2018/11/06

Isosceles rectangular triangles joined at their sides are called Polytans or Polyabolos. If connection at the corners are also allowed we have pseudo polytans. To get real pieces the connection between the corners must be made by little bridges and the other corners must be rounded to allow the bridges to pass between them. Therefore these pieces are called bridged or rounded polytans.

If all touching corners were bridged, we would get too many pieces with holes. Therefore it seems to be better to remove unneccessary bridges. From the pseudo tritan in the picture you can derive two two-sided bridged tritans and three one-sided bridged tritans.

The left bridges must produce a spanning tree with ordinary polytans as vertices. In the example the vertices are single triangles.

I learned about the concept from the Logelium site more than a decade ago and wrote a program to create the pieces. My counts were:

Number of Triangles | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

Two-sided pseudo pieces | 1 | 10 | 91 | 1432 | 23547 | 416177 |

Two-sided bridged pieces | 1 | 10 | 95 | 1574 | 27553 | 517828 |

One-side pseudo pieces | 1 | 15 | 171 | 2799 | 46933 | |

One-sided bridged pieces | 1 | 15 | 179 | 3083 | 54948 |

The program also provided SVG-files for the shapes and I used them to order laser cut one-sided bridged ditans from 6mm acrylic. A figure with this set is shown above.

At the Logelium site some bridged tritans were missing and I looked for constructions with the complete set of 95 pieces. I found solutions for 12x12 squares with a tritan hole inside but afterwards I stopped exploring the set, because laser cutting all pieces seemed to be too expensive. A decade later I produced the pieces with a 3d-printer and digged out the old program to search for more constructions. One of the old solutions is shown below, for others ones click the sets in the table.

Set | Number of Triangles | Number of Pieces | Total Area | Constructions |
---|---|---|---|---|

Bridged Ditans (two-sided) |
2 | 10 | 10 | some 3-fold replicas with one piece left; some tetromino replicas with two pieces left |

Bridged Ditans (one-sided) |
2 | 15 | 15 | lots of symmetric figures |

Bridged Tritans (two-sided) |
3 | 95 | 142.5 | 12x12 square with tritan hole; convex symmetric polygons with up to 8 corners; similar hole triangle |

Bridged Tritans (one-sided) |
3 | 179 | 268.5 | convex symmetric polygons with up to 8 corners |

Bridged Polytans of Order 1..3 (two-sided) |
1..3 | 106 | 153 | figures with two axes of symmetry; two simultaneous replicas |

Bridged Polytans of Order 1..3 (one-sided) |
1..3 | 195 | 284 | two congruent figures or one similar one |

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