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There is one related
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, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...
. There are one triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colli ...
, two square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
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'' h ...
on each vertex
Vertex, vertices or vertexes may refer to:
Science and technology Mathematics and computer science
*Vertex (geometry), a point where two or more curves, lines, or edges meet
*Vertex (computer graphics), a data structure that describes the position ...
. It has Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mor ...
of rr.
John Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
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 truncated triangular tiling).
English mathema ...
by Alicia Boole Stott
Alicia Boole Stott (8 June 1860 – 17 December 1940) was an Irish mathematician. Despite never holding an academic position, she made a number of valuable contributions to the field, receiving an honorary doctorate from the University of Gron ...
's operational language.
There are 3 regular
The term regular can mean normal or in accordance with rules. It may refer to:
People
* Moses Regular (born 1971), America football player
Arts, entertainment, and media Music
* "Regular" (Badfinger song)
* Regular tunings of stringed instrum ...
and 8 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 differ ...
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 Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advant ...
. 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 and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
, Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mor ...
s2. The bicolored square can be distorted into isosceles trapezoid
In Euclidean geometry, an isosceles trapezoid (isosceles trapezium in British English) 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 defi ...
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
A ''k''-uniform tiling is a tiling of Tessellation, tilings of the plane by convex regular polygons, connected edge-to-edge, with ''k'' types of vertices. The 1-uniform tiling include 3 regular tilings, and 8 semiregular tilings. A 1-uniform tilin ...
, having hexagons dissected into 6 triangles. The ''rhombitrihexagonal tiling'' is also related to the truncated trihexagonal tiling
In geometry, the truncated trihexagonal tiling is one of eight semiregular tilings of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex. It has Schläfli symbol of ''tr''.
Names
Uniform colorings
The ...
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 de ...
, placing equal diameter circles at the center of every point. Every circle is in contact with 4 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 6 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 space, hyperbolic plane. Uniform tilings ar ...
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 8 forms, 7 which are 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 fa ...
figures have (*n32) reflectional symmetry.
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
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* Conway, Iowa
* Conway, Kansas
* Conway, Louisiana
* Conway, Massachusetts
* Conway, Michigan
* Conway Town ...
calls 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 truncated triangular tiling).
English mathema ...
. Each kite
A kite is a tethered heavier than air flight, heavier-than-air or lighter-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. ...
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 or lighter-than-air craft with wing surfaces that react against the air to create lift and drag forces. A kite consists of wings, tethers and anchors. Kites often have a bridle and tail to guide the face ...
.
:
Related polyhedra and tilings
It is one of 7 dual uniform tilings in hexagonal symmetry, including the regular duals. This tiling hasface 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 faces are the same. More specifically, all faces must be not merely congruent ...
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 3 mirrors meeting at a point, and 3-fold rotation points.Tilings and Patterns
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 configurationCrystallography ...
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 faces are the same. More specifically, all faces must be not merely congrue ...
figures have (*n32) reflectional symmetry.
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. It has Schläfli symbol of meaning it has 4 squares around every vertex.
Conway called it a quadrille.
The internal angle of th ...
, 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
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 faces are the same. More specifically, all faces must be not merely congruent ...
tiling with kite faces, also a topological variation of a square tiling and with face configuration
In geometry, a vertex configurationCrystallography ...
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 '' Harmonices Mundi'' ( Latin: ''The Harmony of the World'', 1619).
Notation of ...
* 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 dual ...
Notes
References
* (Chapter 2.1: ''Regular and uniform tilings'', p. 58-65) * p40 * John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ''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 educator known for her educational books about mathematics.
Career
Britton was born on 6 November 1944. She taught for many years, at Dawson College in Westmount ...
, ''Introduction to Tessellations'', 1989, , pp. 50–56, dual p. 116
{{Tessellation
Euclidean tilings
Isohedral tilings
Semiregular tilings