Similar Polyaboloes Forming a Convex Shape

Introduction

A polyabolo, or polytan, is a plane figure formed by joining equal isosceles right triangles along equal edges.

How few similar (scaled) copies of a given polyabolo can form a convex shape? For most polyaboloes, the tilings with the fewest tiles use equal tiles, as with this triabolo:

Here I show the only polyaboloes I know of for which the convex tiling with the fewest tiles uses tiles of different sizes. If you find a tiling with fewer tiles than shown, please write.

So far as I know, these polyaboloes cannot tile any convex shape at a uniform scale.

Andrew Bayly points out that polyominoes are also polyaboloes, and that some polyominoes have this property. I do not consider polyominoes here.

See also Similar Polyaboloes Tiling a Triangle, Similar Polyaboloes Tiling a Square, Similar Polyaboloes Tiling an Octagon, and Similar Polyiamonds Forming a Convex Shape.

Small Polyaboloes

Bigger Polyaboloes

In 2025, Andrew Bayly furnished these examples. They use polyaboloes with from 9 to 115 cells.

Decabolo and Dodecabolo

After studying Andrew's examples, I found these:

 

Last revised 2025-04-20.

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Col. George Sicherman [ HOME | MAIL ]