Tiling question

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Offline TofaraMoyo

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« on: April 17, 2018, 08:37:08 PM »
I was wondering what can be said about the following type of fractal....this is a tiling excersize and there are "n" tiles. each with different shapes. But these n tiles can be placed alongside each other in such a way that they can form the shape of any individual member of those n tiles just bigger...what can be said about how you would generate something like that for a given "n"...thankyou!

Offline TGlad

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« Reply #1 on: April 18, 2018, 04:12:05 AM »
You mean n differently shaped tiles can be arranged to form n larger shapes, which are scaled up versions of the n tiles?

It doesn't sound easy. Let's just look at the simplest non-trivial case of two tiles.

We can't use rectangles, because square+rectangle can never make a square, and rectangle+different rectangle can only make one rectangle, not two. Two of the same rectangles cannot be arranged to form the same shape.
Triangles, and quadrilaterals seem unlikely. So the only candidates might be recursively defined shapes, if there are any at all.

Do you have any examples, or links to existing tiling with this property? It's an interesting idea.

Offline TofaraMoyo

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« Reply #2 on: April 18, 2018, 09:36:03 AM »
Hi

Thanks for the reply...here is some background.. i have what is a cellular automaton, what is special about this cellular automaton is that each cell is "adjacent" to cells that are not near it. So the top right cell could be next to the bottom right, which will be next to the center...and so on. So all the cells are connected, and there is a path from one cell to any other one, its just through other cells. What is interesting is that this describes exactly what i am talking about. If we start of with a particular cell and draw a graph showing its connections , its adjacent cells will be on the next level lower , then those cells adjacenbt to those ones will be yet lower...what will happen is that we will eventually list the first node once more at a lower level (as there is a path to it form other cells)...this means that it is fractal like in that at a lower level the pattern repeats...whats more there are an infinite of lower levels...and since this is true of every cell then that means that each cells fractal is made of the other cells fractals...which is what i described about the tiling exercise.....

So how could we create these tiles from the cellular automata....i was thinking we list out a tree starting from any node..then as each node appears we refine its shape untill at a limit at infinity we have generated the shapes we want..

Offline TGlad

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« Reply #3 on: April 19, 2018, 03:57:05 AM »
For the two-tile case, you're looking for mutual Gnomons. That is two shapes that are gnomons of each other.

One almost answer are these two triangles:

They are the same shape but different sizes, and the combined triangle is an upscale of both of them.

Another very similar one is the Ammann Chair tile pair:

It also doesn't quite work, in this case the combination is only a mirror of the single red tile (which should be flipped in the image).

Another almost answer uses the following two tiles, an acute isosceles triangle and the 'golden gnomon' triangle:

 
In the shown configuration it makes a larger acute isosceles triangle, and with one more of the golden gnomon on the left it makes a new golden gnomon triangle.
So you can almost make both, but it requires an extra of one tile to do so.

Another approach to an answer is to start with mutual gnomons that are the same shape, then try to add a small perturbation and see if the shapes can be made a little different. The two cases I can think of are the A4 shaped rectangle, and the 45 degree right-angled triangle. 
« Last Edit: April 19, 2018, 06:27:37 AM by TGlad »

Offline TGlad

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« Reply #4 on: April 19, 2018, 08:18:00 AM »
Another almost is this triangle duo tiling


Notice this produces larger versions of the acute isosceles and the scalene triangles, by arranging in two different ways. However, notice that the scalene triangle has been flipped in the bottom right case; so it isn't really the same shape.

Offline TofaraMoyo

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« Reply #5 on: April 19, 2018, 08:29:08 AM »
Thanks for the awesome examples!

i think my task is not going to be succesful, i am doing research in AI and in particular natural language processing. So the connection is the hypothesis that words mean someting because of the meaning of other words. In this case a shape is the words meaning, which can be made of the other words shapes.

Offline TGlad

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« Reply #6 on: April 20, 2018, 01:18:34 PM »
Well either way it is an interesting question. In fact you've inspired me to write a blog on the idea.

There are solutions for the two shape case, but they're fractal limit sets rather than simple dense shapes:



white is shape A, yellow is shape B, placed so together they make a larger shape A.



Pink is shape B placed so white and pink together make a larger shape B.



So yellow and pink are the same shape, shape B.



I'm not certain if dense (as in solid) shapes are possible but connected shapes are possible. Some of the above are close to being connected.
« Last Edit: April 24, 2018, 04:07:17 AM by TGlad »

Offline TGlad

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« Reply #7 on: April 24, 2018, 03:45:27 AM »
Here are some more. They're definitely different to standard limit sets:















Offline TGlad

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« Reply #8 on: April 25, 2018, 12:44:51 PM »
And a few more:










It's a big search space to explore, 8 dimensional, but those are some of the nice locations. A few more are on the blog post.  O0

Offline TGlad

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« Reply #9 on: April 27, 2018, 01:41:45 PM »
Finally, here's the closest I found to a dense shape in this category:


In fact I'm only looking at the special case where shape A+B=kA and A+B=kB for same scale k (if that makes sense). I call these twin-tiles because if you mass produced loads of tile A (white) and tile B (yellow), you could make bigger and bigger copies of each tile.

So maybe a dense shape exists in the less constrained case of A+B=kA and a+B=jB for different scales k, j.

Pseudocode is in the blog:D
« Last Edit: April 28, 2018, 12:20:10 PM by TGlad »

Offline TGlad

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« Reply #10 on: May 05, 2018, 11:32:49 AM »
I've found the answer to your original question, there has been some work on it.They're called self-tiling tile sets or setisets for short:

https://en.wikipedia.org/wiki/Self-tiling_tile_set

However, their solution for the two tile case includes reflections. If you exclude reflections then the results are fractal like the examples I gave above. You could say these shapes are co-similar rather than self-similar.
« Last Edit: May 06, 2018, 04:07:48 AM by TGlad »


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