Deltoidal Trihexagonal Tiling on:
[Wikipedia]
[Google]
[Amazon]
In
There is one related 2-uniform tiling, having hexagons dissected into six triangles. The ''rhombitrihexagonal tiling'' is also related to the truncated trihexagonal tiling by replacing some of the hexagons and surrounding squares and triangles with dodecagons:
The deltoidal trihexagonal tiling is a dual of the semiregular tiling known as the rhombitrihexagonal tiling. 
The ''deltoidal trihexagonal tiling'' is a part of a set of uniform dual tilings, corresponding to the dual of the rhombitrihexagonal tiling.
(Chapter 21, Naming Archimedean and Catalan polyhedra and tilings. * * * * Keith Critchlow, ''Order in Space: A design source book'', 1970, p. 69-61, Pattern N, Dual p. 77-76, pattern 2 * Dale Seymour and
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane
In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...
. There are one triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
, two square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
s, and one hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A regular hexagon is de ...
on each vertex. It has Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...
of rr.
John Conway calls it a rhombihexadeltille.Conway, 2008, p288 table It can be considered a cantellated by Norman Johnson's terminology or an expanded hexagonal tiling
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a Truncation (geometry), truncated triangular tiling ...
by Alicia Boole Stott
Alicia Boole Stott (8 June 1860 – 17 December 1940) was a British mathematician. She made a number of contributions to the field and was awarded an honorary doctorate from the University of Groningen. She grasped four-dimensional geometry from ...
's operational language.
There are three regular and eight semiregular tilings in the plane.
Uniform colorings
There is only oneuniform coloring
In geometry, a uniform coloring is a property of a uniform figure ( uniform tiling or uniform polyhedron) that is colored to be vertex-transitive. Different symmetries can be expressed on the same geometric figure with the faces following diff ...
in a rhombitrihexagonal tiling. (Naming the colors by indices around a vertex (3.4.6.4): 1232.)
With edge-colorings there is a half symmetry form (3*3) orbifold notation
In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by John Horton Conway, John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curv ...
. The hexagons can be considered as truncated triangles, t with two types of edges. It has Coxeter diagram
Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century.
Coxeter was born in England and educated ...
, Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...
s2. The bicolored square can be distorted into isosceles trapezoid
In Euclidean geometry, an isosceles trapezoid is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides. It is a special case of a trapezoid. Alternatively, it can be defined as a trapezoid in which both legs and bo ...
s. In the limit, where the rectangles degenerate into edges, a triangular tiling
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilater ...
results, constructed as a snub triangular tiling, .
Examples
Related tilings
There is one related 2-uniform tiling, having hexagons dissected into six triangles. The ''rhombitrihexagonal tiling'' is also related to the truncated trihexagonal tiling by replacing some of the hexagons and surrounding squares and triangles with dodecagons:
Circle packing
The rhombitrihexagonal tiling can be used as acircle packing
In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that no circle can be enlarged without creating an overlap. The associated ''packing den ...
, placing equal diameter circles at the center of every point. Every circle is in contact with four other circles in the packing (kissing number
In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere. For a given sphere packing (arrangement o ...
).Order in Space: A design source book, Keith Critchlow, p.74-75, pattern B The translational lattice domain (red rhombus) contains six distinct circles.
:
Wythoff construction
There are eightuniform tiling
In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive.
Uniform tilings can exist in both the Euclidean plane and hyperbolic plane. Uniform tilings are related to t ...
s that can be based from the regular hexagonal tiling (or the dual triangular tiling
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilater ...
).
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are eight forms, seven topologically distinct. (The ''truncated triangular tiling'' is topologically identical to the hexagonal tiling.)
Symmetry mutations
This tiling is topologically related as a part of sequence of cantellated polyhedra with vertex figure (3.4.n.4), and continues as tilings of thehyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P' ...
. These vertex-transitive
In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face i ...
figures have (*n32) reflectional symmetry
Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
.
Deltoidal trihexagonal tiling
The deltoidal trihexagonal tiling is a dual of the semiregular tiling known as the rhombitrihexagonal tiling. Conway
Conway may refer to:
Places
United States
* Conway, Arkansas
* Conway County, Arkansas
* Lake Conway, Arkansas
* Conway, Florida
* Conway, Iowa
* Conway, Kansas
* Conway, Louisiana
* Conway, Massachusetts
* Conway, Michigan
* Conway Townshi ...
called it a tetrille. The edges of this tiling can be formed by the intersection overlay of the regular triangular tiling
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilater ...
and a hexagonal tiling
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a Truncation (geometry), truncated triangular tiling ...
. Each kite
A kite is a tethered heavier than air flight, heavier-than-air craft with wing surfaces that react against the air to create Lift (force), lift and Drag (physics), drag forces. A kite consists of wings, tethers and anchors. Kites often have ...
face of this tiling has angles 120°, 90°, 60° and 90°. It is one of only eight tilings of the plane in which every edge lies on a line of symmetry of the tiling.
The deltoidal trihexagonal tiling is a dual of the semiregular tiling rhombitrihexagonal tiling. Its faces are deltoids or kites
A kite is a tethered heavier than air flight, heavier-than-air craft with wing surfaces that react against the air to create Lift (force), lift and Drag (physics), drag forces. A kite consists of wings, tethers and anchors. Kites often have ...
.
:
Related polyhedra and tilings
It is one of seven dual uniform tilings in hexagonal symmetry, including the regular duals. This tiling has face transitive variations, that can distort the kites into bilateral trapezoids or more general quadrilaterals. Ignoring the face colors below, the fully symmetry is p6m, and the lower symmetry is p31m with three mirrors meeting at a point, and threefold rotation points.Tilings and patterns
''Tilings and patterns'' is a book by mathematicians Branko Grünbaum and Geoffrey Colin Shephard published in 1987 by W.H. Freeman. The book was 10 years in development, and upon publication it was widely reviewed and highly acclaimed.
Structu ...
This tiling is related to the trihexagonal tiling
In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons. See in particular Theorem 2.1.3, p. 59 (classification of uniform tilings); Figure 2.1.5, p.63 (illustration of this tiling), Theorem 2 ...
by dividing the triangles and hexagons into central triangles and merging neighboring triangles into kites.
:
The ''deltoidal trihexagonal tiling'' is a part of a set of uniform dual tilings, corresponding to the dual of the rhombitrihexagonal tiling.
Symmetry mutations
This tiling is topologically related as a part of sequence of tilings withface configuration
In geometry, a vertex configuration is a shorthand notation for representing a polyhedron or Tessellation, tiling as the sequence of Face (geometry), faces around a Vertex (geometry), vertex. It has variously been called a vertex description, vert ...
s V3.4.n.4, and continues as tilings of the hyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P' ...
. These face-transitive
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its Face (geometry), faces are the same. More specifically, all faces must be not ...
figures have (*n32) reflectional symmetry
Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
.
Other deltoidal (kite) tiling
Other deltoidal tilings are possible. Point symmetry allows the plane to be filled by growing kites, with the topology as asquare tiling
In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane consisting of four squares around every vertex. John Horton Conway called it a quadrille.
Structure and properties
The square tili ...
, V4.4.4.4, and can be created by crossing string of a dream catcher. Below is an example with dihedral hexagonal symmetry.
Another face transitive tiling with kite faces, also a topological variation of a square tiling and with face configuration
In geometry, a vertex configuration is a shorthand notation for representing a polyhedron or Tessellation, tiling as the sequence of Face (geometry), faces around a Vertex (geometry), vertex. It has variously been called a vertex description, vert ...
V4.4.4.4. It is also vertex transitive
In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...
, with every vertex containing all orientations of the kite face.
See also
*Tilings of regular polygons
Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his (Latin: ''The Harmony of the World'', 1619).
Notation of Euclidean tilings
Eucl ...
* List of uniform tilings
This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings.
There are three regular and eight semiregular tilings in the plane. The semiregular tilings form new tilings from their du ...
Notes
References
* (Chapter 2.1: ''Regular and uniform tilings'', p. 58-65) * p40 * John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008,(Chapter 21, Naming Archimedean and Catalan polyhedra and tilings. * * * * Keith Critchlow, ''Order in Space: A design source book'', 1970, p. 69-61, Pattern N, Dual p. 77-76, pattern 2 * Dale Seymour and
Jill Britton
Jill E. Britton (6 November 1944 – 29 February 2016) was a Canadian mathematics education, mathematics educator known for her educational books about mathematics.
Career
Britton was born on 6 November 1944. She taught for many years, at Dawson ...
, ''Introduction to Tessellations'', 1989, , pp. 50–56, dual p. 116
{{Tessellation
Euclidean tilings
Isohedral tilings
Semiregular tilings