

A269630


Number of nisohedral edgetoedge colorings of regular polygons.


0




OFFSET

1,1


COMMENTS

An nisohedral coloring has n transitivity classes (or "orbits") of faces with respect to the colorpreserving symmetry group of the coloring.
An nisohedral coloring may use anywhere from 1 color (a tiling) to n colors (a "full coloring").
Two colorings are considered identical (and hence not counted twice) if one can be obtained from the other through some permutation or reassignment of colors.
In "Tilings and Patterns" by Branko Grünbaum and G. C. Shephard, 1986, pp. 102107, the authors choose to require, in their enumeration of uniform colorings, that "tiles in different transitivity classes... have different colors." However, this is more restrictive than the most general definition of a coloring, or "colored tiling", given on p. 102, which states that "to each tile t of a given tiling T we assign one of a finite set of colors." Furthermore, some other studies of colorings actually require that some tiles in different classes share a single color (see the "Transitive perfect colorings" link below for an example). Hence, the enumerations in this sequence adhere solely to the most general coloring definition, with the only restriction being the prohibition of color permutations between colorings, as described in the preceding paragraph.


REFERENCES

Branko Grünbaum and G. C. Shephard, Tilings and Patterns, 1986.


LINKS

Table of n, a(n) for n=1..6.
D. Chavey, Periodic Tilings and Tilings by Regular Polygons I, Thesis, 1984, pp. 165172 gives the 13 2isohedral edgetoedge tilings of regular polygons. Each of these tilings corresponds to two 2isohedral edgetoedge colorings of regular polygons (the tiling itself, plus the analogous "full coloring").
Brian Galebach, nIsohedral EdgetoEdge Colorings of Regular Polygons, Facebook
Junmar Gentuya and René Felix, Transitive perfect colorings of the nonregular Archimedean tilings, arXiv:1507.05153 [math.GR], 2013, finds edgetoedge colorings of regular polygons satisfying certain criteria. All of the colorings found in this paper require that tiles in multiple transitivity classes share colors.


EXAMPLE

The three 1isohedral colorings are the regular tilings (triangles, squares, hexagons).
The 49 2isohedral colorings comprise the 13 2isohedral tilings given in D. Chavey, 1984, the corresponding "full coloring" version of each of those 13 tilings, where each uses two colors (one for each tile type); plus 7 2isohedral colorings of triangles, 9 2isohedral colorings of squares, and 7 2isohedral colorings of hexagons.
The 359 3isohedral colorings comprise the 29 3isohedral tilings, 126 full colorings (which use three colors each), and 204 colorings that use two colors each. These 359 colorings are illustrated in the Facebook link given above.


CROSSREFS

The nisohedral edgetoedge colorings of regular polygons comprise:
The nisohedral edgetoedge tilings of regular polygons (A268184), which use the same color for all face classes (1 color);
The nisohedral edgetoedge "full colorings" of regular polygons (A268591), which use a different color for each face class (n colors); and
All nisohedral edgetoedge colorings of regular polygons using between 2 and n1 colors (future sequence).
Sequence in context: A212651 A225317 A288527 * A067788 A167601 A061185
Adjacent sequences: A269627 A269628 A269629 * A269631 A269632 A269633


KEYWORD

hard,more,nice,nonn


AUTHOR

Brian Galebach, Mar 01 2016


STATUS

approved



