Talk:Hexagonal tiling

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Optical illusion[edit]

The yellow hexagons don't look regular in this image, though I know it's just an optical illusion. Uniform tiling 63-t12.png Double sharp (talk) 15:42, 8 August 2009 (UTC)

You can think of the hexagons as truncated equilateral triangles, since that's what they are in this coloring. Tom Ruen (talk) 00:53, 9 August 2009 (UTC)
Here's a chart of some of the different tessellations related to this one. I based it off of this table.
Construction of Archimedean Tessellations in Euclidean Space with Hexagonal Symmetry
Starting tessellation
Operation
Symbol
{p,q}
CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png
Hexagonal tiling
{6,3}
Uniform tiling 63-t0.png
Triangular tiling
{3,6}
Uniform tiling 63-t2.png
Truncation (t) t{p,q}
CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png
Truncated hexagonal tiling
Uniform tiling 63-t01.png
Truncated triangular tiling
Uniform tiling 63-t12.png
Rectification (r)
Ambo (a)
r{p,q}
CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png
Trihexagonal tiling
Uniform polyhedron-63-t1.png
Bitruncation (2t)
Dual kis (dk)
2t{p,q}
CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png
Truncated triangular tiling
Uniform tiling 63-t12.png
Truncated hexagonal tiling
Uniform tiling 63-t01.png
Birectification (2r)
Dual (d)
2r{p,q}
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png
Triangular tiling
Uniform tiling 63-t2.png
Hexagonal tiling
Uniform tiling 63-t0.png
cantellation (rr)
Expansion (e)
rr{p,q}
CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png
Rhombitrihexagonal tiling
Uniform tiling 63-t02.png
Snub rectified (sr)
Snub (s)
sr{p,q}
CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png
Snub trihexagonal tiling
Uniform tiling 63-snub.png
Cantitruncation (tr)
Bevel (b)
tr{p,q}
CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png
Truncated trihexagonal tiling
Uniform tiling 63-t012.png
However, it might be more helpful to see it as simply another column in a table of tilings. After all, the other columns can simply be thought of as spherical tilings, and the columns to the right as hyperbolic tilings.
Construction of Archimedean Solids and Tessellations
Symmetry Tetrahedral
Tetrahedral reflection domains.png
Octahedral
Octahedral reflection domains.png
Icosahedral
Icosahedral reflection domains.png
Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg H2checkers 237.png
Starting solid
Operation
Symbol
{p,q}
CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png
Tetrahedron
{3,3}
Uniform polyhedron-33-t0.png
Cube
{4,3}
Uniform polyhedron-43-t0.png
Octahedron
{3,4}
Uniform polyhedron-43-t2.png
Dodecahedron
{5,3}
Uniform polyhedron-53-t0.png
Icosahedron
{3,5}
Uniform polyhedron-53-t2.png
Hexagonal tiling
{6,3}
Uniform tiling 63-t0.png
Triangular tiling
{3,6}
Uniform tiling 63-t2.png
Heptagonal tiling
{7,3}
H2 tiling 237-1.png
Order-7 triangular tiling
{3,7}
H2 tiling 237-4.png
Truncation (t) t{p,q}
CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png
truncated tetrahedron
Uniform polyhedron-33-t01.png
truncated cube
Uniform polyhedron-43-t01.png
truncated octahedron
Uniform polyhedron-43-t12.png
truncated dodecahedron
Uniform polyhedron-53-t01.png
truncated icosahedron
Uniform polyhedron-53-t12.png
Truncated hexagonal tiling
Uniform tiling 63-t01.png
Truncated triangular tiling
Uniform tiling 63-t12.png
Truncated heptagonal tiling
H2 tiling 237-3.png
Truncated order-7 triangular tiling
H2 tiling 237-6.png
Rectification (r)
Ambo (a)
r{p,q}
CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png
tetratetrahedron
Uniform polyhedron-33-t1.png
cuboctahedron
Uniform polyhedron-43-t1.png
icosidodecahedron
Uniform polyhedron-53-t1.png
Trihexagonal tiling
Uniform polyhedron-63-t1.png
Triheptagonal tiling
H2 tiling 237-2.png
Bitruncation (2t)
Dual kis (dk)
2t{p,q}
CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png
truncated tetrahedron
Uniform polyhedron-33-t12.png
truncated octahedron
Uniform polyhedron-43-t12.png
truncated cube
Uniform polyhedron-43-t01.png
truncated icosahedron
Uniform polyhedron-53-t12.png
truncated dodecahedron
Uniform polyhedron-53-t01.png
truncated triangular tiling
Uniform tiling 63-t12.png
truncated hexagonal tiling
Uniform tiling 63-t01.png
Truncated order-7 triangular tiling
H2 tiling 237-6.png
Truncated heptagonal tiling
H2 tiling 237-3.png
Birectification (2r)
Dual (d)
2r{p,q}
CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png
tetrahedron
Uniform polyhedron-33-t2.png
octahedron
Uniform polyhedron-43-t2.png
cube
Uniform polyhedron-43-t0.png
icosahedron
Uniform polyhedron-53-t2.png
dodecahedron
Uniform polyhedron-53-t0.png
triangular tiling
Uniform tiling 63-t2.png
hexagonal tiling
Uniform tiling 63-t0.png
Order-7 triangular tiling
H2 tiling 237-4.png
Heptagonal tiling
H2 tiling 237-1.png
cantellation (rr)
Expansion (e)
rr{p,q}
CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png
rhombitetratetrahedron
Uniform polyhedron-33-t02.png
rhombicuboctahedron
Uniform polyhedron-43-t02.png
rhombicosidodecahedron
Uniform polyhedron-53-t02.png
rhombitrihexagonal tiling
Uniform tiling 63-t02.png
Rhombitriheptagonal tiling
H2 tiling 237-5.png
Snub rectified (sr)
Snub (s)
sr{p,q}
CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png
snub tetratetrahedron
Uniform polyhedron-33-s012.png
snub cuboctahedron
Uniform polyhedron-43-s012.png
snub icosidodecahedron
Uniform polyhedron-53-s012.png
snub trihexagonal tiling
Uniform tiling 63-snub.png
Snub triheptagonal tiling
Uniform tiling 73-snub.png
Cantitruncation (tr)
Bevel (b)
tr{p,q}
CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png
truncated tetratetrahedron
Uniform polyhedron-33-t012.png
truncated cuboctahedron
Uniform polyhedron-43-t012.png
truncated icosidodecahedron
Uniform polyhedron-53-t012.png
truncated trihexagonal tiling
Uniform tiling 63-t012.png
Truncated triheptagonal tiling
H2 tiling 237-7.png
As you can see, the truncation of {3, n} for any natural number n always has two hexagons and one n-gon meeting at each vertex, and the dual of {3, n} always has three n-gons meeting at each vertex. When n=6, the truncation is two hexagons plus one hexagon (three hexagons), and the dual is three hexagons, so the truncation and dual of {3, 6} are the same as each other; each has three hexagons meeting at a vertex.2601:2C1:C001:1F20:80B9:A1B0:4862:7768 (talk) 23:36, 16 July 2017 (UTC)

Hexagonal tiling is rigid?[edit]

The article says that the hexagonal tiling is "rigid" unlike the square tiling, which can be deformed into parallelograms. It seems to me that the hexagons could be flattened while keeping the edges the same length and connected in the same fashion. Pciszek (talk) 04:58, 5 September 2011 (UTC)

Agreed, like brick pattern Wallpaper group-cmm-1.jpg is a flattened hexagonal tiling, still equal-edge lengths. SockPuppetForTomruen (talk) 23:58, 5 September 2011 (UTC)

3 types of monohedral convex hexagonal tilings[edit]

(This discussion has been moved from WP:RD/Mathematics#Hexagonal tiling.)

The Guardian writes[1] that "It was proved in 1963 that there are exactly three types of convex hexagon that tile the plane.", but our article treats "hexagonal tiling" as a synonym to the honeycomb. What are the other two? Or is that quote a misunderstanding, and the proof was actually about the colorings? — Sebastian 18:49, 20 August 2015 (UTC)

There might be something in Tilings and Patterns. I do get confused about what it means to have types. Some isohedral variations are here Hexagonal_tiling#Topologically_equivalent_tilings, taken from tilings and Patterns (list of 107 isohedral tilings, p.473-481). Tom Ruen (talk) 19:00, 20 August 2015 (UTC)
This webpage shows the three:[2]. The constraints look like they imply a symmetry. Tom Ruen (talk) 19:07, 20 August 2015 (UTC)
Reinhardt showed that there are only three types of convex hexagons which tessellate:
Type 1: B+C+D=A+E+F=360°, a=d (p2, 2222 symmetry?) lattice has 2 tiles
Type 2, A+B+D=C+E+F=360°, a=d, c=e (pgg, 22× symmetry?) lattice has 4 tiles
Type 3, A=C=E=120°, a=a', c=c', e=e' (p3, 333 symmetry?) lattice has 3 tiles
Thank you, that answers my question. It seems they are all isohedral, although type 2 involves mirroring. If one colored that, it would give a wavy pattern. (I can do that with the SVG file, but unfortunately I can't post it here for copyright reasons.) I can imagine these might also have some practical relevance as the resonances of a graphite lattice. — Sebastian 19:49, 20 August 2015 (UTC)
Interestingly, none of them are constrained as parallelogons, but these have single tile lattices. By symmetry, I'd guess these would be a special case of type 1 when all opposite edges are equal. Tom Ruen (talk) 20:05, 20 August 2015 (UTC)
(edit conflict) Right, but type 1 could be called a semi- or more exactly, a -parallelogon, since 2 pairs of sides are parallel, and one has the same length. Type 2 looks like a||d, but that's just they way they did the picture. As for the special case, yes, that would be the p2 (2222) picture in our article, or, further degenerated, the stretched hexagon. — Sebastian 20:22, 20 August 2015 (UTC)
Here's a best quick guess, taking hexagonal 1-isohedral tilings from http://www.jaapsch.net/tilings/, but there are actually 6 cases listed there. Tom Ruen (talk) 20:53, 20 August 2015 (UTC)
3 types of monohedral convex hexagonal tilings
1 2 3
p2, 2222 pgg, 22× p3, 333
P6-type1.png P6-type2.png P6-type3.png
Prototile p6-type1.png
b=e
B+C+D=360°
Prototile p6-type2.png
b=e, d=f
B+C+E=360°
Prototile p6-type3.png
a=f, b=c, d=e
B=D=F=120°
Lattice p6-type1.png
2-tile lattice
Lattice p6-type2.png
4-tile lattice
Lattice p6-type3.png
3-tile lattice
Thank you for the great reply; this far exceeds what I had hoped for. This would be a great addition to the article; I am tempted to add it to the beginning. But there is one problem: The article is currently laid out for one specific tiling, the one based on the regular hexagon. If we change this article to cover all hexagonal tilings, some parts of the article will need adjustment, and it will not be possible to adjust the sidebar. Maybe I shouldn't worry about that, seeing that the articles Hexagon and Pentagon also mostly cover the regular case, and their sidebar also is only dedicated to that case. Another point to consider is that we already have a big overlap between this article and Hexagonal lattice, which might open the option of merging that article with the regular hexagonal tiling. However, I don't think there's a precedence for that, and I don't know what such an article should be called; I just wanted to mention it now, so we can brain storm on the best solution. — Sebastian 23:16, 20 August 2015 (UTC)
I think it belongs here, same topology, lower symmetry to the regular form, but I've not seen any original sources for it. Tom Ruen (talk) 00:00, 22 August 2015 (UTC)
By "[no] original sources", I understand you mean that you're afraid your source from euler.slu.edu is not original. But there's no need to to refer to original sources; sources need to be reliable, and I don't think there's reason to assume that Saint Louis University is not reliable for this case. We can just write "Reinhardt (1918), cited in SLU ...". (Of course you're using a different order for the naming, but it's mathematically the same; I checked it.) BTW, I just looked at the other source you cite, http://www.jaapsch.net/tilings/, and I don't see where you see six cases. — Sebastian 00:59, 22 August 2015 (UTC)
I'm glad you checked. I used the Java Application at www.jaapsch.net. In it, there are 6 hexagonal/convex/isohedral cases listed. It looks like they are regrouped from 13 isohedral cases Named (P6-1&8&10&12&13 (pg); P6-2&9 (pg); p6-3&7 (p2); p6-4 (pgg); p6-5 (pgg); p6-6&11 (p3)) Tom Ruen (talk) 01:21, 22 August 2015 (UTC)
You mean the tiling viewer at http://www.jaapsch.net/tilings/tilingapp.htm? That just says "A plugin is needed to display this content", which is very unhelpful, since it gives no hint which plugin. But the many new tilings at http://www.jaapsch.net/tilings, such as "k-Isohedral tilings, k>3", give me a clue what the confusion is about: There are just 3 different tiles, but you can arrange them differently to get more than 3 different tilings. I now realize that I didn't have to limit my question at Talk:Pentagonal tiling#Roses and other distinct tilings from special cases just to special cases. — Sebastian 01:46, 22 August 2015 (UTC)
Yes, it's Java. I had to run the Java outside my browser because of security stupidity. Here's the 13 isohedral tilings, from Tilings and Patterns. p.481. Some of them are already in the article. (OOPS, I called it isogonal, should say isohedral!!) Tom Ruen (talk) 02:13, 22 August 2015 (UTC)
13-isogonal hexagonal tilings tilings and patterns p481.png

Terminology[edit]

I think we have a serious terminology problem. When we speak of n tilings or n types of tilings, do we mean

  1. (as the article seems to assume) n different sets of conditions for the sides and angles of the underlying tile, or
  2. (as we seem to be using it in this section) n different symmetries. (Where we are only looking at monohedral arrangements.)? — Sebastian 02:56, 22 August 2015 (UTC)
I completed the images of the 13-isohedral tilings as defined in Tilings and Patterns. I'm not sure how charactize this list besides isoehedral. I'm not sure what to call the list of 3 except "3 types of convex monohedral hexagonal tilings" or whatever the sources say. Tom Ruen (talk) 03:40, 22 August 2015 (UTC)

Is it error?[edit]

Uniform colorings

In table two rows Wythoff in first column, but first linked with Schläfli symbol, second - with Wythoff symbol. Jumpow (talk) 13:37, 25 March 2017 (UTC)

Fixed. Thanks! Tom Ruen (talk) 02:05, 26 March 2017 (UTC)

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