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.

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

George Sicherman
11

George Sicherman
   
1
9

George Sicherman
       
1
3
5

Mike Reid
5

Mike Reid
11

Mike Reid
11

Mike Reid
1
9

George Sicherman
9

George Sicherman
9

George Sicherman
   
3
5

George Sicherman
7

George Sicherman
11

George Sicherman
(after Mike Reid)
29

George Sicherman
29

George Sicherman
3
3
7

George Sicherman
11

George Sicherman
23

George Sicherman
29

George Sicherman
1
1
1
1
59

George Sicherman
 
3
3
5

George Sicherman
7

Mike Reid
29

George Sicherman
 
3
3
5

Mike Reid
9

George Sicherman
17

George Sicherman
35

George Sicherman
3
3
5

Mike Reid
9

Mike Reid
17

George Sicherman
23

George Sicherman
3
3
3
5

Mike Reid
17

George Sicherman
29

George Sicherman
3
3
5

George Sicherman
7

George Sicherman
11

Mike Reid
11

Mike Reid
5
1
11

George Sicherman
15

George Sicherman
77

George Sicherman
 
3
5
7

George Sicherman
11

George Sicherman
23

George Sicherman
35

George Sicherman
7

George Sicherman
3
1
7

George Sicherman
   
9

George Sicherman
1
       
3
1
23

George Sicherman
23

George Sicherman
   
3
1
7

George Sicherman
7

George Sicherman
35

George Sicherman
47

George Sicherman
7

George Sicherman
(squashed by Mike Reid)
1
9

George Sicherman
9

George Sicherman
65

George Sicherman
65

George Sicherman
1
1
1
1
101

George Sicherman
 
3
5
7

George Sicherman
9

George Sicherman
17

George Sicherman
17

George Sicherman
5
5
7

George Sicherman
15

George Sicherman
17

George Sicherman
17

George Sicherman

Holeless Variants

Rowwise Bilateral

Columnwise Bilateral

Birotary

Double Bilateral

Sextuple Rotary

Full

Last revised 2012-07-27.


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