Tag: Norton

  • 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 […]

  • Borcherds’ favourite numbers

    Whenever I visit someone’s YouTube or Twitter profile page, I hope to see an interesting banner image. Here’s the one from Richard Borcherds’ YouTube Channel. Not too surprisingly for Borcherds, almost all of these numbers are related to the monster group or its moonshine. Let’s try to decode them, in no particular order. 196884 John…

  • a SNORTgo endgame

    SNORT, invented by Simon NORTon is a map-coloring game, similar to COL. Only, this time, neighbours may not be coloured differently. SNORTgo, similar to COLgo, is SNORT played with go-stones on a go-board. That is, adjacent stones must have the same colour. SNORT is a ‘hot’ game, meaning that each player is eager to move…

  • a non-commutative Jack Daniels problem

    At a seminar at the College de France in 1975, Tits wrote down the order of the monster group \[ \# \mathbb{M} = 2^{46}.3^{20}.5^9.7^6.11^2.13^3.17·19·23·29·31·41·47·59·71 \] Andrew Ogg, who attended the talk, noticed that the prime divisors are precisely the primes $p$ for which the characteristic $p$ super-singular $j$-invariants are all defined over $\mathbb{F}_p$. Here’s Ogg’s…

  • the 171 moonshine groups

    Monstrous moonshine associates to every element of order $n$ of the monster group $\mathbb{M}$ an arithmetic group of the form \[ (n|h)+e,f,\dots \] where $h$ is a divisor of $24$ and of $n$ and where $e,f,\dots$ are divisors of $\frac{n}{h}$ coprime with its quotient. In snakes, spines, and all that we’ve constructed the arithmetic group…

  • The defining property of 24

    From Wikipedia on 24: “$24$ is the only number whose divisors, namely $1, 2, 3, 4, 6, 8, 12, 24$, are exactly those numbers $n$ for which every invertible element of the commutative ring $\mathbb{Z}/n\mathbb{Z}$ is a square root of $1$. It follows that the multiplicative group $(\mathbb{Z}/24\mathbb{Z})^* = \{ \pm 1, \pm 5, \pm…