# Polyhex Oddities

A polyhex oddity is a plane figure with binary symmetry formed by joining an odd number of copies of a polyhex. Here are the minimal known oddities for the trihexes, tetrahexes, and pentahexes. Please write if you find a smaller solution or solve an unsolved case.

For hexahexes, see Hexahex Oddities.

## Trihexes

Rowwise
Bilateral
Columnwise
Bilateral
BirotaryDouble
Bilateral
Ternary on Cell
Rowwise
Bilateral
Ternary on Cell
Columnwise
Bilateral
Ternary on Vertex
Rowwise
Bilateral
1
9
11
11
3
9
1
1
1
1
1
3
9
3
3
1
5
5
9
3
3

## Tetrahexes

Mike Reid proved that the O and S tetrahexes have no sexirotary oddities.

Rowwise
Bilateral
Columnwise
Bilateral
BirotaryDouble
Bilateral
Sextuple
Rotary
Full
1
1
1
1
9
9
3
3
3
3
3
3
1
1
1
1
None None
3
3
3
3
3
3
3
3
1
3
None None
1
3
3
3
3
3
None 1
None None None None

## Pentahexes

Pentahexes are tricky, so I got help from Mike Reid. Click on the gray figures to expand them.

Rowwise Bilateral Columnwise Bilateral BirotaryDouble
Bilateral
Sextuple
Rotary
Full
1
9
11
11

1
9

1
3
5

Mike Reid
5

Mike Reid
11

Mike Reid
11

Mike Reid
1
9
9
9

3
5
7
11

(after Mike Reid)
29
29
3
3
7
11
23
29
1
1
1
1
59

3
3
5
7

Mike Reid
29

3
3
5

Mike Reid
9
17
35
3
3
5

Mike Reid
9

Mike Reid
17
23
3
3
3
5

Mike Reid
17
29
3
3
5
7
11

Mike Reid
11

Mike Reid
5
1
11
15
41

3
5
7
11
23
35
7
3
1
7

9
1

3
1
23
23

3
1
7
7
35
47
7

(squashed by Mike Reid)
1
9
9
53
65
1
1
1
1
101

3
5
7
9
17
17
5
5
7
15
17
17

### Composite Solutions

Some pentahexes without oddities for certain symmetries can be paired to form oddities.

### Nontrivial Variants

These tilings are irreducible and have more than one tile.

#### Birotary

Last revised 2019-04-13.

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Col. George Sicherman [ HOME | MAIL ]