How To Really Tile Rep-Tile Triangles?

The rep-tile art of splitting a triangle into ’n’ congruent tiles!

Illustration created by the author (figures not to scale)

In my essay on the rep-tile puzzle, I introduced the concept of the rep-tile, covered how Solomon W. Golomb defined a rep-k polygon and demonstrated what it means to construct a rep-3 triangle. If you haven’t read that essay, I would recommend you to do so. This is because I will be building on top of those concepts in this essay.

You see, the rep-3 triangle is a special case for which I couldn’t find any structured approach. However, I’m happy to report that Golomb figured out structured approaches to tiling higher-order rep-k triangles. I will be demonstrating how we could tile the various possible higher-order rep-k triangles in this essay. Let us begin.

The Conditions for Tiling Rep-Tile Triangles

In his 1964 paper titled “Replicating Figures in the Plane” (linked in the references at the end of this essay), Golomb stated three specific conditions for tiling rep-tile triangles. In other words, if these conditions are not met, it would not be possible to tile the corresponding triangle.

Given the intention to tile an arbitrary triangle into n-tiles, Golomb’s conditions are as follows:

1. A triangle can be tiled…