Category: geometry

  • Designer Maths

    This fall, I’ll be teaching ‘Mathematics for Designers’ to first year students in Architecture. The past few weeks I’ve been looking around for topics to be included in such as course, relevant to architects/artists (not necessarily to engineers/mathematicians). One of the best texts I’ve found on this (perhaps in need of a slight update) is […]

  • 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…

  • Escher’s stairs

    Stairways feature prominently in several drawings by Maurits Cornelis (“Mauk”) Escher, for example in this lithograph print Relativity from 1953. Relativity (M. C. Escher) – Photo Credit From its Wikipedia page: In the world of ‘Relativity’, there are three sources of gravity, each being orthogonal to the two others. Each inhabitant lives in one of…

  • Ghost metro stations

    In the strange logic of subways I’ve used a small part of the Parisian metro-map to illustrate some of the bi-Heyting operations on directed graphs. Little did I know that this metro-map gives only a partial picture of the underground network. The Parisian metro has several ghost stations, that is, stations that have been closed…

  • 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…

  • Witt and his Niemeier lattices

    Sunday, January 28th 1940, Hamburg Ernst Witt wants to get two papers out of his system because he knows he’ll have to enter the Wehrmacht in February. The first one, “Spiegelungsgruppen und Aufzählung halbeinfacher Liescher Ringe”, contains his own treatment of the root systems of semisimple Lie algebras and their reflexion groups, following up on…

  • de Bruijn’s pentagrids (2)

    Last time we’ve seen that de Bruijn’s pentagrids determined the vertices of Penrose’s P3-aperiodic tilings. These vertices can also be obtained by projecting a window of the standard hypercubic lattice $\mathbb{Z}^5$ by the cut-and-project-method. We’ll bring in representation theory by forcing this projection to be compatible with a $D_5$-subgroup of the symmetries of $\mathbb{Z}^5$, which…

  • de Bruijn’s pentagrids

    In a Rhombic tiling (aka a Penrose P3 tiling) we can identify five ribbons. Opposite sides of a rhomb are parallel. We may form a ribbon by attaching rhombs along opposite sides. There are five directions taken by sides, so there are five families of ribbons that do not intersect, determined by the side directions.…

  • 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…