## Penrose Tiling the Plane.

• 5.5k
Everything that follows should be understood as prefaced with the phrase, "as I understand it." Unfortunately, I don't. So, HELP! Can someone please make it all a little simpler and maybe more accessible for someone who isn't getting it.

The title of this video, https://www.youtube.com/watch?v=48sCx-wBs34&t=535s is provocative. "The Infinite Pattern That Never Repeats."

The idea is that the (infinite) plane can be tiled with a non-repeating pattern. Hmm. That it can be done with a finite number of tiles (i.e., tile shape/designs). How many shapes of tiles? Two. The meat of the matter between about 4 :55 and 8:30 in the video, but the whole is watchable.

In sum (as I understand it) given two shapes, a finite set of colorings of those shapes, a few rules on combining them (why even bother?), and the plane is (not can be but must be) tiled without repeat.

Sounds very wrong to me.
• 3.4k
I saw that video when it came out and found that penrose tiling fascinating. I don’t know why you think it sounds so wrong but I look forward to discussion about it with people more knowledgeable than me.
• 10
It is mind boggling. Perhaps it is helpful to draw an analogy between the non-repeating sequences of the decimal representations of numbers like Pi and Phi and the non-repeating infinite pattern of Penrose tiling. I recall the video mentions Phi or the Golden Ratio emerging from the geometry of Penrose Tiling. If you believe that the decimal representation of Pi and Phi are non-repeating sequences maybe it is easier to believe the non-repeating pattern of Penrose tiling.

Hmmm, I wonder if there are certain combinations of solids that can fill three dimensional spaces with infinite non-repeating patterns.
• 769
Seems like Penrose has contributed, via studying these patterns, to further discovery (et al like Steinhardt and Levine) of the unique properties of quasicrystals. So there is a practical yield to the wrongness (absurdity) of this kind of rigorous study of science/mathematics.

The concept of quasicrystals — along with the term — was first introduced in 1984 by Steinhardt and Dov Levine, both then at the University of Pennsylvania. — Wikipedia: Icosahedrite

Quasicrystals: The Thrill of the Chase

They realized that by drawing parallel lines on a Penrose tiling, they could prove that the tiles are arranged quasiperiodically, producing five-fold symmetry. This was the breakthrough they needed. The leap to three dimensions produced Steinhardt’s long-imagined icosahedral quasicrystal. — Quasicrystals: The Thrill of the Chase
• 9.6k
This should put an end to the old adage that in an infinite universe there are an infinite number of copies of you.

But it won't.
• 5.5k

To say that the decimal expansion of some numbers, like π, never repeats begs the question of the definition of "repeat." And that because they do repeat, an infinite number of times in an infinite number of ways. Consider: the representing decimal numeral is made of ten fundamental parts, 0 through 9, and these repeated (an infinite number of times - infinity times - IT). Also the two-numeral combinations, 00 through 99, also repeated IT. And the three, 000-999, IT, and the four, 0000 through 9999 IT, and so on.

Thus for any finite sequence, it repeats, and repeats IT.

With the tiles, given two shapes, a finite coloring system, and the rules (again, why bother), taking first one tile, and then any that can legally lie next to it, there are but finitely many ways those two tiles can be laid down. Add the rest of the finite set of tiles one-by-one, and again there are but a finite number of ways they can be laid down. Likely a very large number of ways. And whatever pattern of any finite number of tiles will be repeated IT.

It may be that a clue lies in distinguishing "repeats" from "repeated." Like this. Sub-sequences in π, while they reoccur IT, do not when concatenated equal π. The only sequence that equals π is π itself.

And perhaps similarly with the tiling. While all the sub-tilings repeat IT, none combined with just itself can constitute - reproduce - the whole.

I find in this a kind of "diagonal" argument. Which I somewhat understand. But the notion of non-repeatability, of necessary "aperiodicity," is at the least on its face very misleading (to me).

So as to infinity Bannos, yes (better let Cooper's know!). But maybe different girlfriends, wives, children, and they with their own varieties of sub-patterns, repeating and repeatedly, IT. But never in such a way that the whole repeats, because this infinity will fill eternity. (Cooper's up to that?)
• 834
Thus for any finite sequence, it repeats, and repeats IT.

This idea appears to be connected to normal numbers, and I don't think pi has been proven to be such.
• 1.4k
You are overthinking it. I haven't looked at the video, but the wiki on Penrose tiling explains the sense in which the pattern doesn't repeat: it just means that the entire tiling doesn't possess translational symmetry. If you shift it by some amount in some direction, you can never get the same picture - unlike patterns printed on fabrics, wallpapers, etc. The connection with numbers like pi is that those numbers are irrational, which means that their decimal expansion also does not include an infinite periodic part (semi-infinite in this case).

If you can identify a repeating part in a pattern, you can just do sort of a variable substitution, designating that repeating part as one super-tile. Then the entire tiling is simply that one super-tile repeated infinitely in all directions, like squares on a chessboard or hexagons in a honeycomb. With an aperiodic tiling you cannot do that.
• 7.6k
What a coincidence. I watched this video too not a while ago. I also didn't get it but the takeaway for me was that the pattern doesn't repeat. I suppose it means that no matter how you cut it (in a mathematical sense), a given configuration of tiles will not be found in a different location in the tiling even if you extend the tiling to infinity in all directions. By the way, the guy in the video doesn't actually make an effort to explain why this is the case.

I'm just curious, how does one know that the tiling is patternless?
• 1.3k
To say that the decimal expansion of some numbers, like π, never repeats begs the question of the definition of "repeat." And that because they do repeat, an infinite number of times in an infinite number of ways. Consider: the representing decimal numeral is made of ten fundamental parts, 0 through 9, and these repeated (an infinite number of times - infinity times - IT). Also the two-numeral combinations, 00 through 99, also repeated IT. And the three, 000-999, IT, and the four, 0000 through 9999 IT, and so on.

This is covered in the video. Local features may occur an infinite number of times. This is not what periodicity is, however. For something to be periodic, the whole thing must be a single repeated pattern. So 110011110011110011... is periodic because the sequence as a whole is just the sequence 110011 repeated forever. Pi is not periodic.
• 5.2k
There's a bit where the guy is playing with his tiles, and points out 2 positions that can have alternative tiles, but once one is chosen, there is no choice for the other if the plane is going to be tiled.

There are infinite ways to tile the plane with Penrose tiles, but the finite amount that one can see, or that one has already placed, is insufficient to determine the plane. And it occurs to me that one can regard the tiles already placed as the past, the edge as the present, and the un-layed area as the future. The past does not entirely determine the future (as it would in a periodic tiling), but it constrains it quite tightly.

Sufficiently tightly to produce a geometrical equivalence to entanglement between two 'tile events' that are widely separated.

I am not clever enough to work out if this is just a fun analogy, or there is some real connection such that particle entanglement implies indeterminacy (working it all backwards as it were).
• 446
Once you have an infinite number of tiles you can create a non periodic pattern. Working with only 1 and 2 you can have-

121121112111121111121111112...
How many ways can you arrange 1 and 2? You have

1, 2 and 2, 1

How many ways can you arrange 1, 2, (1, 2) and (2, 1)?

you have-

1, 2, (1, 2), (2, 1)
2, 1, (1, 2), (2, 1)
2, (1, 2), 1, (2, 1)
2, (1, 2), (2, 1), 1 etc and the combinations become infinite as you get more ways of combining the new sets that arise out of previous combinations.
• 7.6k
It seems that if we have two tiles, one a rhombus and the other an equilateral triangle such that the sides of the rhobmus = the sides of the triangle, it's possible to tile without a repeating pattern. For every pattern you detect in the tiling, you can always prevent it from reemerging by fiddling around with the number of triangles or rhombuses. The logic? To my simple mind, every pattern that's possible with these two tiles will consist of a specific number of rhombuses or triangles. Alter their number and the pattern is broken. :chin:
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