - tut". Editable printable net of a truncated tetrahedron with interactive 3D view The Uniform Polyhedra Virtual Reality Polyhedra The...

George W. "Virtual Polyhedra: Zonohedra". Weisstein, Eric W. "Primary Parallelohedron". MathWorld. Bulatov, Vladimir. "Zonohedral Polyhedra Completion". Centore...

There are many relations among the uniform polyhedra. Some are obtained by truncating the vertices of the regular or quasi-regular polyhedron. Others...

folded onto several polyhedra. To be a valid common net, there should not exist any non-overlapping sides, and the resulting polyhedra must be connected...

regular p-gons ({p}), the cells are regular polyhedra of type {p,q}, the vertex figures are regular polyhedra of type {q,r}, and the edge figures are regular...

one of a family of duals to the uniform polyhedra related to the cube and regular octahedron. It is a polyhedra in a sequence defined by the face configuration...

polyhedron, with Schläfli symbol {5⁄2, 3}. It is one of four nonconvex regular polyhedra. It is composed of 12 intersecting pentagrammic faces, with three pentagrams...

non-convex hexahedra exist, with their number depending on how polyhedra are defined. Two polyhedra are "topologically distinct" if they have intrinsically different...

In geometry, a golden rectangle is a rectangle with side lengths in golden ratio 1 + 5 2 : 1 , {\displaystyle {\tfrac {1+{\sqrt {5}}}{2}}:1,} or ⁠ φ :...

in Penrose tilings. Pentagrams are facets inside Kepler–Poinsot star polyhedra and Schläfli–Hess star polychora. There are five regular Platonic solids...

to points. If regular it has notation t2{p,q,...} or 2r{p,q,...}. For polyhedra, a birectification creates a dual polyhedron. Higher degree rectifications...

In geometry, the octagonal prism is a prism comprising eight rectangular sides joining two regular octagon caps. The octagonal prism can also be seen as...

geometry, isotoxal polyhedra and tilings are defined by the property that they have symmetries taking any edge to any other edge. Polyhedra with this property...

"A class of multi-symmetric polyhedra". Tohoku Mathematical Journal. 43: 104–108. Hart, George (2012). "Goldberg Polyhedra". In Senechal, Marjorie (ed...

be concentric, including circles, spheres, regular polygons, regular polyhedra, parallelograms, cones, conic sections, and quadrics. Geometric objects...

There are many relations among the uniform polyhedra. Here they are grouped by the Wythoff symbol. All the faces are identical, each edge is identical...

and some multiple of 10 hexagons. There are three classes of Goldberg polyhedra, one of which is constructed by repeatedly truncating all of its vertices...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

In geometry, an angle of a polygon is formed by two adjacent sides. For a simple polygon (non-self-intersecting), regardless of whether it is convex or...

faces. The tetrahedron and the Szilassi polyhedron are the only two known polyhedra in which each face shares an edge with each other face. The 14 vertices...

Dimensional family of quasiregular polyhedra and tilings: (7.n)2 v t e Symmetry *7n2 [n,7] Hyperbolic... Paracompact Noncompact *732 [3,7] *742 [4,7]...

example of a deltahedron, composite polyhedron, and Johnson solid. Many polyhedra are related to the triangular bipyramid, such as similar shapes derived...

books about polyhedra. Jenkins, Gerald; Bear, Magdalen (1998). Paper Polyhedra in Colour. Tarquin. ISBN 1-899618-23-6. Advanced Polyhedra 1: The Final...

Perihelion Software also produced an in-memory database system called Polyhedra. The group responsible for this product was set up as a subsidiary, Perihelion...

of the Platonic solids other than the cube. Five space-filling convex polyhedra can tessellate 3-dimensional euclidean space using translations only....

Platonic solids - regular polyhedra (all faces of the same type) Archimedean solids - polyhedra with more than one polygon face type. Catalan solids -...

figure exist on an (n−1)-sphere. The term isogonal has long been used for polyhedra. Vertex-transitive is a synonym borrowed from modern ideas such as symmetry...

Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra VRML model Archived 2018-10-11 at the Wayback Machine Conway Notation for Polyhedra Try:...

constituents are identical (although possibly enantiomorphous) uniform polyhedra, in an arrangement that is also uniform, i.e. the symmetry group of the...

convex pseudoregular polyhedra. Summarizing the examples above, the deltahedra can be conclusively defined as the class of polyhedra whose faces are equilateral...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

is the third in the infinite family of trapezohedra, face-transitive polyhedra. Its dual polyhedron is the pentagonal antiprism. As a decahedron it has...

truncated octahedron". Virtual Polyhedra: The Encyclopedia of Polyhedra. Retrieved 2006-09-08. Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge....

A hemi-cuboctahedron is an abstract polyhedron, containing half the faces of a semiregular cuboctahedron. It has 4 triangular faces and 3 square faces...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

Dover, 1973). Cromwell, P.; Polyhedra, CUP hbk (1997), pbk. (1999). Grünbaum, B.; Are your polyhedra the same as my polyhedra? Discrete and comput. geom:...

and stellated polyhedra from the book Polyhedron Models, by Magnus Wenninger. The book was written as a guide book to building polyhedra as physical models...

lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells. The 4-polytopes were discovered...

4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons. There are 47 non-prismatic convex uniform...

the polygonal faces must have at least three sides. When considering polyhedra as a spherical tiling, this restriction may be relaxed, since digons (2-gons)...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

In geometry, the great icosicosidodecahedron (or great icosified icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U48. It has 52 faces...

Børge Jessen, who studied it in 1967. In 1971, a family of nonconvex polyhedra including this shape was independently discovered and studied by Adrien...

In geometry, the truncated order-4 heptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{7,4}. There are two uniform...

In geometry, the regular skew polyhedra are generalizations to the set of regular polyhedra which include the possibility of nonplanar faces or vertex...

In geometry, the truncated order-6 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{6,6}. It can also be identically...

vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive. Their dual figures...

In geometry, a heptagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using septa- (an elision of septua-)...

polyhedron. In face transitive polyhedra, symmetry can be used to require all faces have the same diagram shading. Semiregular polyhedra like the Archimedean solids...

polyhedra. Below are the three prismatic uniform polyhedra containing regular dodecagrams (there are no other dodecagram-containing uniform polyhedra)...

expressed with these prefixes. Platonic solid Dice List of polygons, polyhedra and polytopes Circle Ellipse Shape In Greek usage, τετράγωνον (romanised:...

These polyhedra include bipyramids, gyroelongated bipyramids, deltahedra (wherein the faces are equilateral triangles, and Kleetope of polyhedra. The simplicial...

set of what are nowadays known as the regular star or Kepler–Poinsot polyhedra. In 1876 Edmund Hess used stellation diagrams and discovered the remaining...

third problem by proving that certain polyhedra with equal volume cannot be dissected into each other. Two polyhedra have a dissection into polyhedral pieces...

snub polyhedra, as they are obtained by this construction from a degenerate "polyhedron" with only two faces (a dihedron). Chiral snub polyhedra do not...

polyhedron because by slicing it with a plane, the resulting smaller convex polyhedra do not have regular faces. The pentagonal rotunda can be regarded as half...

In geometry, the truncated order-4 pentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{5,4}. A half symmetry...

vertex arrangement with four uniform polyhedra as follows: Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the...

Uniform Polyhedra Virtual Reality Polyhedra The Encyclopedia of Polyhedra octet truss animation Review: A. F. Wells, Three-dimensional nets and polyhedra, H...

polyhedral compounds that can also be considered as degenerate uniform star polyhedra; respectively, the small complex rhombicosidodecahedron, great complex...

six faces in this sense. In more modern treatments of the geometry of polyhedra and higher-dimensional polytopes, a "face" is defined in such a way that...

rectangular, but others use cuboid to refer to a more general class of polyhedra with six quadrilateral faces. A rectangular cuboid is a convex polyhedron...

In geometry, the order-6 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,6} and is self-dual. This tiling represents...

triacontahedron is somewhat special in being one of the nine edge-transitive convex polyhedra, the others being the five Platonic solids, the cuboctahedron, the icosidodecahedron...

In geometry, the tetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t1{4,5} or r{4,5}. A half symmetry [1+,4...

(this is also true in higher dimensions). The first examples of flexible polyhedra were discovered by Raoul Bricard (1897), which are called Bricard octahedra...

interstitial symmetry polyhedra in a face-centered cubic lattice. The actual interstitial atom would ideally be in the middle of one of the polyhedra....

24200/squjs.vol21iss2pp139-149. The Uniform Polyhedra Virtual Reality Polyhedra The Encyclopedia of Polyhedra Weisstein, Eric W., "Cuboctahedron" ("Archimedean...

can be considered as polyhedra, categorized by their coordination number and bond angles. For example, four-coordinate polyhedra are classified as tetrahedra...

In differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying...

these uniform duals Catalan tilings, in parallel to the Catalan solid polyhedra. Uniform tilings are listed by their vertex configuration, the sequence...

four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions. There...

friendship. They collaborated in discovering infinite warped polyhedra and (finite) warped polyhedra in the fourth dimension, analogous to the previous ones...

In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with seven faces, fifteen edges, and ten vertices. If...

In geometry, the dodecagonal prism is the tenth in an infinite set of prisms, formed by square sides and two regular dodecagon caps. If faces are all regular...

uniform. The triangular prism can be used as the core of constructing other polyhedra, examples are some of the Johnson solids and Schönhardt polyhedron. It...

In geometry, the truncated order-8 hexagonal tiling is a semiregular tiling of the hyperbolic plane. It has Schläfli symbol of t{6,8}. This tiling can...

The regular tetrahedron is also one of the five regular Platonic solids—polyhedra in which all of their faces are regular polygons. Known since antiquity...

In geometry, the tritetragonal tiling or alternated octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbols of {(4,3,3)}...

In geometry, the order-8 triangular tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {3,8}, having eight regular...

In geometry, the snub infinite-order triangular tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of s{3,∞}. John H. Conway, Heidi...

In geometry, the snub pentahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{6,5}. Drawn in chiral pairs, with...

In geometry, the rhombipentahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,2{6,5}. John H. Conway, Heidi Burgiel...

In geometry, the pentahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of r{6,5} or t1{6,5}. John H. Conway, Heidi...

In geometry, the truncated order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{6,4}. A secondary construction...

In geometry, the alternated order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of (3,4,4), h{6,4}, and hr{6,6}...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

higher-genus polyhedra and for immersions of toroidal polyhedra, although some authors only include those with genus 1. Self-crossing toroidal polyhedra are determined...

In geometry, the Reeve tetrahedra are a family of polyhedra with vertices at ( 0 , 0 , 0 ) , ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) , ( 1 , 1 , r ) , {\displaystyle...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

uniform polyhedra o3x5o - id". Editable printable net of an icosidodecahedron with interactive 3D view The Uniform Polyhedra Virtual Reality Polyhedra The...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

In organic chemistry, a Platonic hydrocarbon is a hydrocarbon whose structure matches one of the five Platonic solids, with carbon atoms replacing its...

The infinite-order apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,∞}, which means it has countably infinitely...

commonly occurs in the mathematical treatment of polyhedra. (However, for non-orientable polyhedra, density can only be considered modulo 2 and hence...

In geometry, the tetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol r{6,4}. There are for uniform constructions...

In geometry, the small rhombihexahedron (or small rhombicube) is a nonconvex uniform polyhedron, indexed as U18. It has 18 faces (12 squares and 6 octagons)...

In geometry, the rhombitetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,2{4,5}. The dual is called the...

In geometry, the truncated infinite-order square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,∞}. In (*∞44) symmetry...

In geometry, the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one dodecagon...

In geometry, the snub tritetratrigonal tiling or snub order-8 triangular tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbols of...

In geometry, the snub hexahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{6,6}. Drawn in chiral pairs, with...

In geometry, the truncated order-7 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{4,7}. John H. Conway, Heidi...

Macmillan. p. 53. Retrieved December 1, 2018. Robertson, S. A. (1983). "Polyhedra and symmetry". The Mathematical Intelligencer. 5 (4): 57–60. doi:10.1007/BF03026511...

only even-sided faces. All truncated rectified polyhedra can be snubbed, not just from regular polyhedra. The snub square antiprism is an example of a...

with H. S. M. Coxeter and M. S. Longuet-Higgins enumerated the uniform polyhedra in 1954). This is the only non-degenerate uniform polyhedron with more...

vertex arrangement with four uniform polyhedra as follows: Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the...

In mathematics a linear inequality is an inequality which involves a linear function. A linear inequality contains one of the symbols of inequality: <...

In geometry, the snub order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of s{(4,4,3)} or s{4,6}. The symmetry is...

smaller polyhedra visible within the print also include all of the five Platonic solids and the rhombic dodecahedron. In order to depict polyhedra accurately...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

In geometry, the truncated order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,6}. The dual tiling represents...

faces. The tetrahedron and the Császár polyhedron are the only two known polyhedra (having a manifold boundary) without any diagonals: every two vertices...

In geometry, the tritetratrigonal tiling or shieldotritetragonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t1,2(4,3...

In geometry, the Waterman polyhedra are a family of polyhedra discovered around 1990 by the mathematician Steve Waterman. A Waterman polyhedron is created...

Honeycombs in 3-Space VRML models The Uniform Polyhedra Virtual Reality Polyhedra The Encyclopedia of Polyhedra Prisms and antiprisms Weisstein, Eric W. "Hexagonal...

In geometry, the heptagonal prism is a prism with heptagonal base. This polyhedron has 9 faces (2 bases and 7 sides), 21 edges, and 14 vertices. The area...

are isogonal; thus they are uniform polyhedra. They form one of the two infinite series of semiregular polyhedra, the other series being formed by the...

In geometry, the rhombitetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{6,4}. It can be seen as constructed...

In geometry, the truncated hexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one dodecagon, and one hexakaidecagon...

Dimensional family of quasiregular polyhedra and tilings: (8.n)2 v t e Symmetry *8n2 [n,8] Hyperbolic... Paracompact Noncompact *832 [3,8] *842 [4,8]...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

polyhedron will have two faces on each edge and creates new polyhedra or compounds of polyhedra. Faceting is the reciprocal or dual process to stellation...

In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any...

In geometry, the great ditrigonal icosidodecahedron (or great ditrigonary icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U47. It has...

Pentagonal pyramids occur as pieces and tools in the construction of many polyhedra. They also appear in the field of natural science, as in stereochemistry...

lines meet to form an angle and the point where edges of polygons and polyhedra meet are vertices. The term vertex is also used for a different type of...

non-prismatic uniform polyhedra, which incorporate regular polyhedra, semiregular polyhedra, and star polyhedra, 75 uniform compound polyhedra, inclusive of seven...

In geometry, the cantic order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{(4,4,3)} or h2{6,4}. John...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

Richard. "3D convex uniform polyhedra s3s4s - snic". The Uniform Polyhedra Virtual Reality Polyhedra The Encyclopedia of Polyhedra Editable printable net of...

"hemi" prefix. This prefix is also used to refer to certain projective polyhedra, such as the hemi-cube, which are the image of a 2 to 1 map of a spherical...

In geometry, the truncated order-5 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{6,5}. John H. Conway, Heidi...

polygons (for example, the square or the regular pentagon) and regular polyhedra (for example, the cube). The strong symmetry of the regular polytopes...

regular toroidal polyhedra {p,q}b,c can be defined if either p or q are even, although only euclidean ones above can exist as toroidal polyhedra in 4-dimensions...

John Skilling notes, in his enumeration of uniform compounds of uniform polyhedra, that this compound of 20 tetrahemihexahedra is unique in that it cannot...

In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group. It was discovered by Thorold Gosset...

vertex arrangement with four uniform polyhedra as follows: Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the...

Convex Polyhedra is a book on the mathematics of convex polyhedra, written by Soviet mathematician Aleksandr Danilovich Aleksandrov, and originally published...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

In geometry, a rotunda is any member of a family of cyclic-symmetric polyhedra. They are similar to a cupola but, instead of alternating squares and triangles...

dodecahedron). Other space-filling polyhedra include the square pyramid, plesiohedra, and stereohedra, polyhedra whose tilings have symmetries taking...

rhombus faces. Copies of these can be assembled to form other convex polyhedra with golden rhombus faces, including the Bilinski dodecahedron and rhombic...

spin networks. A twisted geometry can be visualized as collections of polyhedra dual to the nodes of the spin network's graph. Intrinsic and extrinsic...

In geometry, the truncated tetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1,2{4,5} or tr{4,5}. There...

whether or not the polyhedra are congruent. The number is tight, meaning that less than E measurements are not enough if the polyhedra are generic among...

Ruddlesden-Popper phases (a) Sr2RuO4 (n = 1) and (b) Sr3Ru2O7 (n = 2). The polyhedra are part of the perovskite-like layers. In these examples A = A' = Sr2+...

the familiar division of Euclidean space into cubes. However, not all polyhedra have ideal versions – a polyhedron can be ideal only when it can be represented...

vertex arrangement with four uniform polyhedra as follows: Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the...

In geometry, the elongated hexagonal bipyramid is constructed by elongating a hexagonal bipyramid (by inserting a hexagonal prism between its congruent...

In geometry, the nonconvex great rhombicuboctahedron is a nonconvex uniform polyhedron, indexed as U17. It has 26 faces (8 triangles and 18 squares), 48...

triangles. As an antiprism, the square antiprism belongs to a family of polyhedra that includes the octahedron (which can be seen as a triangle-capped antiprism)...

symmetry and centrally symmetric faces, characterize a broader class of polyhedra, the zonohedra, so every parallelohedron is a zonohedron. In any zonohedron...

In geometry, the snub hexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are three triangles, one hexagon, and one octagon on...

Richard. "3D convex uniform polyhedra x3o4x - sirco". The Uniform Polyhedra Virtual Reality Polyhedra The Encyclopedia of Polyhedra Editable printable net...

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with pentagonal faces, starting with the dodecahedron, with...

Matthais; Hosten, Serkan; Schymura, Matthias (2019). "Lonely Runner Polyhedra" (PDF). Integers: The Electronic Journal of Combinatorial Number Theory...

the quadratic lower bound of complexity on the decomposition of convex polyhedra in three dimensions. The later applications are used regarding the problem...

noble polyhedra: Regular polyhedra, that is, the five Platonic solids and the four Kepler–Poinsot polyhedra. Disphenoid tetrahedra. Crown polyhedra, also...

In geometry, the truncated order-6 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{8,6}. A secondary construction...

In geometry, the order-6 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,6}. This regular tiling can also be...

In geometry, the truncated order-8 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{8,8}. This tiling can also...

polyhedron has a midsphere, but the uniform polyhedra, including the regular, quasiregular and semiregular polyhedra and their duals (Catalan solids) all have...

These 10, along with the great circles from projections of two other polyhedra, form the 31 great circles of the spherical icosahedron used in construction...

there are 75 uniform polyhedra, including the five regular convex polyhedra, five polyhedral compounds, four Kepler-Poinsot polyhedra, and thirteen Archimedean...

by axis-parallel faces. Polyhedra constructed in this way from finitely many prisms provide examples of axis-parallel polyhedra with n {\displaystyle n}...

Together with Robert Fricke, they found the five genera of Leonardo polyhedra. Some colleagues further discovered the locally regular and the genus...

called pseudo-uniform polyhedra due to their resemblance to some true uniform polyhedra. There are two known pseudo-uniform polyhedra: the pseudorhombicuboctahedron...

In geometry, the quarter order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of q{4,6}. It is constructed from *3232...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

within nested loops as lattice points inside mathematical objects called polyhedra, performs affine transformations or more general non-affine transformations...

Like many of the related boron hydrides, these clusters are polyhedra or fragments of polyhedra. Carboranes are one class of heteroboranes. In terms of scope...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

possible cube that can pass through a hole in a unit cube. Many other convex polyhedra, including all five Platonic solids, have been shown to have the Rupert...

customary to drop the prefix regular. For instance, all the faces of uniform polyhedra must be regular and the faces will be described simply as triangle, square...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

characterizes the multisets of polygons that can form the faces of simple convex polyhedra. It states that, for given numbers of triangles, quadrilaterals, pentagons...

vertex arrangement with three uniform polyhedra as follows: Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the...

In geometry, the truncated order-4 apeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{∞,4}. A half symmetry coloring...

classify the regular polyhedra. The second chapter uses combinations of regular polyhedra and their duals to generate related polyhedra, including the semiregular...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

This tiling is topologically related as a part of sequence of deltoidal polyhedra with face figure (V3.4.n.4), and continues as tilings of the hyperbolic...

and rings. Three-dimensional forms tend to be regular polyhedra or tessellations of simple polyhedra. Modular origami techniques can be used to create a...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

of 31 regular skew polyhedra in hyperbolic 3-space. J. Richard Gott in 1967 published a larger set of seven infinite skew polyhedra which he called regular...

vertices. The tetrahedron is the simplest of all the ordinary convex polyhedra. The tetrahedron is the three-dimensional case of the more general concept...

member of a family of flexible polyhedra constructed by Raoul Bricard in 1897. The overall shape of one of these polyhedra may change in a continuous motion...

In geometry, the Hessian polyhedron is a regular complex polyhedron 3{3}3{3}3, , in C 3 {\displaystyle \mathbb {C} ^{3}} . It has 27 vertices, 72 3{} edges...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

In geometry, the truncated order-4 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{8,4}. A secondary construction...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

all be realized as spherical polyhedra – they tessellate the sphere. Four examples of non-traditional abstract polyhedra are the Hemicube (shown), Hemi-octahedron...

In geometry, the small dodecicosahedron (or small dodekicosahedron) is a nonconvex uniform polyhedron, indexed as U50. It has 32 faces (20 hexagons and...

Treks into Intuitive Geometry: The World of Polygons and Polyhedra is a book on geometry, written as a discussion between a teacher and a student in the...

In geometry, H. S. M. Coxeter called a regular polytope a special kind of configuration.[citation needed] Other configurations in geometry are something...

In geometry, the 4-5 kisrhombille or order-4 bisected pentagonal tiling is a semiregular dual tiling of the hyperbolic plane. It is constructed by congruent...

convex uniform polyhedra x3x4x - girco". Editable printable net of a truncated cuboctahedron with interactive 3D view The Uniform Polyhedra Virtual Reality...

For uniform polyhedra, there is only one vertex type and therefore the vertex configuration fully defines the polyhedron. (Chiral polyhedra exist in mirror-image...

The dihedral angles for the edge-transitive polyhedra are: Coxeter, Regular Polytopes (1963), Macmillan Company Regular Polytopes, (3rd edition, 1973)...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

Pyramid". MathWorld. Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra Conway Notation for Polyhedra Try: "Y6" [1] Hexagonal pyramid...

In geometry, the great dodecacronic hexecontahedron (or great lanceal ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform...

compounds, and the only regular polyhedral compound composed of only two polyhedra. They form the only fully symmetric stellation of the octahedron, and...

solids into pairs of dual polyhedra. Graph duality is a topological generalization of the geometric concepts of dual polyhedra and dual tessellations, and...

In the mathematical field of Riemannian geometry, M. Gromov's systolic inequality bounds the length of the shortest non-contractible loop on a Riemannian...

and J. C. P. Miller were the first to publish the full list of uniform polyhedra (1954).[P54] He worked for 60 years at the University of Toronto and published...

3) geodesic polyhedron by k5k6. The whirled dodecahedron creates more polyhedra by basic Conway polyhedron notation. The zip whirled dodecahedron makes...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

It is also called a great strombic hexecontahedron. List of uniform polyhedra Maeder, Roman. "67: great rhombicosidodecahedron". MathConsult. Wenninger...

though the snub dodecahedron has more faces. Of all vertex-transitive polyhedra, it occupies the largest percentage (89.80%) of the volume of a sphere...

In mathematics, the plastic ratio is a geometrical proportion, given by the unique real solution of the equation x3 = x + 1. Its decimal expansion begins...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

Origami Polyhedra Design is a book on origami designs for constructing polyhedra. It was written by the origami artist and mathematician John Montroll...

In geometry, an equilateral polygon is a polygon which has all sides of the same length. Except in the triangle case, an equilateral polygon does not need...

2015) Coxeter (1973), Chapter 1, Polygons and Polyhedra, p.4 Coxeter (1973), Chapter 1, Polygons and Polyhedra, pages 4 and 12. Alex Degtyarev, Topology of...

zonohedra. Some failed Johnson solid candidates have coplanar faces. These polyhedra can be perturbed to become convex with faces that are arbitrarily close...

that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes. Research in polyhedral combinatorics...

triangles and one apeirogon at each vertex. Similarly to the uniform polyhedra and the uniform tilings, eight uniform tilings may be based from the regular...

prisms, the uniform antiprisms form an infinite class of vertex-transitive polyhedra. For n = 2, one has the digonal antiprism (degenerate antiprism), which...

what is now called Conway polyhedron notation, which can be applied to polyhedra and tilings. Conway calls Coxeter's operation a semi-snub. In this notation...

meaning that it cannot be separated by a plane into two small regular-faced polyhedra. The surface area of a triangular hebesphenorotunda of edge length a {\displaystyle...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

The gyroelongated bipyramids are the polyhedra constructed with a bipyramid and an antiprism. Their dual polyhedra are the truncated trapezohedra. The...

^{n}} . Many examples of bounded convex polytopes can be found in convex polyhedra and convex polygon. In the 2-dimensional case the full-dimensional examples...

known as the edges. Polyhedra in some cases can be classified, judging from the shape of their faces. For example, when polyhedra have all equilateral...

hexagonal bipyramid is the fourth in an infinite series of bifrustum polyhedra. It has 12 trapezoid and 2 hexagonal faces. This polyhedron can be constructed...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

In geometry, the great disdyakis dodecahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform great truncated cuboctahedron. It has...

can be used to construct cylindrical polyhedra. Mathematics portal Bricard octahedron, a member of flexible polyhedra. Erofeev & Ivanov (2020). Kokotsakis...

In geometry, the great hexacronic icositetrahedron is the dual of the great cubicuboctahedron. Its faces are kites. Part of each kite lies inside the solid...

In geometry, the great deltoidal icositetrahedron (or great sagittal disdodecahedron) is the dual of the nonconvex great rhombicuboctahedron. Its faces...

regular-faced polyhedra. Repeating this process until these pieces cannot produce any convex, regular-faced polyhedra results in elementary polyhedra. The triaugmented...

icositetrahedron refers to a polyhedron with 24 faces, none of which are regular polyhedra. However, many are composed of regular polygons, such as the triaugmented...

uniform polyhedron. It was proven in 1970 that there are only 75 uniform polyhedra other than the infinite families of prisms and antiprisms. John Skilling...

Catalan polyhedra and tilings, p. 285, Rhombic dodecahedron ) Weisstein, Eric W., "Rhombic dodecahedron", MathWorld Virtual Reality Polyhedra – The Encyclopedia...

There are no face-transitive polyhedra with five sides, and there are two distinct topological types. Notable polyhedra with regular polygon faces are:...

regular polyhedra of index two with vertices on one orbit. Regular Polyhedra of Index Two, I Anthony M. Cutler, Egon Schulte, 2010 Regular Polyhedra of Index...

given circle is described by Richmond and further discussed in Cromwell's Polyhedra. The top panel shows the construction used in Richmond's method to create...

In geometry, the tetrated dodecahedron is a near-miss Johnson solid. It was first discovered in 2002 by Alex Doskey. It was then independently rediscovered...

In geometry, the truncated order-6 pentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t1,2{6,5}. The dual of this...

Solution for Uniform Polyhedra", Geometriae Dedicata, 47: 57–110, doi:10.1007/BF01263494. Grünbaum, Branko (2003), ""New" uniform polyhedra", in Bezdek, Andras...

structures. Square pyramids are often used in the construction of other polyhedra and as the cell of a four-dimensional polytope called cubic pyramid. Square...

topologically distinct convex heptahedra, excluding mirror images. (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements...

Foundation. Messer, Peter W. (2002). "Closed-Form Expressions for Uniform Polyhedra and Their Duals". Discrete & Computational Geometry. 27 (3). Springer:...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

domain of a lens space L(p,q). Dihedra have also been called bihedra, flat polyhedra, or doubly covered polygons. As a spherical tiling, a dihedron can exist...

the three-circle figure is called a Reuleaux triangle. Some spherical polyhedra with edges along great circles can be stereographically projected onto...

a Johnson solid. It is the only Johnson solid outside of the uniform polyhedra that can tile three-dimensional space. The name of the gyrobifastigium...

P is called the inradius of P. All regular polyhedra have inscribed spheres, but most irregular polyhedra do not have all facets tangent to a common sphere...

it is an extension of concepts introduced by the Goldberg polyhedra and geodesic polyhedra. The GC construction is primarily studied in organic chemistry...

golden rectangle. Rhombi with this shape form the faces of several notable polyhedra. The golden rhombus should be distinguished from the two rhombi of the...

infinitely many prime numbers, and the construction of regular polygons and polyhedra. Often referred to as the most successful textbook ever written, the Elements...

uniform polyhedra s3s5s - snid". Editable printable net of a Snub Dodecahedron with interactive 3D view The Uniform Polyhedra Virtual Reality Polyhedra The...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

an anomalous role in that it is self-dual, whereas the four remaining polyhedra are mutually dual in pairs... Horne, Jeremy (19 May 2017). Philosophical...

φ(74) = φ(σ(74)). a sad number There are 74 different non-Hamiltonian polyhedra with a minimum number of vertices. Sloane, N. J. A. (ed.). "Sequence A001358"...

solids, the 13 Catalan solids, the 4 Kepler–Poinsot polyhedra, the 2 convex quasiregular polyhedra, and the 2 infinite families of bipyramids and trapezohedra...

dodecahedron. Further truncation, and alternation operations creates two more polyhedra: Rectified truncated tetrahedron Rectified truncated octahedron Rectified...

arrangement with the convex truncated cube and two other nonconvex uniform polyhedra. It additionally shares its edge arrangement with the nonconvex great...

tilings Uniform tilings in hyperbolic plane List of uniform polyhedra List of uniform polyhedra by Schwarz triangle Coxeter Regular Polytopes, Third edition...

subdivision of a tetrahedron. The name "tetrakis" is used for the Kleetopes of polyhedra with square faces. Hence, the tetrakis hexahedron can be considered as...

Schönhardt polyhedra have been used as gadgets in a proof that testing whether a polyhedron has a triangulation is NP-complete. Several other polyhedra, including...

regular and semiregular polyhedra (as a fundamental property); all of these solids also have unique representations as spherical polyhedra, or spherical tilings...

Eric W. "Hexahemioctacron". MathWorld. Weisstein, Eric W., "Cubohemioctahedron" ("Uniform polyhedron") at MathWorld. Uniform polyhedra and duals v t e...

triples, square triangular numbers, Pell numbers, the octagon, and six polyhedra with octahedral symmetry. If the ratio of two quantities a > b > 0 is...

include the bipyramid and the bicone as a surface of revolution. Convex polyhedra with rhombi include the infinite set of rhombic zonohedrons, which can...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

Symmetries of Things 2008, ISBN 978-1-56881-220-5 George Hart's Conway interpreter: generates polyhedra in VRML, taking Conway notation as input v t e...

edge-direct distance of 3 steps. It is in an infinite sequence of Goldberg polyhedra: Near-miss Johnson solid Truncated tetrakis cube C180 Isomers Deza, A...

Polyhedra, CUP, Hbk. 1997, ISBN 0-521-66432-2. Pbk. (1999), ISBN 0-521-66405-5. p.175 Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra"...

a unit distance from the vertex, as does Coxeter (1948). For uniform polyhedra the Dorman Luke construction cuts each connected edge at its midpoint...

Antiparallelograms occur as the vertex figures of certain nonconvex uniform polyhedra. In the theory of four-bar linkages, the linkages with the form of an...

with the same dihedral angles at the shared edges. Faces of semi-regular polyhedra often have different types of faces, which define exspheres of different...

Geometric Folding Algorithms: Linkages, Origami, Polyhedra is a monograph on the mathematics and computational geometry of mechanical linkages, paper...

Peter R. (1997), Polyhedra, Cambridge University Press, p. 91, ISBN 978-0-521-55432-9. Johnson, Norman W. (1966), "Convex polyhedra with regular faces"...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

small stellated dodecahedron and great stellated dodecahedron as regular polyhedra. However, Louis Poinsot in 1809 rediscovered two more, the great icosahedron...

In graph theory, a perfect graph is a graph in which the chromatic number equals the size of the maximum clique, both in the graph itself and in every...

The Symmetries of Things 2008, ISBN 978-1-56881-220-5 George Hart's Conway interpreter: generates polyhedra in VRML, taking Conway notation as input...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n). This hyperbolic tiling is related...

and runcinated forms (5 and 6) contain the faces of two regular skew polyhedra: {4,10|3} and {10,4|3}. There are 15 forms, generated by ring permutations...

his 2003 paper Are Your Polyhedra the Same as My Polyhedra? This tiling is topologically part of sequence of regular polyhedra and tilings with Schläfli...

C_{4v}} of order 8. The square cupola can be found in many constructions of polyhedra. An example is the rhombicuboctahedron, which can be seen as eight overlapping...

there are seven uniform and uniform dual polyhedra named as ditrigonal. There are five uniform ditrigonal polyhedra, all with icosahedral symmetry. The three...

{\displaystyle \textstyle \{(1,\alpha ),(0,1)\}} give rise to two Klein polyhedra, each of which is bounded by a sequence of adjoining line segments. Define...

60 vertices. As the name indicates, it belongs to the family of snub polyhedra. Let ⁠ ρ = {\displaystyle \rho =} ⁠ 1.324717957244746... be the real zero...

within the solid, hence is invisible in solid models. List of uniform polyhedra Maeder, Roman. "68: great truncated icosidodecahedron". MathConsult. Wenninger...

pentagons are made coplanar, making the others concave. List of uniform polyhedra Maeder, Roman. "30: small ditrigonal icosidodecahedron". MathConsult....

face-transitive. The face-transitive polyhedra comprise a set of 9 regular polyhedra, two finite sets comprising 66 non-regular polyhedra, and two infinite sets: 5...

edge-direct distance of 3 steps. Geodesic polyhedra are constructed by subdividing faces of simpler polyhedra, and then projecting the new vertices onto...

uniform polyhedra o3x4x - tic". Editable printable net of a truncated cube with interactive 3D view The Uniform Polyhedra Virtual Reality Polyhedra www.georgehart...

onto B. For this reason, convex isohedral polyhedra are the shapes that will make fair dice. Isohedral polyhedra are called isohedra. They can be described...

other 60 triangular faces occur in the other enantiomer. List of uniform polyhedra Maeder, Roman. "64: great snub dodecicosidodecahedron". MathConsult. Weisstein...

together with two regular polyhedra discovered by Louis Poinsot, as the Kepler–Poinsot polyhedra. He describes polyhedra in terms of their faces, which...

edges, which for convex polyhedra can only be quadrilaterals. An enumeration of the dual graphs of quadrilateral-faced polyhedra is given by Broersma, H...

three-dimensional integral polytopes may be called integral polygons or integral polyhedra, respectively. An n {\displaystyle n} -dimensional regular simplex can...

structure showing the packing of the Na+ and Cl- ions and visualizing the polyhedra, octahedra in this case, defined by the placement of these ions....

ISBN 978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, p 292-298, includes...

and almost all polyhedra do not have such bisectors. Nevertheless, the theorem of the three geodesics can be extended to convex polyhedra by considering...

dodecadodecahedron. It has 60 intersecting irregular pentagonal faces. List of uniform polyhedra Inverted snub dodecadodecahedron Maeder, Roman. "40: snub dodecadodecahedron"...

A Hoberman sphere is a kinetic structure patented by Chuck Hoberman that resembles a geodesic dome, but is capable of folding down to a fraction of its...

a polyhedron is also rectifying its rectification. Cantellation (for polyhedra and tilings) is also called expansion by Alicia Boole Stott: it corresponds...

Petrie polygon for these higher dimensional regular, uniform and dual polyhedra and polytopes, shown in these skew orthogonal projections: A principal...

tetradecahedra, excluding mirror images, having at least 9 vertices. (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements...

triangular bifrustum is the second in an infinite series of bifrustum polyhedra. It has 6 trapezoid and 2 triangle faces. It may also be called the truncated...

In geometry, the polyhedral groups are the symmetry groups of the Platonic solids. There are three polyhedral groups: The tetrahedral group of order 12...

For such clusters, the structures are based on deltahedra, which are polyhedra in which every face is triangular. The 4n clusters are classified as closo-...

cube. Further truncation, and alternation operations creates two more polyhedra: Rectified truncated tetrahedron Rectified truncated octahedron Rectified...

Ana Paula; Campos, Helena (2019). "Interactive Expansion of Achiral Polyhedra". In Cocchiarella, Luigi (ed.). ICGG 2018 - Proceedings of the 18th International...

Goodman-Strauss, Chaim (2008), "Chapter 21: Naming Archimedean and Catalan polyhedra and tilings", The Symmetries of Things, AK Peters, p. 288, ISBN 978-1-56881-220-5...

regular polyhedra and higher polytopes. Coxeter explained in 1937 how he and Petrie began to expand the classical subject of regular polyhedra: One day...

In geometry, the elongated pyramids are an infinite set of polyhedra, constructed by adjoining an n-gonal pyramid to an n-gonal prism. Along with the...

polyhedron used is a Platonic solid or Archimedean solid. However, other polyhedra can be used: the AuthaGraph projection makes use of a polyhedron with...

on a plane or for non-convex polyhedra in R 3 {\displaystyle \mathbb {R} ^{3}} : there exist non-convex flexible polyhedra that have one or more degrees...

deltohedron is the fourth in an infinite series of trapezohedra which are dual polyhedra to the antiprisms. It has twelve faces which are congruent kites. It can...

2008, ISBN 978-1-56881-220-5 George Hart's Conway interpreter: generates polyhedra in VRML, taking Conway notation as input Variations on a Rhombic Theme...

support towers continue above the aqueduct and are topped by two compound polyhedra, revealing Escher's interest in mathematics as an artist. The one on the...

this tiling by combining triangles: It is topologically related to a polyhedra sequence; see discussion. This group is special for having all even number...

hexecontahedron and the great pentagrammic hexecontahedron. List of uniform polyhedra Great snub icosidodecahedron Great retrosnub icosidodecahedron Wenninger...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

triangular prisms. Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79...

icosahedron. Further truncation, and alternation operations creates two more polyhedra: Near-miss Johnson solid Rectified truncated tetrahedron Rectified truncated...

truncated cubes. Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

infinity, reflection, symmetry, perspective, truncated and stellated polyhedra, hyperbolic geometry, and tessellations. Although Escher believed he had...

hexecontahedron and the great pentagrammic hexecontahedron. List of uniform polyhedra Great inverted snub icosidodecahedron Great retrosnub icosidodecahedron...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of...

dodecahemidodecahedron (having the decagrammic faces in common). List of uniform polyhedra Maeder, Roman. "71: great icosihemidodecahedron". MathConsult. Weisstein...

rectified prism (also rectified bipyramid) is one of an infinite set of polyhedra, constructed as a rectification of an n-gonal prism, truncating the vertices...

deltahedra, one of which is the snub disphenoid. More generally, the convex polyhedra in which all faces are regular polygon are the Johnson solids, and every...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

edge-transitive. Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79...

symmetry Equilateral triangle p6m symmetry The planar tilings are related to polyhedra. Putting fewer triangles on a vertex leaves a gap and allows it to be...

have 90 edges. A further four uniform star polyhedra (U37, U55, U58, U66) and four uniform compound polyhedra (UC32, UC34, UC36, UC55) contain 90 edges...

angles exceeds a full turn, as occurs in some vertices of many non-convex polyhedra, then the defect is negative. The concept of defect extends to higher...

Roman. "51: small dodecahemidodecahedron". MathConsult. Weisstein, Eric W. "Small dodecahemidodecahedron". MathWorld. Uniform polyhedra and duals v t e...

In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the vertices and edges of a convex polyhedron...

environment of these other atoms can be thought of as distorted Friauf polyhedra composed of chromium atoms, if next-nearest neighbors are included. Examples...

snub polyhedra and tilings with face configuration V3.3.4.3.n. The pentagonal icositetrahedron is one of a family of duals to the uniform polyhedra related...

Icosahedron hosohedron dihedron Henagon#In spherical geometry Kepler–Poinsot polyhedra Small stellated dodecahedron Great dodecahedron Great stellated dodecahedron...

These will generate uniform tilings. Three mirrors can generate uniform polyhedra; including rational numbers gives the set of Schwarz triangles. Three...

Adventures Among the Toroids: A study of orientable polyhedra with regular faces is a book on toroidal polyhedra that have regular polygons as their faces. It...

counts between elements are shown. Klitzing, Richard. "3D convex uniform polyhedra x3o5o - ike"., circumradius sqrt[(5+sqrt(5))/8 = 0.951057 Klitzing, Richard...

Libellus de quinque corporibus regularibus) is a book on the geometry of polyhedra written in the 1480s or early 1490s by Italian painter and mathematician...

A pentaprism is a five-sided reflecting prism used to deviate a beam of light by a constant 90°, even if the entry beam is not at 90° to the prism. The...

geometric graph theory, toric geometry, and combinatorial topology. Polyhedra and tessellations had been studied for many years by people such as Kepler...

List of circle topics List of curves List of surfaces List of polygons, polyhedra and polytopes List of regular polytopes and compounds Solid geometry,...

tetrahemihexahedra Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79...

dodecicosahedron (having the decagonal faces in common). List of uniform polyhedra Maeder, Roman. "43: small ditrigonal dodecicosidodecahedron". MathConsult...

ten octahedra Skilling, J. (May 1976). "Uniform compounds of uniform polyhedra". Mathematical Proceedings of the Cambridge Philosophical Society. 79...

tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron,...

Eric W. "Isohedron". MathWorld. The Uniform Polyhedra Virtual Reality Polyhedra The Encyclopedia of Polyhedra VRML models (George Hart) <3> Archived 27...

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...

polygons, such as the pentagonal numbers, as well as higher-dimensional polyhedra, such as the tetrahedral numbers. Taking T 0 = 0 {\displaystyle T_{0}=0}...

meaning it cannot be separated by a plane into two small regular-faced polyhedra. The surface area of a hebesphenomegacorona with edge length a {\displaystyle...

"faces" of the resulting object to curve. This means that Scutoids are not polyhedra, because not all of their faces are planar. ... For the computational...

stellated and uniform polyhedra, which he documented in his book Vielecke und Vielflache: Theorie und Geschichte (Polygons and polyhedra: Theory and History...

Hart's Conway interpreter: generates polyhedra in VRML, taking Conway notation as input, [1] VRML model Convex Polyhedra containing Golden Rhombi: 2. Expanded...

three tab/pocket flaps protruding from the base. This particularly suits polyhedra that have equilateral triangular faces: Sonobe modules can replace each...

polyhedron: it cannot be separated by a plane into two small regular-faced polyhedra. One way to construct a bilunabirotunda is by attaching two wedges and...

icosidodecahedron. Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79...

Polyhedrins are a type of viral protein that form occlusion bodies (also called polyhedra), large structures that protect the virus particles from the outside environment...

subareas such as polyhedral combinatorics (the study of faces of convex polyhedra), convex geometry (the study of convex sets, in particular combinatorics...

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n). This tiling is topologically related...

George W. (1996), "The Polyhedra of M.C. Escher", Virtual Polyhedra. Hart, George W., "Max Brücknerʼs Wunderkammer of Paper Polyhedra", Bridges 2019 Conference...

Compound of four cubes (Animation) Type Compound Convex hull Chamfered cube Polyhedra 4 cubes Faces 32 squares Edges 48 Vertices 32 (8 + 24) Symmetry group...

Steinitz's theorem, can always be represented as the graphs of convex polyhedra. The convex polyhedron representing an Apollonian network is a 3-dimensional...

The truncated tetrakis cube, or more precisely an order-6 truncated tetrakis cube or hexatruncated tetrakis cube, is a convex polyhedron with 32 faces:...

4.3.5. The snub tetrapentagonal tiling is fourth in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.n. Wikimedia Commons has media...

honeycombs Wythoff symbol - a symbol for the Wythoff construction of uniform polyhedra and uniform tilings Coxeter Regular Polytopes, Third edition, (1973),...

The programs contain a large library of polyhedra which can be manipulated and altered in various ways. Polyhedra in Great Stella's library include the...

In geometry, a hemi-icosahedron is an abstract regular polyhedron, containing half the faces of a regular icosahedron. It can be realized as a projective...

triangles: The snub tetrahexagonal tiling is fifth in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.n. John H. Conway, Heidi Burgiel...

This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbol {3,p}. This tiling is a part of regular series {n...

fear-response neutral stimuli (mushrooms, flowers, physical representation of polyhedra, firearms, and electrical outlets) on human subjects and found that ophidiophobia...

systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov...

tiling. This semiregular tiling is a member of a sequence of snubbed polyhedra and tilings with vertex figure (3.3.3.3.n) and Coxeter–Dynkin diagram...

vertices. Crossed polygons are common as vertex figures of uniform star polyhedra. Crossed quadrilaterals are most common, including: crossed parallelogram...

thesis and a series of extensions characterized hyperbolic 3-dimensional polyhedra in terms of their dihedral angles, resolving a long-standing open question...

Z=2 TbB41Si1.2 orthorhombic Pbam ferromagnetic < 18 K; B12Si3 and B12 polyhedra Tb9B3Si13.83 R32 a = 6.668 c = 12.405 Z=1 Tb3-xC2Si8(B12)3 Tb1−xB12Si3...

performed later by John C. Slater. There are only five topologically distinct polyhedra which tile three-dimensional space, R 3 {\displaystyle \mathbb {R} ^{3}}...

Joseph (July 2007). Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge University Press. pp. Part II. ISBN 978-0-521-85757-4. Beasley...

Retrieved 2026-06-11. Satterly, John (1958). "The Moments of Inertia of Some Polyhedra". The Mathematical Gazette. 42 (339). Mathematical Association: 11–13...

icosidodecahedron. Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79...

pentagonal bipyramid is the third in an infinite series of bifrustum polyhedra. It has 10 trapezoidal and 2 pentagonal faces. The pentagonal bifrustum...

In geometry, the small hexagrammic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the small retrosnub icosicosidodecahedron. It...

the octahedron. Further truncation, and alternation creates two more polyhedra: Rectified truncated tetrahedron Rectified truncated cube Rectified truncated...

In mathematics, a polyhedral complex is a set of polyhedra in a real vector space that fit together in a specific way. Polyhedral complexes generalize...

Perspective of the Regular Solids) is a book of perspective drawings of polyhedra by German Renaissance goldsmith Wenzel Jamnitzer, with engravings by Jost...

extended Schläfli symbol notation t1,2{p,q,...} or 2t{p,q,...}. For regular polyhedra (i.e. regular 3-polytopes), a bitruncated form is the truncated dual....

tetrahedron. Further truncation, and alternation operations creates two more polyhedra: Rectified truncated cube Rectified truncated octahedron Rectified truncated...

polyhedra Maeder, Roman. "62: small dodecahemicosahedron". MathConsult. Weisstein, Eric W. "Small dodecahemicosahedron". MathWorld. Uniform polyhedra...

accurate filling. In the neo filling, orientable polyhedra are filled traditionally, but non-orientable polyhedra have their faces filled with the modulo-2 method...

long diagonals), or of a rhombohedron or parallelepiped (less symmetric polyhedra that still have the same combinatorial structure as a cube). In the case...

gyroelongated pyramids (also called augmented antiprisms) are an infinite set of polyhedra, constructed by adjoining an n-gonal pyramid to an n-gonal antiprism....

uniform polyhedra with cubic symmetry Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). "21. Naming Archimedean and Catalan Polyhedra and Tilings"...

In geometry, the medial icosacronic hexecontahedron (or midly sagittal ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform...

Polytetrahedron is a term used for three distinct types of objects, all based on the tetrahedron: A regular convex 4-polytope made up of 600 tetrahedral...

The truncated triakis octahedron, or more precisely an order-8 truncated triakis octahedron, is a convex polyhedron with 30 faces: 8 sets of 3 pentagons...

In geometry, a dissection problem is the problem of partitioning a geometric figure (such as a polytope or ball) into smaller pieces that may be rearranged...

symmetry, and icosahedral symmetry, named after the triangle-faced regular polyhedra with these symmetries. All of the discrete point symmetries are subgroups...

In mathematics Nef polygons and Nef polyhedra are the sets of polygons and polyhedra which can be obtained from a finite set of halfplanes (halfspaces)...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

published a paper titled Convex Polyhedra with Regular Faces.[A66] Inside, it contained the list of 92 convex polyhedra with regular faces, along with...

solid is one of 92 strictly convex polyhedra that are composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids...

flattened horizontal and vertical hexagons, and squares on the perpendicular polyhedra. With regular faces, the elongated square bipyramid can form a tessellation...

truncated hexahedra. Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79...

computing pioneer. He worked in number theory and on geometry, particularly polyhedra, where Miller's monster is a nickname of the great dirhombicosidodecahedron...

This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbol {n,3}. From a Wythoff construction there are eight...

C={\frac {1}{2}}a{\sqrt {5+2{\sqrt {2}}}}.} Rajwade, A. R. (2001), Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem, Texts and Readings...

York: W. H. Freeman & Co., (1987), ISBN 0-7167-1193-1. Grünbaum, B.; Polyhedra with Hollow Faces, Proc of NATO-ASI Conference on Polytopes ... etc. (Toronto...

In geometry, the great hexagonal hexecontahedron (or great astroid ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform...

angles of the polyhedra to which it belongs is equal to 4π ; for each edge in the gluing the sum of the dihedral angles of the polyhedra to which it belongs...

dodecicosidodecahedron". MathConsult. Coxeter, H. S. M. (May 13, 1954). "Uniform Polyhedra". Philosophical Transactions of the Royal Society of London. Series A...

Geometry Algorithms Library which has support for Boolean operations on Polyhedra; but no sweep, revolve or NURBS. Open CASCADE is an open-source modeling...

until those pyramids cannot be sliced into more convex, regular-faced polyhedra again. Because its faces are regular polygons, such a pentagonal bipyramid...

of the circumscribed circle. All regular polyhedra have circumscribed spheres, but most irregular polyhedra do not have one, since in general not all...

edges. Conversely, the graphs that are skeletons of three-dimensional polyhedra can be characterized by Steinitz's theorem as being exactly the 3-vertex-connected...

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (5n). Wikimedia Commons has media related...

In geometry, the elongated cupolae are an infinite set of polyhedra, constructed by adjoining an n-gonal cupola to a 2n-gonal prism. There are three elongated...

octadecahedra, excluding mirror images, having at least 11 vertices. (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements...

results to certain classes of non-convex polyhedra. It is also possible to specify three-dimensional polyhedra uniquely by the direction and perimeter...

decomposition of this type can be obtained in which all the cuboids are convex polyhedra with flat faces. A tetragonal trapezohedron appears in the upper left...

and the sphinx tiling as a special case of the polyiamond. Classes of polyhedra with equilateral triangles as their faces are deltahedra and the family...

similar role for the medial pentagonal hexecontahedron. List of uniform polyhedra Snub dodecadodecahedron Wenninger, Magnus (1983), Dual Models, Cambridge...

other uniform polyhedra: the convex rhombicuboctahedron, the small rhombihexahedron, and the small cubicuboctahedron. List of uniform polyhedra Weisstein...

In geometry, the small icosacronic hexecontahedron (or small lanceal trisicosahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform...

(solid). Shapes studied in five dimensions include counterparts of regular polyhedra and of the sphere. In five or more dimensions, only three regular polytopes...

In geometry, the small ditrigonal dodecacronic hexecontahedron (or fat star) is a nonconvex isohedral polyhedron. It is the dual of the uniform small ditrigonal...