Aperiodic set of prototilesA 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.
Regular polygonEdges and verticesn{\displaystyle n}Schläfli symbol{n}{\displaystyle \{n\}}Coxeter–Dynkin diagramSymmetry groupDn, order 2nDual polygonSelf-dualArea (with side length s{\displaystyle s})A=14ns2cot(πn){\displaystyle A={\tfrac {1}{4}}ns^{2}\cot \left({\frac {\pi }{n}} ight)}Internal angle(n−2)×πn{\displaystyle (n-2)\times {\frac {\pi }{n}}}Internal angle sum(n−2)×π{\displaystyle \left(n-2 ight)\times {\pi }}Inscribed circle diameterdIC=scot(πn){\displaystyle d_{\text{IC}}=s\cot \left({\frac {\pi }{n}} ight)}Circumscribed circle diameterdOC=scsc(πn){\displaystyle d_{\text{OC}}=s\csc \left({\frac {\pi }{n}} ight)}PropertiesConvex, cyclic, equilateral, isogonal, isotoxalIn Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length).
Aperiodic tilingIn the mathematics of tessellations, a non-periodic tiling is a tiling that does not have any translational symmetry. An aperiodic set of prototiles is a set of tile-types that can tile, but only non-periodically. The tilings produced by one of these sets of prototiles may be called aperiodic tilings. The Penrose tilings are a well-known example of aperiodic tilings.
Wallpaper groupClassification of a two-dimensional repetitive patternPrimitive cells of the seventeen wallpaper groupsA wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, especially in textiles, tiles, and wallpaper.
Plane (mathematics)In mathematics, a plane is a two-dimensional space or flatsurface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. When working exclusively in two-dimensional Euclidean space, the definite article is used, so theEuclidean plane refers to the whole space. Several notions of a plane may be defined.
Euclidean tilings by convex regular polygonsTilings of the Euclidean plane by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his Harmonice Mundi (Latin: The Harmony of the World, 1619).
SurfaceA surface, as the term is most generally used, is the outermost or uppermost layer of a physical object. It is the portion or region of the object that can first be observed and with which other objects first interact. The concept of surface has been abstracted and formalized in mathematics, specifically in geometry.
MathematicsMathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.
Honeycomb (geometry)In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space. Honeycombs are usually constructed in ordinary Euclidean ("flat") space. They may also be constructed in non-Euclidean spaces, such as hyperbolic honeycombs.
ShapeA shape is a graphical representation of an object's form or its external boundary, outline, or external surface. It is distinct from other object properties, such as color, texture, or material type. In geometry, shape excludes information about the object's position, size, orientation and chirality. A figure is a representation including both shape and size (as in, e.g., figure of the Earth).
DimensionProperty of a mathematical space From left to right: a square, a cube and a tesseract. The square is two-dimensional (2D) and bounded by one-dimensional line segments; the cube is three-dimensional (3D) and bounded by two-dimensional squares; the tesseract is four-dimensional (4D) and bounded by three-dimensional cubes. The first four spatial dimensions, represented in a two-dimensional picture.
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