The Most Powerful Diagram in Mathematics

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⬣ ABOUT ⬣
I have loved this diagram ever since I first saw it on the coffee cup of one of my lecturers / colleagues. But I was shocked to discover that its extraordinary properties weren’t very well-known! In this video, I build up some theory necessary to understand the MOG, then demonstrate how to use it.

⬣ TIMESTAMPS ⬣
00:00 - Intro
02:49 - Motivation
07:14 - Steiner Systems
18:23 - Three Big Questions
29:21 - S(5,8,24) and the MOG
44:47 - Outro

⬣ HINT ⬣
Why might S(2,3,10) be impossible to construct? Try and prove the following lemma: If S(t,k,n) exists, then S(t-1,k-1,n-1) exists. Then use the contrapositive of this statement together with what we know about the number of blocks. In fact, one can prove that: If S(t,k,n) exists, then S(t-m,k-m,n-m) exists for integer m such that t-m is non-negative.

⬣ INVESTIGATORS ⬣
Nothing for you here. Sorry!

⬣ REFERENCES ⬣
R. T. Curtis, A New Combinatorial Approach to M24. Math. Proc. Camb. Phil. Soc. (1976), 79, 25.
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices, and Groups. Springer Science (1991).

⬣ CREDITS ⬣
Music by Danjel Zambo.
Pythagorean Theorem diagram:
https://upload.wikimedia.org/wikipedi...
Jakob Steiner:
https://upload.wikimedia.org/wikipedi...
Gino Fano:
https://en.wikipedia.org/wiki/Gino_Fano