Beyond Rep-tiles

A rep-tile is a shape that can be cut into smaller copies of the same shape - https://en.wikipedia.org/wiki/Rep-tile

Doubt: Is there a triangle T such that a certain number of T's can tile a triangle dissimilar to T?

Trivial example: 2 copies of right triangles tile an isosceles triangle dissimilar to the right triangle. An equilateral triangle is tiled by 3 mutually identical triangles which are not equilateral.

So the question is whether there are more non-trivial such examples, say, a triangle that can be tiled by 5 or 6 or n (but not less) mutually identical triangles that are dissimilar to itself.

Is there any quadrilateral that can tile a quad that is dissimilar to itself?

Again there is the trivial example of two copies of a trapezium with two angles = 90 degrees together forming a rectangle. Again, are there more non-trivial examples?

An equilateral triangle can be cut into 3 identical quadrilaterals that meet at its centroid. So one can ask, is there any triangle that allows partition into some other finite number of identical quadrilaterals.

And these questions have obvious 3d generalizations. Trivial examples can again be made by lofting the 2d trivial examples into prisms. And a regular tetrahedron can be cut into 4 identical tets that are dissimilar to the regular one.

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A doubt about a theorem by Kantarovich:

Theorem (as I understand it): Given any planar convex region P and a set of n points, we can shoose weights for each point such that a weighted Voronoi partition of P with these points as nuclei is an equipartition. And if these points are moved continuously, the weights for which they yield an equipartition also change continuously.

Doubt: Is it guaranteed that if one insists all weights to be 1, there is a set of nuclei that gives an equipartition? Do Voronoi partitions with such special sets of points have any further properties?