Английская Википедия:33344-33434 tiling
33344-33434 tilings | ||
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Файл:2-uniform n16.svgФайл:2-uniform n17.png Faced colored by their symmetry positions | ||
Type | 2-uniform tiling | |
Designation[1] | [33.42; 32.4.3.4]1 | [33.42; 32.4.3.4]2 |
Vertex configurations | 3.3.4.3.4 and 3.3.3.4.4 | |
Symmetry | p4g, [4,4+], (4*2) | pgg, [4+,4+], (22×) |
Rotation symmetry | p4, [4,4]+, (442) | p2, [4+,4+]+, (2222) |
Properties | 4-isohedral, 5-isotoxal | 3-isohedral, 6-isotoxal |
In geometry of the Euclidean plane, a 33344-33434 tiling is one of two of 20 2-uniform tilings of the Euclidean plane by regular polygons. They contains regular triangle and square faces, arranged in two vertex configuration: 3.3.3.4.4 and 3.3.4.3.4.[2]
The first has triangles in groups of 3 and square in groups of 1 and 2. It has 4 types of faces and 5 types of edges.
The second has triangles in groups of 4, and squares in groups of 2. It has 3 types of face and 6 types of edges.
Geometry
Its two vertex configurations are shared with two 1-uniform tilings:
Файл:Snub square tiling vertfig.png 3.3.4.3.4 |
Файл:Tiling 33344-vertfig.png 3.3.3.4.4 |
Файл:1-uniform n9.svg snub square tiling |
Файл:1-uniform n8.svg elongated triangular tiling |
Circle Packings
These 2-uniform tilings can be used as a circle packings.
In the first 2-uniform tiling (whose dual resembles a key-lock pattern): cyan circles are in contact with 5 other circles (3 cyan, 2 pink), corresponding to the V33.42 planigon, and pink circles are also in contact with 5 other circles (4 cyan, 1 pink), corresponding to the V32.4.3.4 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 (mini-vertex configuration polygons; one dimensional duals to the respective planigons). Both images coincide.
In the second 2-uniform tiling (whose dual resembles jagged streams of water): cyan circles are in contact with 5 other circles (2 cyan, 3 pink), corresponding to the V33.42 planigon, and pink circles are also in contact with 5 other circles (3 cyan, 2 pink), corresponding to the V32.4.3.4 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 (mini-vertex configuration polygons; one dimensional duals to the respective planigons). Both images coincide.
C[33.42; 32.4.3.4]1 | a33.42; 32.4.3.4]1 | C[33.42; 32.4.3.4]2 | a[33.42; 32.4.3.4]2 |
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Файл:Circle Packing of 2-Uniform 3.3.4.3.4 and 3.3.3.4.4.png | Файл:Ambo Operation on 2-Uniform 3.3.4.3.4 and 3.3.3.4.4.png | Файл:Circle Packing of 2-Uniform 3.3.4.3.4 and 3.3.3.4.4 II.png | Файл:Ambo Operation of 2-Uniform 3.3.4.3.4 and 3.3.3.4.4 II.png |
Dual tilings
The dual tilings have right triangle and kite faces, defined by face configurations: V3.3.3.4.4 and V3.3.4.3.4, and can be seen combining the prismatic pentagonal tiling and Cairo pentagonal tilings.
Faces | 1-uniform | 2-uniform | ||
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V3.3.3.4.4 | V3.3.4.3.4 | V3.3.3.4.4 and V3.3.4.3.4 | ||
Файл:Tiling face 3-3-3-4-4.svg V3.3.3.4.4 80px V3.3.4.3.4 |
Файл:1-uniform 8 dual.svg prismatic pentagonal tiling |
Файл:1-uniform 9 dual.svg Cairo pentagonal tiling |
Файл:2-uniform 16 dual.svg Dual tiling I |
Файл:2-uniform 17 dual.svg Dual tiling II |
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 #15
- Шаблон:The Geometrical Foundation of Natural Structure (book) pp. 35–43
- Sacred Geometry Design Sourcebook: Universal Dimensional Patterns, Bruce Rawles, 1997. pp. 36–37 [1]
- Introduction to Tessellations, Dale Seymour, Jill Britton, (1989), p.57, Fig 3-24 Tessellations of regular polygons that contain more than one type of vertex point
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 1 of 20
- ↑ Шаблон:Cite book p. 65-67
- ↑ Chavey (1989)