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 ]

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

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
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

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 ]

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
65
65
1
1
1
1
101
 
3
5
7
9
17
17
5
5
7
15
17
17

Holeless Variants

Rowwise Bilateral

Columnwise Bilateral

Birotary

Double Bilateral

Sextuple Rotary

Full

Last revised 2017-01-31.


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Col. George Sicherman [ HOME | MAIL ]