Category: groups

  • The monster prime graph

    Here’s a nice, symmetric, labeled graph: The prime numbers labelling the vertices are exactly the prime divisors of the order of the largest sporadic group: the monster group $\mathbb{M}$. \[ \# \mathbb{M} = 2^{46}.3^{20}.5^9.7^6.11^2.13^3.17.19.23.29.31.41.47.59.71 \] Looking (for example) at the character table of the monster you can check that there is an edge between two […]

  • Mamuth to Elephant (3)

    Until now, we’ve looked at actions of groups (such as the $T/I$ or $PLR$-group) or (transformation) monoids (such as Noll’s monoid) on special sets of musical elements, in particular the twelve pitch classes $\mathbb{Z}_{12}$, or the set of all $24$ major and minor chords. Elephant-lovers recognise such settings as objects in the presheaf topos on…

  • Mamuth to Elephant (2)

    Last time, we’ve viewed major and minor triads (chords) as inscribed triangles in a regular $12$-gon. If we move clockwise along the $12$-gon, starting from the endpoint of the longest edge (the root of the chord, here the $0$-vertex) the edges skip $3,2$ and $4$ vertices (for a major chord, here on the left the…

  • From Mamuth to Elephant

    Here, MaMuTh stands for Mathematical Music Theory which analyses the pitch, timing, and structure of works of music. The Elephant is the nickname for the ‘bible’ of topos theory, Sketches of an Elephant: A Topos Theory Compendium, a two (three?) volume book, written by Peter Johnstone. How can we get as quickly as possible from…

  • Do we need the sphere spectrum?

    Last time I mentioned the talk “From noncommutative geometry to the tropical geometry of the scaling site” by Alain Connes, culminating in the canonical isomorphism (last slide of the talk) Or rather, what is actually proved in his paper with Caterina Consani BC-system, absolute cyclotomy and the quantized calculus (and which they conjectured previously to…

  • Imagination and the Impossible

    Two more sources I’d like to draw from for this fall’s maths for designers-course: 1. Geometry and the Imagination A fantastic collection of handouts for a two week summer workshop entitled ’Geometry and the Imagination’, led by John Conway, Peter Doyle, Jane Gilman and Bill Thurston at the Geometry Center in Minneapolis, June 1991, based…

  • Lockdown reading : SNORT

    In this series I’ll mention some books I found entertaining, stimulating or comforting during these Corona times. Read them at your own risk. This must have been the third time I’ve read The genius in by basement – The biography of a happy man by Alexander masters. I first read it when it came out…

  • a monstrous unimodular lattice

    An integral $n$-dimensional lattice $L$ is the set of all integral linear combinations \[ L = \mathbb{Z} \lambda_1 \oplus \dots \oplus \mathbb{Z} \lambda_n \] of base vectors $\{ \lambda_1,\dots,\lambda_n \}$ of $\mathbb{R}^n$, equipped with the usual (positive definite) inner product, satisfying \[ (\lambda, \mu ) \in \mathbb{Z} \quad \text{for all $\lambda,\mu \in \mathbb{Z}$.} \] But…

  • Know thy neighbours

    Two lattices $L$ and $L’$ in the same vector space are called neighbours if their intersection $L \cap L’$ is of index two in both $L$ and $L’$. In 1957, Martin Kneser gave a method to find all unimodular lattices (of the same dimension and signature) starting from one such unimodular lattice, finding all its…

  • The Leech lattice neighbour

    Here’s the upper part of Kneser‘s neighbourhood graph of the Niemeier lattices: The Leech lattice has a unique neighbour, that is, among the $23$ remaining Niemeier lattices there is a unique one, $(A_1^{24})^+$, sharing an index two sub-lattice with the Leech. How would you try to construct $(A_1^{24})^+$, an even unimodular lattice having the same…