Polyabolo Irreptiles

Introduction

A polyabolo or polytan is a plane figure formed by joining equal isosceles right triangles along equal edges. An irreptile is a geometric shape that can be divided into smaller pieces, not necessarily equal, with the same shape as the original.

The term irreptile was coined by Dr. Karl Scherer, who studied such tilings extensively. His results appear in chapters 3 and 4 of his book A Puzzling Journey to the Reptiles and Related Animals, and in some of his Wolfram Demonstrations, notably Irreptiles and Some Irreptiles of Order Greater Than 20.

Erich Friedman's Math Magic for October 2002 presents irreptilings for various kinds of polyforms, discovered by Scherer, Friedman, and others. It shows minimal known polyabolo irreptilings for polyaboloes with up to seven cells.

Here I present minimal known irreptilings for polyaboloes, including all those that Dr. Scherer showed to have irreptilings. Polyaboloes with the same shapes as smaller polyaboloes are omitted. Proper reptilings are shown in bright green. Tiles with shaded backgrounds are polyominoes.

  • Monabolo
  • Diaboloes
  • Triaboloes
  • Tetraboloes
  • Pentaboloes
  • Hexaboloes
  • Heptaboloes
  • Octaboloes
  • Enneaboloes
  • Decaboloes
  • Larger Polyaboloes
  • Monabolo

    2

    Diaboloes

    4
    2

    Triaboloes

    8

    Karl Scherer
    3

    Tetraboloes

    4

    Karl Scherer
    8
    34
    4

    Pentaboloes


    Karl Scherer
    5

    Karl Scherer
    34
    14

    Hexaboloes

    4
    4
    17
    4
    16
    5
    4
    5
    16

    Karl Scherer
    8

    Heptaboloes

    10
    8
    18

    Octaboloes

    4
    16
    4
    6
    14
    13
    4
    6

    Karl Scherer
    14
    6
    6
    10
    23
    8

    Enneaboloes

    20
    21

    Decaboloes

    4

    Karl Scherer
    10

    Karl Scherer
    40
    4
    4
    4
    17
    14
    17
    55
    18
    13
    35
    12

    Karl Scherer
    27
    10

    Larger Polyaboloes

    This 16-bolo irreptiling by Dr. Scherer has 19 tiles. The book and demo state that it has 21.

    Last revised 2018-09-07.


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    Col. George Sicherman [ HOME | MAIL ]