Geometric dissection theory

Question

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: https://en.wikipedia.org/wiki/Wallace%E2%80%93Bolyai%E2%80%93Gerwien_theorem — Yemon Choi, May 20 '19 at 07:13
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. — Gerry Myerson, 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: https://hsm.stackexchange.com/questions/3227/on-einsteins-proof-of-the-so-called-pythagorean-theorem — José Hdz. Stgo., May 20 '19 at 16:11

score 5 · Answer 1 · edited May 20 '19 at 17:58

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.

score 3 · Answer 2 · answered May 20 '19 at 15:49

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.

Geometric dissection theory

2 Answers2