Talk:Tessellation/Archive 1

March 2011

Given the Latin, and the spelling given in the Concise Oxford Dictionary, I'd say tesselation isn't correct.

Charles Matthews 07:18, 30 Nov 2003 (UTC)

So what'd you say is correct? I ignore Latin and have not the C.O.D. ... --euyyn 22:03, 7 June 2006 (UTC)

Yet another trans-Atlantic difference! Collins English Dictionary and Random House prefer tessellation. --Yitzhak1995 (talk) 04:43, 1 March 2011 (UTC)

Generalization to higher dimensions

um, instead of just putting this out there with no explanation of what that means, could you at least attempt to define what generalization to higher dimensions means, as well as the impact of tiling on these higher dimensions? --65.113.35.130 (talk) 21:45, 4 November 2009 (UTC)

I agree. For example, "generalization to higher dimensions" leaves one to wonder. Does the author mean the surface of planes warped into other dimensions, like the 2D surface of a ball or a platonic solid (tetrahedron, dodecahedron, icosahedron, etc.)? Or, filling a 2d area, 3d space, 4d space, etc. without gaps or overlaps, in the same way that stacking identical boxes in a room can fill that space... a 3D area... without gaps and overlaps?

In general, I think the definition for tessellation that we're using is like a miniskirt: It reveals both too much and too little. For non-mathematicians, of what use is a definition that doesn't even make it clear that the area should not have gaps nor overlaps? And for everyone, of what use is a definition that contains vagueness like "...generalization to higher dimensions"?

Perhaps we should start with a quote from M. C. Escher: "Tessellation is the regular division of [a]plane." and then give an expanded explanation that doesn't leave ambiguity about how it applies to higher dimensions.Sethnessatwikipedia (talk) 03:15, 20 May 2012 (UTC)

Tiling

What is the difference between tiling and tessellation? --Henrygb 00:35, 25 Mar 2005 (UTC)

According to Mathworld, a tessellation is formally a periodic tiling, and, as such, this entry is factually inaccurate. (See: http://mathworld.wolfram.com/Tiling.html, http://mathworld.wolfram.com/Tessellation.html)

The definition of a tiling is "a collection of disjoint open sets, the closures of which cover the plane" which makes no claims about periodicity or lack thereof. I think this should be corrected and another entry written for 'Tiling'. Also in such a case the graphic on this page would be wholly inappropriate, being an aperiodic tiling. Perhaps a mosaic would be more fitting. --Caliprincess 18:17, 18 August 2005 (UTC)

According to Grünbaum & Shepherd (which I trust more than MathWorld), tiling and tessellation are used synonymously or with similar meanings in the mathematical literature. -- Jitse Niesen (talk) 16:01, 31 August 2005 (UTC)

"average number of sides meeting at a vertex"

This phrase, in the section Tessellation#Number of sides of a polygon vs. number of sides at a vertex, needs clarification - it is easy to assume this means that you should count the lines meeting at each of the 5 vertices of the bathroom tiling, and divide by 5, to get an average of 3.4.

Is it sufficient just to add "over the entire plane"? Or should it tell the reader how to determine what the average is? I have only a vague idea how to do the latter, and am not at all sure that it generalises to more complex tessellations. Hv 10:44, 6 September 2005 (UTC)

I think you need to take the harmonic mean of the number of lines meeting at each vertex. The statement seems to be true with some caveats (I'm not sure whether the average always exist or what happens if an edge of the tessellation consists of more than one edge of the polygon). A reference would be nice to settle the matter. -- Jitse Niesen (talk) 14:08, 6 September 2005 (UTC)

For the bathroom tiling we have 2x4+4x3, divided by 6, the ordinary mean value.--Patrick 15:47, 6 September 2005 (UTC)

I added some details and a proviso for extreme cases.--Patrick 16:28, 6 September 2005 (UTC)

This section gives a version of Euler's Theorem for Tilings (Grünbaum and Shephard, 3.3.3) but not expressed very precisely. The theorem only applies for normal tilings (3.2), which excludes the case of "tiles getting smaller and smaller outwardly", while limits may "depend on how the region is expanded to infinity" whenever the tiling is not metrically balanced (3.6); note the theorem can still be expressed (with more care) for tilings which are not metrically balanced. More discussion of these concepts and the precise form of the theorem might be good. Joseph Myers 17:47, 6 September 2005 (UTC)

You're right, I found the statement in the form 1/a + 1/b = 1/2 (notation as in the article) in (G&S, 3.5.13). There they say it's true for each "strongly balanced" tiling, which means that the averages are well defined. -- Jitse Niesen (talk) 18:06, 6 September 2005 (UTC)

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

In my opinion this section needs to be completely rewritten. From the perspective of someone unfamiliar with the subject, they're given a list of numbers with no explination for how they were chosen, and then told that bricks are hexagons and squares are pentagons.

Meekohi 15:45, 5 March 2006 (UTC)

I completely agree. I was so puzzled by the captions of the images that I read the section just to find out who and for what counted those rectangular bricks as hexagons and those rectangular tiles as pentagons. Of course I refered to the section by chance, as the title (N1 versus N2!!) is surrealist to me. In the end, I only found a formula which stands for those bricks and tiles when you treat the word side in an odd way... --euyyn 22:01, 7 June 2006 (UTC)

Definitely, this section needs a rewrite. I added to a caption (for the "hexagonal" bricks) what I thought was the the correct answer but frankly a BS addition and I'm not terribly happy with it. If you interpret where the the lines where bricks join as edges, the corners as vertexes (as in a planar graph), and the areas enclosed as "shapes" the "count" makes sense, though.

To make this section more understandable, alot more background information, like some basic definitions need to be added. Root⁴(one) 19:23, 3 February 2007 (UTC)

I rewrote the captions, linked to the hexagonal tiling and Cairo pentagonal tiling of topologically identical tilings. I admit more explanation could help still. Tom Ruen 21:02, 3 February 2007 (UTC)

Seven Colours?

It appears that the caption for Image:Torus-with-seven-colours.png on the Tessellation page is wrong, it inadvertantly claims the ability to violate the Four color theorem. The statement about this pattern requiring 7 colours only holds if it is wrapped onto a torus, in an infinite tiling the pattern still only needs 4 colours. Even if I heave the wrong end of the stick here, perhaps the caption could be clearer. --Cs01ab 14:14, 4 April 2006 (UTC)

The four colour theorem only hold for the plane, or equivalently the sphere. Both there surfaces have Euler characteristic two. The Euler characteristic of the torus is zero so it is topologically distinct, and yield a different number for the minimal colouring. See Four color theorem#Generalizations. --Salix alba (talk) 14:33, 4 April 2006 (UTC)

But the caption doesn't talk about joining the arrows to form a torus (as is expected by the arrows, which are, so, unnecesary and misleading), it talks about "infinitely repeating the region [in the plane]". The article's section mentions the number 7 by saying that "to produce a colorization that respects the symmetries of the tessellation, you may need as many as seven colors". The caption says that for this concrete tiling, you have to use no less than seven colors. So... where does the 7 come from? Is 7 the maximum number of colors that can be required? How does the 7 relate to the fact that the tile in question is "formed" by 7 parallelograms? I feel the 7 must be related to the torus thing, but doesn't know how. --euyyn 22:18, 7 June 2006 (UTC)

There are seven regions (each consisting of an infinite number of separate pieces), each of which touches all six others; they thus require seven colors. —Tamfang (talk) 07:23, 6 November 2010 (UTC)

I'm sorry, but the regions don't all touch each other. They only touch each other if you wrap the rectangle into a torus. This article is about tesselation of the plane, not toruses (torii?) The graphic should be deleted. In addition, it's quite easy to verify that they can be colored with 4 colors, which also renders this graphic unnecessary. Tomelmar (talk) 04:34, 9 January 2011 (UTC)

A wallpaper pattern (in simplest form) is a tiling of the plane by a finite number of regions each consisting of an infinite number of disjoint pieces (in a regular relation to each other). Why disqualify such patterns from consideration, just because its fundamental domain can also be considered a torus tiling?

torii is a kind of Japanese ceremonial gate. I'm all for Latin plurals (particularly because it grates me to add a plural ending to an explicitly singular ending like –us), but there is no Latin word in which singular –us becomes plural –ii. Consider focus, foci, radius, radii. —Tamfang (talk) 05:25, 9 January 2011 (UTC)

Well, indeed I do know a bit about how this relates to the torus, as a rectangular region in which opposite sides can be considered adjacent (and that's what happens if you are restricted to repeat the tile ad infinitum) isn't but a torus unfolded. So the answers to my questions are "From the generalization of the theorem to toruses", "Yes", and "The tile pictured is formed by 7 parallelogram just to show the minimum number of colors needed". Someone add an explanation in the article, please? (I'm not confident of my own English) --euyyn 22:27, 7 June 2006 (UTC)

I think a caption is not suited to deal with an argument such as "generalization of 4 color theorem to non-planar surfaces" and it lacks a clear argument that states that the whole torus thing it's related at all with a coloration which preserve symmetries. I suggest changing completely the caption so that it refers to article's main text, and if deemed necessary put there all explanations. --85.18.188.72 (talk) 09:54, 4 November 2010 (UTC)

Hyperbolic Square?

I know very little about hyperbolic shapes or hyperbolic geometry, but a recent edit comment I found kinda interesting... There do appear to be such entities as hyperbolic squares as this online paper Chapter 5 (of what?): The Farey tessellation and circle packing. But what M. C. Escher drew, if it was a hyperbolic square, I wouldn't know (and frankly I don't care to know at the moment.) The current edit may better reflect what should be the intended meaning as I think I can reasonably guess that those entities drawn next to the triangles are quadrilaterals in hyperbolic space. But again, I know hardly anything... like practically Zilch, with respect to hyperbolic geometry. Root⁴(one) 05:30, 7 May 2007 (UTC)

Russian interwiki is put incorrect

Please, correct somebody who knows how to do it.

Seven colours reprise

Image:Torus-with-seven-colours.png on the Tessellation page.. claims if painted before tiling then only four colours are needed, could you please explain this? it clearly requires seven, or my eyes decieve me. Please help ! SkyInTheSea 12:57 9/12/07

When the caption says "If we tile before coloring ...", it means that if you tile the plane with an uncoloured pattern of paralleograms, and then colour in the parallelograms afterwards without requiring the colouring to respect the symmetry of the repeating rectangular lattice (this is the key point) then you can colour the pattern of uncoloured parallelograms with only 4 colours. However, if you require the parallelograms that are in the same place in the repeating rectangle to always have the same colour (which has to be the case if you are going to wrap the colouring around a torus) then you have to use at least 7 colours, as is shown in the diagram.

As a 1 dimensional analogy, think of colouring the integers so that no two consecutive integers have the same colour. Obviously you only need two colours - you can colour odd numbers Red and even numbers Blue, so your repeating pattern is RB. However, if you also require integers that leave the same remainder when divided by 5 to have the same colour (or, equivalently, wrap the number line around a circle) then you need at least 3 colours, with a repeating pattern such as RBRBG. Gandalf61 (talk) 15:28, 12 December 2007 (UTC)

By the way, the article is not quite accurate when it states: Note that the coloring guaranteed by the four-color theorem will not in general respect the symmetries of the tessellation. To produce a coloring which does, as many as seven colors may be needed, as in the picture at right. The most reasonable way to interpret "respect the symmetries of the tessellation" seems to be that any symmetry of the uncolored tessellation must also take like colors to like colors when applied to the colored tessellation. In general this is not compatible with the "no tiles of equal colors meet" citerion. For example, the tiling of the plane with regular hexagons has symmetries that take any tile to any other tile, so in order to "respect the symmetries of the tesselation" all tiles must have the same color. (What's worse, this is also the case for the uncolored version of the example in the figure!)

I think that the "as many as seven colors" claim is true if we restrict our attention to a symmetry group of translations such that a tile never shares an edge with its image under a nontrivial symmetry. But I'm not sure how to express this condition succinctly. (By the way, the proper reference for 7 being sufficient would be the Heawood conjecture, as applied to a torus). –Henning Makholm 00:32, 14 January 2008 (UTC)

etymology

tessela may be immediately from Latin, but it's obviously from a Greek word for a four-sided piece. I'm just sayin'. —Tamfang (talk) 05:59, 6 January 2008 (UTC)

?

How do they work? —Preceding unsigned comment added by 74.142.116.75 (talk) 20:41, 2 March 2008 (UTC)

How do what work? —Tamfang (talk) 08:00, 17 March 2008 (UTC)

Tesselators maybe —Preceding unsigned comment added by 67.53.37.221 (talk) 15:17, 30 December 2008 (UTC)

RE the article for Tesselation: Grief, this is a poorly written entry. Someone with a knack for writing coherent English needs to rewrite it. —Preceding unsigned comment added by 97.124.239.62 (talk) 09:33, 21 October 2009 (UTC)

Tiling, Space-filling, Fractal

I began with looking at the Sierpinksi Triangle, Sierpinski Carpet where it appears evident that any suitable tiling will lend itself to a Sierpinski space-filling pattern. The Hausdorf dimension page leads into the fractal dimension of such curves and shapes. Futher discussion required. JK-Salisbury —Preceding unsigned comment added by 86.160.138.236 (talk) 12:38, 2 June 2008 (UTC)

What's the Bathroom Floor Tiling?

I can't figure out what the "bathroom floor tiling" is. It is mentioned in this sentence: "Similarly, for the bathroom floor tiling we have (5, 3 1/3)." What does this tiling look like? Does it have some other name?Fluoborate (talk) 09:29, 17 January 2009 (UTC)

I assume it refers to the basketweave tiling as showed in File:Wallpaper_group-p4g-1.jpg. The numbers seem right, I have seen these tiling on bathroom floors, and that page calls refers to bathroom floors. I tried to make the text clearer. -- Jitse Niesen (talk) 13:03, 17 January 2009 (UTC)

half a picture

A tesselation can also be half of a picture repeated so it becomes a whole.

I removed this anonymous addition because I can't make sense of it. Egotistical of me, eh? —Tamfang (talk) 04:11, 18 February 2009 (UTC)

Origin?

I think a brief description of the origin of tessellation might be worth including.Gosox5555 (talk) 00:14, 16 March 2009 (UTC)

Which sense of "origin". The raw idea of covering a plane surface with identical tiles is surely lost in prehistory. Do you mean the etymology of the word "tessellation"? –Henning Makholm (talk) 12:49, 16 March 2009 (UTC)

Disambiguation page needed

Tessellation can refer to something in computer graphics that's totally unrelated to what's discussed here. I guess it can be summarized as "to create vertexes dynamically at run time". I was actually looking for that usage when running into this page. freeman (talk) 22:24, 21 March 2009 (UTC)

It's not totally unrelated. —Tamfang (talk) 01:55, 23 March 2009 (UTC)

Well of course it's not literally totally unrelated, otherwise they won't share a common name... however I would say they are quite different. The computer graphics technique tessellation emphasizes on creating new vertexes to make a curve or surface more smooth, and most of the time there is no repetition involved at all. freeman (talk) 16:30, 24 March 2009 (UTC)

Wouldn't need a whole disambiguation page, just a link up top to a full article on the other term (if such a thing existed) or a brief description in this article. DreamGuy (talk) 15:18, 23 March 2009 (UTC)

Self-dual tessellations

The picture in this section is impossible to see if you have even very mild colorblindness. It should be fixed or deleted. —Preceding unsigned comment added by 24.61.14.7 (talk) 03:31, 13 October 2009 (UTC)

Earl Nightingale used this word to describe the process of putting together a well crafted talk. —Preceding unsigned comment added by 74.131.225.15 (talk) 15:15, 26 October 2009 (UTC) helped by me (not saying my name)!! —Preceding unsigned comment added by 81.98.243.5 (talk) 18:28, 11 January 2010 (UTC)

This article does not define what "self-dual" means. Given that this is a technical term, "self-dual" deserves at least an appositive to describe it.Yitzhak1995 (talk) 04:40, 1 March 2011 (UTC)

The colour part is still confusing

Ok, after tiling with the parallellograms, can't I just paint one line with alternating red and yellow and the next line with alternating green and blue, and so on? Or is this for some reason not symmetric? Moberg (talk) 09:33, 12 May 2010 (UTC)

Are you referring to the seven-color picture? You can't color that patch as you describe, if you want its opposite sides to match. —Tamfang (talk) 18:02, 15 May 2010 (UTC)

And why would I like the opposite sides to matchh? Moberg (talk) 19:33, 16 May 2010 (UTC)

Because otherwise some of the parallelograms will be broken into two or more colors. —Tamfang (talk) 00:49, 17 May 2010 (UTC)

No, because the sides aren't touching each other, it's a rectangle. Moberg (talk) 17:45, 19 May 2010 (UTC)

Note the first words of the caption: "If this parallelogram pattern is colored before tiling it over a plane...." In other words, the rectangle is a fundamental domain of a wallpaper pattern. Each side touches the opposite side of an identical rectangle if you prefer. You can repeat the pattern and then color it, but then the rectangle in question is not a unit of repetition. —Tamfang (talk) 18:29, 20 May 2010 (UTC)

I think there's simply confusion about what's being colored. Moberg, the spaces are being colored, not the lines. Imagine that the lines are simply infinitely thin, colorless separators between the parallelograms.Sethnessatwikipedia (talk) 03:18, 20 May 2012 (UTC)

think about it

Imagine if you are a quilter or a tiler or you must have a good sense of geometry and imaginationFdkmx236 (talk) 18:23, 14 May 2010 (UTC)

One might reasonably think so. My mom is a quilter and I was recently surprised to learn that she does not understand the geometry of the moon's phases. —Tamfang (talk) 18:00, 15 May 2010 (UTC)

External links

The external links section is being used to advertise products. I suggest that interested editors find some common ground on useful, stable, appropriate links and start removing advertising.

For now, I removed everything. Maybe someone could add back well-supported appropriate sites? I don't know the whole rules on links, but I don't think wikipedia articles are supposed to be for advertising your products. --Kleopatra (talk) 05:17, 28 October 2010 (UTC)

The ENTIRE external link section was removed from the article, copied below for consideration. Tom Ruen (talk) 06:32, 2 November 2010 (UTC)

Yes, as I said above, "I removed everything." (I've added italics to my original.) This means essentially the same thing as "the ENTIRE link section has been removed." Spam is not allowed in wikipedia either in articles or on article talk pages. I think, because there are so many spam links, that it's okay to make editors search the history for viable links to be returned to the article. If it were not spam, I would have simply copied it here. Please return only links that are not spam. Thanks. --Kleopatra (talk) 07:35, 4 November 2010 (UTC)

I've restored the link to my site, Tessellations.org, a purely educational tessellation history-art-makeyourown-and-easy-geometry site that's been at the top Google & Yahoo search results rankings since 2005. If you don't mind my saying so, I think it was rash to simply delete ALL the entries that were there. Why not simply have clicked on each one, determined quickly whether it was spam, and deleted only the spam / commercially motivated ones? I'm jus' sayin'...--User:SethnessatWikipedia Sethnessatwikipedia (talk) 11:15, 15 May 2012 (UTC)

parametric drive

Algorithm generating mesh is driven by the parameters.

Does this sentence mean anything nontrivial? —Tamfang (talk) 17:05, 20 April 2011 (UTC)

For children

Comment moved from article--Salix (talk): 16:06, 8 June 2011 (UTC)

Dear Wikipedia , Some of the infomation you give is very useful , however some is not ! Some ieas whch would improve WIKIPEDIA is making readible for any ages such as children ! Me being a child i find it hard to undisatand some content in Wikipedia so mor eor less you should make it child friendly ! Your grayefully Chubbs !xx

There is Wikipedia in Simple English. Maybe that one is easier to understand for you? Toshio Yamaguchi (talk) 10:18, 10 June 2011 (UTC)

New!... maybe...

Tessellation of the first quadrant by Carnot cycles of equal areas, beginning.

I recently "find" a tessellation of part of the first quadrant. If you take the logarithms of the variables, it is very simple: x + y = lnt with t = 1, 1.2, 1.4, 1.6 etc. and 1.667 * x + y = s with s = 0, 0.2, 0.4, 0.6 etc.. If someone is interested to create the "whole" image, I will be very happy and I will be very grateful to him. Blogbreather (talk) 09:49, 6 February 2012 (UTC)

Wikipedia is an encyclopedia, not a newspaper. Because you say it's "new", then by definition it doesn't belong in Wikipedia. We report things that others have already reported; we do not publish original material. An encyclopedia quotes the authorities; it does not publish new, original research. Instead, try taking your "find" to a group like the Tilings listserv at http://tiling.uttyler.edu/read/all_forums/ where tessellation experts can review and discuss your "find". Sethnessatwikipedia (talk) 03:06, 20 May 2012 (UTC)

Tiling (again)

The above section on Tiling correctly concluded that "tiling" and "tessellation" are synonymous. (Shepherd & Grünbaum, 1987, is currently the definitive source, though a new edition is due any day now.) However the article is currently defining neither word in the sense used by Shepherd & Grünbaum. This article describes "tessellation" as an action (a "process"). The word "tiling" is defined as an everyday object in the opening, and then is used in the sense of Shepherd & Grünbaum in the rest of the article without explanation. The opening should be reworded so that both "tessellation" and "tiling" are given their modern mathematical meanings (as per Shepherd & Grünbaum). --seberle (talk) 21:00, 8 May 2012 (UTC)