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There's absolute no reason to exclude tiling images. I've done my best to keep it minimal.
These are expressed only in this article:
Visual templates wiki-links for related articles:
Tom Ruen ( talk) 16:06, 4 June 2015 (UTC)
All of this material was removed by David Eppstein, his fourth removal effort. His arguments are above, and I can't possibly agree with this removal. In my mind the connect between things is just as important as what things are, and pictures can be a 1000 words to people who want to see how things are connected.
If anyone else thinks this is important material to keep, perhaps they'll speak up? Tom Ruen ( talk) 22:21, 5 June 2015 (UTC)
There are eight uniform tilings based on Wythoff constructions from the regular hexagonal tiling. The truncated triangular tiling, t{3,6} is topologically equivalent to the hexagonal tiling. [1] The vertices of the trihexagonal tiling are positioned at the mid-edges of both the hexagonal tiling and the triangular tiling.
Uniform hexagonal/triangular tilings | ||||||||
---|---|---|---|---|---|---|---|---|
Fundamental domains |
Symmetry: [6,3], (*632) | [6,3]+, (632) | ||||||
{6,3} | t{6,3} | r{6,3} | t{3,6} | {3,6} | rr{6,3} | tr{6,3} | sr{6,3} | |
Config. | 63 | 3.12.12 | (6.3)2 | 6.6.6 | 36 | 3.4.6.4 | 4.6.12 | 3.3.3.3.6 |
The trihexagonal tiling or kagome lattice can be used as a periodic circle packing, placing equal diameter circles at the center of every vertex. This circle packing is not a true lattice because there are 3 circle positions within each lattice cell. Every circle is in contact with 4 other circles in the packing ( kissing number). [2] [3] The gap inside each hexagon allows for one circle, creating the densest packing from the triangular tiling, with each circle contact with the maximum of 6 circles.
The trihexagonal tiling is a series of symmetry mutations of quasiregular tilings sharing with vertex configurations (3.n)2, moving from the sphere to the Euclidean plane and into the hyperbolic plane. All of these tilings are identical colorings of their fundamental domain of symmetry, with orbifold notation *n32. [4]
*n32 orbifold symmetries of quasiregular tilings: (3.n)2 | |||||||
---|---|---|---|---|---|---|---|
Construction |
Spherical | Euclidean | Hyperbolic | ||||
*332 | *432 | *532 | *632 | *732 | *832... | *∞32 | |
Quasiregular figures |
|||||||
Vertex | (3.3)2 | (3.4)2 | (3.5)2 | (3.6)2 | (3.7)2 | (3.8)2 | (3.∞)2 |
References
I removed the following section as original research. In this case, the material looks on-topic and the images are not excessive. And the sources given are adequate to prove the existence and 2-uniformity of these three tilings. However, they do not single them out as a group nor do they source any relationship between these tilings and the trihexagonal tiling. We need sources saying not only that the tilings exist, but that they are connected to the trihexagonal tiling, in order to say in our article that they are connected. And if we can't say they are connected, there is no point in including them. — David Eppstein ( talk) 18:56, 6 June 2015 (UTC)
Quasiregular | Johnson solid | |||
---|---|---|---|---|
Icosidodecahedron (Pentagonal gyrobirotunda) |
Pentagonal orthobirotunda |
Elongated pentagonal orthobirotunda |
Elongated pentagonal gyrobirotunda |
Gyroelongated pentagonal birotunda |
There are 3 related 2-uniform tilings (having two types of vertices), and 5 related 3-uniform tilings. All 9 tilings contain 3.6.3.6 vertices. The first tiling offsets rows of hexagons of the trihexagonal tiling changing half of the 3.6.3.6 vertices into 3.3.6.6. [1] The other two are elongations of those two, inserting an infinite prism, and separating 3.6.3.6 or 3.3.6.6 vertices into 3.6.4.4. [2] Similarly there are 3 tilings with only alternate rows elonagated or offset, with 2 vertex types, but being 3-uniform. Finally two gyorelongated forms extend the first two by triangles as infinite antiprisms, have two types of vertices, but 3 symmetry orbits, so they are 3-uniform. [3]
The offset and elongated offset tilings 2-uniform tilings are similar to the leno weave [4], each has sets of one or two straight parallel lines and in one direction and two curves alternating positions in the perpendicular direction.
References
The following Wikimedia Commons file used on this page or its Wikidata item has been nominated for deletion:
Participate in the deletion discussion at the nomination page. — Community Tech bot ( talk) 03:53, 21 March 2023 (UTC)
This article is rated Stub-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
There's absolute no reason to exclude tiling images. I've done my best to keep it minimal.
These are expressed only in this article:
Visual templates wiki-links for related articles:
Tom Ruen ( talk) 16:06, 4 June 2015 (UTC)
All of this material was removed by David Eppstein, his fourth removal effort. His arguments are above, and I can't possibly agree with this removal. In my mind the connect between things is just as important as what things are, and pictures can be a 1000 words to people who want to see how things are connected.
If anyone else thinks this is important material to keep, perhaps they'll speak up? Tom Ruen ( talk) 22:21, 5 June 2015 (UTC)
There are eight uniform tilings based on Wythoff constructions from the regular hexagonal tiling. The truncated triangular tiling, t{3,6} is topologically equivalent to the hexagonal tiling. [1] The vertices of the trihexagonal tiling are positioned at the mid-edges of both the hexagonal tiling and the triangular tiling.
Uniform hexagonal/triangular tilings | ||||||||
---|---|---|---|---|---|---|---|---|
Fundamental domains |
Symmetry: [6,3], (*632) | [6,3]+, (632) | ||||||
{6,3} | t{6,3} | r{6,3} | t{3,6} | {3,6} | rr{6,3} | tr{6,3} | sr{6,3} | |
Config. | 63 | 3.12.12 | (6.3)2 | 6.6.6 | 36 | 3.4.6.4 | 4.6.12 | 3.3.3.3.6 |
The trihexagonal tiling or kagome lattice can be used as a periodic circle packing, placing equal diameter circles at the center of every vertex. This circle packing is not a true lattice because there are 3 circle positions within each lattice cell. Every circle is in contact with 4 other circles in the packing ( kissing number). [2] [3] The gap inside each hexagon allows for one circle, creating the densest packing from the triangular tiling, with each circle contact with the maximum of 6 circles.
The trihexagonal tiling is a series of symmetry mutations of quasiregular tilings sharing with vertex configurations (3.n)2, moving from the sphere to the Euclidean plane and into the hyperbolic plane. All of these tilings are identical colorings of their fundamental domain of symmetry, with orbifold notation *n32. [4]
*n32 orbifold symmetries of quasiregular tilings: (3.n)2 | |||||||
---|---|---|---|---|---|---|---|
Construction |
Spherical | Euclidean | Hyperbolic | ||||
*332 | *432 | *532 | *632 | *732 | *832... | *∞32 | |
Quasiregular figures |
|||||||
Vertex | (3.3)2 | (3.4)2 | (3.5)2 | (3.6)2 | (3.7)2 | (3.8)2 | (3.∞)2 |
References
I removed the following section as original research. In this case, the material looks on-topic and the images are not excessive. And the sources given are adequate to prove the existence and 2-uniformity of these three tilings. However, they do not single them out as a group nor do they source any relationship between these tilings and the trihexagonal tiling. We need sources saying not only that the tilings exist, but that they are connected to the trihexagonal tiling, in order to say in our article that they are connected. And if we can't say they are connected, there is no point in including them. — David Eppstein ( talk) 18:56, 6 June 2015 (UTC)
Quasiregular | Johnson solid | |||
---|---|---|---|---|
Icosidodecahedron (Pentagonal gyrobirotunda) |
Pentagonal orthobirotunda |
Elongated pentagonal orthobirotunda |
Elongated pentagonal gyrobirotunda |
Gyroelongated pentagonal birotunda |
There are 3 related 2-uniform tilings (having two types of vertices), and 5 related 3-uniform tilings. All 9 tilings contain 3.6.3.6 vertices. The first tiling offsets rows of hexagons of the trihexagonal tiling changing half of the 3.6.3.6 vertices into 3.3.6.6. [1] The other two are elongations of those two, inserting an infinite prism, and separating 3.6.3.6 or 3.3.6.6 vertices into 3.6.4.4. [2] Similarly there are 3 tilings with only alternate rows elonagated or offset, with 2 vertex types, but being 3-uniform. Finally two gyorelongated forms extend the first two by triangles as infinite antiprisms, have two types of vertices, but 3 symmetry orbits, so they are 3-uniform. [3]
The offset and elongated offset tilings 2-uniform tilings are similar to the leno weave [4], each has sets of one or two straight parallel lines and in one direction and two curves alternating positions in the perpendicular direction.
References
The following Wikimedia Commons file used on this page or its Wikidata item has been nominated for deletion:
Participate in the deletion discussion at the nomination page. — Community Tech bot ( talk) 03:53, 21 March 2023 (UTC)