A268591 Number of n-isohedral edge-to-edge "full colorings" of regular polygons.

3, 36, 126, 313, 484, 966

Offset: 1

Author

Brian Galebach, Feb 07 2016

Keywords

• hard
• more
• nice
• nonn

Comments

An n-isohedral coloring has n transitivity classes (or "orbits") of faces with respect to the color-preserving symmetry group of the coloring.

A n-isohedral "full coloring" requires that each face class uses a different color, so is therefore an n-colored, n-isohedral coloring. (The term "full coloring" appears to possibly be a new concept, not previously explored or documented in the theory of colorings.)

The full colorings (n colors) are arguably the most fundamental set of n-isohedral colorings -- even more so than the tilings (1 color) -- because all n-isohedral "sub-colorings" (1 to n-1 colors) can be derived from the n-isohedral full colorings.

In "Tilings and Patterns" by Branko Grünbaum and G.C. Shephard, 1986, pp. 102-107, the authors enumerate (as 32) the uniform edge-to-edge colorings of regular polygons (where uniform means that there is a single vertex transitivity type). Despite stating the requirement that "tiles in different transitivity classes... have different colors", they still include six colorings (three with triangles, and three with squares) in which two separate classes (within the coloring) use a single color. It is reasonable to assume they may have been referring to transitivity classes in the original uncolored tiling, not the coloring itself. However, this seems an arguably less consistent (and useful) treatment than enumerating only "full colorings", in which we refer to the face classes relative to the color-preserving symmetry group of the coloring itself.

Examples

			The three 1-isohedral full colorings are the regular tilings (triangles, squares, hexagons).  The 36 2-isohedral full colorings are composed of the 13 2-isohedral tilings given in D. Chavey, 1984, where each of those 13 tilings use two colors (one for each tile type); plus 7 2-isohedral colorings of triangles, 9 2-isohedral colorings of squares, and 7 2-isohedral colorings of hexagons.
		

References

• Branko Grünbaum, G. C. Shephard, Tilings and Patternsm, 1986, pp. 102-107

Links

• D. Chavey, Periodic Tilings and Tilings by Regular Polygons I, Thesis, 1984, pp. 165-172 gives the 13 2-isohedral edge-to-edge tilings of regular polygons. Each of these tilings corresponds to one 2-isohedral edge-to-edge full coloring of regular polygons.
• Brian Galebach, Full coloring results announcement, Facebook.

Crossrefs

Analogous to the n-isohedral edge-to-edge tilings of regular polygons (A268184), which use the same color for all face classes (1 color), as opposed to a different color for each face class (n colors).