Set of tile shapes that can create nonrepeating patterns
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A periodic tiling with a fundamental unit (triangle) and a primitive cell (hexagon) highlighted. A tiling of the entire plane can be generated by fitting copies of these triangular patches together. To do this, the basic triangle must be rotated 60 degrees to fit edge-to-edge to a neighboring triangle. Thus a triangular tiling of fundamental units is generated that is mutually locally derivable from the tiling by the colored tiles. The other figure drawn onto the tiling, the white hexagon, represents a primitive cell of the tiling. Copies of the corresponding patch of coloured tiles can be translated to form an infinite tiling of the plane. It is not necessary to rotate this patch to achieve this.
The Penrose tiles are an aperiodic set of tiles, since they admit only non-periodic tilings of the plane (see next image).All of the infinitely many tilings by the Penrose tiles are aperiodic. That is, the Penrose tiles are an aperiodic set of prototiles.
A set of prototiles is aperiodic if copies of the prototiles can be assembled to create tilings, such that all possible tessellation patterns are non-periodic. The aperiodicity referred to is a property of the particular set of prototiles; the various resulting tilings themselves are just non-periodic.
A given set of tiles, in the Euclidean plane or some other geometric setting, admits a tiling if non-overlapping copies of the tiles in the set can be fitted together to cover the entire space. A given set of tiles might admit periodic tilings — that is, tilings that remain invariant after being shifted by a translation (for example, a lattice of square tiles is periodic). It is not difficult to design a set of tiles that admits non-periodic tilings as well as periodic tilings. (For example, randomly arranged tilings using a 2×2 square and 2×1 rectangle are typically non-periodic.)
However, an aperiodic set of tiles can only produce non-periodic tilings.[1][2] Infinitely many distinct tilings may be obtained from a single aperiodic set of tiles.[3]
The best-known examples of an aperiodic set of tiles are the various Penrose tiles.[4][5] The known aperiodic sets of prototiles are seen on the list of aperiodic sets of tiles. The underlying undecidability of the domino problem implies that there exists no systematic procedure for deciding whether a given set of tiles can tile the plane.
^Senechal, Marjorie (1996) [1995]. Quasicrystals and geometry (corrected paperback ed.). Cambridge University Press. ISBN 978-0-521-57541-6.
^Grünbaum, Branko; Geoffrey C. Shephard (1986). Tilings and Patterns. W.H. Freeman & Company. ISBN 978-0-7167-1194-0.
^A set of aperiodic prototiles can always form uncountably many different tilings, even up to isometry, as proven by Nikolaï Dolbilin in his 1995 paper The Countability of a Tiling Family and the Periodicity of a Tiling
A setofprototiles is aperiodic if copies of the prototiles can be assembled to create tilings, such that all possible tessellation patterns are non-periodic...
arbitrarily large periodic regions or patches. A set of tile-types (or prototiles) is aperiodic if copies of these tiles can form only non-periodic tilings...
einstein problem asks about the existence of a single prototile that by itself forms an aperiodicsetofprototiles; that is, a shape that can tessellate...
"non-periodic". An aperiodic tiling uses a small setof tile shapes that cannot form a repeating pattern (an aperiodicsetofprototiles). A tessellation of space...
technique to obtain an aperiodicsetof just six prototiles. The first Penrose tiling (tiling P1 below) is an aperiodicsetof six prototiles, introduced by Roger...
if it has exactly one prototile. A setofprototiles is said to be aperiodic if every tiling with those prototiles is an aperiodic tiling. In March 2023...
chapters survey a variety of advanced topics in tiling theory: colored patterns and tilings, polygonal tilings, aperiodic tilings, Wang tiles, and tilings...
prototiles. In 1977 Robert Ammann discovered a number ofsetsofaperiodicprototiles, i.e., prototiles with matching rules forcing nonperiodic tilings; in...
ratio, its cross-sections form aperiodic sequences with a similar recursive structure to the Fibonacci word. Generalizations of this tiling to three dimensions...
substitution tiling created from L-tromino prototiles. These prototiles are examples of rep-tiles and so an iterative process of decomposing the L tiles into smaller...
two-dimensional shape that forms the prototile for an aperiodic tiling, but not for any periodic tiling? Falconer's conjecture: setsof Hausdorff dimension greater...
for an AperiodicPrototile. PhD dissertation, Rutgers University. Gardner, Martin (August 1975). "More about tiling the plane: the possibilities of polyominoes...
The A1 tiles are one of five setsof tiles discovered by Ammann and described in Tilings and patterns. The A1 tile set is aperiodic, i.e. they tile the...
Several variations of this tiling have been studied, all of whose prototiles exhibit the golden ratio: Penrose's original version of this tiling used four...
Rhoads, Glenn C. (2003). Planar Tilings and the Search for an AperiodicPrototile. PhD dissertation, Rutgers University. Grünbaum and Shephard, section...
5, 6, 7, 8, 9, and 13 allow parametric possibilities with nonconvex prototiles. Periodic tilings are characterised by their wallpaper group symmetry...
there is a finite (though very large) setof so-called prototiles, with each being obtained by coloring the sides of T {\displaystyle T} , so that the pinwheel...
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