# Hextals

I introduce the name hextal (polyHex fracTAL) to include fractal IFS tiles in which the elements are laid out on a hexagonal grid. (Alternative names that could have been introduced instead are hextiles - putting the emphasis on the tiling property rather than the fractal property - and Loeschian fractals or tiles - they are Perron tiles in which the corresponding Perron number is an Eisenstein integer whose magnitude is a Loeschian number.)

Hextals have m2 + mn + n2 elements, where m is an integer greater than equal to 1, n is an integer greater than or equal to 0, and m2 + mn + n2 is greater than 1.

Hextals and their derivatives are the richest subset of Perron and pseudo-Perron tiles.

There are hextals with 3 (trihextals), 4 (tetrahextals), 7 (heptahextals), 9 (enneahextals), 12 (dodecahextals), 13 (tridecahextals), 16 (hexadecahextals) and so on elements.

There are also constructions which produce tiles with varying numbers of elements.

• linear, caterpillar and allied hextals (n)
• bilinear and tetragonal hextals (n)
• diamond, chevron and square hextals (n2)
• equilateral triangles (n2+2n+1)
• fudgeflakes (3n2)
• flowsnakes (3n2+3n+1)
• starflakes (9n2+3n+1)
• alterflakes
• terhexagonoids (9n2+6n+1)
• hepterhexagonoids
• triangular interstitial hextals

As part of the richness of the set of hextals, there are many pairs of (appropriate scaled) tiles where composition of their IFSs produces new tiles. These include

• a core set of composites based on diamond hextals, fudgeflakes, flowsnakes, starflakes, skewflakes, alterflakes, terhexagonoids, etc.
• composite linear hextals
• composites of linear trihextals and fudgeflakes
• composite fudgeflakes
• composite flowsnakes
• composite cyclic tetrahextals

© 2017, 2019 Stewart R. Hinsley