Rep-tiles

Feel free to add your own thoughts to the Discord!


Problem:

A rep-tile is a shape that can be dissected into smaller copies of the same shape. Which shapes are definitely rep-tiles? Which shapes are definitely not? Which shapes do you wonder about?


Known results:

All parallelograms (squares, rectangles, and rhombuses) are rep-tiles. Fit 4 of them together and you've got a larger copy of the original.

All triangles are rep-tiles. Again, fit 4 of them together to obtain a larger copy.

Here are three trapezoids that are rep-tiles:


Conjectures:

These are the only 3 trapezoids which are rep-tiles.

The only trapezoid with two 90 deg angles is the 45-90-90-135 trapezoid pictured above.

Questions that interest me:

Which polyominoes are rep-tiles?

Which polyabolos are rep-tiles?

Which kites, if any, are rep-tiles?


Known results:

No regular n-gon, where \(n \ge 5,\) can be a rep-tile. Each exterior angle, for a regular \(n\)-gon, is given by \(360/n.\) Each interior angle is given by \(180(n - 2)/n.\) We find that \(360/n \lt 180(n - 2)/n\) for \(n \ge 5.\) Hence, attempting to put some regular n-gon in a larger copy of itself will create two angles which are too small to be filled.

To produce a rep-tile, every interior angle in the larger copy must be filled. However, every polyomino has at least 1 interior angle of \(90^\circ.\) Thus, the X pentomino (pictured below) cannot be a rep-tile, as it cannot fill a single \(90^\circ\) angle.

Sierpiński's triangle, Sierpiński's carpet, and the dragon curve, are all rep-tiles.

Wikipedia states "any fractal defined by an iterated function system of n contracting maps of the same ratio is rep-n." I'm not sure what any of that means, but I wonder which other named fractals must then also be rep-tiles. You can find a list of named fractals here and here.

If a shape is a rep-tile, then it must also tile the plane. Just build recursively larger copies, ad infinitum.

Call the building block of a polyanimal its "primitive shape." For example, for polyominoes, the primitive shape is a square. For polyaboloes, it's a 45-45-90 triangle. If a polyanimal can tile its "primitive shape," then it must be a rep-tile.

Every rectangle with whole number side lengths can tile some square. If the rectangle has dimensions mxn, then the obvious square it can tile has all sides of length mn. The smallest square an mxn rectangle can tile has all sides of length lcm(m, n). Hence, if we can use a polyomino to tile a rectangle, then that polyomino must be a rep-tile.


Question:

Can all polyomino rep-tiles tile some rectangle?

Can any shape with a curved edge be a rep-tile? My gut says no, but I don't even know how to get started on such a proof. It might take some upper undergraduate-level, or maybe even graduate-level math to prove such a thing. This is not something I will be attempting. Just throwing it out there in case it interests someone else.

A kite can't be a rep-tile, unless it's a rhombus. See below for a visual proof. The first image is a kite with one acute angle, and one obtuse, along its line of symmetry. The second image is a kite with two acute angles. The third has two obtuse angles. My strategy was to focus on the smallest acute angle in each kite, as the smallest acute angle can be filled by exactly one angle, in exactly one way, except in the case of two obtuse angles on the line of symmetry, in which case there are two ways. For non-rhombus kites having two obtuse angles along their line of symmetry, start by placing one kite in one of the acute angles of the larger copy. Notice that two acute angles are formed between the small kite and the large kite housing it. These angles cannot be filled by any combination of the kite's angles, and so the kite can't be a rep-tile.