# Geometric dissection theory

A few days ago, i realized that one way to prove the Pythagorean Theorem is to dissect the given right-angled triangle into 2 triangles similar to it, and apply well-known properties of ratios of areas. So this fundamental theorem can be viewed as a consequence of the possibility to dissect a given right-triangle into 2 similar sub-triangles.

My question is: Which branch of geometry systematically studies dissections of general polygons (or other shapes) into other sub-polygons (sub-shapes) of a given type, and the metrical properties that emerge as consequences of these dissections?

My search is on-going, i have seen some relevant results & books, for example the problem of dissecting a random triangle into n similar triangles, and books on discrete & combinatorial geometry that border on the subject (but not studied them yet), so i'm asking for some further guidance towards a systematic exposition of the relevant theory, in 2D, 3D or more dimensions..

• Not an answer to your question, but possibly relevant to your interests: en.wikipedia.org/wiki/… May 20 '19 at 7:13
• The proof you give in your first paragraph, I call the one line proof of Pythagoras' Theorem, the one line being the one you have to draw to dissect the triangle. May 20 '19 at 13:08
• @Gerry Myerson: A relevant discussion in hsm.stackexchange at which you may want to take a look later on: hsm.stackexchange.com/questions/3227/… May 20 '19 at 16:11

You might search for 'scissors congruence', which discusses which polyhedra (= higher-dimensional analogues of polygons) can be cut up to make which other ones. The question of whether you can cut a cube into finitely many pieces and reassemble to make a regular tetrahedron of the same volume was on Hilbert's problem list of around 1900, and was one of the first to be solved (when Dehn showed that it can't). I am not sure that this is exactly the sort of thing that you are looking for, but it is related and it's an interesting area of mathematics.

The following paper proves (1) that every polygon may be partitioned (dissected) into (a finite number of) non-obtuse triangles, and (2) into acute triangles:

Maehara, Hiroshi. "Acute triangulations of polygons." European Journal of Combinatorics 23, no. 1 (2002): 45-55. Dissections specifically into similar triangles:

Jones, C. A., P. Jones, and A. B. Bolt. "Dissections of triangles into five similar triangles." The Mathematical Gazette 82, no. 494 (1998): 225-234. 