# the tilings of the plane

### the seventeen groups of classical tilings

See the page dedicated to the discrete groups of plane symmetries, largely inspired by
DISCRETE GROUPS OF PLANE ISOMETRIES - A new classification and their representations as Wallpaper groups. by Georges Lion (2010)

There are many examples of paving with figurative drawings in Maurits Cornelis Escher's work (end of the page).

### the Penrose tilings

By assembling two identical golden triangles we get two pairs of quadrilaterals (dart and kite - skinny and fat rhombi) which allow to carry out nice tilings with 5-fold symmetry (stars and suns below).

To get non-periodic tilings of the plane one must use a simple matching rule: tiles have to meet in a way such that each arc of each tile is connected with the arcs on the neighbored tiles. Since both sets use golden triangles, it makes sense that a kite and dart tiling can be translated into a rhombi tiling, and vice versa.
An other example of a tilling of this type designed by Nicolas Hannachi (one my notice an assembling condition).

Discovered by Roger Penrose in 1974 these tilings have curious properties; among them:
• any finite region of a Penrose tiling appears an infinity of times in any Penrose tiling,
• there are small regions with a 5-fold symmetry,
• if the tiling grows infinitely the ratio of the numbers of tiles in each type tends toward φ!
• a rhombi tilling reveals a pattern of two types of overlapping decagons; the ratio of the two populations is φ!

 references: •  Penrose tilings  by Eric Hwang   -   Penrose tilings of the plane  by Ianiv Schweber •  Penrose tiling applet ( download penrose.jar then open it with Java) •  a Penrose tiling generator by J.-L. Sigrist

### an aperiodic tiling by John Conway (1937-2020)

 Divide a right-angle triangle of sides 1, 2 and √5 into five isometric triangles (and similar to the big triangle). Enlarge (factor 2) and divide each of these five triangles in the same way; iterate the process. We thus obtain a nice aperiodic tiling (no invariance by translation) whose triangles can take an infinity of different directions. Simplicity and beauty! Thank you John Conway.     reference: deux minutes pour Jonh Conway (youtube, in French)

### aperiodic tilings with a single tile ("einstein" from German ein Stein, one stone)

reference: "La quête du pavé apériodique unique"  by Jean-Paul Delahaye (in French)  Pour la Science n°433, November 2013

 In the 1970s Roger Penrose showed that it was possible to tile the plane aperiodically by combining only two different tiles. But do aperiodic tilings with a single tile exist? Half a century later, in March 2023, a geometry enthusiast, David Smith, created an "einstein", a discovery confirmed by a team of four mathematicians: the "hat", a 13-sided polygon, assembly of eight kites arranged in a hexagonal network. One detail, however, bothered the purists: these aperiodic tilings used hats, some of which were turned over and which could therefore be considered as a second tile. A few weeks later the problem was solved with the creation of a pure "einstein" linked to the hat that its inventors named "spectre". This polygon has 14 sides of the same length and its angles measure 90°, 120° or 180° (starting from the vertex of the flat angle, we have alternately 120° and 90°). All tilings formed with this tile or its reflection are aperiodic, and no tiling composed of the spectre and its reflection is.     reference: A chiral aperiodic monotile (David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss)

### tilings by Maurits Cornelis Escher

 The Cairo tiling in two drawings by M.C. Escher (1967)
 Fishes by M.C. Escher (1938, 1940 & 1943) Circle Limit III  (woodcut by M.C. Escher - 1959) This drawing also depicts a tiling, but here we are in the Poincaré disk, a model of the hyperbolic plane. Another world... • Hyperbolic tessellations  by David E. Joyce •  Hyperbolic planar tessellations  by Don Hatch

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