Uniform honeycombs in hyperbolic space
In hyperbolic geometry, a uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells. Honeycombs are divided between compact and paracompact forms defined by Coxeter groups, the first category only including finite cells and vertex figures, the nine compact Coxeter groups are listed here with their Coxeter diagrams, in order of the relative volumes of their fundamental simplex domains. These 9 families generate a total of 76 unique uniform honeycombs, the full list of hyperbolic uniform honeycombs has not been proven and an unknown number of non-Wythoffian forms exist. One known example is cited with the family below, only two families are related as a mirror-removal halving, ↔. There are just two radical subgroups with nonsimplectic domains that can be generated by removing a set of two or more mirrors separated by all other mirrors by even-order branches. One is, represented by Coxeter diagrams an index 6 subgroup with a trigonal trapezohedron fundamental domain ↔, the other is, index 120 with a dodecahedral fundamental domain.
There are 23 paracompact Coxeter groups of rank 4 that produce paracompact uniform honeycombs with infinite or unbounded facets or vertex figure, other paracompact Coxeter groups exists as Vinberg polytope fundamental domains, including these triangular bipyramid fundamental domains as rank 5 graphs including parallel mirrors. Uniform honeycombs exist as all permutations of rings in these graphs, the bitruncated and runcinated forms contain the faces of two regular skew polyhedrons, and. There are 15 forms, generated by ring permutations of the Coxeter group and this family is related to the group by a half symmetry, or ↔, when the last mirror after the order-4 branch is inactive, or as an alternation if the third mirror is inactive ↔. There are 9 forms, generated by ring permutations of the Coxeter group, there are 11 forms, generated by ring permutations of the Coxeter group, or. If the branch ring states match, a symmetry can double into the family. There are 9 forms, generated by ring permutations of the Coxeter group, there are 9 forms, generated by ring permutations of the Coxeter group, The bitruncated and runcinated forms contain the faces of two regular skew polyhedrons, and.
There are 6 forms, generated by ring permutations of the Coxeter group, there are 4 extended symmetries possible based on the symmetry of the rings, and. This symmetry family is related to a radical subgroup, index 6, ↔, constructed by. The truncated forms contain the faces of two regular skew polyhedrons, there are 9 forms, generated by ring permutations of the Coxeter group, The truncated forms contain the faces of two regular skew polyhedrons, and. There are 6 forms, generated by ring permutations of the Coxeter group, there are 4 extended symmetries possible based on the symmetry of the rings, and. The truncated forms contain the faces of two regular skew polyhedrons and this is the complete enumeration of the 76 Wythoffian uniform honeycombs. The alternations are listed for completeness, but most are non-uniform, uniform tilings in hyperbolic plane List of regular polytopes#Tessellations of hyperbolic 3-space James E. ed. Dover Publications,1973
In geometry, a regular icosahedron is a convex polyhedron with 20 faces,30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most sides and it has five equilateral triangular faces meeting at each vertex. It is represented by its Schläfli symbol, or sometimes by its vertex figure as 18.104.22.168.3 or 35 and it is the dual of the dodecahedron, which is represented by, having three pentagonal faces around each vertex. A regular icosahedron is a pentagonal bipyramid and a biaugmented pentagonal antiprism in any of six orientations. The name comes from Greek εἴκοσι, meaning twenty, and ἕδρα, the plural can be either icosahedrons or icosahedra. The surface area A and the volume V of a regular icosahedron of edge length a are, note that these vertices form five sets of three concentric, mutually orthogonal golden rectangles, whose edges form Borromean rings. If the original icosahedron has edge length 1, its dual dodecahedron has edge length √5 − 1/2 = 1/ϕ = ϕ −1, the 12 edges of a regular octahedron can be subdivided in the golden ratio so that the resulting vertices define a regular icosahedron.
The locations of the vertices of a regular icosahedron can be described using spherical coordinates, if two vertices are taken to be at the north and south poles, the other ten vertices are at latitude ±arctan ≈ ±26. 57°. These ten vertices are at evenly spaced longitudes, alternating between north and south latitudes and this projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane, an icosahedron has 43,380 distinct nets. To color the icosahedron, such that no two adjacent faces have the color, requires at least 3 colors. A problem dating back to the ancient Greeks is to determine which of two shapes has larger volume, an icosahedron inscribed in a sphere, or a dodecahedron inscribed in the same sphere, the problem was solved by Hero and Fibonacci, among others. Apollonius of Perga discovered the result that the ratio of volumes of these two shapes is the same as the ratio of their surface areas. Both volumes have formulas involving the golden ratio, but taken to different powers, as it turns out, the icosahedron occupies less of the spheres volume than the dodecahedron.
The following construction of the icosahedron avoids tedious computations in the number field ℚ necessary in more elementary approaches, the existence of the icosahedron amounts to the existence of six equiangular lines in ℝ3. Indeed, intersecting such a system of lines with a Euclidean sphere centered at their common intersection yields the twelve vertices of a regular icosahedron as can easily be checked. Conversely, supposing the existence of an icosahedron, lines defined by its six pairs of opposite vertices form an equiangular system. In order to such an equiangular system, we start with this 6 ×6 square matrix
Uniform star polyhedron
In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are sometimes called nonconvex polyhedra to imply self-intersecting, each polyhedron can contain either star polygon faces, star polygon vertex figures or both. The complete set of 57 nonprismatic uniform star polyhedra includes the 4 regular ones, called the Kepler–Poinsot polyhedra,5 quasiregular ones, there are two infinite sets of uniform star prisms and uniform star antiprisms. The nonconvex forms are constructed from Schwarz triangles, all the uniform polyhedra are listed below by their symmetry groups and subgrouped by their vertex arrangements. Regular polyhedra are labeled by their Schläfli symbol, other nonregular uniform polyhedra are listed with their vertex configuration or their Uniform polyhedron index U. Note, For nonconvex forms below an additional descriptor Nonuniform is used when the convex hull vertex arrangement has same topology as one of these, for example an nonuniform cantellated form may have rectangles created in place of the edges rather than squares.
There is one form, the tetrahemihexahedron which has tetrahedral symmetry. There are two Schwarz triangles that generate unique nonconvex uniform polyhedra, one triangle, and one general triangle. The general triangle generates the octahemioctahedron which is given further on with its octahedral symmetry. There are 8 convex forms, and 10 nonconvex forms with octahedral symmetry, there are four Schwarz triangles that generate nonconvex forms, two right triangles and two general triangles. There are 8 convex forms and 46 nonconvex forms with icosahedral symmetry, some of the nonconvex snub forms have reflective vertex symmetry. Coxeter identified a number of star polyhedra by the Wythoff construction method. It is counted as a uniform polyhedron rather than a uniform polyhedron because of its double edges. Star polygon List of uniform polyhedra List of uniform polyhedra by Schwarz triangle Coxeter, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, a proof of the completeness on the list of elementary homogeneous polyhedra, Ukrainskiui Geometricheskiui Sbornik, 139–156, MR0326550 Skilling, J.
The complete set of polyhedra, Philosophical Transactions of the Royal Society of London. Mathematical and Physical Sciences,278, 111–135, doi,10. 1098/rsta.1975.0022, ISSN 0080-4614, JSTOR74475, MR0365333 HarEl, zvi Har’El, Kaleido software, dual images Mäder, R. E. Messer, Peter W. Closed-Form Expressions for Uniform Polyhedra and Their Duals
Paracompact uniform honeycombs
In geometry, uniform honeycombs in hyperbolic space are tessellations of convex uniform polyhedron cells. These families can produce uniform honeycombs with infinite or unbounded facets or vertex figure, including ideal vertices at infinity, of the uniform paracompact H3 honeycombs,11 are regular, meaning that their group of symmetries acts transitively on their flags. These have Schläfli symbol and are shown below and this is a complete enumeration of the 151 unique Wythoffian paracompact uniform honeycombs generated from tetrahedral fundamental domains. The honeycombs are indexed here for cross-referencing duplicate forms, with brackets around the nonprimary constructions, the alternations are listed, but are either repeats or dont generate uniform solutions. Single-hole alternations represent a removal operation. If an end-node is removed, another family is generated. These nonsimplectic Coxeter groups are not enumerated on this page, except as special cases of groups of the tetrahedral ones.
The complete list of nonsimplectic paracompact Coxeter groups was published by P. Tumarkin in 2003, the smallest paracompact form in H3 can be represented by or, or which can be constructed by a mirror removal of paracompact hyperbolic group as, =. The doubled fundamental domain changes from a tetrahedron into a quadrilateral pyramid, another pyramid is or, constructed as =, =. Removing a mirror from some of the cyclic hyperbolic Coxeter graphs become bow-tie graphs, = or, = or, another nonsimplectic half groups is ↔. A radial nonsimplectic subgroup is ↔, which can be doubled into a triangular prism domain as ↔, there are 11 forms,4 unique to this family, generated by ring permutations of the Coxeter group, with ↔. There are 9 forms, generated by ring permutations of the Coxeter group, there are 5 forms,1 unique, generated by ring permutations of the Coxeter group. Repeat constructions are related as, ↔, ↔, and ↔, there are 9 forms, generated by ring permutations of the Coxeter group. There are 11 forms,4 unique, generated by ring permutations of the Coxeter group,7 are half symmetry forms of, ↔.
There are 11 forms,4 unique, generated by ring permutations of the Coxeter group,7 are half symmetry forms of, ↔. There are 11 forms,4 unique, generated by ring permutations of the Coxeter group,7 are half symmetry forms of, ↔. There are 11 forms,4 unique, generated by ring permutations of the Coxeter group,7 are half symmetry forms of, ↔. There are 8 forms,1 unique, generated by ring permutations of the Coxeter group, two are duplicated as ↔, two as ↔, and three as ↔
In geometry, a uniform 4-polytope is a 4-polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons. 47 non-prismatic convex uniform 4-polytopes, one set of convex prismatic forms. There are a number of non-convex star forms. Regular star 4-polytopes 1852, Ludwig Schläfli found 4 of the 10 regular star 4-polytopes, discounting 6 with cells or vertex figures and this construction enumerated 45 semiregular 4-polytopes. 1912, E. L. Elte independently expanded on Gossets list with the publication The Semiregular Polytopes of the Hyperspaces, polytopes with one or two types of semiregular facets, Convex uniform polytopes,1940, The search was expanded systematically by H. S. M. Coxeter in his publication Regular and Semi-Regular Polytopes,1966 Norman Johnson completes his Ph. D. dissertation The Theory of Uniform Polytopes and Honeycombs under advisor Coxeter, completes the basic theory of uniform polytopes for dimensions 4 and higher. 1986 Coxeter published a paper Regular and Semi-Regular Polytopes II which included analysis of the unique snub 24-cell structure, 1998-2000, The 4-polytopes were systematically named by Norman Johnson, and given by George Olshevskys online indexed enumeration.
Johnson named the 4-polytopes as polychora, like polyhedra for 3-polytopes, from the Greek roots poly,2004, A proof that the Conway-Guy set is complete was published by Marco Möller in his dissertation, Vierdimensionale Archimedische Polytope. Möller reproduced Johnsons naming system in his listing,2008, The Symmetries of Things was published by John H. He used his own ijk-ambo naming scheme for the indexed ring permutations beyond truncation and bitruncation, nonregular uniform star 4-polytopes, 2000-2005, In a collaborative search, up to 2005 a total of 1845 uniform 4-polytopes had been identified by Jonathan Bowers and George Olshevsky. Regular 4-polytopes are a subset of the uniform 4-polytopes, which satisfy additional requirements, Regular 4-polytopes can be expressed with Schläfli symbol have cells of type, faces of type, edge figures, and vertex figures. The existence of a regular 4-polytope is constrained by the existence of the regular polyhedra which becomes cells, there are 64 convex uniform 4-polytopes, including the 6 regular convex 4-polytopes, and excluding the infinite sets of the duoprisms and the antiprismatic hyperprisms.
5 are polyhedral prisms based on the Platonic solids 13 are polyhedral prisms based on the Archimedean solids 9 are in the self-dual regular A4 group family,9 are in the self-dual regular F4 group family. 15 are in the regular B4 group family 15 are in the regular H4 group family,1 special snub form in the group family. 1 special non-Wythoffian 4-polytopes, the grand antiprism, TOTAL,68 −4 =64 These 64 uniform 4-polytopes are indexed below by George Olshevsky. Repeated symmetry forms are indexed in brackets, in addition to the 64 above, there are 2 infinite prismatic sets that generate all of the remaining convex forms, Set of uniform antiprismatic prisms - sr× - Polyhedral prisms of two antiprisms. Set of uniform duoprisms - × - A product of two polygons, the 5-cell has diploid pentachoric symmetry, of order 120, isomorphic to the permutations of five elements, because all pairs of vertices are related in the same way. Facets are given, grouped in their Coxeter diagram locations by removing specified nodes, there is one small index subgroup +, order 60, or its doubling +, order 120, defining a omnisnub 5-cell which is listed for completeness, but is not uniform
Convex uniform honeycomb
In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells. They can be considered the three-dimensional analogue to the uniform tilings of the plane, the Voronoi diagram of any lattice forms a convex uniform honeycomb in which the cells are zonohedra. 1905, Alfredo Andreini enumerated 25 of these tessellations,1991, Norman Johnsons manuscript Uniform Polytopes identified the complete list of 28. 1994, Branko Grünbaum, in his paper Uniform tilings of 3-space, independently enumerated all 28 and he found the 1905 paper, which listed 25, had 1 wrong, and 4 being missing. Grünbaum states in this paper that Norman Johnson deserves priority for achieving the same enumeration in 1991, alexeyev of Russia had contacted him regarding a putative enumeration of these forms, but that Grünbaum was unable to verify this at the time. Only 14 of the uniform polyhedra appear in these patterns. This set can be called the regular and semiregular honeycombs and it has been called the Archimedean honeycombs by analogy with the convex uniform polyhedra, commonly called Archimedean solids.
Recently Conway has suggested naming the set as the Architectonic tessellations, the individual honeycombs are listed with names given to them by Norman Johnson. For cross-referencing, they are given with list indices from Andreini, Johnson, and Grünbaum. Coxeter uses δ4 for a honeycomb, hδ4 for an alternated cubic honeycomb, qδ4 for a quarter cubic honeycomb. The fundamental infinite Coxeter groups for 3-space are, The C ~3, The B ~3, alternated cubic, The A ~3 cyclic group, or, There is a correspondence between all three families. Removing one mirror from C ~3 produces B ~3 and this allows multiple constructions of the same honeycombs. If cells are colored based on positions within each Wythoff construction. In addition there are 5 special honeycombs which dont have pure reflectional symmetry and are constructed from reflectional forms with elongation and gyration operations, the total unique honeycombs above are 18. The total unique honeycombs above are 10. Combining these counts,18 and 10 gives us the total 28 uniform honeycombs, the regular cubic honeycomb, represented by Schläfli symbol, offers seven unique derived uniform honeycombs via truncation operations.
The reflectional symmetry is the affine Coxeter group, There are four index 2 subgroups that generate alternations, and +, with the first two generated repeated forms, and the last two are nonuniform. The B ~4, group offers 11 derived forms via truncation operations, There are 3 index 2 subgroups that generate alternations, and +