TITLE: Decomposable Families of Shapes
QUESTION [16 upvotes]: There are two types of golden triangles in the world, as shown in the following picture:

Here $\varphi = \dfrac{1+\sqrt{5}}{2}$ denotes the golden ratio.  Each of these golden triangles can be decomposed into two smaller golden triangles in two different ways:

(More generally, if we fix a natural number $n \geq 5$, then the set of all similarity classes of triangles whose angles are multiples of $\pi/n$ has a similar property.  The golden triangles are the case where $n=5$.)
In general, we say that a finite family of (similarity classes of) polygons is decomposable if each polygon in the family can be decomposed into two polygons from the family in at least two different ways.
So here are my questions:

What are some other decomposable families of polygons?
Does this property (or something similar) already have a name in the literature?  Where can I look this up?
Are there any decomposable families of shapes in the plane whose members are not polygons?  For example, do there exist any finite decomposable families of fractal shapes?
Are there any decomposable families of polyhedra in $\mathbb{R}^3$?

Motivation:  I have an interesting way of making an infinite discrete group associated to any decomposable family, and I want more examples of such groups.

REPLY [7 votes]: Consider the following family of 4 similarity classes: squares, 1-by-2 rectangles, 1-by-3 rectangles, and 1-by-4 rectangles. Each polygon in the family can be decomposed into two polygons from the family in at least two different ways:

squares can be decomposed to two 1-by-2 rectangles either horizontally or vertically.
1-by-2 rectangles can be decomposed to two squares vertically, or to two 1-by-4 rectangles horizontally.
1-by-3 rectangles can be decomposed vertically to a square and a 1-by-2 rectangle, or vice versa.
1-by-4 rectangles can be decomposed vertically to a square and a 1-by-3 rectangle, or vice versa, or two 1-by-2 rectangles.

This set can be extended arbitrarily by adding 1-by-N rectangles for every integer $N\geq 5$. This can also be extended to polyhedra in $R^3$, by making each rectangle a hypercuboid of size 1-by-1-by-N.

Another family, which I describe in the same way I built it (hopefully this will some day develop into an algorithm):

Take a right-angled isosceles triangle (RAIT). Cut it using the perpendicular bisector to one of its shorter sides. The result is a smaller RAIT, and a right-angled quadrilateral (trapezoid). This can be done in two ways (one for each shorter-side).
Take the trapezoid. Cut it to two RAITs of different sizes. Cut it in another way, to a square and a RAIT.
Take the square. Cut it to two RAITs in two ways (one for each diagonal).

In conclusion, we have a family of 3 decomposable shapes: RAIT, trapezoid and square.

So, currently we have a family of 2 decomposable shapes (in the OP), a family of 3 (above) and families of 4 or more (above the above).