# Tiling a Shape with Ternary Symmetry with
the Heptiamonds and the Tetrahexes

## Introduction

A *heptiamond* is a figure made of seven equilateral triangles joined
edge to edge.
There are 24 such figures, not distinguishing reflections and rotations.
The nomenclature is taken from K. Ishino.

A *tetrahex* is a figure made of four regular hexagons joined edge
to edge.
There are 7 such figures, not distinguishing reflections and rotations.

Todor Tchervenkov has defined
*bi-tileability*
as the ability to tile a shape with either of two sets of tiles.
At this page
he presents some results in bi-tileability using polyiamonds and polyhexes.

## Heptiamonds and Tetrahexes

If we identify the monohex with the hexagonal hexiamond, the 24 heptiamonds
have the same area as the 7 tetrahexes: 168 iamond cells.
This suggests that some shapes can be tiled with the heptiamonds
and with the tetrahexes.
See Tchervenkov's page for general examples.
The general problem of tiling a shape with the heptiamonds
and with the tetrahexes has too many solutions to present.
Tchervenkov has reduced the problem by adding conditions of
symmetry, large holes, or numerous holes.

## Ternary Symmetry

Here I present all such shapes that have ternary (3-way) symmetry.
Some also have mirror symmetry.
Click on a 28-hex to see its tilings by the heptiamonds and the tetrahexes.
The tilings are not necessarily unique.

*Last revised 2020-10-10.*

Back to Polyform Tiling
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Polyform Curiosities

Col. George Sicherman
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