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Notation for tesselations
In geometry, the Wythoff symbol is a notation representing a Wythoff construction of a uniform polyhedron or plane tiling within a Schwarz triangle. It
Wythoff_symbol
In geometry, method for constructing a uniform polyhedron or plane tiling
A vertex is placed at the point A. This produces a polyhedron with Wythoff symbol a|b c, where a equals π divided by the angle of the triangle at A, and
Wythoff_construction
grouped by the Wythoff symbol. All the faces are identical, each edge is identical and each vertex is identical. They all have a Wythoff symbol of the form
List of uniform polyhedra by Wythoff symbol
List_of_uniform_polyhedra_by_Wythoff_symbol
There are different notations for expressing these uniform solutions, Wythoff symbol, Coxeter diagram, and Coxeter's t-notation. Simple tiles are generated
Lists of uniform tilings on the sphere, plane, and hyperbolic plane
Lists_of_uniform_tilings_on_the_sphere,_plane,_and_hyperbolic_plane
complete. All reflectional forms can be made by Wythoff constructions, represented by Wythoff symbols, or Coxeter-Dynkin diagrams, each operating upon
List of Euclidean uniform tilings
List_of_Euclidean_uniform_tilings
hexagon passing through the origin The Wythoff symbol relates the polyhedron to spherical triangles. Wythoff symbols are written p|q r, p q|r, p q r| where
List of uniform polyhedra by vertex figure
List_of_uniform_polyhedra_by_vertex_figure
Dutch mathematician
geometry, Wythoff is known for the Wythoff construction of uniform tilings and uniform polyhedra and for the Wythoff symbol used as a notation for these geometric
Willem_Abraham_Wythoff
Vertex-transitive tiling of the plane by regular polygons
domain mirrors meeting at 90 degrees generate no new mirrors). The Wythoff symbol takes the three integers and separates them by a vertical bar (|). If
Uniform_tiling
Uniform star polyhedron whose faces pass through its center
figure. The nine forms, listed with their Wythoff symbols and vertex configurations are: Note that Wythoff's kaleidoscopic construction generates the nonorientable
Hemipolyhedron
Polyhedron with 24 faces
spherical icosahedron used in construction of geodesic domes. It has four Wythoff constructions between four Schwarz triangle families: 2 | 5 5/2, 2 | 5
Dodecadodecahedron
Polyhedron with two kinds of faces
triangular faces. Coxeter defines a quasiregular polyhedron as one having a Wythoff symbol in the form p | q r, and it is regular if q=2 or q=r. The Coxeter-Dynkin
Quasiregular_polyhedron
Isogonal polyhedron with regular faces
only its Wythoff symbol. The 57 nonprismatic nonconvex forms, with exception of the great dirhombicosidodecahedron, are compiled by Wythoff constructions
Uniform_polyhedron
Uniform star polyhedron with 124 faces
polyhedron that cannot be made by the Wythoff construction from a spherical triangle. It has a special Wythoff symbol | 3⁄2 5⁄3 3 5⁄2, relating it to a spherical
Great dirhombicosidodecahedron
Great_dirhombicosidodecahedron
Any of the five regular polyhedra
directly from their symmetry groups. They are listed for reference Wythoff's symbol for each of the Platonic solids. The tetrahedron, cube, and octahedron
Platonic_solid
Polyhedron with 7 faces
only non-prismatic uniform polyhedron with an odd number of faces. Its Wythoff symbol is 3/2 3 | 2, but that represents a double covering of the tetrahemihexahedron
Tetrahemihexahedron
Archimedean solid with 62 faces
notation eD or aaD Schläfli symbols rr{5,3} or r { 5 3 } {\displaystyle r{\begin{Bmatrix}5\\3\end{Bmatrix}}} t0,2{5,3} Wythoff symbol 3 5 | 2 Coxeter diagram
Rhombicosidodecahedron
of uniform polyhedra by vertex figure List of uniform polyhedra by Wythoff symbol List of uniform polyhedra by Schwarz triangle Coxeter, Harold Scott
List_of_uniform_polyhedra
Tiling of a plane by regular hexagons and equilateral triangles
by the wallpaper group p6mm, (*632), and the tiling can be derived as a Wythoff construction within the reflectional fundamental domains of this group
Trihexagonal_tiling
abstract vertex figure is the same. This tiling may be denoted by the Wythoff symbol 3 4 | 4, and is depicted on the right. Alternatively and more subtly
Small_cubicuboctahedron
Polyhedron with 10 faces
vertices. Its vertex figure is a crossed quadrilateral. It is given Wythoff symbol 4⁄3 4 | 3, although that is a double-covering of this figure. A nonconvex
Cubohemioctahedron
All omnitruncated polyhedra are considered as zonohedra. They have Wythoff symbol p q r | and vertex figures as 2p.2q.2r. More generally, an omnitruncated
Omnitruncated_polyhedron
Polyhedron resulting from the snub operation
are all point groups. For example, the snub cube: Snub polyhedra have Wythoff symbol | p q r and by extension, vertex configuration 3.p.3.q.3.r. Retrosnub
Snub_polyhedron
spherical triangle groups them by the Wythoff symbol. The vertex figure can be discovered by considering the Wythoff symbol: p|q r - 2p edges, alternating q-gons
List of uniform polyhedra by spherical triangle
List_of_uniform_polyhedra_by_spherical_triangle
Regular tiling of the plane
at a point occupy a full 360 degrees. The triangular tiling has Schläfli symbol of {3,6}. English mathematician John Conway called it a deltille, named
Triangular_tiling
Symmetric subdivision in hyperbolic geometry
group. Each symmetry family contains 7 uniform tilings, defined by a Wythoff symbol or Coxeter-Dynkin diagram, 7 representing combinations of 3 active mirrors
Uniform tilings in hyperbolic plane
Uniform_tilings_in_hyperbolic_plane
Polyhedron with 32 faces
Schläfli symbol a{5,3}, as an altered dodecahedron, and Coxeter diagram or . It is constructed from Schwarz triangle (3 3 5⁄2) with Wythoff symbol 3 | 5⁄2
Small ditrigonal icosidodecahedron
Small_ditrigonal_icosidodecahedron
Semiregular tiling of the Euclidean plane
triangle, two squares, and one hexagon on each vertex. It has Schläfli symbol of rr{3,6}. John Conway calls it a rhombihexadeltille. It can be considered
Rhombitrihexagonal_tiling
Prism with a 7-sided base
Heptagons 7 squares Edges 21 Vertices 14 Vertex configuration 7.4.4 Wythoff symbol 2 7 | 2 Coxeter diagram Symmetry group D7h, [7,2], (*722), order 28
Heptagonal_prism
Geometric tiling
an order-3 heptagonal tiling and an order-7 triangular tiling. From a Wythoff construction there are eight hyperbolic uniform tilings that can be based
Rhombitriheptagonal_tiling
Pictorial representation of symmetry
subsymmetries, each with a uniquely marked up Coxeter–Dynkin diagram. The Wythoff symbol represents a special case of the Coxeter diagram for rank 3 graphs,
Coxeter–Dynkin_diagram
Tiling of the hyperbolic plane
related as a part of sequence of regular polyhedra with Schläfli symbol {n,3}. From a Wythoff construction there are eight hyperbolic uniform tilings that
Heptagonal_tiling
Polyhedron with 24 faces
Schläfli symbol b{5,5⁄2}, as a blended great dodecahedron, and Coxeter diagram . It has 4 Schwarz triangle equivalent constructions, for example Wythoff symbol
Ditrigonal_dodecadodecahedron
Isogonal polytope with uniform facets
The Wythoffian uniform polyhedra and tilings can be defined by their Wythoff symbol, which specifies the fundamental region of the object. An extension
Uniform_polytope
Type of polyhedron
all with icosahedral symmetry. The three uniform star polyhedron with Wythoff symbol of the form 3 | p q or 3/2 | p q are ditrigonal, at least if p and
Ditrigonal_polyhedron
Polyhedron with 42 faces
extra double-covered triangles) (with extra double-covered pentagons) Wythoff symbol 2 5 (3/2 5/2) | Symmetry group Ih, [5,3], *532 Index references U39
Small_rhombidodecahedron
Polyhedron with 32 faces
vertices. Its vertex figure is a crossed quadrilateral. It has a composite Wythoff symbol, 3 5⁄3 (3⁄2 5⁄2) |, requiring two different Schwarz triangles to generate
Great_dodecicosahedron
Semiregular tiling of the plane
There are three triangles and two squares on each vertex. Its Schläfli symbol is s{4,4}. Conway calls it a snub quadrille, constructed by a snub operation
Snub_square_tiling
Degenerate uniform star polyhedron
(30x2) V = 12 (χ = -16) Faces by sides 20{3}+12{5/2} Coxeter diagram Wythoff symbol 5 | 3 5/3 Symmetry group Ih, [5,3], *532 Index references U-, C-, W-
Great complex icosidodecahedron
Great_complex_icosidodecahedron
Subdivision of the plane into polygons that are all regular
with rectangles Uniform tilings in hyperbolic plane Wallpaper group Wythoff symbol Cundy, H.M.; Rollett, A.P. (1981). Mathematical Models;. Stradbroke
Euclidean tilings by convex regular polygons
Euclidean_tilings_by_convex_regular_polygons
Kepler–Poinsot polyhedron with 20 faces
four Kepler–Poinsot polyhedra (nonconvex regular polyhedra), with Schläfli symbol {3,5⁄2} and Coxeter-Dynkin diagram of . It is composed of 20 intersecting
Great_icosahedron
Semiregular tiling of the plane
It is also the only convex uniform tiling that can not be created as a Wythoff construction. It can be constructed as alternate layers of apeirogonal
Elongated_triangular_tiling
Variously-defined concept in geometry
use the term semiregular polyhedra to classify uniform polyhedra with Wythoff symbol of the form p q | r, a definition encompassing only six of the Archimedean
Semiregular_polyhedron
Antiprism with 6-sided caps
= 24 V = 12 (χ = 2) Faces by sides 12{3}+2{6} Schläfli symbol s{2,12} sr{2,6} Wythoff symbol | 2 2 6 Coxeter diagram Symmetry group D6d, [2+,12], (2*6)
Hexagonal_antiprism
Geometric polyhedral group
order 10 20 × rotoreflection by 60°, order 6 15 × reflection, order 2 Wythoff symbol List of spherical symmetry groups Coxeter, H. S. M. Regular Polytopes
Polyhedral_group
Semiregular tiling of a plane
triangles at the original vertex locations. It is given an extended Schläfli symbol of t{6,3}. Conway calls it a truncated hextille, constructed as a truncation
Truncated_hexagonal_tiling
12-sided box
36, V = 24 (χ = 2) Faces by sides 12{4}+2{12} Schläfli symbol t{2,12} or {12}×{} Wythoff symbol 2 12 | 2 2 2 6 | Coxeter diagrams Symmetry D12h, [12,2]
Dodecagonal_prism
15 V = 10 (χ = 2) Faces by sides 5{4}+2{5/2} Schläfli symbol t{2,5/2} or {5/2}×{} Wythoff symbol 2 5/2 | 2 Coxeter diagram Symmetry D5h, [5,2], (*522)
Pentagrammic_prism
Index Name Picture Dual name Dual picture Wythoff symbol Vertex figure and Schläfli symbol Symmetry group U# K# V E F Faces by type 1 Tetrahedron Tetrahedron
List of Wenninger polyhedron models
List_of_Wenninger_polyhedron_models
20 V = 10 (χ = 2) Faces by sides 10{3}+2{5/2} Schläfli symbol s{2,10/3} sr{2,5/3} Wythoff symbol | 2 2 5/3 Coxeter diagram = Symmetry D5d, [5,2], (*522)
Pentagrammic crossed-antiprism
Pentagrammic_crossed-antiprism
Polyhedron with 22 faces
E = 60 V = 30 (χ = −8) Faces by sides 12{5/2}+10{6} Coxeter diagram Wythoff symbol 5/3 5/2 | 3 (double covering) Symmetry group Ih, [5,3], *532 Index references
Small_dodecahemicosahedron
Prism with an 8-sided base
= 24, V = 16 (χ = 2) Faces by sides 8{4}+2{8} Schläfli symbol t{2,8} or {8}×{} Wythoff symbol 2 8 | 2 2 2 4 | Coxeter diagrams Symmetry D8h, [8,2], (*822)
Octagonal_prism
Polyhedron with 54 faces
pentagrams), 120 edges and 60 vertices. It is given a Schläfli symbol t0,2{5⁄2,5}, and by the Wythoff construction this polyhedron can also be named a cantellated
Rhombidodecadodecahedron
Nonconvex uniform polyhedron with 20 faces
48 V = 24 (χ = −4) Faces by sides 8{3}+6{4}+6{8/3} Coxeter diagram Wythoff symbol 3 4 | 4/3 4 3/2 | 4 Symmetry group Oh, [4,3], *432 Index references
Great_cubicuboctahedron
(30x2) V = 12 (χ = −16) Faces by sides 20{3}+12{5} Coxeter diagram Wythoff symbol 5 | 3/2 5 Symmetry group Ih, [5,3], *532 Index references U-, C-, W-
Small complex icosidodecahedron
Small_complex_icosidodecahedron
Archimedean solid with 62 faces
notation bD or taD Schläfli symbols tr{5,3} or t { 5 3 } {\displaystyle t{\begin{Bmatrix}5\\3\end{Bmatrix}}} t0,1,2{5,3} Wythoff symbol 2 3 5 | Coxeter diagram
Truncated_icosidodecahedron
Uniform star polyhedron with 204 faces
by sides 120{3}+60{4}+24{5/2} Coxeter diagram {{{Skilling-Coxeter}}} Wythoff symbol | (3/2) 5/3 (3) 5/2 Symmetry group Ih, [5,3], *532 Index references
Great disnub dirhombidodecahedron
Great_disnub_dirhombidodecahedron
Semiregular tiling of the hyperbolic plane
tiling, order 3-7 kisrhombille, represent a fundamental domain of the Wythoff construction for the symmetry group [7,3]. This tiling can be considered
Truncated triheptagonal tiling
Truncated_triheptagonal_tiling
Semiregular tiling
tiling by regular convex polygons which contains an octagon. It has Schläfli symbol of t{4,4}. Conway calls it a truncated quadrille, constructed as a truncation
Truncated_square_tiling
Polyhedron with 22 faces
= −8) Faces by sides 12{5}+10{6} Coxeter diagram (double covering) Wythoff symbol 5/4 5 | 3 (double covering) Symmetry group Ih, [5,3], *532 Index references
Great_dodecahemicosahedron
Polyhedron with 22 faces
Decagrams 20 triangles Edges 40 Vertices 20 Vertex configuration 10⁄3.3.3.3 Wythoff symbol -2 2 10⁄3 Symmetry group D10d, [2+,20], (2*10), order 40 Dual polyhedron
Decagrammic_antiprism
many isotoxal polygonal tilings of the hyperbolic plane, including the Wythoff constructions from the regular hyperbolic tilings {p,q}, and non-right
List of isotoxal polyhedra and tilings
List_of_isotoxal_polyhedra_and_tilings
Kepler–Poinsot polyhedron
great stellated dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol {5⁄2, 3}. It is one of four nonconvex regular polyhedra. It is composed
Great_stellated_dodecahedron
Polyhedron with 44 faces
V = 60 (χ = −16) Faces by sides 20{3}+12{5}+12{10} Coxeter diagram Wythoff symbol 3/2 5 | 5 3 5/4 | 5 Symmetry group Ih, [5,3], *532 Index references
Small_dodecicosidodecahedron
Uniform star polyhedron with 26 faces
six decagonal faces passing through the model center. It is given a Wythoff symbol, 3⁄2 3 | 5, but that construction represents a double covering of this
Small_icosihemidodecahedron
Geometric figure
180 V = 60 (χ = −8) Faces by sides (40+60){3}+12{5/2} Coxeter diagram Wythoff symbol | 5/2 3 3 Symmetry group Ih, [5,3], *532 Index references U32, C41,
Small snub icosicosidodecahedron
Small_snub_icosicosidodecahedron
Semiregular tiling of the Euclidean plane
each vertex. It has Schläfli symbol sr{3,6}. The snub tetrahexagonal tiling is a related hyperbolic tiling with Schläfli symbol sr{4,6}. Conway calls it a
Snub_trihexagonal_tiling
Prism with an infinite-sided polygon base
Type Semiregular tiling Vertex configuration 4.4.∞ Schläfli symbol t{2,∞} Wythoff symbol 2 ∞ | 2 Coxeter diagram Symmetry [∞,2], (*∞22) Rotation symmetry
Apeirogonal_prism
Regular tiling of the hyperbolic plane
tilings with Schläfli symbol {n,3}. And also is topologically part of sequence of regular tilings with Schläfli symbol {8,n}. From a Wythoff construction there
Octagonal_tiling
Uniform star polyhedron with 18 faces
= −12) Faces by sides 12{5}+6{10} Coxeter diagram (double covering) Wythoff symbol 5/4 5 | 5 (double covering) Symmetry group Ih, [5,3], *532 Index references
Small_dodecahemidodecahedron
Archimedean solid with 26 faces
notation bC or taC Schläfli symbols tr{4,3} or t { 4 3 } {\displaystyle t{\begin{Bmatrix}4\\3\end{Bmatrix}}} t0,1,2{4,3} Wythoff symbol 2 3 4 | Coxeter diagram
Truncated_cuboctahedron
3d shape
E = 20 V = 10 (χ = 2) Faces by sides 10{3}+2{5/2} Schläfli symbol sr{2,5/2} Wythoff symbol | 2 2 5/2 Coxeter diagram Symmetry D5h, [5,2], (*552), order
Pentagrammic_antiprism
Polyhedron with 13 faces
11}; can be used in a Coxeter-Dynkin diagram to represent it; its Wythoff symbol is 2 11 | 2; in Conway polyhedron notation it can be represented by
Tridecahedron
Polyhedron with 2 faces
dihedron. A regular dihedron, with Schläfli symbol {n,2}, is made of two regular polygons, each with Schläfli symbol {n}. A spherical dihedron is made of two
Dihedron
Semiregular tiling of the hyperbolic plane
be seen in a sequence of quasiregular polyhedrons and tilings: From a Wythoff construction there are eight hyperbolic uniform tilings that can be based
Triheptagonal_tiling
Semi-regular arrangement of squares, decagons, and dodecagons
and white, and mirrors exist on the boundaries between colors. From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be
Truncated pentahexagonal tiling
Truncated_pentahexagonal_tiling
Polyhedron with 42 faces
extra double-covered triangles) (with extra double-covered pentagrams) Wythoff symbol 2 5/3 (3/2 5/4) | Symmetry group Ih, [5,3], *532 Index references U73
Great_rhombidodecahedron
Uniform tiling of the hyperbolic plane
octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbols of {(4,3,3)} or h{8,3}. Although a sequence of edges seem to represent
Alternated_octagonal_tiling
Polyhedron
V = 60 (χ = −8) Faces by sides 20{3}+12{5/2}+20{6} Coxeter diagram Wythoff symbol 5/2 3 | 3 Symmetry group Ih, [5,3], *532 Index references U31, C40,
Small_icosicosidodecahedron
Spherical polyhedron composed of lunes
Schläfli symbol {2,n}, with each spherical lune having internal angle 2π/nradians (360/n degrees). For a regular polyhedron whose Schläfli symbol is {m
Hosohedron
Polyhedron with 44 faces
V = 60 (χ = −16) Faces by sides 12{5}+12{5/2}+20{6} Coxeter diagram Wythoff symbol 5/3 5 | 3 5/2 5/4 | 3 Symmetry group Ih, [5,3], *532 Index references
Icosidodecadodecahedron
Polyhedron with 62 faces
is also called the quasirhombicosidodecahedron. It is given a Schläfli symbol rr{5⁄3,3}. Its vertex figure is a crossed quadrilateral. This model shares
Nonconvex great rhombicosidodecahedron
Nonconvex_great_rhombicosidodecahedron
Uniform star polyhedron with 12 faces
uniform polyhedron Faces 8 triangles 4 hexagons Edges 24 Vertices 12 Wythoff symbol 3 2 3 ∣ 3 {\textstyle {\frac {3}{2}}\,3\mid 3} Dual polyhedron octahemioctacron
Octahemioctahedron
Regular tiling of the hyperbolic plane
square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,5}. This tiling is topologically related as a part of sequence of
Order-5_square_tiling
apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,5}. The dual to this tiling represents the fundamental domains of
Order-5_apeirogonal_tiling
Polyhedron with 12 faces
Decagrams 10 squares Edges 30 Vertices 20 Vertex configuration 10/3.4.4 Wythoff symbol 2 10/3 | 2 Symmetry group D10h, [2,10],(*2.10.10), order 40 Dual polyhedron
Decagrammic_prism
There are many relationships among the uniform polyhedra. The Wythoff construction is able to construct almost all of the uniform polyhedra from the acute
List of uniform polyhedra by Schwarz triangle
List_of_uniform_polyhedra_by_Schwarz_triangle
Polyhedron with 32 faces
extra double-covered triangles) (with extra double-covered pentagrams) Wythoff symbol 3 5 (3/2 5/4) | Symmetry group Ih, [5,3], *532 Index references U50
Small_dodecicosahedron
Polyhedron with 32 faces
equivalent constructions, for example Wythoff symbol 3 | 3 5⁄4 gives Coxeter diagram = . It has extended Schläfli symbol a{5⁄2,3} or c{3,5⁄2}, as an altered
Great ditrigonal icosidodecahedron
Great_ditrigonal_icosidodecahedron
= 28 V = 14 (χ = 2) Faces by sides 14{3}+2{7} Schläfli symbol s{2,14} sr{2,7} Wythoff symbol | 2 2 7 Coxeter diagram Symmetry group D7d, [2+,14], (2*7)
Heptagonal_antiprism
Prism with a 5-sided base
can be seen as a truncated pentagonal hosohedron, represented by Schläfli symbol t{2,5}. Alternately it can be seen as the Cartesian product of a regular
Pentagonal_prism
Polyhedron with 18 faces
extra double-covered triangles) (with extra double-covered squares) Wythoff symbol 2 4/3 (3/2 4/2) | Symmetry group Oh, [4,3], *432 Index references U21
Great_rhombihexahedron
Semiregular tiling of the hyperbolic plane
octagon, and one dodecagon on each vertex. It has Schläfli symbol of tr{6,4}. From a Wythoff construction there are fourteen hyperbolic uniform tilings
Truncated tetrahexagonal tiling
Truncated_tetrahexagonal_tiling
Semiregular tiling of the hyperbolic plane
vertex configurations (n.6.6), and [n,3] Coxeter group symmetry. From a Wythoff construction there are eight hyperbolic uniform tilings that can be based
Truncated order-7 triangular tiling
Truncated_order-7_triangular_tiling
Polyhedron with 52 faces
120 V = 60 (χ = −8) Faces by sides 20{3}+12{5}+20{6} Coxeter diagram Wythoff symbol 3/2 5 | 3 3 5/4 | 3 Symmetry group Ih, [5,3], *532 Index references
Great_icosicosidodecahedron
Polyhedron with 10 faces
squares Edges 24 Vertices 16 Vertex configuration 8/3.4.4 Wythoff symbol 2 8/3 | 2 Schläfli symbol s{2,16/3} sr{2,8/3} Coxeter diagram Symmetry group D8h
Octagrammic_prism
Polyhedron with 18 faces
extra double-covered triangles) (with extra double-covered squares) Wythoff symbol 2 4 (3/2 4/2) | Symmetry group Oh, [4,3], *432 Index references U18
Small_rhombihexahedron
Polyhedron with 24 faces
icositrigonal pyramids have a regular icositrigon as a base, and its Schläfli symbol is {}∨{23}. The surface area S {\displaystyle S} and volume V {\displaystyle
Icositetrahedron
Regular tiling of the hyperbolic plane
fundamental square domains of this symmetry. This bicolored tiling has a wythoff construction t1{(4,4,3)}. A second 6-color symmetry can be constructed
Order-6_square_tiling
Antiprism with a five-sided base
= 20 V = 10 (χ = 2) Faces by sides 10{3}+2{5} Schläfli symbol s{2,10} sr{2,5} Wythoff symbol | 2 2 5 Coxeter diagram Symmetry group D5d, [2+,10], (2*5)
Pentagonal_antiprism
Polyhedron with 32 faces
pentagrams and 20 hexagons), 90 edges, and 60 vertices. It is given a Schläfli symbol t{3,5⁄2} or t0,1{3,5⁄2} as a truncated great icosahedron. Cartesian coordinates
Truncated_great_icosahedron
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