New type of pentagonal tile

[jaap]

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]

Hm, that's really cool. I've never even thought about something like this.

[Flumble]

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]

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.

[Flumble]

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.

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.