# Talk:Tessellation

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## Uncited section, should probably be a separate article

The following uncited mathematical essay seems vaguely related, but probably doesn't belong in the article, even if citations were found for it, so I've moved it here. Anyone like to work out what to do with it? Chiswick Chap (talk) 09:40, 10 January 2013 (UTC)

Number of sides of a polygon versus number of sides at a vertex

For an infinite tiling, let $a$ be the average number of sides of a polygon, and $b$ the average number of sides meeting at a vertex. Then $( a - 2 ) ( b - 2 ) = 4$. For example, we have the combinations (3, 6), (31/3, 5), (33/4, 42/7), (4, 4), (6, 3), for the tilings in the article Tilings of regular polygons.

A continuation of a side in a straight line beyond a vertex is counted as a separate side. For example, the bricks in the picture are considered hexagons, and we have combination (6, 3). Similarly, for the basketweave tiling often found on bathroom floors, we have (5, 31/3).

For a tiling which repeats itself, one can take the averages over the repeating part. In the general case the averages are taken as the limits for a region expanding to the whole plane. In cases like an infinite row of tiles, or tiles getting smaller and smaller outwardly, the outside is not negligible and should also be counted as a tile while taking the limit. In extreme cases the limits may not exist, or depend on how the region is expanded to infinity.

For finite tessellations and polyhedra we have

$( a - 2 ) ( b - 2 ) = 4 \left( 1 - \frac{\chi}{F} \right) \left( 1 - \frac{\chi}{V} \right)$

where $F$ is the number of faces and $V$ the number of vertices, and $\chi$ is the Euler characteristic (for the plane and for a polyhedron without holes: 2), and, again, in the plane the outside counts as a face.

The formula follows observing that the number of sides of a face, summed over all faces, gives twice the total number of sides in the entire tessellation, which can be expressed in terms of the number of faces and the number of vertices. Similarly the number of sides at a vertex, summed over all vertices, also gives twice the total number of sides. From the two results the formula readily follows.

In most cases the number of sides of a face is the same as the number of vertices of a face, and the number of sides meeting at a vertex is the same as the number of faces meeting at a vertex. However, in a case like two square faces touching at a corner, the number of sides of the outer face is 8, so if the number of vertices is counted the common corner has to be counted twice. Similarly the number of sides meeting at that corner is 4, so if the number of faces at that corner is counted the face meeting the corner twice has to be counted twice.

A tile with a hole, filled with one or more other tiles, is not permissible, because the network of all sides inside and outside is disconnected. However it is allowed with a cut so that the tile with the hole touches itself. For counting the number of sides of this tile, the cut should be counted twice.

For the Platonic solids we get round numbers, because we take the average over equal numbers: for $( a - 2 ) ( b - 2 )$ we get 1, 2, and 3.

From the formula for a finite polyhedron we see that in the case that while expanding to an infinite polyhedron the number of holes (each contributing −2 to the Euler characteristic) grows proportionally with the number of faces and the number of vertices, the limit of $( a - 2 ) ( b - 2 )$ is larger than 4. For example, consider one layer of cubes, extending in two directions, with one of every 2 × 2 cubes removed. This has combination (4, 5), with $( a - 2 ) ( b - 2 ) = 6 = 4 (1 + 2/10) (1 + 2/8)$, corresponding to having 10 faces and 8 vertices per hole.

Note that the result does not depend on the edges being line segments and the faces being parts of planes: mathematical rigor to deal with pathological cases aside, they can also be curves and curved surfaces.

 An example tessellation of the surface of a sphere by a truncated icosidodecahedron A torus can be tiled by a repeating matrix of isogonal quadrilaterals.

As well as tessellating the 2-dimensional Euclidean plane, it is also possible to tessellate other n-dimensional spaces by filling them with n-dimensional polytopes. Tessellations of other spaces are often referred to as honeycombs. Examples of tessellations of other spaces include:

• Tessellations of n-dimensional Euclidean space. For example, 3-dimensional Euclidean space can be filled with cubes to create the cubic honeycomb.
• Tessellations of n-dimensional hyperbolic space. For example, M. C. Escher's Circle Limit III depicts a tessellation of the hyperbolic plane (using the Poincaré disk model) with congruent fish-like shapes. The hyperbolic plane admits a tessellation with regular p-gons meeting in q's whenever $\tfrac{1}{p}+\tfrac{1}{q} < \tfrac{1}{2}$; Circle Limit III may be understood as a tiling of octagons meeting in threes, with all sides replaced with jagged lines and each octagon then cut into four fish.

See (Magnus 1974) for further non-Euclidean examples.

There are also abstract polyhedra which do not correspond to a tessellation of a manifold because they are not locally spherical (locally Euclidean, like a manifold), such as the 11-cell and the 57-cell. These can be seen as tilings of more general spaces.

## Another uncited essay: Self-dual tessellations

Here's another math essay, this time so brief as to be incomprehensible to the rest of us. Needs citations, and perhaps a suitable home if any kind owners can be found. Chiswick Chap (talk) 09:44, 10 January 2013 (UTC)

I believe blurb below is already discussed at Dual polyhedron#Self-dual polytopes and tessellations and at Uniform tiling#Self-dual tilings. It is on topic, but there seems no point to including it in a third article. --Mark viking (talk) 09:43, 1 February 2013 (UTC)
Self-dual tessellations
Self-dual square tiling

Tilings and honeycombs can also be self-dual. All n-dimensional hypercubic honeycombs with Schlafli symbols {4,3n−2,4} are self-dual.

I'm not an expert on all that, but there's certainly no point in having it spammed all over. Feel free to remove as many copies of it as you see fit, specially if they're uncited. Chiswick Chap (talk) 10:16, 1 February 2013 (UTC)

## Minor technicalities

Isn't the Greek for four ″tetra" and not "tessera"? 86.151.117.141 (talk) 20:30, 5 February 2013 (UTC)

There are different forms, see Etymology Dictionary for explanation. You'll find that -tt- and -ss- are pretty much the same thing in ancient Greek, like thalassa/thalatta for the sea. Chiswick Chap (talk) 21:03, 5 February 2013 (UTC)

## Tesselation in math vs art vs computer graphics

In watching this page for a few months, I've noticed a tension between the artistic aspects of tesselation and the mathematical aspects of tesselation. On one hand, tesselation originates from the laying of mosaics. Artists and craftspersons create tesselations all the time in laying floors, creating stained glass windows, crafting of quilts and much more. The tesselation article should reflect the history and variety of these activities and creations.

On the other hand, tesselations have a long history in mathematics, too. There is a well-developed theory of tesselation in mathematics, on more than just a flat 2D plane. The math definition of tesselation has some conceptual overlap with, but is very different from, the artistic definition.

On the gripping hand, tesselation in computer graphics is an important topic, too. The word is used to refer both to the triangular decomposition of a polygon or a surface mesh of an object and also to the sort of tiling used in texture generation and mapping. These definitions overlap with, but are both different than the mathematical and artistic definitions.

Right now the article is an uneasy amalgam of all three fields, but doesn't cover any of the fields adequately. I am wondering if the users are best served by a single tesselation article, or would it be better to break it up into tessellation (mathematics), tessellation (computer graphics) and tessellation as the main article concentrating on the art and craft aspect? I favor the art and craft aspect in the main article because it is the most accessible of the three approaches but we could also have tessellation as a WP:DAB or WP:DABCONCEPT pointing to art, math and computer graphics versions of the concept. I note there is already a patterns in nature article that helps in this fourth field.

What do you all think? Better alternatives?

--Mark viking (talk) 00:05, 25 April 2013 (UTC)

I'm not sure there are many editors around to help here, it has felt a bit lonely for a long while. WP is a global encyclopedia, so while math is welcome, it MUST be explained for the million, not just for the mathematician; and the math-types have been very remiss in adding citations, which has left open the possibility that there's quite a bit of WP:OR in there. FWIW I believe the main article (here) must cover all bases - after all, the math underlies and explains a lot of the effect of T in art (think Escher). Therefore, a good, integrated article here is surely our goal; it would be utterly inappropriate (ok, unencyclopedic) to make Tessellation into a mere disambiguation page.
That said, if you want to write some sub-articles, the topic is certainly big enough. On the art side, you are surely right that this is the accessible end of the telescope, but it should be used to lead gently into the math. I wrote patterns in nature, by the way, and it does gently hint at science and math, but it says very little about tessellations. Since you mention related articles, Symmetry is also in a sad state, and like this article here, it needs to connect math and reality much more strongly. That needs of course a human mathematician who has strong art and communication skills, and can cite sources --- or an arty type with unusually strong math, natch. Chiswick Chap (talk) 06:46, 25 April 2013 (UTC)
Thanks for your thoughts and recommendations on this. I agree that if the problem is not a tug-of-war among different interests, but a lack of clarity and good referencing on the technical end, then a well-integrated article is a possibility. If/when I add material, probably more on the technical end, I'll do so in this article. --Mark viking (talk) 16:10, 26 April 2013 (UTC)
Thanks for joining in. Would it be a good thing to have a pair of images in the computer graphics section to explain and illustrate how one drapes a tessellation over a modelled object (stage 1, say) and then applies shading, colours, texture etc to the triangles (stage 2, or maybe stages 2..5)? I suspect that would make the subject a lot more approachable. Chiswick Chap (talk) 20:51, 27 April 2013 (UTC)
Thanks for your comments. An explanatory figure or two would help here. I've seen some good figures in the literature, but nothing yet that would pass WP's stringent copyright policies. I'll see what I can come up with later tonight. --Mark viking (talk) 21:01, 27 April 2013 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── Hey guys, I don't know that much about the mathematical definition of tesselation, but I know enough about the artistic and computer graphics definitions to realise that they're not at all the same thing. They are two separate definitions that happen to share the same word:

Tesselation: Multiple identical two-dimensional shapes that can be slotted together with no gaps and repeated in two dimensions.

Tesselation: The process of removing polygons from a 3D model as it moves away from the viewpoint, in order to avoid rendering individual polygons that are too far away to discern.

They may not be the most accurate definitions, but I think they convey the distinction well enough. I made a separate article for the computer graphics definition. What do you guys think?

InternetMeme (talk) 14:01, 25 May 2013 (UTC)

Sounds promising. How about putting a link to your article here so we can read it? Chiswick Chap (talk) 15:07, 25 May 2013 (UTC)
While tesselation is used to control level of detail in computer graphics and especially video games, that is not the only use for it. With more development, breaking out the CG tesselation topic into its own article could work, but I think we need a short (say, one paragraph) CG section in this article with a pointer to the main article, like what was done with tessellation in nature. Thanks, --Mark viking (talk) 16:03, 25 May 2013 (UTC)
IM has already moved the old CG section from here to a new sub-article, Tessellation (computer graphics). At the least, we need a paragraph marking the topic, as you say. Chiswick Chap (talk) 16:07, 25 May 2013 (UTC)

## Parallelogram diagram

Regarding the 7-colour parallelogram diagram... it does not make any sense to me whatsoever. No matter how its tiled, its always a planar tesselation and so the 4 color theorem always applies to it. If the before-tiling version is colored with 7 colors so each numbered parallelogram is given a distinct color that has nothing to do with tiling and will be the same after tiling. Maybe I am missing something huge here but I don't think so. It seems to be a copyvio from http://books.google.com/books?id=X2ue6oE56hQC&pg=PA112&lpg=PA112&dq=seven+colors+tiled+parallelogram&source=bl&hl=en&redir_esc=y#v=onepage&q=seven%20colors%20tiled%20parallelogram&f=false but that is beside the point, except for the fact that the version here has omitted an important sentence although it doesn't seem to make any more sense to me.

## Image of tiling

I'm really not sure I see why we should be adding more images of tilings to the mathematics section. There are an infinite number that could be created, so there's plainly no hope of covering all bases. The image is of no special beauty or historical significance, at least none is asserted. It does not relate in any obvious way to the existing text. The proper place for a gallery of additional images of interest is Wikimedia Commons, not here. I've removed it once, and am quite ready to do so again as soon as we've discussed this. There appears to be no good reason to keep the image. Chiswick Chap (talk) 11:33, 19 June 2013 (UTC)

Beauty is subjective…  And then, you seem afraid of an infinite number of images, but actually there are two images in the current section "In mathematics".  This periodic tiling by regular polygons has only two kinds of tiles: triangular and hexagonal tiles.  However, when several centers of hexagons lies exactly on a straight line, the common orientation of these hexagons relative to the line is not obvious.
Aughost (talk) 12:17, 19 June 2013 (UTC)
Hi, I half agree with each of you. One illustration is enough at this point and the existing image is a clearer diagram because it shows the edges. But User:Aughost's image is interesting because it is a snub, as well as being beautiful 9aren't they all?) so I have placed it in the Gallery. Hope everybody is happy with this? — Cheers, Steelpillow (Talk) 12:45, 19 June 2013 (UTC)
Many thanks for finding a solution which makes sense mathematically and in terms of readers' needs from the article. Chiswick Chap (talk) 13:58, 19 June 2013 (UTC)
Thank you.  Art and mathematics can support each the other…
Aughost (talk) 14:50, 19 June 2013 (UTC)