Polyiamond Irreptiling

Introduction

A polyiamond is made by joining equal equilateral triangles along equal edges.

A reptiling of a polyform is a dissection of it into two or more equal pieces that are similar to the original polyform. An irreptiling of a polyform is a dissection of it into pieces that are all similar to the original polyform, but not necessarily of equal size. Some shapes have irreptilings and no reptilings.

For more information about irreptiles, see Karl Scherer's book A Puzzling Journey to the Reptiles and Related Animals.

Here I omit polyiamonds that are similar to polyiamonds with fewer cells, and I show only tilings with the fewest possible pieces. If you find a tiling with fewer pieces, or solve an unsolved case, please write.

Dr. Karl Scherer coined the term irreptile and was the first to study irreptiles. Many of these tilings are taken from his book A Puzzling Journey to the Reptiles and Related Animals.

See also Polydrafter Irreptiling. For irreptilings of polyforms other than polyiamonds, see Erich Friedman's Math Magic for October 2002.

  • Moniamond
  • Diamond
  • Triamond
  • Tetriamonds
  • Pentiamonds
  • Hexiamonds
  • Heptiamonds
  • Octiamonds
  • Enneiamonds
  • Dekiamonds
  • Hendekiamonds
  • Dodekiamonds

    • Moniamond

    4

    Proper Irreptiling

    5

    Diamond

    4

    Proper Irreptiling

    5

    Triamond

    4

    Proper Irreptiling

    5

    Tetriamonds

    4

    Proper Irreptiling

    5

    Pentiamonds

    8

    Andrew Bayly
    55

    Hexiamonds


    Carl Schwenke and Johann Schwenke
    4

    Karl Scherer
    6

    Karl Scherer
    16

    Karl Scherer
    6
    4

    Proper Irreptiling


    Carl Schwenke and Johann Schwenke
    6
    6

    Heptiamonds

    14

    Octiamonds

    4
    4
    16
    4

    Karl Scherer
    10
    9

    Karl Scherer
    10

    Proper Irreptilings

    6
    7
    19
    7
    10

    Enneiamonds

    18

    Karl Scherer
    30

    Karl Scherer
    25
    20

    Karl Scherer
    14

    Dekiamonds

    4
    13
    13

    Carl Schwenke and Johann Schwenke
    27
    11
    11

    Carl Schwenke and Johann Schwenke
    20
    10
    14

    Proper Irreptiling

    6

    Hendekiamonds


    Carl Schwenke and Johann Schwenke
    18

    Dodekiamonds

    4
    12
    12
    21
    6
    9

    Carl Schwenke and Johann Schwenke
    22
    18
    27
    18

    Carl Schwenke and Johann Schwenke
    58
    114

    Carl Schwenke and Johann Schwenke
    30
    18

    Carl Schwenke and Johann Schwenke
    22
    18
    27
    6
    9
    15
    4
    85
    8

    Carl Schwenke and Johann Schwenke
    72
    12

    Carl Schwenke and Johann Schwenke
    18

    Carl Schwenke and Johann Schwenke
    18
    184
    27
    12
    12
    8
    18
    18
    11
    18
    27
    14
    27
    27
    27
    27

    Carl Schwenke and Johann Schwenke
    31

    Karl Scherer
    5

    Carl Schwenke and Johann Schwenke
    55

    Proper Irreptilings

    6
    6

    Last revised 2025-11-24.

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    Col. George Sicherman [ HOME | MAIL ]