### Rep-tiles - April 1995

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(A repeat of our May 1994 challenge)

(Thanks to Doris Schattschneider for contributing this project.)

A rep-tile is a tile that can be used to tile a larger scale copy of itself (or, to put it another way, a rep-tile is a tile that can be subdivided into a finite number of congruent tiles that are each similar to the original tile). Solomon W. Golomb coined the name in 1964.

Any square is a rep-tile - you can split any square into smaller, congruent squares, and you can build a square with smaller squares. For example,

Any triangle is also a rep-tile. So are rectangles and parallelograms.

Questions:

1. Can you find a way to divide any triangle into 4 congruent similar triangles?

2. How would you divide squares, rectangles, and parallelograms?

3. Can you find other quadrilaterals that are rep-tiles? There is at least one kind of trapezoid (which has an neat subdivision).

4. How about other polygons? Are there things that make a polygon a good candidate for being a rep-tile?

There is no general theory about rep-tiles, and there are certainly a lot of undiscovered ones, so this is your chance to make mathematical history!

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3 July 1995