# An Aperiodic Monotile

Curve 208 is a doozie!

Aperiodic_Monotile2.2.gh (572.5 KB)

and these curves are far from random - there are often several of the same length! Very like a dragon curve.

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This is fascinating stuff indeed!
Just to help us beginners… When you design the decorations for the tiles, is there some sort of logic that you apply in terms of how the decorations meet the edges of the tile? I can see that equal length edges should probably have the same number of curves meeting them at the same distances from the ends of the edges, in order that the would join together when tiled, but is there anything else that you consider?

As with penrose, if we overlay simpler geometry rather than more complex geometry, we see lines begin to emerge, and we also see curves emerge in an odd, hallucinatory way, where at times it’s easier to see them by not looking directly at them.

Aperiodic_Monotile2_RE.gh (582.4 KB)

Another simplifier, triangles really pop here, partial circles vanish: minimum spanning tree of the centroids, plus multipipe:

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There is an excellent overview of this at An aperiodic monotile exists! | The Aperiodical.

Basically, the tiles here are internal to one of four ‘meta-tiles’, which have fairly restrictive possibilities for meeting adjacent meta-tiles. Larger clusters can be created by an expansion rule.

What Daniel has done is to create a thread pattern that meets the edges of the meta-tile normally, so as to line up gracefully with neighbours, and which obeys the matching rules for the meta-tile edges.

I assume that the initial triangle pattern Daniel used comes from https://cs.uwaterloo.ca/~csk/hat/app.html, which I have yet to explore, although it is possible that Daniel did the triangle expansion himself.

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I used the H7/H8 substitution system from the paper to generate this patch myself

The numbers are a bit confusing at first because the count changes in the first level.
for step 1, H7 is made up of 7 hats, and H8 is made up of 8 hats.
for step 2, to get one larger cluster you join 6 of the previous level clusters(5 H8s and 1 H7), and to get the other larger cluster you join 7 (6 H8s and 1 H7).
Then you iterate step 2
It converges to this fractal shape

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As for the matching rules if you want to make patterns continuous across tiles-

You can use just the 6 long edges, directed away from the 120 degree vertex

You can use just the 8 short edges (and note that topologically the bottom horizontal side is actually 2 short edges, since the sides of other tiles form a T-junction when they meet it)

Or you can use all 14 edges

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I wanted to get my mind round where the orienting triangles came from too, so I made up a grasshopper definition that made them from whole cloth, rather than relying on making patches offline.

It was a bit messy, in that the fractal dimension between one expansion and the next should be TAU**2, which is an irrational number, but I was using integer coordinates for the triangles, which forced me to work out the six new centres explicitly for each further iteration, rather than passing in a scaling ratio. I’m sure someone could do it more elegantly!

Monotile1.5.gh (20.1 KB)

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This was my next question!

Hi daniel, can I ask how you drew the target triangles?
I can’t understand the logic behind this triangle pattern to map into?

I generated this patch with the H7/H8 substitution system described in the paper. Here are the first few steps:

This process converges to the fractal shape shown in my animation above.

I used the triangles only as a way of mapping other shapes to the tiles - as these are rigid transformations one could also use planes here if you ensure the input geometry is the right scale.

There’s another different substitution method described in the paper that might be interesting to try using 4 different tiles which it names H,P,T,F.

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Max, would you be so kind to share how you got the second pattern?

I’ve been looking for something like this…

Cheers

I understand. Thank you

Hi Xavier,

If by ‘second’ you mean the weave pattern, here is my attempt at doing it, based on Daniel’s wonderful animation algorithm. I made the base weave pattern using Grasshopper, which is ridiculous; I really must learn how to use base Rhino some time!

Tapeweave1.gh (41.6 KB)

Bob

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There are many ways to connect between edges of the tiles that result in complex loops of various lengths. One of them I found interesting is this one, which connects into infinite curves in 3 directions.

We can move the connection points closer to the corners (though not exactly on the corner as then we’d have 4 coincident points and wouldn’t be able to connect the continuous curves up)

and the curves become these stepped lines

I find these interesting because they suggest to me that there might be an analogue to be found to the Ammann bars in Penrose tilings.

A nice article that covers these a bit

Which includes this quote from Grunbaum and Shephard
“it is the system of bars which are fundamental and the only
function of the tiles is to give a practical realization to them.”

The stepped curves do not give us these nice matching rules and alternative generation methods like the Ammann bars, but perhaps there is a way of decorating the hat tile or one of the edge length variants with segments in a way that does?

step_sequence.gh (124.3 KB)

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Thanks Bob and Daniel for the amazing graphics and explanations!

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I like the almost 3d effect of these tile decorations…

Also quite satisfying is control point curve from the polyline…

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Hi Xavier,

Re-reading this thread, I realize too late that you were actually asking about Max Allstadt’s second pattern, not Daniel’s. Max said “minimum spanning tree of the centroids, plus multipipe”, which took me a while to get my mind around, since I have been thinking in two dimensions!

So I had a go at it! I did not manage exactly his effect, but connecting all centroid points to their two nearest neighbours and then using multipipe indeed produces much his effect, which is very striking:

Tree1.gh (30.6 KB)

Upping the pipe radius until there is more contact might even give something rigid enough for 3-D printing - I could see this being a nice broach, if made in silver!

Bob

Hello Bob,

Thanks very much for looking further.

This was the pattern im looking for

I tried to tweek your models but did not get there sooner or later…

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Close, but no cigar! Sorry.

My one actually looks better with thinner lines.

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damn that’s good!