The classic problem of constructing a perfect squared square, i.e. tiling a square with a finite number of squares of pairwise distinct sidelengths (which must be rational multiples of the tiled square's, so we need only look for integral tilings) has solutions (one is shown below). However, it has been proved impossible to tile an equilateral triangle with equilateral triangles of distinct sizes (elegant proof here). It is also impossible to do it for other (more-sided) regular polygons since the angle between a corner piece and the side of the tiled polygon will be acute.
In three dimensions, one can prove very elegantly that it is impossible to divide a cube into a finite number of cubes of distinct sizes. A natural generalization is to ask whether it is possible to divide a regular tetrahedron into tetrahedra of distinct sizes (I assume the other Platonic solids will not work)? This problem appears more complicated, and I don't know if either impossibility proof (for cube or equilateral triangle) will generalize to tetrahedra. (This question appears to be asking something somewhat different, related to a specific type of division).
It is possible however to tile an equilateral triangle with equilateral triangles which can have the same size if their orientation is different; an example is shown below. It is worth wondering whether such a dissection might be possible for the tetrahedron.
Thus I have the following questions:
- Is it possible to divide a regular tetrahedron into regular tetrahedra of distinct sizes?
- If it is possible, can a proof like this be extended to show that all dissections are rational?
- Is the dissection possible if we only force tetrahedra of the same orientation to have distinct sizes?
Perfect squared square:
Equilateral triangle dissection: