Atoms of Penrose

Prefix

1. In this post, you will read the results of my trials with the aperiodic tiling. Since it is written by a mechanical engineer who only fools around with the tiles and writes his findings with the same way he figure things out (which you may find very primitive), language of this post will not be scientific. However, if you are not interested with geometry and/or not familiar with the subject, you probably won’t like the post either. So I expect this post will be only interesting to a few people. Will this stop me writing this post? Hell no.

2. This blog has been initiated in Turkish and in future I intend to continue writing posts in Turkish. The reason that this post is in English is due to what I mentioned in 1^st item above. There will be few people who might be interested in this post and I didn’t want them to miss it because it is not in English. (Why it is not in another language is mostly due to my lack of other language skills and also almost all the source I see about the subject in internet is in English. So hopefully it can reach to maximum people who are curious about it.

My first and only post about Penrose Tilings was written about 2 years ago. I will make a recap since it was written in Turkish. If you are not interested you can pass to the next part. 

History

I am a person who easily gets distracted and finds himself procrastinating very often. In one of these moments, I’ve got interested in Penrose Tilings after seeing Roger Penrose’s name in it. His name was mentioned in Stephen Hawking’s “Brief History of Time” many times and I wondered the connection. Then I learnt what are the Penrose tiles and what they mean in aperiodic tilings world.

This is not a post which will explain comprehensively what the basics of the subject are. You can find many sources in internet which explains it way better than I can. You can try this or this one if you are new to the subject.

What intrigued me after getting into it was the challenge to have the aperiodic tiling with the least number of prototiles. To have the aperiodic tiling with one prototile was still a debate and it was called Ein stein Problem even though it has nothing to do with Einstein. (Ein Stein means one tile in German as we learn from Wikipedia.) So finding an aperiodic tiling with one prototile was a big deal and there was nothing holding me to try it.

One good side of being Mechanical Engineer in this computer age is that you probably have ability to draw what you want in 2 or 3D with one of many wonderful drawing tools in your computer. This gives you the chance to virtually create shapes and delete, copy, modify endlessly and maybe hundreds of times faster than the conventional drawing styles. So, after learning the challenge I mentioned above, it was like shouting at me to give it a try.

So I daftly started to draw like I can find what the most genius mathematicians couldn’t find since decades. I have read that the golden ratio (φ=1,6180339887…) has been involved within the Penrose Prototiles and it must also be involved in my prototile. Why? Because the golden ratio was cool as fuck. It was so cool that, it must be everywhere in my shape. All edges, angles etc. must involve some sort of golden ratio in it. I was like, the more golden ratio I put in it, the better the result I get (Yes, I was that silly when started to deal with it). Then I came up with this;

All I did was drawing 4 lines with golden ratio between each other and connecting them to have a tetragon. When the internal angles came out with the 36° and its multiples, I was totally amazed with it. I was even dumb enough to be fascinated by finding golden ration with the sin and cos of the internal angles (What do you expect when the sin and cos is related with the ratio between the lengths of the edges, you idiot. You already set the lengths with golden ratio.). I even named it as “Golden Trapezoid”.

That must be it, the ultimate shape that will lead me to my aperiodic tiling. Then I started to multiply and align the shape to create my tile without any space or overlapping. It didn’t take long to realize that it leads me to these 2 tilings:

That wasn’t what I expected. Since all the edges have golden ratio, the corners of prototiles does not intersect between the planes which have different orientation and always keep the same periodic structure. If you go far enough from the center, all you can see would be the different rotations of this:

I was defeated very quickly (like a part of me expected it since the very beginning). One thing though, was still a bit of a question; the (not happening) intersections that I marked below.

Since the golden ratio is an irrational number, it means that these intersections will only occur at infinity and each of them occurs on a different place on the adjacent shape. So can we call it aperiodic? I had no idea and seems like there is no way to find it for me.  (I assume these kind of technical explanations all along this post will be very painful for me and also for the reader since I am neither a native English speaker nor a mathematician who studied in English. I promise to write technically more correct explanations in future if this post will mean anything to the tiling world. But for now, we have to live with them.) So I wrote all these things in this blog and gave it up there. If it means anything, somebody could see it and take it further. If not, then stays there with no harm.

Eren Strikes Back

So we came to present day. Without any meaningful reason, I recently got involved in the same challenge again. I opened my previous drawings and tried to figure out how I can have some different results. Looking at the Penrose Tiling made with rhombuses, I realized that having golden ratio in all edges of the shape is not that good idea. Instead, I better keep fewer edges with the golden ratio and give rational lengths to some other edges of the shape. With this principle I could have some intersections and break the loop before my tile elongates into eternity with periodical patterns. So this time I came up with this:

With some weird hope, I started to tile the plane with my new prototile. It looked promising in the beginning; there were the intersections I expected:

But then when I continued, I have faced with the results of golden ratio again. Seems like our fancy ratio has become my biggest problem due to its irrational nature. But I also feel that the internal angles made of 36° are also mandatory and I needed at least one edge to have the ratio for that. Anyway soon I start to have this:

It seems like I again ended up with the same endless periodical planes only with the difference of a starting pentagon and weird thin small planes between my endless periodical planes.

This was predictable of course, 1,618... long irrational edges would eventually join with the rational ones and their corners would only intersect at infinity, so my planes will be endless periodical ones. I was defeated again.

But then I decided to concentrate on the points which are closer to intersect. There might be a pattern in them which may help me. I took only the not intersecting part, turned it a bit to make it horizontal then I marked the closest intersections. I showed below where φ unit long edges (blue), gets close to intersect with 1 unit long ones (green). I also showed how many φ long edges it takes to reach that point with blue (8^th, 13^th etc.). It looks like this:

Attack of the Fibonacci

It didn’t take long to realize that all these figures are consequent numbers in Fibonacci sequence. I once again was daftly amazed with this mystery but then realized that it was a natural result of the Fibonacci series since each consequent number in Fibonacci has a ratio between them which is about φ and it approximates to φ when the number gets higher. Below is a basic explanation of this;

So these are the first 10 numbers in Fibonacci Series.

F1	F2	F3	F4	F5	F6	F7	F8	F9	F10
1	1	2	3	5	8	13	21	34	55

Let’s have a look at the ratio between each consequent number;   

    1 / 1    = 1

    2 / 1   = 2

    3 / 2   = 1,5

    5 / 3   = 1,666666  

    8 / 5   = 1,6

  13 / 8   = 1,625

21 / 13   = 1,615384  

34 / 21   = 1,619047  

55 / 34   = 1,617647  

And can show this on a graph like that:

Source

So we can formulize this as;  

and therefor;

F_n+1  /  F_n ≈ φ

And also can be written as 

F_n  x  φ ≈  F_n+1   

If we apply this to our case:

  8 x φ = 12,9442672 ≈ 13

13 x φ = 21,0344342 ≈ 21

21 x φ = 33,9787014 ≈ 34

34 x φ = 55,0131356 ≈ 55

So instead of finding something mysterious, I found a nice visualization of the Fibonacci Numbers and the golden ratio among them.

Anyway, finally I found my pattern but how can this be useful for me? Since my first trial 2 years ago, I have wondered what it will look like when these corners intersect at infinity and where it will go from there.

So since I finally proved that they will eventually intersect, I decided to try these intersections like they happened at infinity. I could take one of these points above and neglect the differences then continue from there.  So I had the following shape;

 And the corner marked with the circle looks like that when you look closer;

So instead of that, I will pretend like I repeated it endlessly and the corners finally intersected like that:

When I continued with this principle, I started to have this shape;

Click to Enlarge

It looks pretty aperiodic to me but still cannot say exactly. Since I couldn’t come up with any matching rule, it was also becoming difficult to decide which way to proceed after each intersection point. But these shapes were somehow familiar. It looked like a a distorted Penrose tiling to me and I decided to try something simpler. Instead of the previous starting shape and thin lines, I wanted to try this:

Then I continued;

This definitely looked like Penrose tiling made with rhombuses. Then I decided to give them colors like the Penrose prototiles and got this:

Click to Enlarge

And this is the original Penrose tiling picture from Wikipedia:

They pretty much look like the same with only one small difference. While the original Penrose tiling is made of 2 different rhombuses with each having 1 unit long edges, my tiling was made with same rhombuses but they had infinitely long edges and they were made of the same single shape repeated endlessly.

I finally did it. Since the Penrose is a proven aperiodic tiling that means my shape also leads to an aperiodic tiling. The method I used to find it could also be used to create tiles in real world with finite numbers of prototiles. The neglected difference at the points where they are close to intersect could be eleminated in joint gaps of the tiles and it can practically be applied.

 The only question mark is since my tiling will only show itself at infinity, will this be accepted as an aperiodic tiling or not. But anyway, now I know what it will look like at infinity and it is definitely a Penrose tiling. Wow, just wow.

Now it is time to check what will happen if I try the same thing with my 2 years old golden trapezoid.

So I started as usual, but this time 0,618.. long edge corner should intersect with (2,618… + 1.618…) long edges corner. But this wasn’t enough. For the second rhombus shape, 1 unit long edge corner should intersect with (2,618… + 1.618…) long one, as well. So all of their first neglect able intersection would be after 144 and 21 and 89 repeats. A bit complex when you say like that. Never mind, just look at below shape. ;

Click to Enlarge

We all know where we will go from here; Penrose Tiling again. That means as long as you keep the internal angles as 36, 72, 108, 144 and top and bottom edges parallel, then you will reach to a Penrose tiling, doesn’t matter how long your edges are. Only thing matter will be the number of trapezoids that you will use to see the first neglect able intersection point. But since our real intersection occurs at infinity, that actually doesn’t mean anything.

So I can say that I did not only find the shape that will tile the plane aperiodically, I have found the rule to the shapes that can do it. Here are the rules.

Arrows indicate that the edge lengths can vary as long as keeping the angles same. The top and bottom edges need to be parallel because there are also shapes which has the same angles and doesn’t have top and bottom edges parallel, like the bottom shape, but I couldn’t manage to make an aperiodical tiling with them. They either end up overlapping or with a periodical tiling. Maybe that’s because I wasn’t able to do it but for now, the rule requires them to be parallel.

 I suppose to write a conclusion here. All I can write for now is that even though with infinite number of shapes, I believe that I have showed an aperiodical tiling with the single prototile(s) is possible. Hope someone can confirm this one day so it can be useful. I believe that it might be useful for crystallography and quasicrystal researchers, as well. 

Besides, I must say that it also amazes me. The planes which were going to different directions endlessly at the begining is somehow changing their direction at some point which you cannot exactly foresee smacks of the coincidences in our lifes and how it goes to somewhere that we previously could never imagine. And having a pattern like Penrose tiling from something which seem unpredictable at the beginning is just poetic. Maybe it also means that what seems like caotic to us also is a tiny part of a beautiful structure that we are not able to see yet. It resembles so many patterns that I cannot even describe properly. Anyway that much romance for now is enough I guess.

My next plan is to have a real aperiodical tiling on a wall or something else with this single shape by eleminating these neglect able difference inside the joint gaps. So I can have a real life example of it. Why? Because as I said in the title of my first Turkish post about the subject; Loving Penrose Tilings Inexplicably!