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or higher, or a polytope of dimension 3 (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all...

ratio) as its faces. The great rhombic triacontahedron is a nonconvex isohedral, isotoxal polyhedron with 30 intersecting rhombic faces. The rhombic hexecontahedron...

(edge-transitive) but not isohedral (face-transitive). Semi-regular if every face is a regular polygon but it is not isohedral (face-transitive) or isotoxal...

In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other...

edge-to-edge. There are also 2-isohedral tilings by special cases of type 1, type 2, and type 4 tiles, and 3-isohedral tilings, all edge-to-edge, by special...

and isotoxal, but not isohedral. Their duals, including the rhombic dodecahedron and the rhombic triacontahedron, are isohedral and isotoxal, but not...

ISBN 0-7167-1193-1. (Page 476, Tilings by polygons, #19 of 56 polygonal isohedral types by quadrangles) Media related to Herringbone patterns at Wikimedia...

They are all isohedral. Each has parametric variations within a fixed symmetry. Type 2 contains glide reflections, and is 2-isohedral keeping chiral...

is said to be anisohedral if it admits a tiling, but no such tiling is isohedral (tile-transitive); that is, in any tiling by that shape there are two...

PhD) Known for Einstein problem Scientific career Fields Mathematics, Computer Science Institutions University of Waterloo Website https://isohedral.ca/...

Skilling's figure. Dual polyhedra to uniform polyhedra are face-transitive (isohedral) and have regular vertex figures, and are generally classified in parallel...

Euclidean 3-space, but such that no tessellation by this polyhedron is isohedral. Such anisohedral tiles were found by Karl Reinhardt in 1928, but these...

toplogical disk, §1.2 prototile, regular tiling, monohedral tiling, k-isohedral tiling (face-transitive), §1.3 symmetry, isometry, rotation, translation...

Because the disdyakis triacontahedron has the highest sphericity of any isohedral figure, it has been studied for its potential use in constructing discrete...

eleven deltahedra were later added by Olshevsky, Other subclasses are the isohedral deltahedron that was later discovered by both McNeill and Shephard (2000)...

another topological equivalence of a rhombic dodecahedron form. It is isohedral with tetrahedral symmetry order 24, distorting rhombic faces into kites...

Quasiregular polyhedra are isogonal and isotoxal, but not isohedral; their duals are isohedral and isotoxal, but not isogonal. The dual of an isotoxal polyhedron...

congruent kites. It can be described by the Conway notation dA6. It is an isohedral (face-transitive) figure, meaning that all its faces are the same. More...

internal angle of a regular pentagon, ⁠3π/5⁠, is not a divisor of 2π. An isohedral tiling is a special variation of a monohedral tiling in which all tiles...

faces of the three octahedra; it is another of the known isohedral deltahedra. A third isohedral deltahedron sharing the same face planes, the compound...

their faces are transitive under these symmetry transformations, they are isohedral. They are the dual polyhedra of prisms and the prisms are the dual of...

strongly than having all faces congruent, the trigonal trapezohedra are isohedral figures, meaning that they have symmetries that take any face to any other...

in diamond are this shape. The trapezo-rhombic dodecahedral honeycomb Isohedral tilings Sometimes, two or more different polyhedra, not necessarily having...

A noble polyhedron is one which is isohedral (all faces the same) and isogonal (all vertices the same). They were first studied in any depth by Edmund...

Tetrahedron Petrie polygon octagon Coxeter group B4, [3,3,4] Dual 16-cell Properties convex, isogonal, isotoxal, isohedral, Hanner polytope Uniform index 10...

trapezohedron, which has six congruent rhombic faces (also called an isohedral rhombohedron). For parallelepipeds with D2h symmetry, there are two cases:...

of odd parts is a part 1803 = number of decahexes that tile the plane isohedrally but not by translation or by 180-degree rotation (Conway criterion) 1804...

faces in common). The great deltoidal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the nonconvex great rhombicosidodecahedron...

sphere with 100 facets flattened into it. Note that this design is not isohedral; it does not function as a uniform fair die as some results are more likely...

sequence. There are 1987 polyiamonds with 12 cells that tile the plane isohedrally. There are 1987 ways to place four non-attacking chess kings on a 5 ×...

known as k-uniform or k-isogonal; if there are t orbits of tiles, as t-isohedral; if there are e orbits of edges, as e-isotoxal. k-uniform tilings with...

In geometry, the medial disdyakis triacontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform truncated dodecadodecahedron. It...

to the general quintic equation. The regular icosahedron is isogonal, isohedral, and isotoxal: any two vertices, two faces, and two edges of a regular...

Archimedean solids, and vice versa. The Catalan solids are face-transitive or isohedral, meaning that their faces are symmetric to one another, but they are not...

of pentagon found by Karl Reinhardt in 1918 that can tile the plane isohedrally (all tiles symmetric to each other). One of these two families consists...

packing is π 4 ≈ 0.785 {\textstyle {\frac {\pi }{4}}\approx 0.785} . Isohedral tilings have identical faces (face-transitivity) and vertex-transitivity...

two vertices are equivalent under symmetries of the polyhedron) is an isohedral polyhedron (one in which any two faces are equivalent [...]), and vice...

orbit, the figure may be transitive on the orbit. Individually, they are isohedral (or face-transitive, meaning the symmetry transformations involve the...

[4,4+], (4*2) pgg, [4+,4+], (22×) Rotation symmetry p4, [4,4]+, (442) p2, [4+,4+]+, (2222) Properties 4-isohedral, 5-isotoxal 3-isohedral, 6-isotoxal...

two regular n-gonal rings. It has 2n congruent kite faces, and it is isohedral. n = 2. A degenerate trapezohedron: a geometric figure with 6 vertices...

of a non-convex deltahedron, one of the few known deltahedra that are isohedral (meaning that all faces are symmetric to each other). In another of the...

crystallography, a stereohedron is a convex polyhedron that fills space isohedrally, meaning that the symmetries of the tiling take any copy of the stereohedron...

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

Vertex configuration 3.4.6.4 and 4.6.12 Symmetry p6m, [6,3], (*632) Rotation symmetry p6, [6,3]+, (632) Properties 2-uniform, 4-isohedral, 4-isotoxal...

In geometry, the small hexagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform small snub icosicosidodecahedron....

having all first coronas the same implies that the tiling is periodic and isohedral, meaning that all tiles are symmetric to each other. The binary tilings...

across a horizontal plane. A triangular bipyramid is face-transitive (or isohedral). If the tetrahedra are regular, all edges of a triangular bipyramid are...

In geometry, the medial pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the snub dodecadodecahedron. It has 60 intersecting...

In geometry, the great dirhombicosidodecacron is a nonconvex isohedral polyhedron. It is the dual of the great dirhombicosidodecahedron. In Magnus Wenninger's...

symmetry orbit, such that the appearance remains unchanged; these are isohedral, isogonal, and isotoxal respectively. Hence, it is considered a regular...

polyhedron. Every dihedral angle in a polyhedron that is isotoxal and/or isohedral has the same value. This includes the 5 Platonic solids, the 13 Catalan...

hexecontahedron (or midly petaloid ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform inverted snub dodecadodecahedron...

stellapentakis dodecahedron (or great astropentakis dodecahedron) is a nonconvex isohedral polyhedron. It is the dual of the truncated great icosahedron. It has...

In geometry, the small stellapentakis dodecahedron is a nonconvex isohedral polyhedron. It is the dual of the truncated great dodecahedron. It has 60...

Critchlow, p.74-75, pattern 1 Tilings and Patterns, from list of 107 isohedral tilings, p.473-481 Coxeter, Regular Complex Polytopes, pp. 111-112, p...

University. Grünbaum and Shephard, section 9.4 Keating, K.; Vince, A. (1999). "Isohedral Polyomino Tiling of the Plane". Discrete & Computational Geometry. 21...

icosahedron, the complete stellation is a noble polyhedron, because it is both isohedral (face-transitive) and isogonal (vertex-transitive). Coxeter et al. (1999)...

great rhombihexacron (or great dipteral disdodecahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform great rhombihexahedron (U21)...

This shape is called a plesiohedron. The tiling generated in this way is isohedral, meaning that it not only has a single prototile ("monohedral") but also...

and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. list of 107 isohedral tilings, p. 473-481 Fedorov's Five Parallelohedra The surprising 3D parallelo-hexagon...

Petrie polygon pentagon Coxeter group A4, [3,3,3] Dual Self-dual Properties convex, isogonal, isotoxal, isohedral, projectively unique Uniform index 1...

polyhedron which is also noble, being both isogonal (equal vertices) and isohedral (equal faces). Crown polyhedra are self-intersecting and topologically...

{\displaystyle \mathrm {I} _{\mathrm {h} }} Dihedral angle (degrees) 144° Dual polyhedron Icosidodecahedron Properties convex, isohedral, isotoxal, zonohedron Net...

icosidodecahedron is the great rhombic triacontahedron; it is nonconvex, isohedral and isotoxal. It has 30 intersecting rhombic faces. It can also be called...

monohedral tilings it is denoted [3.6.3.6]. It is also one of 56 possible isohedral tilings by quadrilaterals, and one of only eight tilings of the plane...

the family of Platonic solids, is a regular polyhedron. It is isogonal, isohedral, and isotoxal: any two vertices, two faces, and two edges of a regular...

Baronetage of the United Kingdom Noble polyhedron, a polyhedron which is isohedral and isogonal North of Boston Library Exchange, or NOBLE, a cooperative...

In geometry, the order-4 apeirogonal tiling is a regular tiling of the hyperbolic plane. It covers the hyperbolic plane, which is a non-Euclidean surface...

dodecadodecahedron The medial pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the snub dodecadodecahedron. It has 60 intersecting...

hexagonal hexecontahedron (or midly dentoid ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform snub icosidodecadodecahedron...

[3,4,3], order 1152 B4, [4,3,3], order 384 D4, [31,1,1], order 192 Dual Self-dual Properties convex, isogonal, isotoxal, isohedral Uniform index 22...

Circle Limit III is a woodcut made in 1959 by Dutch artist M. C. Escher, in which "strings of fish shoot up like rockets from infinitely far away" and...

called the great disnub dirhombidodecacron. It is a nonconvex infinite isohedral polyhedron. Like the visually identical great dirhombicosidodecacron in...

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

tetradyakis hexahedron (or great disdyakis dodecahedron) is a nonconvex isohedral polyhedron. It has 48 intersecting scalene triangle faces, 72 edges, and...

with oblique form. The right pentagonal bipyramid is face-transitive or isohedral, meaning any mapping of two adjacent faces preserves its symmetrical appearance...

is a dual uniform tiling in the Euclidean plane. It is one of 15 known isohedral pentagon tilings. It can be seen as a stretched hexagonal tiling with...

In geometry, the order-7 heptagrammic tiling is a tiling of the hyperbolic plane by overlapping heptagrams. This tiling is a regular star-tiling, and has...

three symmetry positions, two centered on vertices, and one on midedge. Isohedral variations with the same chiral octahedral symmetry can be constructed...

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

{\displaystyle n<10,000} . In 1946, Alaoglu and John H. Giese constructed uniform, isohedral polyhedra that were topologically equivalent to a torus. Axiom of Choice...

(having the hexagonal faces in common). The rhombicosacron is a nonconvex isohedral polyhedron. It is the dual of the uniform rhombicosahedron, U56. It has...

other words, the second problem asked which tiles, but does not admit an isohedral (tile-transitive) tiling. Such tiles are not equivalent under any symmetry...

rhombidodecacron (or small dipteral ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the small rhombidodecahedron. It is visually...

its Conway polyhedron notation is dA12. Dodecagonal trapezohedra are isohedral figures. There are two examples of Johnson solids which are icositetrahedra...

The only small index subgroup is the alternation, [7,3]+, (732). Three isohedral (regular or quasiregular) tilings can be constructed from this tiling...

In geometry, the order-4 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,4}. It can also be called a pentapentagonal...

there are k orbits of planigons, a tiling is known as k-dual-uniform or k-isohedral; if there are t orbits of dual vertices, as t-isogonal; if there are e...

In geometry, the great triakis icosahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform great stellated truncated dodecahedron....

dodecacronic hexecontahedron (or great lanceal trisicosahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform great ditrigonal dodecicosidodecahedron...

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

known as k-uniform or k-isogonal; if there are t orbits of tiles, as t-isohedral; if there are e orbits of edges, as e-isotoxal. k-uniform tilings with...

dodecahedron, yet results in the same pentagonal hexecontahedron convex hull. Isohedral variations can be constructed with pentagonal faces with 3 edge lengths...

rhombic triacontahedron (or midly rhombic triacontahedron) is a nonconvex isohedral polyhedron. It is a stellation of the rhombic triacontahedron, and can...

stellapentakis dodecahedron (or small astropentakis dodecahedron) is a nonconvex isohedral polyhedron. It is the dual of the truncated great dodecahedron. It has...

In geometry, the infinite-order square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,∞}. All vertices are ideal, located...

In geometry, the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of {3,7}. The symmetry group of the tiling...

Providence, RI, 1998, pp. 207–246. Accompanying web site: Catalog of Isohedral Tilings by Symmetric Polygonal Tiles 2005 Parson Lecturer - Dr. Doris...

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...

pentagrammic hexecontahedron (or great dentoid ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the great retrosnub icosidodecahedron. Its...

shown that this is not the case. Disphenoid tetrahedral honeycomb - an isohedral tessellation of 3-space by irregular tetrahedra Packing problem The triakis...

In geometry, the small rhombidodecacron is a nonconvex isohedral polyhedron. It is the dual of the small rhombidodecahedron. It is visually identical...

hexagonal hexecontahedron (or great astroid ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform great snub dodecicosidodecahedron...

disdyakis triacontahedron (or trisdyakis icosahedron) is a nonconvex isohedral polyhedron. It is the dual of the great truncated icosidodecahedron. Its...

contrast, Tilings and patterns uses square brackets around the symbol for isohedral tilings. This notation represents a sequential count of the number of...

In geometry, the great pentakis dodecahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform small stellated truncated dodecahedron...

0099004435452335} The medial hexagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform snub icosidodecadodecahedron...

hexecontahedron (or great petaloid ditriacontahedron) is a nonconvex isohedral polyhedron and dual to the uniform great snub icosidodecahedron. It has...

semiregular tiling of the Euclidean plane. It is one of the 15 known isohedral pentagon tilings. Its six pentagonal tiles radiate out from a central...

Vertex configuration 3.4.3.12 and 3.12.12 Symmetry p4m, [4,4], (*442) Rotation symmetry p4, [4,4]+, (442) Properties 2-uniform, 3-isohedral, 3-isotoxal...

icosahedron Petrie polygon 30-gon Coxeter group H4, [3,3,5], order 14400 Dual 120-cell Properties convex, isogonal, isotoxal, isohedral Uniform index 35...

symmetries of the graph, these heptadecahedra are plesiohedra and form an isohedral tessellation of three-dimensional space. Other convex polyhedra with 17...

with 17 faces each. They are plesiohedra, polyhedra that tile space isohedrally. Experimenting with the structures formed by these polyhedra led physicist...

pentagonal hexecontahedron (or petaloidal trisicosahedron) is a nonconvex isohedral polyhedron. It is composed of 60 concave pentagonal faces, 150 edges and...

In 2-dimensional hyperbolic geometry, the infinite-order hexagonal tiling is a regular tiling. It has Schläfli symbol of {6,∞}. All vertices are ideal...

group B4, [3,3,4], order 384 D4, order 192 Dual Tesseract Properties convex, isogonal, isotoxal, isohedral, regular, Hanner polytope Uniform index 12...

Elements F = 20, E = 60 V = 32 (χ = -8) F = 20, E = 60 V = 24 (χ = -16) Isohedral faces Duals Great ditrigonal icosidodecahedron Ditrigonal dodecadodecahedron...

the icosahedron. The great stellapentakis dodecahedron is a nonconvex isohedral polyhedron. It is the dual of the truncated great icosahedron. It has...

the rhombicosacron (or midly dipteral ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform rhombicosahedron, U56. It has...

In geometry, the heptagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {7,3}, having three regular heptagons...

In geometry, the heptagrammic-order heptagonal tiling is a regular star-tiling of the hyperbolic plane. It has Schläfli symbol of {7,7/2}. The vertex figure...

In geometry, the order-6 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,6}. This tiling represents a hyperbolic...

faces do not all lie within the same symmetry orbit and they are thus not isohedral. This is a consequence of the pseudo-uniform polyhedra having the same...

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

deltoidal hexecontahedron (or midly lanceal ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the rhombidodecadodecahedron. It has 60...

In geometry, an order-2 apeirogonal tiling, apeirogonal dihedron, or infinite dihedron is a tessellation (gap-free filling with repeated shapes) of the...

In geometry, the order-4 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,4}. This tiling represents a hyperbolic...

In geometry, the tetrakis square tiling is a tiling of the Euclidean plane. It is a square tiling with each square divided into four isosceles right triangles...

In geometry, the great rhombic triacontahedron is a nonconvex isohedral, isotoxal polyhedron. It is the dual of the great icosidodecahedron (U54). Like...

rhombidodecacron (or Great dipteral ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the great rhombidodecahedron. It is visually...

In geometry, the order-4 heptagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {7,4}. This tiling represents a hyperbolic...

hexecontahedron (or great sagittal ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the nonconvex great rhombicosidodecahedron...

In geometry, the infinite-order triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of {3,∞}. All vertices are ideal,...

In geometry, the medial deltoidal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the rhombidodecadodecahedron. Its 60 intersecting...

In geometry, the octagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {8,3}, having three regular octagons...

In geometry, the order-3 apeirogonal tiling is a regular tiling of the hyperbolic plane. It is represented by the Schläfli symbol {∞,3}, having three regular...

In geometry, the order-5 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,5}. This tiling is topologically related...

In geometry, an apeirogonal hosohedron or infinite hosohedron is a tiling of the plane consisting of two vertices at infinity. It may be considered an...

hexecontahedron (or midly sagittal ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform icosidodecadodecahedron. Its...

single polyhedron tiling Euclidean 3-space, such that no tiling by it is isohedral (an anisohedral tile). The problem as stated was solved by Karl Reinhardt...

last four elements. The medial disdyakis triacontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform truncated dodecadodecahedron...

Vertex figure tetrahedron Petrie polygon 30-gon Coxeter group H4, [3,3,5] Dual 600-cell Properties convex, isogonal, isotoxal, isohedral Uniform index 32...

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 order-5 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,5}. The dual to this tiling represents...

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

Isohedral variations Symmetry p6m, [6,3], (*632) p31m, [6,3+], (3*3) Form Faces Kite Half regular hexagon Quadrilaterals...

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...

In geometry, the order-6 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,6}. The dual to this tiling represents...

or of three octagonal prisms. The great rhombihexacron is a nonconvex isohedral polyhedron. It is the dual of the uniform great rhombihexahedron (U21)...

In geometry, the order-4 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,4}. Its checkerboard coloring can be...

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...

In geometry, the order-5 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,5}, constructed from five pentagons...

In geometry, the truncated order-7 heptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{7,7}, constructed from...

In 2-dimensional hyperbolic geometry, the infinite-order pentagonal tiling is a regular tiling. It has Schläfli symbol of {5,∞}. All vertices are ideal...