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.
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
© 2017, 2019 Stewart R. Hinsley