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
    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 | Composite Solutions ] | Nontrivial 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
    53
    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

    Composite Solutions

    Some pentahexes without oddities for certain symmetries can be paired to form oddities.

    Birotary

    9 Tiles

    Double Bilateral

    11 Tiles

    Sextuple Rotary

    11 Tiles
    17 Tiles
    23 Tiles
    29 Tiles
    41 Tiles
    47 Tiles

    Full

    11 Tiles
    17 Tiles
    23 Tiles
    29 Tiles
    35 Tiles
    41 Tiles
    47 Tiles
    53 Tiles
    59 Tiles
    65 Tiles

    Nontrivial Variants

    These tilings are irreducible and have more than one tile.

    Rowwise Bilateral

    Columnwise Bilateral

    Birotary

    Last revised 2020-01-09.


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    Col. George Sicherman [ HOME | MAIL ]