Talk:Euclidean tilings by convex regular polygons

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Title

Shouldn't this be "Tilings by regular polygons" or even "Plane tilings by regular polygons"? Melchoir 01:18, 9 October 2005 (UTC)[reply]

I agree. Feel free to move/rename this title - I don't know how! (besides copying contents and putting old under delete) Tom Ruen 01:46, 9 October 2005 (UTC)[reply]

Done. Oh, for future reference, check out Wikipedia:Merging and moving pages. Melchoir 02:20, 9 October 2005 (UTC)[reply]

Cleanup and topics not covered

I cleaned up the article and removed the cleanup tag. There are still a number of topics discussed in Grünbaum and Shephard which could perhaps be discussed on this page but aren't yet:

More on $k$ -uniform (and $k$ -isohedral, $k$ -isotoxal) tilings, with additional examples and something on the Krötenheerdt tilings; equitransitive tilings.
Non-edge-to-edge tilings: equitransitive unilateral tilings by squares; the problem of tiling the plane with exactly one square of each integer edge length.
Star polygons, both in the style of Kepler (Grünbaum and Shephard section 2.5, the lists there being incomplete) and as hollow self-intersecting polygons (section 12.3); I shouldn't make the call as to notability of the former, having published regarding them.
The duals of the uniform tilings (Laves tilings).
Archimedean and uniform colourings of tilings.

Joseph Myers 00:41, 6 October 2005 (UTC)[reply]

This is wrong, but I'm not sure how to fix it

This part:

In particular, if three polygons meet at a vertex and one has an odd number of sides, the other two polygons must be the same size. If they are not, they would have to alternate around the first polygon, which is impossible if its number of sides is odd.

is correct, but the tilings it makes impossible are uniform (semiregular) tilings.

However, immediately below is a section which says things like "cannot appear in *any* tiling of regular polygons". (Not just uniform tilings).

There is no justification for the claim that it can't appear in *any* tiling. I can't figure out if this is a true claim with no justification, or a false claim that happened because someone confused "any tiling" and "any uniform tiling". Ken Arromdee 18:31, 22 November 2006 (UTC)[reply]

I'm pretty sure it can all be justified. E.g. for a tiling involving 3.7.42: a triangle with a vertex of this type would have to have another vertex of this type at the other endpoint of the edge shared between the 3 and the 42; consequently the third vertex of the 3 would have to be 3.7.7, which doesn't work. The same reasoning shows 3.8.24, 3.9.18, and 3.10.15 impossible. 4.5.20 is the only vertex type involving 20-5 and 4-5, so if a vertex of this type existed, you'd have to have 4's and 20's alternating around the 5, impossible by the same reasoning as you've blockquoted above, and the same reasoning works for 5.5.10.

Changed it to read "cannot appear in any uniform tiling of regular polygons". Double sharp (talk) 12:20, 27 April 2014 (UTC)[reply]

I changed it back to read "cannot appear in any tiling of regular polygons". That statement is true, and is much stronger. Technically, what's true is that those vertex types cannot appear in any edge-to-edge tiling of regular polygons, so I added the phrase "edge-to-edge" at the beginning of that section, since all of these statements apply only in that case. This result can be found in my Ph.D. thesis, but was known long before then. I suspect the first person to state this was Kepler, 1619, in "Harmonice Mundi". The statement you began with, about three polygons meeting at a vertex and one has an odd number of sides, is a very limited argument. As you say, it only eliminates certain uniform tilings, and doesn't apply to the general situation. It's a statement often used for elementary tiling sources that are limiting themselves to the consideration of uniform tilings, but it adds nothing here. It should probably be deleted. Chaveyd (talk) 13:49, 27 April 2014 (UTC)[reply]

What does this mean?

"3.4.3.12 - not uniform, has two different types of vertices 3.4.3.12 and 3.3.4.3.4"

What is claimed to have these two different types of vertices? If it's the tiling, there isn't only one possible tiling with 3.4.3.12; in fact there's a 2-uniform tiling with 3.4.3.12 and 12.12.3, not 3.3.4.3.4 at all. Similarly with much of the rest of this list.... I would edit it, but I'm not 100% sure I'm right in my guess about what the list is trying to say. 91.105.25.26 01:53, 20 August 2007 (UTC)[reply]

Yes there are problems with the list as its far from complete. One way it could be considered is as a list of vertex configs which yield a uniform tiling, and explination as to why the other configs do not yield a unifiorm one. It could do with an expansion listing the other tilings which can contain a particular type of vertex. --Salix alba (talk) 08:16, 20 August 2007 (UTC)[reply]

How many??

The following proves there are infinitely many tilings made out of regular polygons:

A hexagon is always interchangeable with a flat pyramid (6 triangles meeting at a vertex)
A dodecagon is always interchangeable with a flat cupola (a hexagon surrounded by squares on each edge and triangles at each vertex)

But, is the following question known:

How many tilings of regular polygons are there that don't contain in them any flat pyramid or flat cupola?? Georgia guy (talk) 22:55, 10 October 2010 (UTC)[reply]

Still infinitely many (even uncountably many), even if you require the polygons to meet edge-to-edge, because you can tile the plane by any sequence of square strips and equilateral triangle strips. —David Eppstein (talk) 22:59, 10 October 2010 (UTC)[reply]

Added Reference link

The references for this page have long included a link to a 1989 paper of mine. Since that source is difficult to access, I have added a direct link from that reference to my web page from which one can download a copy of the paper. I hope doing so this is not a violation of Wikipedia rules. I was not the person who originally added this link (which I'm sure would be a violation), I only added the link from that reference to my page. Chaveyd (talk) 05:53, 26 June 2013 (UTC)[reply]

It seems unproblematic to me but I fixed the formatting. —David Eppstein (talk) 06:09, 26 June 2013 (UTC)[reply]