Tiling a Shape with Ternary Symmetry with
the Heptiamonds and the Tetrahexes
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
There are 7 such figures, not distinguishing reflections and rotations.
Todor Tchervenkov has defined
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.
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
Col. George Sicherman