math chick | Future-Giraffes

time to get really serial….

we decided to give you an extra piece of super serial wisdom for your TRUE architecture….

so some experience points that are going to be serially appreciated by all of those parammetric/grasshopper freaks out there….

Penrose tilings

A Penrose tiling is a non-periodic tiling generated by an aperiodic set of prototiles. Penrose tilings are named after mathematician and physicist Roger Penrose who investigated these sets in the 1970s. The aperiodicity of the Penrose prototiles implies that a shifted copy of a Penrose tiling will never match the original. A Penrose tiling may be constructed so as to exhibit both reflection symmetry and fivefold rotational symmetry, as in the diagram below.

A Penrose tiling has many remarkable properties, most notably:

It is non-periodic, which means that it lacks any translational symmetry. More informally, a shifted copy will never match the original.
It is self-similar, so the same patterns occur at larger and larger scales. Thus, the tiling can be obtained through “inflation” (or “deflation”) and any finite patch from the tiling occurs infinitely many times.
It is a quasicrystal: implemented as a physical structure a Penrose tiling will produce Bragg diffraction and its diffractogram reveals both the fivefold symmetry and the underlying long range order.

Penrose tilings are simple examples of aperiodic tilings of the plane.A tiling is a covering of the plane by tiles with no overlaps or gaps; the tiles normally have a finite number of shapes, called prototiles, and a set of prototiles is said to admit a tiling or tile the plane if there is a tiling of the plane using only tiles congruent to these prototiles. The most familiar tilings (e.g., by squares or triangles) are periodic: a perfect copy of the tiling can be obtained by translating all of the tiles by a fixed distance in a given direction. Such a translation is called a period of the tiling; more informally, this means that a finite region of the tiling repeats itself in periodic intervals. If a tiling has no periods it is said to be non-periodic. A set of prototiles is said to beaperiodic if it tiles the plane, but every such tiling is non-periodic; tilings by aperiodic sets of prototiles are called aperiodic tilings.

Early aperiodic tables

The subject of aperiodic tilings received new interest in the 1960s when logician Hao Wang noted connections betweendecision problems and tilings. In particular, he introduced tilings by square plates with colored edges, now known as Wang dominoes or tiles, and posed the “Domino Problem“: to determine whether a given set of Wang dominoes could tile the plane with matching colors on adjacent domino edges. He observed that if this problem were undecidable, then there would have to exist an aperiodic set of Wang dominoes. At the time, this seemed implausible, so Wang conjectured no such set could exist.

an aperiodic set of Wang dominoes

Robinson’s six prototiles

The first Penrose tiling(P1 below) is also an aperiodic set of six prototiles, introduced by Roger Penrose in a 1974 paper, but is based on pentagons rather than squares. Any attempts to tile the plane with regular pentagons will necessarily leave gaps, but Johannes Kepler showed, in his 1619 work Harmonices Mundi, that these gaps could be filled using pentagrams (viewed as star polygons), decagons and related shapes. Acknowledging inspiration from Kepler, Penrose was able to find matching rules (which can be imposed by decorations of the edges) for these shapes, in order to obtain an aperiodic set; his tiling can be viewed as a completion of Kepler’s finite Aa pattern, and other traces of these ideas can be found in Albrecht Dürer’s work.

Penrose subsequently reduced the number of prototiles to two, discovering the kite and dart tiling(P2) and the rhombus tiling(P3). The rhombus tiling was independently discovered by Robert Ammann in 1976. Penrose and John H. Conway investigated the properties of Penrose tilings, and discovered that a substitution property explained their hierarchical nature; their findings were publicized by Martin Gardner in his January 1977 “Mathematical Games” column in Scientific American.

In 1981, De Bruijn explained a method to construct Penrose tilings from five families of parallel lines as well as a “cut and project method”, in which Penrose tilings are obtained as two-dimensional projections from a five-dimensional cubic structure. In this approach, the Penrose tiling is viewed as a set of points, its vertices, while the tiles are geometrical shapes obtained by connecting vertices with edges.

The pentagonal Penrose tiling (P1) drawn in black on a colored rhombus tiling (P3) with yellow edges.

A P1 tiling using Penrose’s original set of 6 prototiles

Pentagon with an inscribed thick rhomb (light), acute Robinson triangles (lightly shaded) and a small obtuse Robinson triangle (darker). Dotted lines give additional edges for inscribed kites and darts.