Uniform Tilings of The Euclidean Plane
There are symmetry groups on the Euclidean plane constructed from fundamental triangles: (4 4 2), (6 3 2), and (3 3 3). Each is represented by a set of lines of reflection that divide the plane into fundamental triangles.
These symmetry groups create 3 regular tilings, and 7 semiregular ones. A number of the semiregular tilings are repeated from different symmetry constructors.
A prismatic symmetry group represented by (2 2 2 2) represents by two sets of parallel mirrors, which in general can have a rectangular fundamental domain. It generates no new tilings.
A further prismatic symmetry group represented by (∞ 2 2) which has an infinite fundamental domain. It constructs two uniform tilings, the apeirogonal prism and apeirogonal antiprism.
The stacking of the finite faces of these two prismatic tilings constructs one non-Wythoffian uniform tiling of the plane. It is called the elongated triangular tiling, composed of alternating layers of squares and triangles.
Right angle fundamental triangles: (p q 2)
| (p q 2) | Fund. triangles |
Parent | Truncated | Rectified | Bitruncated | Birectified (dual) |
Cantellated | Omnitruncated |
Snub |
|---|---|---|---|---|---|---|---|---|---|
| Wythoff symbol | q | p 2 | 2 q | p | 2 | p q | 2 p | q | p | q 2 | p q | 2 | p q 2 | | | p q 2 | |
| Schläfli symbol | t0{p,q} | t0,1{p,q} | t1{p,q} | t1,2{p,q} | t2{p,q} | t0,2{p,q} | t0,1,2{p,q} | s{p,q} | |
| Coxeter-Dynkin diagram | |||||||||
| Vertex figure | pq | (q.2p.2p) | (p.q.p.q) | (p.2q.2q) | qp | (p.4.q.4) | (4.2p.2q) | (3.3.p.3.q) | |
| Square tiling (4 4 2) |
V4.8.8 |
{4,4} |
4.8.8 |
4.4.4.4 |
4.8.8 |
{4,4} |
4.4.4.4 |
4.8.8 |
3.3.4.3.4 |
| Hexagonal tiling (6 3 2) |
V4.6.12 |
{6,3} |
3.12.12 |
3.6.3.6 |
6.6.6 |
{3,6} |
3.4.6.4 |
4.6.12 |
3.3.3.3.6 |
General fundamental triangles: (p q r)
| Wythoff symbol (p q r) |
Fund. triangles |
q | p r | r q | p | r | p q | r p | q | p | q r | p q | r | p q r | | | p q r |
|---|---|---|---|---|---|---|---|---|---|
| Coxeter-Dynkin diagram | |||||||||
| Vertex figure | (p.q)r | (r.2p.q.2p) | (p.r)q | (q.2r.p.2r) | (q.r)p | (q.2r.p.2r) | (r.2q.p.2q) | (3.r.3.q.3.p) | |
| Triangular (3 3 3) |
V6.6.6 |
(3.3)3 |
3.6.3.6 |
(3.3)3 |
3.6.3.6 |
(3.3)3 |
3.6.3.6 |
6.6.6 |
3.3.3.3.3.3 |
Non-simplical fundamental domains
The only possible fundamental domain in Euclidean 2-space that is not a simplex is the rectangle (∞ 2 ∞ 2), with Coxeter-Dynkin diagram: . All forms generated from it become a square tiling.
Read more about this topic: Uniform Tiling
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Thus concepts linked like chainmail in the mind.”
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—Friedrich Nietzsche (1844–1900)