Category: GoV

  • 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…

  • 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…

  • 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…

  • GoV 2 : Viruses and quasi-crystals

    If you look around for mathematical theories of the structure of viruses, you quickly end up with the work of Raidun Twarock and her group at the University of York. We’ve seen her proposal to extend the Caspar-Klug classification of viruses. Her novel idea to distribute proteins on the viral capsid along Penrose-like tilings shouldn’t…

  • GoV 1 : Geometry of viruses

    As you may have guessed from the symmetries of Covid-19 post, I did spend some time lately catching up with the literature on the geometric structure and symmetries of viruses. It may be fun to run a little series on this. A virus is a parasite, so it cannot reproduce on its own and needs…

  • The symmetries of Covid-19

    A natural question these days might be: “what are the rotational symmetries of the Covid-19 virus?” Most illustrations show a highly symmetric object, suggesting it might have icosahedral symmetry. In fact, many viruses do have icosahedral symmetry as a result of the ‘genetic economy principle’ proposed by Watson and Crick in 1956, resulting in the…