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https://preview.redd.it/ik1xaier4kjc1.png?width=2000&format=png&auto=webp&s=7ae7c1b2948d8dab1e15b502283c64324a562263

This was made by overlaying two patterns of triangles with angles (90,45,15) degrees. Both patterns were identical, but positioned differently. I had a conjecture that they will line up into a periodic picture, and they did!

But then, to re-create it as a real tiling, I spent many hours creating expressions for lengths and angles of each small tile. This thing has twenty distinct tile shapes!

One way to understand it is to start with a tiling of (90,45,15) triangles, separate the triangles into 6 classes, and then cut each of them in a unique way.

https://preview.redd.it/jxzoducq6kjc1.png?width=2000&format=png&auto=webp&s=25934f1ebcbc63ff41f41f938190459cd05469d7

The secret ingredient of this picture is this: in a right triangle (90,45,15), the longer side is exactly twice the shorter side.

I have write quite a few complex transforms which work wonderfully on periodic tilings because I can simply access the pixels in a modulo fashion. This results in beautiful Escherian figures. Now I'm wondering what these transforms would look like with aperiodic tilings. I'm especially interested of course in the new 'ein-stein'. Like Escher, who made tiles into salamanders and all sorts of animals, I have designed a flying duck for the ein-stein.

The complex transform shaders will try to access verge large coordinates. Nearing infinity actually, but I'll cheat a little and loop the texture when it becomes too small to see. But I'll need a large plane nevertheless. Is there software 1. to make such a large plane of ein-steins? and 2. does it allow for custom drawings/textures on the tile?

Reddit search thinks nobody has asked this. Somebody has to do it, why not me.

Who has their bathroom in (in order of prestige? or does it go in the other direction?)

1. Spectre tiles
2. Penrose tiles
3. Some other aperiodic tiling

?

Other rooms or even exterior tilings would also be acceptable, but I feel bathrooms should win.

Also, for anybody this turns up: how did you source the tiles? (especially if you live in the UK)

I have an idea to create some unique illustrations / art pieces and wondered if the maths in the idea was sound. By unique I mean they would be illustrations of a bit of an aperiodic tiling of the plane, around a set of far off coordinates such that the exact illustration could only be found/reproduced if the starting coordinates were known. Is there a minimum number of tiles needed to ensure that a piece of the plane is unique for a given level of precision?

From what I've grasped from youtube, the coordinates can assembled by building supertiles in a loop & chasing the desired "direction". Is that pointing me in the right direction ? Have I understood enough of the basics of aperiodic tiling and the general idea of a specific bit of the tiling being "unique" is true?

My (probably wrong :) ) intuition is that it's kind of like a public-private keypair and that with the co-ordinates, one could quickly verify the uniqieness of the illustration. But without knowing the coordinates it's NP hard to find where on the plane the illustration came from, thus making it "unique"?

I'm thinking the coordinates could be some massive numbers derived from a SHA256 hash of a poetic phrase or something along those lines for added artsy points, suggestions / better ideas are very welcome :).

I'm sorry, I'm really new to this. I'm an artist and not great at technical or math stuff. I've watched a couple videos about the chiral aperiodic monotile called the spectre, and followed all the links I can find, but the pages the purport to have images or SVGs to download all have thick boarders that extend outside the true edge, making them not actually tile properly from what I can tell. At least, when I bring the SVGs in Zbrush or Blender I can't get them to fit perfectly. Any tips?

Hello, found this subreddit today and I thought I should post something.

I have been always interested in this problematics, focusing on periodic hyperbolic tilings.

A few years back, I've put together an algorithm that can generate tilings, given the list of allowed tile shapes and vertices. I used it for several applications, for example enumeration of k-uniform Euclidean tilings beyond the previously discovered limits (https://oeis.org/A068599), and extended it to the first explicitly constructed 14-Archimedean tiling:

https://preview.redd.it/3574fhr0jtjb1.png?width=1000&format=png&auto=webp&s=4e902328f726f993b5ea42eb1943dd4ca081589d

Of course, there's no need to limit ourselves to regular polygons:

https://preview.redd.it/22wkin7ektjb1.png?width=2000&format=png&auto=webp&s=029de56a14d4bfc23bf4f614edb7442dc30b395b

Or, it can be used to assemble hyperbolic tilings with vertices that do not allow for uniform configurations:

https://preview.redd.it/i980nyxojtjb1.png?width=2000&format=png&auto=webp&s=7779f424e77f4ed9ac0327b4d707da7d504c2e74

(All images are made in the HyperRogue engine.)

The most interesting applications are what I call "hybrid tilings". In hyperbolic geometry, each tuple of 3 or more regular polygons that can fit around a vertex has a unique edge length that allows the polygons to do so. It is not, as far as I know, well-researched which tuples would resolve to the same edge, but I have found an interesting list of solutions:

(3,5,8,8)/(3,4,8,40)

(4,4,5,5)/(3,3,10,10)

(3,5,6,18)/(3,6,6,9)

(3,4,4,5,5,5)/(3,3,4,5,5,20)/(3,3,3,5,20,20)

And when we allow distinct (but commensurate) edge lengths for the polygons, we can get something like this:

https://preview.redd.it/6cfw2kb7ltjb1.png?width=2000&format=png&auto=webp&s=d20e544cb23887f76f6e0a7382fcc84699348aa2

I've posted my results before in other subreddits. I am interested in whether there are other applications where this algorithm could come in handy.

spectre_fish

Tiles created with: Kosovircek/SpectreTileMaker: A web app for deforming edges of spectre tiles. (github.com)

Inpainting with stable diffusion: AUTOMATIC1111/stable-diffusion-webui: Stable Diffusion web UI (github.com)

Using model AnythingV5Ink_ink: 万象熔炉 | Anything V5/Ink - ink | Stable Diffusion Checkpoint | Civitai

Prompt: a fish

If you're retiling your bathroom or some other room, unless you're doing it using an aperiodic tiling, please post in r/HomeImprovement instead.

Following up on an old, now closed-for-comments post on textbooks on tiling:

Some books discussing tiling:

• Martin Gardner's Penrose tiles to trapdoor ciphers...and the return of Dr Matrix. ISBN 0-88385-521-6
• Craig S. Kaplan - Introductory tiling theory for computer graphics. ISBN: 9781608450176 (paperback) / ISBN: 9781608450183 (ebook)
• Branko Grünbaum & G.C. Shephard - Tilings and patterns. ISBN: 0-7167-1193-1

Hi everyone! I'm a student doing my MS in mathematics, and I recently came across some concepts surrounding things like Penrose tilings. I found it very fascinating, to say the least. Can someone please suggest a textbook that I can study to learn more about tilings and tesselations?

Does anyone have any recommendations for software for designing tiling/tessellations? I have been using my mechanical CAD software (because it's what I know) but I assume there is something more efficient out there.