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.
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
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

Holeless Variants
Columnwise Bilateral
Double Bilateral
Pentahexes are tricky, so I got help from
Mike Reid.
Click on the gray figures to expand them.
[ Holeless Variants
 Composite Solutions
 Nontrivial Variants
 MirrorSymmetric 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
Some pentahexes without oddities for certain symmetries
can be paired to form oddities.
Helmut Postl and Johann Schwenke found some of these
fullsymmetric oddities.
Birotary
9 Tiles
Double Bilateral
11 Tiles
Sextuple Rotary
11 Tiles
17 Tiles
23 Tiles
47 Tiles
Full
11 Tiles
17 Tiles
23 Tiles
29 Tiles
35 Tiles
41 Tiles
47 Tiles
53 Tiles
59 Tiles
65 Tiles
These tilings are irreducible and have more than one tile.
Rowwise Bilateral
Columnwise Bilateral
Birotary
Mike Reid found that this fullsymmetry oddity for the Q pentahex
can be tiled with vertical mirror symmetry!
After Mike told me that a smaller solution probably existed,
I found this one:
Last revised 20230123.
Back to
Polyform Oddities
< Polyform Curiosities
Col. George Sicherman
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