(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:
- Can you find a way to divide any triangle into 4 congruent similar triangles?
- How would you divide squares, rectangles, and parallelograms?
- Can you find other quadrilaterals that are rep-tiles? There is at least one kind of trapezoid (which has an neat subdivision).
- 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!
We received only a few solutions to this problem, and unfortunately none of the solutions was what we were looking for.
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