Pentagonal Tiling

I discovered this while thinking about the pentagonal tiling of type 15. Is this a new type of tiling? If it is, then I think I have found several other new pentagonal tilings like this one and the pentagonal tiling of type 15. They all have vertices which lie in the field $ \mathbb{Q} (24) $. The internal angles for a pentagon in the image above are

$ 60,150,90,120,120 $

And the lengths of the edges of a pentagon in the image above are

$1, \sqrt{3} ,1,1,2 $

  • Dang, that actually seems to tessellate. But I feel like there's a "mini-hexagon" in the top right that tessellates by itself... – pie314271 Mar 3 '17 at 5:55
  • 1
    This seems to be just a combination of two tilings. If those two are known, does this count as something new? – Claudius Mar 3 '17 at 6:07
  • According to Wikipedia, there are fifteen types of convex pentagons known to tile the plane monohedrally: en.wikipedia.org/wiki/Pentagonal_tiling This tiling is related to type 15. – six Mar 3 '17 at 6:15
  • I added a new picture which shows how it is a combination of two tilings. – six Mar 3 '17 at 22:18
up vote 3 down vote accepted

About 6 or 7 years ago I did a computer search for various monohedral polygon tilings. I was hoping to find a 15th pentagon type, but got side-tracked into cataloguing all the tilings I found along the way. See my tilings page.

I did not find this one of yours, as it is 4-isohedral and I did not search for those much because my programs were too slow for that. The tile itself however is obviously of a known type, namely type 1 (two adjacent corners sum to 180, or equivalently two parallel sides). It is also of type 2. I would however be interested in seeing any other tilings you have found.

Recently Michaël Rao has announced a proof that the 15 known convex pentagon tile types are the only ones. You can read about it in this article, and read the draft paper here. It is plausible, but as it depends on an exhaustive computer search, it will be a while before it is replicated and confirmed.

Here is a picture of the tiling with the tiles coloured by isohedrality class:

enter image description here

During my searches I had also found a tiling using the exact same tile, but this one is 3-isohedral:

enter image description here

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