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.

Trihexes

Tetrahexes

Pentahexes

Holeless Variants

Composite Solutions

Nontrivial Variants

For hexahexes, see Hexahex Oddities.

Trihexes

Rowwise Bilateral	Columnwise Bilateral	Birotary	Double 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

Holeless Variants

Ternary on Vertex, Rowwise Bilateral

Tetrahexes

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

Rowwise
Bilateral Columnwise
Bilateral Birotary Double
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

Rowwise Bilateral	Columnwise Bilateral	Birotary	Double 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

Holeless Variants

Columnwise Bilateral

Double Bilateral

Pentahexes

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

[ Holeless Variants | Composite Solutions | Nontrivial Variants | Mirror-Symmetric Tilings ]

Rowwise Bilateral Columnwise Bilateral Birotary Double
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
47

Mike Reid

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
53

1
1
1
1
101

3
5
7
9
17
17

5
5
7
15
17
17

Rowwise Bilateral	Columnwise Bilateral	Birotary	Double 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	47 Mike Reid
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	53
1	1	1	1	101
3	5	7	9	17	17
5	5	7	15	17	17