isohedral figure

A polyhedron (or polytope in general) is k-isohedral if it contains k faces within its symmetry fundamental domains.

{{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}} 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).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. ("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).Tilings and patterns, p. 20, 23.

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

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.{{cite web | url=http://www.polytope.net/hedrondude/dice4.htm | title=Four Dimensional Dice up to Twenty Sides }}

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.

• Edge-transitive
• Anisohedral tiling

{{reflist}}

• {{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]

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Category:Polyhedra

Category:4-polytopes