| 释义 |
- Geometry
- Related k-uniform tilings of regular polygons Dual tiling
- Notes
- References
- External links
| 3-4-6-12 tiling | | | Type | 2-uniform tiling | | Vertex configuration |
3.4.6.4 and 4.6.12 | | Symmetry | p6m, [6,3], (*632) | | Rotation symmetry | p6, [6,3]+, (632) | | Properties | 2-uniform, 4-isohedral, 4-isotoxal |
In geometry of the Euclidean plane, the 3-4-6-12 tiling is one of 20 2-uniform tilings of the Euclidean plane by regular polygons, containing regular triangles, squares, hexagons and dodecagons, arranged in two vertex configuration: 3.4.6.4 and 4.6.12.[1][2][3][4] It has hexagonal symmetry, p6m, [6,3], (*632). It is also called a demiregular tiling by some authors. GeometryIts two vertex configurations are shared with two 1-uniform tilings: | rhombitrihexagonal tiling | truncated trihexagonal tiling |
3.4.6.4 |
4.6.12 |
It can be seen as a type of diminished rhombitrihexagonal tiling, with dodecagons replacing periodic sets of hexagons and surrounding squares and triangles. This is similar to the Johnson solid, a diminished rhombicosidodecahedron, which is a rhombicosidodecahedron with faces removed, leading to new decagonal faces. Related k-uniform tilings of regular polygonsThe hexagons can be dissected into 6 triangles, and the dodecagons can be dissected into triangles, hexagons and squares. Dissected polygons| Hexagon | Dodecagon
(each has 2 orientations) |
|---|
| 3-uniform tilings | | 48 | 26 | 18 |
|---|
[36; 3.3.4.3.4; 3.3.4.12] |
[3.4.4.6; (3.4.6.4)2] |
[36; (3.3.4.3.4)2] |
Dual tilingThe dual tiling has right triangle and kite faces, defined by face configurations: V3.4.6.4 and V4.6.12, and can be seen combining the deltoidal trihexagonal tiling and kisrhombille tilings.
Dual tiling |
V3.4.6.4
V4.6.12 |
Deltoidal trihexagonal tiling |
Kisrhombille tiling |
Notes1. ^Critchlow, pp. 62–67
2. ^Grünbaum and Shephard 1986, pp. 65–67
3. ^In Search of Demiregular Tilings #4
4. ^Chavey (1989)
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
- {{The Geometrical Foundation of Natural Structure (book)}} pp. 35–43
- {{cite book | ref=harv | authorlink=Branko Grünbaum | last1=Grünbaum | first1=Branko | last2=Shephard | first2=G. C. | title=Tilings and Patterns | publisher=W. H. Freeman | date=1987 | isbn=0-7167-1193-1}} p. 65
- Sacred Geometry Design Sourcebook: Universal Dimensional Patterns, Bruce Rawles, 1997. pp. 36–37 [https://www.amazon.com/exec/obidos/ASIN/0965640582]
External links - {{cite journal | first=D. |last=Chavey | title=Tilings by Regular Polygons—II: A Catalog of Tilings | url=https://www.beloit.edu/computerscience/faculty/chavey/catalog/ | journal=Computers & Mathematics with Applications | year=1989 | volume=17 | pages=147–165 | doi=10.1016/0898-1221(89)90156-9|ref=harv}}
- {{cite web | author = Dutch, Steve | title = Uniform Tilings | url = http://www.uwgb.edu/dutchs/symmetry/uniftil.htm | accessdate = 2006-09-09 | archive-url = https://web.archive.org/web/20060909053826/http://www.uwgb.edu/dutchs/SYMMETRY/uniftil.htm | archive-date = 2006-09-09 | dead-url = yes | df = }}
- {{MathWorld | urlname=DemiregularTessellation | title=Demiregular tessellation}}
- In Search of Demiregular Tilings, Helmer Aslaksen
- n-uniform tilings Brian Galebach, 2-Uniform Tiling 1 of 20
2 : Euclidean plane geometry|Tessellation |