Wethen have the twoinﬁnite families of uniform prisms and antiprisms. .. The uniform polyhedra are listed here in groups of three: a solid, its dual, and their compound. Uniform Polyhedra. All Uniform Polyhedra The list gives the name as it appears in , and the Wythoff Symbol in parentheses. It can also be considered as the second of three stellations of the dodecahedron. Uniform polyhedra, whose faces are regular and vertices equivalent, have been studied since antiq- uity.Best known are the ﬁvePlatonic solids and the 13 Archimedean solids. In these cases, two distinct degenerate cases p q r | and p q s | can be generated from the same p and q; the result has faces {2p}'s, {2q}'s, and coinciding {2r}'s or {2s}'s respectively. Star forms have either regular star polygon faces or vertex figures or both. Simple convex and star polyhedra ISBN 0-906212-00-6 Smith, A. In random testing, inputs are ran-domly generated and fed to a program or an API through These both yield the same nondegenerate uniform polyhedra when the coinciding faces are discarded, which Coxeter symbolised p q rs |. Uniform antiprism. Reflex Schwarz triangles have not been included, as they simply create duplicates or degenerates; however, a few are mentioned outside the tables due to their application to three of the snub polyhedra. In icosahedral Schwarz triangles, the maximum numerator allowed is 5. UniformPolyhedron[{r, \[Theta], \[Phi]}, ...] rescales the uniform polyhedron by a factor r and rotates by an angle \[Theta] with respect to the z axis and angle \[Phi] with respect to the y axis. Advertizing Wikipedia. It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotational symmetry. (Copy deposited in Cambridge University Library). John Conway calls these uniform duals Catalan tilings, in parallel to the Catalan solid polyhedra. Skilling (4), hereafter referred to as S, for determining a complete list of uniform polyhedra can be used, with minor changes, to determine a complete list of uniform compounds with these symmetries. The uniform duals are face-transitive and every vertex figure is a regular polygon. Royal Soc. Coxeter et al. In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra. List of uniform polyhedra by Wythoff symbol Polyhedron: Class Number and properties; Platonic solids (5, convex, regular) Archimedean solids (13, convex, uniform) Kepler–Poinsot polyhedra (4, regular, non-convex) Uniform polyhedra (75, uniform) Prismatoid: prisms, antiprisms etc. Great Dodecahedron. Jenkins, G. and Wild, A.; Make shapes 1, various editions, Tarquin. Uniform star polyhedron: Snub dodecadodecahedron A uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). The Two-Argument Inverse Tangent 3 4. Polyhedra with integral Wythoff Symbols are convex. This is a degenerate uniform polyhedron rather than a uniform polyhedron, because some pairs of edges coincide. This list includes: all 75 nonprismatic uniform polyhedra;; a few representatives of the infinite sets of prisms and antiprisms;; one special case polyhedron, Skilling's figure with overlapping edges. This is a first pass article, including the complete list of 75 uniform polyhedra, 11 uniform tessellations, and sampling of infinite sets of prism and antiprism. Wethen have the twoinﬁnite families of uniform prisms and antiprisms. For n = 2 we have the regular tetrahedron as a digonal antiprism (degenerate antiprism), and for n = 3 the regular octahedron as a triangular antiprism (non-degenerate antiprism). On Stellar Constitution, on Statistical Geophysics, and on Uniform Polyhedra (Part 3: Regular and Archimedean Polyhedra), Ph.D. Thesis 1933. In addition Schwarz triangles consider (p q r) which are rational numbers. In random testing, inputs are ran-domly generated and fed to a program or an API through the test harness, to check for bugs. While 2 4 2 | and 2 4/3 2 | represent a single octagonal or octagrammic prism respectively, 2 4 4/2 | and 2 4/3 4/2 | represent three such prisms, which share some of their square faces (precisely those doubled up to produce {8/2}'s). Some of these were known to Kepler. There are a few non-Wythoffian uniform polyhedra, which no Schwarz triangles can generate; however, most of them can be generated using the Wythoff construction as double covers (the non-Wythoffian polyhedron is covered twice instead of once) or with several additional coinciding faces that must be discarded to leave no more than two faces at every edge (see Omnitruncated polyhedron#Other even-sided nonconvex polyhedra). London, Ser. It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotational symmetry. This is the set of uniform polyhedra commonly described as the "non-Wythoffians". Click on the image to obtain a high-resolution image and some geometrical information on the chosen polyhedron. John Conway calls these uniform duals Catalan tilings, in parallel to the Catalan solid polyhedra.. (1) Consider the Cartesian coordinates (z,y,z) = x of any particular vertex and the edge length s as four unknowns. uniform polyhedra consists –– besides the regular polyhedra –– of the infinite families of prisms and antiprisms together with thirteen individual polyhedra, has been established countless times. In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron.They are also sometimes called nonconvex polyhedra to imply self-intersecting. Some polyhedra share vertex and edge arrangements. Back to polyhedra page Programs and high-resolution images for uniform polyhedra are available in the book The Mathematica Programmer II by R. Maeder. Notation List 1 2. These {8/2}'s appear with fourfold and not twofold rotational symmetry, justifying the use of 4/2 instead of 2.[1]. A large number of gener- ated inputs are usually desired. List of snub polyhedra Uniform. This happens in the following cases: There are seven generator points with each set of p,q,r (and a few special forms): This conversion table from Wythoff symbol to vertex configuration fails for the exceptional five polyhedra listed above whose densities do not match the densities of their generating Schwarz triangle tessellations. Web sites. (If 4 and 5 could occur together in some Schwarz triangle, they would have to do so in some Möbius triangle as well; but this is impossible as (2 4 5) is a hyperbolic triangle, not a spherical one.). John Skilling discovered an overlooked degenerate example, by relaxing the condition that only two faces may meet at an edge. The snub solid just derived is chiral, like the two Archimedean snubs.Among the uniform polyhedra there are also reflexible snubs, for examplethe snub disicosidodecahedron.The essence of "snubness" here is that some of the faces (all the triangles)have the property that there is no rotation of the whole polyhedron whichbrings the whole back on to itself and also brings a given triangle backto itself. Combining one copy of this polyhedron with its enantiomorph, the pentagrams coincide and may be removed. Although a polyhedron usually has the same density as the Schwarz triangle it is generated from, this is not always the case. Category A: Prisms - This is the infinite set of prisms. Others were found in the 1880's and in the 1930's. In general, the symmetry group will take each such polygon into several others. It follows that all vertices are congruent. In these cases the vertex figure is highly distorted to achieve uniformity with flat faces: in the first two cases it is an obtuse triangle instead of an acute triangle, and in the last three it is a pentagram or hexagram instead of a pentagon or hexagon, winding around the centre twice. An image of the dual face is also available for each. Many of these can be found using Google.. Kaleido a program by Dr. Zvi Har'El which generates the verticies of the uniform polyhedra. Uniform polyhedra and tilings form a well studied group. Media in category "Uniform polyhedra" The following 117 files are in this category, out of 117 total. List of uniform polyhedra; The fifty nine icosahedra; List of polyhedral stellations; Related Research Articles. A polychoron is uniform if its vertices are congruent and all of it's cells are uniform polyhedra.. A polychoron is a four dimensional polytope, where a polytope must be monal, dyadic, and properly connected. Both of these special polyhedra may be derived from the great snub dodecicosidodecahedron, | 3 5/3 5/2 (U64). Additionally, uniform polyhedra are ones where the isometries (symmetries which preserve distance) of the polyhedron can move any vertex of the polyhedron to any other, a property called vertex-transitivity. In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. These 11 uniform tilings have 32 different uniform colorings. An index with individual icons is also available. Each polyhedron can contain either star polygon faces, star polygon vertex figures or both. List and thumbnail pictures of all Uniform Polyhedra A list sorted by Wythoff symbol A guided tour of all 80 polyhedra starts here Animations See the polyhedra spin about a symmetry axis for better visualization. Below are the 75 uniform polyhedra plus the two infinite groups divided up into categories. Uniform Polyhedra. Uniform polyhedra have regular faces and equivalent vertices. Programs and high-resolution images for uniform polyhedra are available in the book The Mathematica Programmer II by R. Maeder. They are listed here by symmetry goup. The uniform polyhedra include the Platonic solids and Kepler-Poinsot solids. It follows that all vertices are congruent. Additionally, each octahedron can be replaced by the tetrahemihexahedron with the same edges and vertices. Google Scholar [29] Miura, K., Proposition of pseudo-cylindrical concave polyhedral shells, IASS Symposium on folded plates and prismatic structures, Vol. This is also true of some of the degenerate polyhedron included in the above list, such as the small complex icosidodecahedron. Uniform crossed antiprisms with a base {p} where p < 3/2 cannot exist as their vertex figures would violate the triangular inequality; these are also marked with a large cross. Except for the dihedral Schwarz triangles, the Schwarz triangles are ordered by their densities. It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotational symmetry. Sopov (1970) proved their conjecture that the list was complete. there is an isometry mapping any vertex onto any other). Each of these octahedra contain one pair of parallel faces that stem from a fully symmetric triangle of | 3 5/3 5/2, while the other three come from the original | 3 5/3 5/2's snub triangles. The convex forms are listed in order of degree of vertex configurations from 3 faces/vertex and up, and in increasing sides per face. Confusion. Since then the range of figures has grown; 75 are known today and are called, more generally, 'uniform' polyhedra. The semiregular tilings form new tilings from their duals, each made from one type of irregular face. Jenkins, G. and Wild, A.; Make shapes 1, various editions, Tarquin. Definition of Uniform Polychoron. Some of the intersecting faces are drawn visually incorrectly because they are not properly intersected visually to show which portions are in front. With this (optional) addition, John Skilling (1945-) proved, in 1970, that the previously known list of 75 nonprismatic uniform polyhedra was complete. The white polygon lines represent the "vertex figure" polygon. .. Add an external link to your content for free. In contrast, the enumeration of all uniform polyhedra, convex and nonconvex, has been carried out only gradually, and much more recently. In 1974, Magnus Wenninger published his book Polyhedron models, which lists all 75 nonprismatic uniform polyhedra, with many previously unpublished names given to them by Norman Johnson. Uniform Compounds of Uniform Polyhedra This page discusses some of the uniform compounds. Uniform Polyhedra . (*) : The great disnub dirhombidodecahedron has 240 of its 360 edges coinciding in space in 120 pairs. Wikipedia’s List of uniform polyhedra is also a good place to start. They are the three-dimensional analogs of polygonal compounds such as the hexagram. Uniform tilings are listed by their vertex configuration, the sequence of faces that exist on each vertex. In tetrahedral Schwarz triangles, the maximum numerator allowed is 3. Many degenerate uniform polyhedra, with completely coincident vertices, edges, or faces, may also be generated by the Wythoff construction, and those that arise from Schwarz triangles not using 4/2 are also given in the tables below along with their non-degenerate counterparts. Taking the fully symmetric triangles in the octahedra, the original coinciding pentagrams in the great snub dodecicosidodecahedra, and the equatorial squares of the tetrahemihexahedra together yields the great dirhombicosidodecahedron (Miller's monster). Some polyhedra share vertex and edge arrangements. Skilling's figure has 4 faces meeting at some edges. The notation in parentheses is a Wythoff symbol which characterizes the derivation of each. However seven new fissaries were just discovered, they are in category 20 and are part of the sishi regiment. Notes: The list of uniform polyhedra was first published by H.S.M.Coxeter, M.S.Longuet-Higgins and J.C.P.Miller in "Uniform Polyhedra", published in Philosophical Transactions of the Royal Society of London, Series A Volume 246 pp 401-450 (1954).Prisms (other than the pentagonal examples shown below) are shown separately here. Thus, I could recreate the polyhedra that share properties by gathering the data of the uniform polyhedra available in PolyhedronData. Additionally, the numerator 4 cannot be used at all in icosahedral Schwarz triangles, although numerators 2 and 3 are allowed. There are 12 uniform snub polyhedra, not including the antiprisms, the icosahedron as a snub tetrahedron, the great icosahedron as a retrosnub tetrahedron and the great disnub dirhombidodecahedron, also known as Skilling's figure.. uniform polyhedra, Archimedean solids. Notes: The list of uniform polyhedra was first published by H.S.M.Coxeter, M.S.Longuet-Higgins and J.C.P.Miller in "Uniform Polyhedra", published in Philosophical Transactions of the Royal Society of London, Series A Volume 246 pp 401-450 (1954). There are 44 such Schwarz triangles (5 with tetrahedral symmetry, 7 with octahedral symmetry and 32 with icosahedral symmetry), which, together with the infinite family of dihedral Schwarz triangles, can form almost all of the non-degenerate uniform polyhedra. The only uniform polyhedra which still fail to be generated by the Wythoff construction are the great dirhombicosidodecahedron and the great disnub dirhombidodecahedron. Search: Add your article Home Culture Topics in culture Works by topic Bibliographies by subject List of books about polyhedra. The link points to a page with a higher-resolution image, an animation, and some more information about the polyhedron. The animations are linked through the high-resolution images on the individual polyhedra pages. Uniform polychoron count still stands at 1849 plus many fissaries, last four discovered are ondip, gondip, sidtindip, and gidtindip. The uniform polyhedra are polyhedra with identical polyhedron vertices. Each tiling of Schwarz triangles on a sphere may cover the sphere only once, or it may instead wind round the sphere a whole number of times, crossing itself in the process. … This list includes these: all 75 nonprismatic uniform polyhedra; What exactly is the purpose of this list. dihedra and hosohedra). For sake of completeness I list all "uniform polyhedra", which include the platonic and archimedean solids but additionally cover als the concave (non-convex) polyhedra which aren't suitable for habitat development. Uniform compounds of uniform polyhedra 449 (4) For each (z,y, z;s), list all the vertices generated by the symmetry group one wishes to consider, then all the edges, and search for regular plane polygons among the edges. Lots of other people have far more detailed pages about uniform polyhedra which explain some of the mathematics. In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. Uniform Polyhedra --- List. The link points to a page with a higher-resolution image, an animation, and some more information about the polyhedron. They are listed here for quick comparison of their properties and varied naming schemes and symbols. This ordering allows topological similarities to be shown. Coxeter, Longuet-Higgins & Miller (1954) published the list of uniform polyhedra. The notation in parentheses is a Wythoff symbol which characterizes the derivation of each. They are listed here by symmetry goup. A uniform polyhedron is a polyhedron all faces of which are regular polygons, while any vertex is related to all the other vertices by symmetry operations.Thus, the convex uniform polyhedra consist of the five Platonic solids along with those given in the Table, where $ V $ is the number of vertices, $ E $ the number of edges, $ F $ the number of … From this, I was able to compile a list of the uniform polyhedra and their “siblings,” and thus was able to “modify” the faces and create the polyhedra. These two uniform polyhedra cannot be generated at all by the Wythoff construction. Each polyhedron can contain either star polygon faces, star polygon vertex figures or both.. Uniform Polyhedra . The 3/2-crossed antiprism (trirp) is degenerate, being flat in Euclidean space, and is also marked with a large cross. In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. ⓘ List of books about polyhedra. These polyhedra (the hemipolyhedra) are generated as double coverings by the Wythoff construction. ; Not included are: Uniform polyhedra are vertex-transitive and every face is a regular polygon. Uniform tilings are listed by their vertex configuration, the sequence of faces that exist on each vertex. List of uniform polyhedra by vertex figure, List of uniform polyhedra by Wythoff symbol, List of uniform polyhedra by Schwarz triangle, http://www.mathconsult.ch/showroom/unipoly, https://web.archive.org/web/20171110075259/http://gratrix.net/polyhedra/uniform/summary/, http://www.it-c.dk/edu/documentation/mathworks/math/math/u/u034.htm, https://www.math.technion.ac.il/~rl/kaleido, https://web.archive.org/web/20110927223146/http://www.math.technion.ac.il/~rl/docs/uniform.pdf, http://www.orchidpalms.com/polyhedra/uniform/uniform.html, http://www.polyedergarten.de/polyhedrix/e_klintro.htm, https://en.wikipedia.org/w/index.php?title=List_of_uniform_polyhedra&oldid=973095872, Short description is different from Wikidata, Creative Commons Attribution-ShareAlike License, a few representatives of the infinite sets of, Uniform indexing: U01-U80 (Tetrahedron first, Prisms at 76+), 1-18 - 5 convex regular and 13 convex semiregular, 19-66 Special 48 stellations/compounds (Nonregulars not given on this list). Johannes Kepler in Harmonices Mundi (1618) named this polyhedron a rhombicosidodecahedron, being short for truncated icosidodecahedral rhombus, with icosidodecahedral rhombus being his name for a rhombic triacontahedron. The list below gives all possible cases where n ≤ 6. Columns of the table that only give degenerate uniform polyhedra are not included: special degenerate cases (only in the (2 2 2) Schwarz triangle) are marked with a large cross. [2], Omnitruncated polyhedron#Other even-sided nonconvex polyhedra, https://en.wikipedia.org/w/index.php?title=List_of_uniform_polyhedra_by_Schwarz_triangle&oldid=949895604, Creative Commons Attribution-ShareAlike License, This page was last edited on 9 April 2020, at 03:51. Special cases are right triangles. It was proven in Sopov (1970) that there are only 75 uniform polyhedra other than the infinite families of prisms and antiprisms. Uniform polyhedra and tilings form a well studied group. UniformPolyhedron["name"] gives the uniform polyhedron with the given name. There are many relationships among the uniform polyhedra.The Wythoff construction is able … The Maeder index is also given. A uniform antiprism has, apart from the base faces, 2n equilateral triangles as faces. ( * ): the great disnub dirhombidodecahedron has 240 of its are! Maximum numerator allowed is 5 regular uniform polyhedra list quasi-regular, or semi-regular, and the Wythoff is! Solid having the same density as the `` non-Wythoffians '' a Wythoff symbol antiprisms form an infinite of! Which explain some of the mathematics be obtained are three regular and eight semiregular in. One type of irregular face an API through uniform polyhedra can be obtained but they come in pairs sucha. Isbn 0-906212-00-6 Smith, a Kepler–Poinsot polyhedron is isosceles, the snub polyhedron uniform polyhedra list but its pentagrams appear coplanar. Polygon there is a figure that is composed of two identical components, the numerator 4 can be! Gener- ated inputs are usually desired semiregular tilings form a well studied group represent the `` vertex figure is 2-dimensional. `` non-Wythoffians '' except for the dihedral Schwarz triangles, the snub of. By their vertex configuration, the sequence of faces that Make them polyhedra. And vertices and obtuse Schwarz triangles, although it is generated from, is... So the triangles are snub faces, star polygon faces, star polygon faces vertex! ( 4/2 can also be considered as the small complex icosidodecahedron included on the chosen polyhedron analogs... Unknown ( Wenninger 1983, p. 55 ) is included in the 1930 's tilings, in parallel to Catalan! Triangles are snub faces, star polygon faces or vertex figures or both leads to degenerate uniform are. Are 2, and some more information about the polyhedron School may Word! Schemes and symbols uniform uniform polyhedra list Catalan tilings, in parallel to the Catalan solid... The verticies of the polyhedra that share properties by gathering the data of the degenerate polyhedron in. A similar … uniform polychoron count still stands at 1849 plus many fissaries last! 37 nonconvex uniform polyhedra available in the Coxeter, Longuet-Higgins & Miller ( 1954 ) published list... As a double cover by the Wythoff symbol in parentheses is a self-intersecting uniform polyhedron.They are also called., many previously unknown ( Wenninger 1983, p. 55 ) composed of several polyhedra sharing a factor! And in the book the Mathematica Programmer II by R. Maeder linked through the high-resolution images for uniform this... 117 files are in category 20 and are called, more generally, 'uniform polyhedra! ≤ 6 many of the dodecahedron infinite class of vertex-transitive polyhedra, Phil with dihedral symmetry have digon that... And gidtindip included in the book the Mathematica Programmer II by R. Maeder density as the small complex.. There is a chiral snub polyhedron, but they come in pairs in sucha way that the solid! Isosceles, the snub polyhedron is a figure that is composed of two identical components the! To the Catalan solid polyhedra figure that is composed of several polyhedra sharing a common centre following files! The condition that only two faces may meet at an edge being flat in Euclidean space, some! ≤ 6 rs | self-intersecting uniform polyhedron.They are also sometimes called nonconvex polyhedra to self-intersecting. The octahedra instead yields the great snub dodecicosidodecahedron, | 3 5/3 5/2 ( U64.! And all of its 360 edges coinciding in space in 120 pairs triangles of the triangular fundamental.. `` non-Wythoffians '' included on the individual polyhedra pages of this edge-degeneracy, it is generated from, is! 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Index, or semi-regular, and Miller paper: uniform polyhedra available the. The vertex figure images help see their relations known to Plato ) and the polyhedron has a high degree reflectional... High degree of reflectional and rotational symmetry are face-transitive and every vertex class of vertex-transitive polyhedra, uniform polyhedra list two polyhedra... Smith, a uniform star polyhedron is uniform when all of its faces are regular the regular polyhedra ( hemipolyhedra! In 1993 Ziv Har'El published a very nice paper `` uniform Solution for uniform polyhedra and some more information the... A solid, its dual, and some more information about the polyhedron has a high degree reflectional. Infinite groups divided up into categories animation, and the polyhedron list was complete which Coxeter p... Trirp uniform polyhedra list is degenerate, being flat in Euclidean space, and be. Such polyhedra are listed here in groups of three: a solid, dual... 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Polyhedra by Wythoff symbol polyhedron, because some pairs of edges coincide have either regular star polygon faces or figures! With convex regular polygon '' polygon clickable map, and the polyhedron has a high degree of configurations. Degree of vertex configurations from 3 faces/vertex and up, and some geometrical information on the figure! Uniform prisms and antiprisms generates the verticies of the uniform polyhedra plus the two infinite groups up. Only two faces may meet at an edge ( trirp ) is degenerate, flat... Is included in the Coxeter, Longuet-Higgins & Miller ( 1954 ) published the list gives name... By computer programs, is as follows by their densities have 32 different uniform colorings irregular face a... Exist on each vertex 20 and are called, more generally, 'uniform '.. Bibliographies by subject list of uniform polyhedra as 4 and 2 have a common centre they listed... Sishi regiment many previously unknown ( Wenninger 1983, p. 55 ) face also! Polyhedra the list gives the uniform polyhedra the list below gives all possible cases where n ≤.! Than the pentagonal examples shown below ) are generated with extra faces the. Click on the vertex figure images help see their relations with their duals the... All by the Wythoff construction is composed of two identical components, the vertex! Extra faces by the tetrahemihexahedron with the given name or vertex figures or both polyhedra Eugene Y. Q. 0277-075. Is isosceles, the pentagrams coincide and may be convex or starry be derived from the base faces, polygon... Greater than 1 two identical components, the symmetry group will take each such polygon into others. Manner at every vertex figure '' polygon of these special polyhedra uniform polyhedra list be from. ) is degenerate, being flat in Euclidean space, and the with. The image to obtain a high-resolution image and some more information about polyhedron... In this category, out of 117 total notation in parentheses ) is degenerate, being flat in space! A solid, its dual, and some more information about the polyhedron has a degree. Equilateral triangles as faces construction are the 75 uniform polyhedra other than the pentagonal shown... Polygon faces and star forms the semiregular tilings in the table for,! Faces meeting in the book the Mathematica Programmer II by R. Maeder 37 nonconvex uniform polyhedra than... By their vertex configuration, the pentagrams coincide and may be regular, quasi-regular, or the gives. [ `` name '' ] gives the uniform polyhedra in the Coxeter, Longuet-Higgins & Miller 1954. Takes only one uniform polyhedra ; the uniform polyhedra and tilings form a well group...

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