Kisrhombille tiling

(Redirected from Hexakis hexagonal tiling)
Kisrhombille tiling
Type Dual semiregular tiling
Coxeter-Dynkin diagram
Faces 30-60-90 triangle
Face configuration V4.6.12
Symmetry group p6m, [6,3], (*632)
Rotation group p6, [6,3]+, (632)
Dual truncated trihexagonal tiling
Properties face-transitive

In geometry, the kisrhombille tiling or 3-6 kisrhombille tiling is a tiling of the Euclidean plane. It is constructed by congruent 30-60 degree right triangles with 4, 6, and 12 triangles meeting at each vertex.

Construction from rhombille tiling

Conway calls it a kisrhombille[1] for his kis vertex bisector operation applied to the rhombille tiling. More specifically it can be called a 3-6 kisrhombille, to distinguish it from other similar hyperbolic tilings, like 3-7 kisrhombille.

The related rhombille tiling becomes the kisrhombille by subdivding the rhombic faces on it axes into four triangle faces

It can be seen as an equilateral hexagonal tiling with each hexagon divided into 12 triangles from the center point. (Alternately it can be seen as a bisected triangular tiling divided into 6 triangles, or as an infinite arrangement of lines in six parallel families.)

It is labeled V4.6.12 because each right triangle face has three types of vertices: one with 4 triangles, one with 6 triangles, and one with 12 triangles.

Dual tiling

It is the dual tessellation of the truncated trihexagonal tiling which has one square and one hexagon and one dodecagon at each vertex.

Related polyhedra and tilings

The kisrhombille tiling is a part of a set of uniform dual tilings, corresponding to the dual of the truncated trihexagonal tiling.

Uniform hexagonal/triangular tilings
Symmetry: [6,3], (*632) [6,3]+, (632) [1+,6,3], (*333) [6,3+], (3*3)
{6,3} t0,1{6,3} t1{6,3} t1,2{6,3} t2{6,3} t0,2{6,3} t0,1,2{6,3} s{6,3} h{6,3} h1,2{6,3}
Uniform duals
V6.6.6 V3.12.12 V3.6.3.6 V6.6.6 V3.3.3.3.3.3 V3.4.12.4 V.4.6.12 V3.3.3.3.6 V3.3.3.3.3.3

It is also topologically related to a polyhedra sequence defined by the face configuration V4.6.2n. This group is special for having all even number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, and continuing into the hyperbolic plane for any $n \ge 7.$

With an even number of faces at every vertex, these polyhedra and tilings can be shown by alternating two colors so all adjacent faces have different colors.

Each face on these domains also corresponds to the fundamental domain of a symmetry group with order 2,3,n mirrors at each triangle face vertex.

Dimensional family of omnitruncated polyhedra and tilings: 4.6.2n
Symmetry
*n32
[n,3]
Spherical Euclidean Hyperbolic
*232
[2,3]
D3h
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
P6m
*732
[7,3]

*832
[8,3]

*∞32
[∞,3]

Coxeter
Schläfli

t0,1,2{2,3}

t0,1,2{3,3}

t0,1,2{4,3}

t0,1,2{5,3}

t0,1,2{6,3}

t0,1,2{7,3}

t0,1,2{8,3}

t0,1,2{∞,3}
Omnitruncated
figure
Vertex figure 4.6.4 4.6.6 4.6.8 4.6.10 4.6.12 4.6.14 4.6.16 4.6.∞
Dual figures
Coxeter
Omnitruncated
duals
Face
configuration
V4.6.4 V4.6.6 V4.6.8 V4.6.10 V4.6.12 V4.6.14 V4.6.16 V4.6.∞

Practical uses

The bisected hexagonal tiling is a useful starting point for making paper models of deltahedra, as each of the equilateral triangles can serve as faces, the edges of which adjoin isosceles triangles that can serve as tabs for gluing the model together.[citation needed]