## New type of pentagonal tile

For the discussion of math. Duh.

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jaap
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### New type of pentagonal tile

A new type of tileable pentagon has been found!
There are many kinds of tiling that use just one shape of pentagonal tile (and its mirror image if necessary). Up to now, these tiles fell into 14 types (see http://www.mathpuzzle.com/tilepent.html ). Now another has been found.

I did a lot of research on this five years ago, and found many tilings that seemed to be new, but no new tile shapes. My computer search hadn't gone far enough to find this new type. I'm a little miffed that I didn't get there, but very happy to see there was still (at least) one to be found. Who knows, there may be even more out there.

Magnanimous
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### Re: New type of pentagonal tile

Hm, that's really cool. I've never even thought about something like this.
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Flumble
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### Re: New type of pentagonal tile

My thoughts, in unspecified order:
1. interesting
2. not all text is readable due to the resolution
3. the tiling pattern is pretty complex
4. this isn't published yet, is it?
5. the tile in the picture has a specific combination of values, no independent variable (except for scale obviously). Is it a class by itself? Is it part of a finite or countable set?

jaap
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### Re: New type of pentagonal tile

Flumble wrote:My thoughts, in unspecified order:
1. interesting
2. not all text is readable due to the resolution
3. the tiling pattern is pretty complex
4. this isn't published yet, is it?
5. the tile in the picture has a specific combination of values, no independent variable (except for scale obviously). Is it a class by itself? Is it part of a finite or countable set?

2. Click the image, or open the pdf version I have attached to this post. Here's a link to another image.
3. Yes. It is based on this isohedral tiling with hexagons, where each hexagon is replaced by a cluster of 3 identical pentagon tiles.
4. No. Casey announced it on a tiling mailing list today.
5. Correct. There is no degree of freedom, and this tile is in a class by itself. Type 14, which was found in 1985, similarly has no degree of freedom. If there are any others to be found, it would not surprise me if they were like that too.
Attachments
type-15.pdf

Flumble
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### Re: New type of pentagonal tile

jaap wrote:2. Click the image, or open the pdf version I have attached to this post. Here's a link to another image.

I feel so stupid right now. Thanks for the pdf nonetheless.

jaap wrote:Type 14, which was found in 1985, similarly has no degree of freedom. If there are any others to be found, it would not surprise me if they were like that too.

Ah, from the way it's described at mathpuzzle, I thought that type 14 had (at least) one degree of freedom.

mPora
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### Re: New type of pentagonal tile

Flumble wrote:
jaap wrote:Type 14, which was found in 1985, similarly has no degree of freedom. If there are any others to be found, it would not surprise me if they were like that too.

Ah, from the way it's described at mathpuzzle, I thought that type 14 had (at least) one degree of freedom.

Me too, why does it not have any degree of freedom. I've spent ages on this and have not been able to find any new pentagonal tiles that don't already exist.
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jaap
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### Re: New type of pentagonal tile

Rogereric wrote:@jaap would you please share with us some evidence based research regarding your work i mean some reference based material because i am also working on not exactly the same thing but something like that. i am looking forward to you! thanks

(I realise that was a spam post, but I might as well give a little follow up post to this thread anyway)

The Mann/McLoud-Mann/von Derau paper is here:
https://arxiv.org/abs/1510.01186

I had basically done the same thing, but without the proofs to show the algorithm works. In short, to generate a tiling, the steps are:
2. Put k copies of that tile together to form a block.
3. Treat that block as a single tile, and put it into an isohedral tiling template.
(Tom McLean's site lists all these, but it seems the site is no longer working. I've mirrored it here.)
4. Set up the system of equations that the side lengths and angles need to satisfy for this tiling to exist (i.e. angles around a vertex sum to 360, lengths on either side of a line must be equal). Discard it is the system is inconsistent, or if it only has solutions where one or more of the angles or side lengths is non-positive.
5. Find a particular solution to the set of equations such that the angles and side lengths are geometrically consistent (the edges form a non-intersecting closed loop). Discard it if there is no such solution.

You have to do step #2 and #3 in all possible ways in order to enumerate all possible tilings. One difficulty is weeding out duplicates. Another is that when you can't find a solution in step #5, proving that there is no solution is hard.

I did a full search with k=2, both convex and non-convex tiles, and with triangles, quadrilaterals, and pentagons. I also catalogued all the tilings that I found.
I did some searches for convex pentagons with k=3, but this took a long time and there were many false positives (i.e. potential tilings that only fail in step #5). I would have found the 15th tiling if I had simply weeded out all (potential) tilings that use a type 1 or type 2 tile, rather than try to catalogue them all. To be honest, I didn't believe there would be another tile type.