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 elongated triangular tiling is a

semiregular tiling 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 ...

of the Euclidean plane. There are three triangles and two squares on each vertex. It is named as 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 ...

elongated by rows of squares, and given

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 ...

:e.

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 ...

calls it a isosnub quadrille.Conway, 2008, p.288 table There are 3

regular Regular may refer to: Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses * Regular character, ...

and 8 semiregular tilings in the plane. This tiling is similar to the

snub square tiling In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex. Its Schläfli symbol is ''s''. Conway calls it a snub quadrille, constructed by a snub operation appli ...

which also has 3 triangles and two squares on a vertex, but in a different order.

Construction

It is also the only convex

uniform 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 ...

that can not be created as a

Wythoff construction In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process ...

. It can be constructed as alternate layers of

apeirogonal prism In geometry, an apeirogonal prism or infinite prism is the arithmetic limit of the family of prisms; it can be considered an infinite polyhedron or a tiling of the plane.Conway (2008), p.263 Thorold Gosset called it a ''2-dimensional semi-chec ...

s and

apeirogonal antiprism In geometry, an apeirogonal antiprism or infinite antiprismConway (2008), p. 263 is the arithmetic limit of the family of antiprisms; it can be considered an infinite polyhedron or a tiling of the plane. If the sides are equilateral triangles, ...

Uniform colorings

There is one

uniform 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 ...

s of an elongated triangular tiling. Two 2-uniform colorings have a single vertex figure, 11123, with two colors of squares, but are not 1-uniform, repeated either by reflection or glide reflection, or in general each row of squares can be shifted around independently. The 2-uniform tilings are also called

Archimedean 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 ...

s. There are infinite variations of these Archimedean colorings by arbitrary shifts in the square row colorings.

Circle packing

The elongated triangular tiling can be used as a

circle 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 5 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, circle pattern F : 1-uniform-8-circlepack

Related tilings

Sections of stacked triangles and squares can be combined into radial forms. This mixes two vertex configurations, 3.3.3.4.4 and 3.3.4.3.4 on the transitions. Twelve copies are needed to fill the plane with different center arrangements. The duals will mix in

cairo pentagonal tiling In geometry, a Cairo pentagonal tiling is a tessellation of the Euclidean plane by congruent convex pentagons, formed by overlaying two tessellations of the plane by hexagons and named for its use as a paving design in Cairo. It is also called Ma ...

pentagons.

Symmetry mutations

It is first in a series of symmetry mutations with hyperbolic uniform tilings with 2*''n''2

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 ...

symmetry,

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...

4.''n''.4.3.3.3, and

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 ...

. Their duals have hexagonal faces in the hyperbolic plane, 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.''n''.4.3.3.3. There are four related 2-uniform tilings, mixing 2 or 3 rows of triangles or squares.

Prismatic pentagonal tiling

The prismatic pentagonal tiling is a dual uniform tiling in the Euclidean plane. It is one of 15 known

isohedral 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 congruen ...

pentagon tiling In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon. A regular tiling, regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a Pentagon#Regular ...

s. It can be seen as a stretched

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 ...

with a set of parallel bisecting lines through the hexagons.

calls it an . Each of its pentagonal

faces The face is the front of the head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affect the ...

has three 120° and two 90° angles. It is related to the

Cairo pentagonal tiling In geometry, a Cairo pentagonal tiling is a tessellation of the Euclidean plane by congruent convex pentagons, formed by overlaying two tessellations of the plane by hexagons and named for its use as a paving design in Cairo. It is also called Ma ...

with

V3.3.4.3.4.

Geometric variations

Monohedral

pentagonal tiling In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon. A regular tiling, regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a Pentagon#Regular ...

type 6 has the same topology, but two edge lengths and a lower p2 (2222)

wallpaper group A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetry, symmetries in the pattern. Such patterns occur frequently in architecture a ...

symmetry:

Related 2-uniform dual tilings

There are four related 2-uniform dual tilings, mixing in rows of squares or hexagons (the prismatic pentagon is half-square half-hexagon).

Notes

References

* (Chapter 2.1: ''Regular and uniform tilings'', p. 58-65) * p37 * John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008,

* Keith Critchlow, ''Order in Space: A design source book'', 1970, p. 69-61, Pattern Q₂, Dual p. 77-76, pattern 6 * 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

External links

* * * {{Tessellation Euclidean tilings Isogonal tilings Semiregular tilings Triangular tilings