The Most Powerful Diagram in Mathematics

⬣ LINKS ⬣ ⬡ PATREON: https://www.patreon.com/anotherroof ⬡ CHANNEL: https://www.youtube.com/c/AnotherRoof ⬡ TWITCH: https://www.twitch.tv/anotherroof ⬡ WEBSITE: https://anotherroof.top ⬡ SUBREDDIT: https://www.reddit.com/r/anotherroof ⬣ 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/wikipedia/commons/d/d2/Pythagorean.svg Jakob Steiner: https://upload.wikimedia.org/wikipedia/commons/5/55/JakobSteiner.jpg Gino Fano: https://en.wikipedia.org/wiki/Gino_Fano