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Basic shape(s) used in a tessellation
In mathematics, a prototile is one of the shapes of a tile in a tessellation. A tessellation of the plane or of any other space is a cover of the space
Prototile
Question about single-shape aperiodic tiling
problem asks about the existence of a single prototile that by itself forms an aperiodic set of prototiles; that is, a shape that can tessellate space
Einstein_problem
Covering by shapes without overlaps or gaps
tile shapes that cannot form a repeating pattern (an aperiodic set of prototiles). A tessellation of space, also known as a space filling or honeycomb
Tessellation
Rule from the theory of the tiling of the plane
English mathematician John Horton Conway, is a sufficient rule for when a prototile will tile the plane. It consists of the following requirements: The tile
Conway_criterion
Polyhedron formed by joining two prisms
}{6}}=150^{\circ }} . The Schmitt–Conway–Danzer biprism, also called an SCD prototile, is a polyhedron combinatorially equivalent to the gyrobifastigium, but
Gyrobifastigium
Form of plane tiling without repeats at scale
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_tiling
Tiling of the plane by pentagons
5, 6, 7, 8, 9, and 13 allow parametric possibilities with nonconvex prototiles. Periodic tilings are characterised by their wallpaper group symmetry
Pentagonal_tiling
Set of tile shapes that can create nonrepeating patterns
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
Aperiodic_set_of_prototiles
Non-periodic tiling of the plane
only a finite number of shapes. These shapes are called prototiles, and a set of prototiles is said to admit a tiling or tile the plane if there is a
Penrose_tiling
Type of space-filling polyhedron
the Voronoi diagram forms a honeycomb in which there is only a single prototile shape, the shape of these Voronoi cells. This shape is called a plesiohedron
Plesiohedron
Method for constructing tilings
tile substitutions generate aperiodic tilings, which are tilings whose prototiles do not admit any tiling with translational symmetry. The most famous of
Substitution_tiling
Natural number
regular octagon. The Ammann–Beenker tiling is a nonperiodic tesselation of prototiles that feature prominent octagonal silver eightfold symmetry, that is the
8
Quadrilateral symmetric across a diagonal
angles. Kites of two shapes (one convex and one non-convex) form the prototiles of one of the forms of the Penrose tiling. Kites also form the faces of
Kite_(geometry)
Mathematical spiral tiling
tessellates the plane with congruent copies of itself. In this case, the prototile is an elongated irregular nonagon, or nine-sided figure. The most interesting
Voderberg_tiling
Number, approximately 1.618
together. Several variations of this tiling have been studied, all of whose prototiles exhibit the golden ratio: Penrose's original version of this tiling used
Golden_ratio
Topics referred to by the same term
Aperiodic frequency Aperiodic graph Aperiodic semigroup Aperiodic set of prototiles Aperiodic tiling Periodic (disambiguation) Strange attractor, a region
Aperiodic_(disambiguation)
Regular tiling of a two-dimensional space
hexagonal tiling has a structure consisting of a regular hexagon only as its prototile, sharing two vertices with other identical ones, an example of monohedral
Hexagonal_tiling
Prism with a 3-sided base
Delone set in order to create a honeycomb. The triangular prism is a prototile of the triangular prismatic honeycomb. The gyrobifastigium, constructed
Triangular_prism
Geometric shape formed from five squares
Rhoads, Glenn C. (2003). Planar Tilings and the Search for an Aperiodic Prototile. PhD dissertation, Rutgers University. Gardner, Martin (August 1975).
Pentomino
Non-periodic tiling of the plane
In geometry, an Ammann A1 tiling is a tiling from the 6-piece prototile set shown on the right. They were found in 1977 by Robert Ammann. Ammann was inspired
Ammann_A1_tilings
Regular tiling of the Euclidean plane
The square tiling has a structure consisting of one type of congruent prototile, the square, sharing two vertices with other identical ones. This is an
Square_tiling
Overview of and topical guide to geometry
Trapezoid Isosceles trapezoid Sangaku Straightedge Symmedian Tessellation Prototile Aperiodic tiling Wang tile Penrose tiling Trapezoid (trapezium) Isosceles
Outline_of_geometry
Tiling of the hyperbolic plane
called a monohedral tiling, and the shape of the tiles is called the prototile of the tiling. The binary tilings are monohedral tilings of the hyperbolic
Binary_tiling
Non-periodic tiling of the plane
\varphi ={\frac {a}{b}}={\frac {1+{\sqrt {5}}}{2}}\approx 1.618.} The prototiles are Robinson triangles, but the relationship is different: The Penrose
Tübingen_triangle
Geometric shape formed from six squares
Rhoads, Glenn C. (2003). Planar Tilings and the Search for an Aperiodic Prototile. PhD dissertation, Rutgers University. Mathematische Basteleien: Hexominos
Hexomino
Shape subdivided into copies of itself
rep-n if the dissection uses n copies. Such a shape necessarily forms the prototile for a tiling of the plane, in many cases a nonperiodic tiling. A rep-tile
Rep-tile
Geometric shape formed from squares
Rhoads, Glenn C. (2003). Planar Tilings and the Search for an Aperiodic Prototile. PhD dissertation, Rutgers University. Grünbaum and Shephard, section
Polyomino
Mathematics book
§1.1 Tiling, Euclidean plane, packing, covering, toplogical disk, §1.2 prototile, regular tiling, monohedral tiling, k-isohedral tiling (face-transitive)
Tilings_and_patterns
Nonperiodic substitution tiling
is a nonperiodic substitution tiling created from L-tromino prototiles. These prototiles are examples of rep-tiles and so an iterative process of decomposing
Chair_tiling
British astronomer and mathematician (1801–1898)
he found may be generated by overlaying a regular square tiling whose prototile is the larger square with a Pythagorean tiling generated by the two smaller
Henry_Perigal
Topics referred to by the same term
from its Italian name Partito Democratico Cristiano Sammarinese) SCD prototile, a space-filling polyhedron South Carolina Department of Public Safety
SCDP
English mathematician (1937–2020)
he devised the Conway criterion which is a fast way to identify many prototiles that tile the plane. He investigated lattices in higher dimensions and
John_Horton_Conway
Non-periodic tiling of the plane
nonperiodic tiling which can be generated either by an aperiodic set of prototiles as done by Robert Ammann in the 1970s, or by the cut-and-project method
Ammann–Beenker_tiling
On lattices and sphere packing in Euclidean space
symmetries.[citation needed] Equivalently, it asks the existence of a single prototile that forms an aperiodic set, a shape that can tessellate space but only
Hilbert's_eighteenth_problem
the sides of a triangle Conway criterion – a criterion for identifying prototiles that admit a periodic tiling Conway group – any of the groups Co0, Co1
List of things named after John Horton Conway
List_of_things_named_after_John_Horton_Conway
Set of shapes that can be tiled with smaller replicas of the same set
their pieces form substitution tilings, or tessellations in which the prototiles can be dissected or combined so as to yield smaller or larger duplicates
Self-tiling_tile_set
2015 mathematics book by Akiyama and Matsunaga
problems, wallpaper groups, pentagonal tilings, the Conway criterion for prototiles and Escher-like tilings of the plane by animal-shaped figures, aperiodic
Treks_into_Intuitive_Geometry
Semiregular tiling of a plane
tessellations, tessellations generated by reflections across each edge of a prototile. It is one of 7 dual uniform tilings in hexagonal symmetry, including
Truncated_hexagonal_tiling
Tiling by squares of two sizes
Rigby found several prototiles, including the Koch snowflake, that may be used to tile the plane only by using copies of the prototile in two or more different
Pythagorean_tiling
Hungarian mathematician (1915–2005)
2-dimensional regular polytope) A semi-regular tessellation with three prototiles: a triangle, a square and a hexagon. The other section, entitled "Genetics
László_Fejes_Tóth
Image Description Dimension Packing constant Comments Monohedral prototiles all 1 Shapes such that congruent copies can form a tiling of space Circle,
List of shapes with known packing constant
List_of_shapes_with_known_packing_constant
Non-periodic tiling in geometry
multiple of π {\displaystyle \pi } . Radin found a collection of five prototiles, each of which is a marking of T {\displaystyle T} , so that the matching
Pinwheel_tiling
the tiles induces another Euclidean graph.) If there are finitely many prototiles in the tessellation, and the tessellation is periodic, then the resulting
Periodic_graph_(geometry)
algorithms for generating the following structures: Polyominos, polyiamond prototiles, and polyhex (mathematics) hydrocarbon molecules. Topological orderings
Reverse-search_algorithm
PROTOTILE
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PROTOTILE
Girl/Female
Indian
Abbreviation of Hindu Lord Shiva Mantra aumn namah shivay
Boy/Male
Tamil
Thought, Meditation
Male
Arthurian
, (Sir), knight of Arthur.
Surname or Lastname
Jewish (Ashkenazic)
Jewish (Ashkenazic) : patronymic from the Yiddish personal name Mikhl (see Michael).English : patronymic from the medieval personal name Michel (see Mitchell).Dutch : patronymic from the personal name Michel, a Dutch variant of Michael.Americanized spelling of Michelsen.
Boy/Male
British, English
Spear-rule
Girl/Female
Hindu, Indian
Symbol of Love; Love; Affection
Boy/Male
English
Lives in the valley of the majestic one.
Surname or Lastname
English
English : variant of Ovett (see Oviatt).
Boy/Male
Celtic American English Irish Welsh
From the fortress.
Surname or Lastname
English
English : possibly a variant of Mares.
PROTOTILE
PROTOTILE
PROTOTILE
PROTOTILE
PROTOTILE