
Pentagon Tiling Proof Solves Century-Old Math Problem - petethomas
https://www.quantamagazine.org/pentagon-tiling-proof-solves-century-old-math-problem-20170711/
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lordnacho
I love how there's a mix of simple things like why you can't tile things with
more than 6 edges and really complex things like what the headline is about.

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vanattab
For those that don't see why it's so simple.

[https://plus.maths.org/content/trouble-
five](https://plus.maths.org/content/trouble-five)

~~~
plopilop
Unless I'm wrong, the link you gave only speaks about impossibility for
regular convex polygons, whereas the original links deals with the more
general impossibility for any convex polygon with more than 6 sides.

~~~
obmelvin
I believe he links it for a bit of general background, but it is older so
naturally it wouldn't include the latest finding

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GregBuchholz
When it came to the description of "einsteins" (a single tile aperiodic
tessellation), I couldn't help but think of the images in the 3rd edition of
The Scheme Programming Language:

[http://www.scheme.com/tspl3/binding.html#./binding:h0](http://www.scheme.com/tspl3/binding.html#./binding:h0)

[http://www.scheme.com/tspl3/examples.html#./examples:h0](http://www.scheme.com/tspl3/examples.html#./examples:h0)

...Are there holes in those "tilings", or are the tiles not all the same
shape, or am I misunderstanding what non-periodic means in this context?

And what is the name for those types of "self-surrounding" tiles on the cover:

[http://www.scheme.com/tspl3/canned/large-
cover.png](http://www.scheme.com/tspl3/canned/large-cover.png)

~~~
markegli
The tiles in an aperiodic tiling must _guarantee_ that the tiling is not
periodic simply by their shape.

[https://en.wikipedia.org/wiki/Aperiodic_tiling](https://en.wikipedia.org/wiki/Aperiodic_tiling)

All three of the Scheme examples _can_ be tiled periodically even though they
aren't tiled periodically in the examples. How to tile them periodically is
left as an exercise to the reader, but it's not hard.

~~~
GregBuchholz
Got to love this quote from the "Einstein Problem" page on Wikipedia:

[https://en.wikipedia.org/wiki/Einstein_problem](https://en.wikipedia.org/wiki/Einstein_problem)

"Depending on the particular definitions of nonperiodicity and the
specifications of what sets may qualify as tiles and what types of matching
rules are permitted, the problem is either open or solved."

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Infernal
There's definitely something Douglas Adams-esque about that phrasing.

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droithomme
Marjorie Jeuck Rice, who was the real key to this breakthrough by finding new
tilings that had been claimed impossible, passed away only last week.

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euyyn
I didn't know about Marjorie Rice; interesting and uplifting "underdog" story.

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DC-3
Me neither - then I googled her and found that she had died just over a week
ago (02/07/17). As sad as this is, it seems that she was a remarkably bright
woman who lived a long and fruitful life. It's satisfying that in the
seemingly ultra-inaccessible world of modern mathematics that (for want of a
better term) normal people can still make headway.

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bmc7505
I was recently watching some Computerphile videos and was surprised to learn
that several geometric problems have fundamental applications outside the
physical sciences, such as geometric sphere packing and error correcting
codes. Does tiling research have any known applications in CS or information
theory?

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DonbunEf7
Wang tiles, mentioned only at the end of the article, encode Turing-equivalent
computation in their simple color-matching rule. It is unknown exactly how
useful Wang tiles are to theory of computation, other than as a wonderful
demonstration of how easily Turing-complete systems arise naturally.

I suspect that tiling may have applications in resource management,
particularly when allocating resources like memory which have an underlying
topological structure.

Edit:
[https://arxiv.org/pdf/1506.06492.pdf](https://arxiv.org/pdf/1506.06492.pdf)
is worth considering, but I don't yet know what its relevance is.

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doctoboggan
Greg Egan explores the concept of Wang tiles as a naturally occurring computer
in his excellent novel "Diaspora":

[http://www.gregegan.net/DIASPORA/DIASPORA.html](http://www.gregegan.net/DIASPORA/DIASPORA.html)

It gets a bit dense at times, but I would highly recommend it, especially if
you are interested in turing complete systems.

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microcolonel
I love that pattern where the pentagons make up hexagons. When I own a house
that's going in the kitchen.

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surement
Are you referring to this?
[https://en.wikipedia.org/wiki/Pentagonal_tiling#Pentagonal.2...](https://en.wikipedia.org/wiki/Pentagonal_tiling#Pentagonal.2Fhexagonal_tessellation)

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pierrebai
When they talk about the einstein, I assume they mean a shape that can _only_
tile the plane non-periodically. If the tile is allowed to tile the plane both
periodically and non-periodically, the solution would be obvious.

~~~
GregBuchholz
>If the tile is allowed to tile the plane both periodically and non-
periodically, the solution would be obvious.

Maybe you can draw a sketch for those of us who aren't immediately seeing the
obvious?

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kevinwang
Squares, but each row of squares has a random offset

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irishsultan
That is a periodic tiling (just think about what happens if you follow the
tiling along the rows instead of along the columns, you'll periodically see
the same pattern).

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kevinwang
No, you wouldn't necessarily

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irishsultan
[https://en.wikipedia.org/wiki/Euclidean_tilings_by_convex_re...](https://en.wikipedia.org/wiki/Euclidean_tilings_by_convex_regular_polygons#Tilings_that_are_not_edge-
to-edge)

It's listed as a periodic tiling. If you move the whole plane one square to
the left or right you'll find that everything fits into place nicely, making
this a periodic tiling.

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kevinwang
Hm, I'm not knowledgeable about this subject but this seems like the offset on
each row is NOT random, but it's instead repeating. Imagine instead that each
time I added a new row, I chose the offset to be, say, a completely new value.

~~~
irishsultan
Doesn't matter, because your new row will still be offset with the same random
value to existing rows after moving one square to the left or right.

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v64
Here's [1] an article from 2015 describing Casey Mann, Jennifer McLoud, and
David Von Derau's discovery of the 15th type of pentagon.

[1] [https://www.theguardian.com/science/alexs-adventures-in-
numb...](https://www.theguardian.com/science/alexs-adventures-in-
numberland/2015/aug/10/attack-on-the-pentagon-results-in-discovery-of-new-
mathematical-tile)

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Aissen
Fascinating. And the fact that it's been confirmed by a competing team makes
it really believable.

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kleer001
I wonder if it has a Conway's Life glider like Penrose tiles.

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Uptrenda
Wow, these programs would make excellent screen savers.

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newtem0
I would really like to see a website dedicated to showcasing beautiful visual
manifestations of tiling

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pierrebai
You could try Taprats:
[https://sourceforge.net/projects/taprats/](https://sourceforge.net/projects/taprats/)

The UI is not perfect, especially the tiling designer (but once you master the
keyboard shortcut it works okay). And sin of sins, it's written in Java! But I
do think some of the built-in example are nice. I'm biased.

