Английская Википедия:3-4-3-12 tiling
| 3-4-3-12 tiling | |
|---|---|
| Файл:2-uniform n2.svg | |
| Type | 2-uniform tiling |
| Vertex configuration | Файл:Vertex type 3-4-3-12.svg Файл:Vertex type 3-12-12.svg 3.4.3.12 and 3.12.12 |
| Symmetry | p4m, [4,4], (*442) |
| Rotation symmetry | p4, [4,4]+, (442) |
| Properties | 2-uniform, 3-isohedral, 3-isotoxal |
In geometry of the Euclidean plane, the 3-4-3-12 tiling is one of 20 2-uniform tilings of the Euclidean plane by regular polygons, containing regular triangles, squares, and dodecagons, arranged in two vertex configuration: 3.4.3.12 and 3.12.12.[1][2][3][4]
The 3.12.12 vertex figure alone generates a truncated hexagonal tiling, while the 3.4.3.12 only exists in this 2-uniform tiling. There are 2 3-uniform tilings that contain both of these vertex figures among one more.
It has square symmetry, p4m, [4,4], (*442). It is also called a demiregular tiling by some authors.
Circle Packing
This 2-uniform tiling can be used as a circle packing. Cyan circles are in contact with 3 other circles (1 cyan, 2 pink), corresponding to the V3.122 planigon, and pink circles are in contact with 4 other circles (2 cyan, 2 pink), corresponding to the V3.4.3.12 planigon. It is homeomorphic to the ambo operation on the tiling, with the cyan and pink gap polygons corresponding to the cyan and pink circles (one dimensional duals to the respective planigons). Both images coincide.
| Circle Packing | Ambo |
|---|---|
| Файл:Circle Packing of Small Star Square Dodecagonal Tiling.png | Файл:Ambo of Small Square Dodecagonal Tiling.png |
Dual tiling
The dual tiling has kite ('ties') and isosceles triangle faces, defined by face configurations: V3.4.3.12 and V3.12.12. The kites meet in sets of 4 around a center vertex, and the triangles are in pairs making planigon rhombi. Every four kites and four isosceles triangles make a square of side length <math>2+\sqrt{3}</math>.
Dual tiling |
Файл:Semiplanigon V.3.4.3.12 (Desmos Generated).png V3.4.3.12 Semiplanigon Файл:Tiling face 3-12-12.svg V3.12.12 Planigon |
This is one of the only dual uniform tilings which only uses planigons (and semiplanigons) containing a 30° angle. Conversely, 3.4.3.12; 3.122 is one of the only uniform tilings in which every vertex is contained on a dodecagon.
Related tilings
It has 2 related 3-uniform tilings that include both 3.4.3.12 and 3.12.12 vertex figures:
| Файл:3-uniform 8.svg 3.4.3.12, 3.12.12, 3.4.6.4 |
Файл:3-uniform 9.svg 3.4.3.12, 3.12.12, 3.3.4.12 |
V3.4.3.12, V3.12.12, V3.4.6.4 |
V3.4.3.12, V3.12.12, V3.3.4.12 |
This tiling can be seen in a series as a lattice of 4n-gons starting from the square tiling. For 16-gons (n=4), the gaps can be filled with isogonal octagons and isosceles triangles.
| 4 | 8 | 12 | 16 | 20 |
|---|---|---|---|---|
| Файл:Square lattice with squares.svg Square tiling Q |
Файл:Square lattice with octagons.svg Truncated square tiling tQ |
Файл:Square lattice with dodecagons.svg 3-4-3-12 tiling |
Файл:Square lattice with 16-gons.svg Twice-truncated square tiling ttQ |
Файл:Square lattice with 20-gons.svg 20-gons, squares trapezoids, triangles |
Notes
References
- Keith Critchlow, Order in Space: A design source book, 1970, pp. 62–67
- Ghyka, M. The Geometry of Art and Life, (1946), 2nd edition, New York: Dover, 1977. Demiregular tiling
- Шаблон:The Geometrical Foundation of Natural Structure (book) pp. 35–43
- Шаблон:Cite book p. 65
- Sacred Geometry Design Sourcebook: Universal Dimensional Patterns, Bruce Rawles, 1997. pp. 36–37 [1]
External links
- Шаблон:Cite journal
- Шаблон:Cite web
- Шаблон:MathWorld
- In Search of Demiregular Tilings, Helmer Aslaksen
- n-uniform tilings Brian Galebach, 2-Uniform Tiling 2 of 20
- ↑ Critchlow, pp. 62–67
- ↑ Grünbaum and Shephard 1986, pp. 65–67
- ↑ In Search of Demiregular Tilings #1
- ↑ Chavey (1989)
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