Isohedral figure

{{Short description|Generalisation of dice with identical faces}} {{redirect-distinguish|isohedron|icosahedron}} {{For|the related '''Isohedral numbers'''|Anisohedral tiling}} thumb|A set of isohedral dice

In geometry, a tessellation of dimension {{math|2}} (a plane tiling) or higher, or a polytope of dimension {{math|3}} (a polyhedron) or higher, is '''isohedral''' or '''face-transitive''' if all its faces are the same. More specifically, all faces must be not merely congruent but must be ''transitive'', i.e. must lie within the same ''symmetry orbit''. In other words, for any two faces {{mvar|A}} and {{mvar|B}}, there must be a symmetry of the ''entire'' figure by translations, rotations, and/or reflections that maps {{mvar|A}} onto {{mvar|B}}. For this reason, convex isohedral polyhedra are the shapes that will make fair dice.<ref>{{citation|title=Dungeons, dragons, and dice|first=K. Robin|last=McLean|journal=The Mathematical Gazette|volume=74|issue=469|year=1990|pages=243–256|doi=10.2307/3619822|jstor=3619822|s2cid=195047512 }}.</ref>

Isohedral polyhedra are called '''isohedra'''. They can be described by their face configuration. An isohedron has an even number of faces.

The dual of an isohedral polyhedron is vertex-transitive, i.e. isogonal. The Catalan solids, the Platonic Solids, the bipyramids, and the trapezohedra are all isohedral. They are the duals of the (isogonal) Archimedean solids, Platonic Solids, prisms, and antiprisms, respectively. The Platonic solids, which are either self-dual or dual with another Platonic solid, are vertex-, edge-, and face-transitive (i.e. isogonal, isotoxal, and isohedral).

A form that is isohedral, has regular vertices, and is also edge-transitive (i.e. isotoxal) is said to be a quasiregular dual. Some theorists regard these figures as truly quasiregular because they share the same symmetries, but this is not generally accepted.

A polyhedron which is isohedral and isogonal is said to be noble.

Not all isozonohedra<ref>{{Cite web|url=http://mathworld.wolfram.com/Isozonohedron.html|title=Isozonohedron|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-26}}</ref> are isohedral.<ref>{{Cite web|url=http://mathworld.wolfram.com/Isohedron.html|title=Isohedron|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-21}}</ref> For example, a rhombic icosahedron is an isozonohedron but not an isohedron.<ref>{{Cite web|url=http://mathworld.wolfram.com/RhombicIcosahedron.html|title=Rhombic Icosahedron|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-21}}</ref>

==Examples== {| class="wikitable" style="width:720px;" !colspan=3|Convex||Concave |- valign=top |180px Hexagonal bipyramids, V4.4.6, are nonregular isohedral polyhedra. |240px The Cairo pentagonal tiling, V3.3.4.3.4, is isohedral. |240px The rhombic dodecahedral honeycomb is isohedral (and isochoric, and space-filling). |240px A square tiling distorted into a spiraling H tiling (topologically equivalent) is still isohedral. |}

==Classes of isohedra by symmetry== {| class="wikitable sortable" |- !Faces||Face config.||Class||Name||Symmetry||Order||Convex||Coplanar||Nonconvex |- align=center !4 |V33 ||Platonic ||tetrahedron tetragonal disphenoid rhombic disphenoid |Td, [3,3], (*332) D2d, [2+,2], (2*) D2, [2,2]+, (222) |24 4 4 4 ||60px|Tetrahedron60px60px|| || |- align=center !6 |V34 ||Platonic |cube trigonal trapezohedron asymmetric trigonal trapezohedron |Oh, [4,3], (*432) D3d, [2+,6] (2*3) D3 [2,3]+, (223) |48 12 12 6 ||60px|Cube30px60px|| || |- align=center !8 |V43 ||Platonic |octahedron square bipyramid rhombic bipyramid square scalenohedron |Oh, [4,3], (*432) D4h,[2,4],(*224) D2h,[2,2],(*222) D2d,[2+,4],(2*2) |48 16 8 8 ||60px|Octahedron60px60px60px60px60px|| ||60px

|- align=center !12 |V35 ||Platonic |regular dodecahedron pyritohedron tetartoid |Ih, [5,3], (*532) Th, [3+,4], (3*2) T, [3,3]+, (*332) |120 24 12 ||60px|Dodecahedron60px60px||60px60px||60px60px |- align=center !20 |V53 ||Platonic ||regular icosahedron |Ih, [5,3], (*532) |120 ||60px|Icosahedron|| || |- align=center ! 12 |V3.62 | Catalan | triakis tetrahedron |Td, [3,3], (*332) |24 | 60px|Triakis tetrahedron||60px60px||60px |- align=center ! 12 |V(3.4)2 | Catalan | rhombic dodecahedron deltoidal dodecahedron |Oh, [4,3], (*432) Td, [3,3], (*332) |48 24 | 60px|Rhombic dodecahedron60px60px||60px||60px60px |- align=center ! 24 |V3.82 | Catalan | triakis octahedron |Oh, [4,3], (*432) |48 | 60px|Triakis octahedron|| ||60px60px |- align=center ! 24 |V4.62 | Catalan | tetrakis hexahedron |Oh, [4,3], (*432) |48 | 60px|Tetrakis hexahedron60px||60px60px||60px60px |- align=center ! 24 |V3.43 | Catalan | deltoidal icositetrahedron |Oh, [4,3], (*432) |48 | 60px|Deltoidal icositetrahedron60px||60px60px60px||60px |- align=center ! 48 |V4.6.8 | Catalan | disdyakis dodecahedron |Oh, [4,3], (*432) |48 | 60px|Disdyakis dodecahedron||60px60px60px||60px60px |- align=center ! 24 |V34.4 | Catalan | pentagonal icositetrahedron |O, [4,3]+, (432) |24 | 60px|Pentagonal icositetrahedron || || |- align=center ! 30 |V(3.5)2 | Catalan | rhombic triacontahedron |Ih, [5,3], (*532) |120 | 60px|Rhombic triacontahedron || || |- align=center ! 60 |V3.102 | Catalan | triakis icosahedron |Ih, [5,3], (*532) |120 | 60px|Triakis icosahedron|| || 60px60px60px60px |- align=center ! 60 |V5.62 | Catalan | pentakis dodecahedron |Ih, [5,3], (*532) |120 | 60px|Pentakis dodecahedron|| || 60px60px60px60px60px |- align=center ! 60 |V3.4.5.4 | Catalan | deltoidal hexecontahedron |Ih, [5,3], (*532) |120 | 60px|Deltoidal hexecontahedron||120px||60px |- align=center ! 120 |V4.6.10 | Catalan | disdyakis triacontahedron |Ih, [5,3], (*532) |120 | 60px|Disdyakis triacontahedron||60px60px60px||60px60px60px |- align=center ! 60 |V34.5 | Catalan | pentagonal hexecontahedron |I, [5,3]+, (532) |60 | 60px|Pentagonal hexecontahedron|| || |- align=center !2''n'' |V33.''n'' ||Polar |trapezohedron asymmetric trapezohedron |D''n''d, [2+,2''n''], (2*''n'') D''n'', [2,''n'']+, (22''n'') |4''n'' 2''n'' ||30px60px60px60px 60px60px|| || |- align=center !2''n'' 4''n'' |V42.''n'' V42.2''n'' V42.2''n'' ||Polar |regular ''n''-bipyramid isotoxal 2''n''-bipyramid 2''n''-scalenohedron |D''n''h, [2,''n''], (*22''n'') D''n''h, [2,''n''], (*22''n'') D''n''d, [2+,2''n''], (2*''n'') |4''n'' ||60px60px60px60px|| ||60px60px60px60px60px60px60px |}

==''k''-isohedral{{anchor|monohedral}} figure== A polyhedron (or polytope in general) is '''''k''-isohedral''' if it contains ''k'' faces within its symmetry fundamental domains.<ref>{{cite journal |last=Socolar |first=Joshua E. S. |year=2007 |title=Hexagonal Parquet Tilings: ''k''-Isohedral Monotiles with Arbitrarily Large ''k'' |journal=The Mathematical Intelligencer |volume=29 |issue=2 |pages=33–38 |doi=10.1007/bf02986203 |arxiv=0708.2663 |s2cid=119365079 |url=http://www.phy.duke.edu/~socolar/hexparquet.pdf |access-date=2007-09-09 |format=corrected PDF |archive-date=2016-03-03 |archive-url=https://web.archive.org/web/20160303191142/http://www.phy.duke.edu/~socolar/hexparquet.pdf |url-status=dead }}</ref> Similarly, a '''''k''-isohedral tiling''' has ''k'' separate symmetry orbits (it may contain ''m'' different face shapes, for ''m'' = ''k'', or only for some ''m'' < ''k'').<ref>Craig S. Kaplan, [https://books.google.com/books?id=OPtQtnNXRMMC "Introductory Tiling Theory for Computer Graphics"] {{Webarchive|url=https://web.archive.org/web/20221208000331/https://books.google.com/books?id=OPtQtnNXRMMC |date=2022-12-08 }}, 2009, Chapter 5: "Isohedral Tilings", p. 35.</ref> ("1-isohedral" is the same as "isohedral".)

A '''monohedral''' polyhedron or monohedral tiling (''m'' = 1) has congruent faces, either directly or reflectively, which occur in one or more symmetry positions. An '''''m''-hedral''' polyhedron or tiling has ''m'' different face shapes ("''dihedral''", "''trihedral''"... are the same as "2-hedral", "3-hedral"... respectively).<ref>Tilings and patterns, p. 20, 23.</ref>

Here are some examples of ''k''-isohedral polyhedra and tilings, with their faces colored by their ''k'' symmetry positions:

{| class=wikitable width=640 !3-isohedral !4-isohedral !isohedral !2-isohedral |- !colspan=2|2-hedral regular-faced polyhedra !colspan=2|Monohedral polyhedra |- |160px |160px |160px |160px |- |The rhombicuboctahedron has 1 triangle type and 2 square types. |The pseudo-rhombicuboctahedron has 1 triangle type and 3 square types. |The deltoidal icositetrahedron has 1 face type. |The pseudo-deltoidal icositetrahedron has 2 face types, with same shape. |} {| class=wikitable width=400 !2-isohedral !4-isohedral !Isohedral !3-isohedral |- !colspan=2|2-hedral regular-faced tilings !colspan=2|Monohedral tilings |- |160px |160px |160px |right|160px |- |The Pythagorean tiling has 2 square types (sizes). |This 3-uniform tiling has 3 triangle types, with same shape, and 1 square type. |The herringbone pattern has 1 rectangle type. |This pentagonal tiling has 3 irregular pentagon types, with same shape. |}

==Related terms== A '''cell-transitive''' or '''isochoric''' figure is an ''n''-polytope (''n'' ≥ 4) or ''n''-honeycomb (''n'' ≥ 3) that has its cells congruent and transitive with each others. In 3 dimensions, the catoptric honeycombs, duals to the uniform honeycombs, are isochoric. In 4 dimensions, isochoric polytopes have been enumerated up to 20 cells.<ref>{{cite web | url=http://www.polytope.net/hedrondude/dice4.htm | title=Four Dimensional Dice up to Twenty Sides }}</ref>

A '''facet-transitive''' or '''isotopic''' figure is an ''n''-dimensional polytope or honeycomb with its facets ((''n''−1)-faces) congruent and transitive. The dual of an ''isotope'' is an isogonal polytope. By definition, this isotopic property is common to the duals of the uniform polytopes. *An isotopic 2-dimensional figure is isotoxal, i.e. edge-transitive. *An isotopic 3-dimensional figure is isohedral, i.e. face-transitive. *An isotopic 4-dimensional figure is isochoric, i.e. cell-transitive.

==See also== * Edge-transitive * Anisohedral tiling

==References== {{reflist}}

==External links== * {{GlossaryForHyperspace | anchor=Isotope | title=Isotope}} * {{MathWorld | urlname=IsohedralTiling | title=Isohedral tiling}} * {{MathWorld | urlname = Isohedron | title = Isohedron}} * [http://loki3.com/poly/isohedra.html isohedra] 25 classes of isohedra with a finite number of sides * [http://mathartfun.com/thedicelab.com/DiceDesign.html Dice Design at The Dice Lab]

{{DEFAULTSORT:Isohedral Figure}} Category:Polyhedra Category:4-polytopes