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.

For heptahexes, see Heptahex 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 2016-06-06.


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Col. George Sicherman [ HOME | MAIL ]