Tag: Conway

  • 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…

  • 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…

  • Penrose’s aperiodic tilings

    Around 1975 Sir Roger Penrose discovered his aperiodic P2 tilings of the plane, using only two puzzle pieces: Kites (K) and Darts (D) The inner angles of these pieces are all multiples of $36^o = \tfrac{180^o}{5}$, the short edges have length $1$, and the long edges have length $\tau = \tfrac{1+\sqrt{5}}{2}$, the golden ratio. These…

  • Conway’s musical sequences (2)

    A Conway musical sequence is an infinite word in $L$ and $S$, containing no two consecutive $S$’s nor three consecutive $L$’s, such that all its inflations remain musical sequences. We’ve seen that such musical sequences encode an aperiodic tiling of the line in short ($S$) and long ($L$) intervals, and that such tilings are all…

  • 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…

  • Conway’s musical sequences

    Before we’ll come to applications of quasicrystals to viruses it is perhaps useful to illustrate essential topics such as deflation, inflation, aperiodicity, local isomorphism and the cut-and project method in the simplest of cases, that of $1$-dimensional tilings. We want to tile the line $\mathbb{R}^1$ with two kinds of tiles, short ($S$) and ($L$) long…

  • We sit in our ivory towers and think

    I’m on vacation, and re-reading two ‘metabiographies’: Philippe Douroux : Alexandre Grothendieck : Sur les traces du dernier génie des mathématiques and Siobhan Roberts : Genius At Play: The Curious Mind of John Horton Conway . Siobhan Roberts’ book is absolutely brilliant! I’m reading it for the n-th time, first on Kindle, then hardcopy, and…

  • Monstrous dessins 2

    Let’s try to identify the $\Psi(n) = n \prod_{p|n}(1+\frac{1}{p})$ points of $\mathbb{P}^1(\mathbb{Z}/n \mathbb{Z})$ with the lattices $L_{M \frac{g}{h}}$ at hyperdistance $n$ from the standard lattice $L_1$ in Conway’s big picture. Here are all $24=\Psi(12)$ lattices at hyperdistance $12$ from $L_1$ (the boundary lattices): You can also see the $4 = \Psi(3)$ lattices at hyperdistance $3$…