Family of shapes that can be tiled into one another Okay, I'm trying to ask a question which hasn't been asked before, it may be futile, but let's see.
So let's take a square, this will be our shape A. We can tile a 2x1 rectangle by using shapes congruent to A. So we take 2 squares to build a 2x1 rectangle. But... now we need to go back too. Can we do so?
Sure we can. So now we take the 2x1 rectangle as our shape B, and we tile shapes congruent to shape B to make shape A.
Hope it's clear enough whan I'm getting at. Now, we have 2 members in this family, but in fact, this family is infinite. It also includes the 3x1 rectangle, 4x1 rectangle, and so forth. It also includes several polyominoes, like the L-tetromino and the T-tetromino. They can all be transformed into a square, and into each other.
Another infinite family is composed of parallellograms. (If one would consider that as a different family)
(And I suppose you could also make a distinction between rectangles and squares, since not all rectangles can be tiled into a square, since one of its sides could be an irrational number)
But aside from all this, there's another, perhaps finite, family, which has the equilateral triangle as a member.
This trapezium is also a member, they can be coverted into one another.
https://math.stackexchange.com/questions/603586/can-you-make-an-equilateral-triangle-from-3identical-trapezoids
The sphinx, made up by 6 equilateral triangles, is also a member:
https://math.stackexchange.com/questions/3953136/which-equilateral-triangles-does-the-p-hexiamond-the-sphinx-tile
And then there's this one, a longer sphinx, made up by 8 equilateral triangles, from a book by Karl Scherer:
http://www.recmath.com/PolyPages/PolyPages/index.htm?Polyiamonds.htm
So now we have a family of 4 members. Are there any other? It seems to be unknown.
But what about other families? Can you find another family? I don't know any. I'm assuming that a family has to have at least 2 members.
 A: Any member of a nontrivial family like this has to be a rep-tile; looking among those will give you many examples.
Some specific examples:

*

*The family of all rectifiable polyominoes, which includes all rectangles with rational ratios but also many other polyominoes, like the T-tetromino (though classifying them is a hard problem in general). While it's not inconceivable to have a set of polyominoes which mutually tile one another that aren't rectifiable, to my knowledge every known reptilic polyomino is rectifiable, so no examples are known at least.


*The family of all polyiamonds that can tile an equilateral triangle, which again includes those you list but also many more.


*Let $S$ be the set of polyominoes which tile a rectangle via only translations and $180^\circ$ rotations - this includes all rectangles of rational ratios, the L tromino, the P pentomino, and many others (though certainly not all rectifiable polyominoes). Applying any affine transformation to the members of this set preserves their ability to tile one another.


*The Sierpinski carpet, a domino formed from two Sierpinski carpets, and an L-tetromino formed from four Sierpinski carpets all mutually tile one another, to use one of the more fractally rep-tiles. (Many more examples in this vein are possible by subdividing the fractal at smaller scales.)


*The Sierpinski triangle is a member of a family along with a row of three Sierpinski triangles; here’s a diagram.

A: There is quite a lot known about polyominoes made of unit squares which tile some rectangle (which would have integer sides) For example this 6 square tile fills a
$23\times 24$ rectangle but no smaller.

Hence it tiles a square region of side $552=23\cdot 24$ and thus any polyomino at all.
