# Hexiamond Pair Tri-Oddities

## Introduction

A hexiamond is a plane figure made of six equilateral triangles joined edge to edge. There are 12 such figures, not distinguishing reflections and rotations. They were first enumerated by T. H. O'Beirne.

A tri-oddity is an arrangement of a number of copies of a polyform, not a multiple of 3, with ternary symmetry. It is a variant of an oddity.

Many polyforms cannot have ternary symmetry. Polyiamonds can, along with polyhexes and polycubes.

Here I show minimal known tri-oddities formed by copies of two hexiamonds, using at least one of each. If you find a smaller solution, or solve an unsolved case, please let me know.

• Arbitrary Solutions
• Holeless Solutions
• ## Arbitrary Solutions

### Table

This table shows the fewest tiles known to construct a tri-oddity with two given hexiamonds.

AEFHILOPSUVX
A * 5 4 8 4 4 10 13 13 13
E 5 * 5 5 5 5 4 5 5 5 5 5
F 4 5 * 7 7 4 4 5 8 7 7 11
H 8 5 7 * 10 4 4 7 7 4 7 10
I 5 7 10 * 7 4 4 8 8 4
L 4 5 4 4 7 * 4 5 7 2 7 7
O 4 4 4 4 4 4 * 4 4 4 4 4
P 10 5 5 7 4 5 4 * 4 4 4 7
S 13 5 8 7 8 7 4 4 * 7 7 13
U 13 5 7 4 8 2 4 4 7 * 5 8
V 5 7 7 7 4 4 7 5 * 4
X 13 5 11 10 4 7 4 7 13 8 4 *

### Solutions

These solutions are minimal. They are not necessarily uniquely minimal.

## Holeless Solutions

### Table

This table shows the fewest tiles known to construct a holeless tri-oddity with two given hexiamonds.

AEFHILOPSUVX
A * 5 4 8 4 4 10 16 13 ?
E 5 * 5 5 5 5 4 5 5 5 5 5
F 4 5 * 7 7 4 4 5 10 7 11 16
H 8 5 7 * 10 4 4 7 7 4 7 14
I 5 7 10 * 7 4 4 13 13 4
L 4 5 4 4 7 * 4 7 7 4 7 7
O 4 4 4 4 4 4 * 4 4 4 4 4
P 10 5 5 7 4 7 4 * 4 4 4 7
S 16 5 10 7 13 7 4 4 * 7 10 19
U 13 5 7 4 13 4 4 4 7 * 7 10
V 5 11 7 7 4 4 10 7 * 4
X ? 5 16 14 4 7 4 7 19 10 4 *

### 19 Tiles

Last revised 2023-04-28.

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Col. George Sicherman [ HOME | MAIL ]