# Talk:Hexagonal tiling

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## Optical illusion

The yellow hexagons don't look regular in this image, though I know it's just an optical illusion. 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}
Hexagonal tiling
{6,3}
Triangular tiling
{3,6}
Truncation (t) t{p,q}
Truncated hexagonal tiling
Truncated triangular tiling
Rectification (r)
Ambo (a)
r{p,q}
Trihexagonal tiling
Bitruncation (2t)
Dual kis (dk)
2t{p,q}
Truncated triangular tiling
Truncated hexagonal tiling
Birectification (2r)
Dual (d)
2r{p,q}
Triangular tiling
Hexagonal tiling
cantellation (rr)
Expansion (e)
rr{p,q}
Rhombitrihexagonal tiling
Snub rectified (sr)
Snub (s)
sr{p,q}
Snub trihexagonal tiling
Cantitruncation (tr)
Bevel (b)
tr{p,q}
Truncated trihexagonal tiling
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
Octahedral
Icosahedral
Starting solid
Operation
Symbol
{p,q}
Tetrahedron
{3,3}
Cube
{4,3}
Octahedron
{3,4}
Dodecahedron
{5,3}
Icosahedron
{3,5}
Hexagonal tiling
{6,3}
Triangular tiling
{3,6}
Heptagonal tiling
{7,3}
Order-7 triangular tiling
{3,7}
Truncation (t) t{p,q}
truncated tetrahedron
truncated cube
truncated octahedron
truncated dodecahedron
truncated icosahedron
Truncated hexagonal tiling
Truncated triangular tiling
Truncated heptagonal tiling
Truncated order-7 triangular tiling
Rectification (r)
Ambo (a)
r{p,q}
tetratetrahedron
cuboctahedron
icosidodecahedron
Trihexagonal tiling
Triheptagonal tiling
Bitruncation (2t)
Dual kis (dk)
2t{p,q}
truncated tetrahedron
truncated octahedron
truncated cube
truncated icosahedron
truncated dodecahedron
truncated triangular tiling
truncated hexagonal tiling
Truncated order-7 triangular tiling
Truncated heptagonal tiling
Birectification (2r)
Dual (d)
2r{p,q}
tetrahedron
octahedron
cube
icosahedron
dodecahedron
triangular tiling
hexagonal tiling
Order-7 triangular tiling
Heptagonal tiling
cantellation (rr)
Expansion (e)
rr{p,q}
rhombitetratetrahedron
rhombicuboctahedron
rhombicosidodecahedron
rhombitrihexagonal tiling
Rhombitriheptagonal tiling
Snub rectified (sr)
Snub (s)
sr{p,q}
snub tetratetrahedron
snub cuboctahedron
snub icosidodecahedron
snub trihexagonal tiling
Snub triheptagonal tiling
Cantitruncation (tr)
Bevel (b)
tr{p,q}
truncated tetratetrahedron
truncated cuboctahedron
truncated icosidodecahedron
truncated trihexagonal tiling
Truncated triheptagonal tiling
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?

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 is a flattened hexagonal tiling, still equal-edge lengths. SockPuppetForTomruen (talk) 23:58, 5 September 2011 (UTC)

## 3 types of monohedral convex hexagonal tilings

(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 ${\displaystyle 2+1 \over 6}$-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

b=e
B+C+D=360°

b=e, d=f
B+C+E=360°

a=f, b=c, d=e
B=D=F=120°

2-tile lattice

4-tile lattice

3-tile lattice
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)

### Terminology

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?

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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