
Attack on the pentagon results in discovery of new mathematical tile (2015) - cfedenko
https://www.theguardian.com/science/alexs-adventures-in-numberland/2015/aug/10/attack-on-the-pentagon-results-in-discovery-of-new-mathematical-tile
======
steiza_
For those looking to create patterns like these in real life, I've had some
success describing aperiodic tiling expansion rules in PostScript:
[https://github.com/steiza/postscript_fractals](https://github.com/steiza/postscript_fractals)

From there you can send it to a laser cutter (see link for pictures), vinyl
cutter, plasma cutter, CNC machine, etc.

PostScript is particularly great because it's a stack based language, and
aperiodic tilings are often defined by expansion rules that are basically
recursive algorithms. Just don't go too deep or you'll blow the stack!

~~~
specialist
Nicely done. I envy your etched windows.

I've long fantasized about using penrose tiling for my bathroom. Alas, my
craftsmanship is terrible and I can barely manage subway style tiling.

I think someones could make some side money doing these one-offs for local
builders, remodelers. Every service bureau I've ever met with their own laser
(CNC, 3D printer) has spare capacity.

I've also long wanted to have client-side procedurally generated backgrounds
for web pages, and user interfaces in general. Wood grain, marbling, penrose
tiling, fractals... I got something kinda working using applets with Sun's
HotJava browser, once upon a time.

I just saw that CSS now has a paint image hook. Made me think it's time to
rescratch that itch. Unless someone beats me to it... :)

~~~
miluge
It really isn't that hard to make custom shape tiles by hand using the average
Joe tools but it's a long, precise and tedious process. I kinda miss doing it
with my step father blasting Chris Rhea on the stereo and slapping my head
whenever I was "unattentive" and had a 1mm difference from the model, it isn't
a huge difference but add it up to all the tiles and you can get a pretty
messy job in the end...I miss those days.

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rufugee
It's fascinating to me that Marjorie Rice, a housewife in her 50s, essentially
taught herself, created her own mathematical notation, and found four
pentagons on her own.
([https://en.wikipedia.org/wiki/Marjorie_Rice](https://en.wikipedia.org/wiki/Marjorie_Rice)).
I'm astonished and encouraged by her work.

~~~
dekhn
I think there is a lot of latent amateur mathematical talent going
underutilized.

~~~
jerf
There's a number of Numberphile videos I've seen where there are computer
checks being done on various conjectures famous enough to on a video series
like Numberphile, and even IIRC one case where a counterexample was found by
an amateur with a computer. I've often thought there must be a lot of much
less famous conjectures that are probably something an amateur with a computer
could attack, and could well be the first to try. Or be the first one to try
with gigabytes and teraflops, if the last person tried in an era where "kilo"
still sounded pretty impressive.

Math is well covered, but the frontier is larger than ever; it grows every
day. There's just a certain art to finding them.

Another field I've wondered about that may be relevant to HN interests is
quantum algorithms. There's a lot of very smart people working the theory and
the engineering and such, but I wouldn't be all that surprised there's a lot
of room for even a relative amateur to learn about what quantum computers can
do, and find _practical_ speedups to existing problems, not because nobody is
trying but because the field of existing problems is just so large that there
almost has to be low hanging fruit still available. Theorists right now are
mostly interested in solutions that change O() classes, but someone just
having some fun will care if they can find a 100x speedup that is technically
not in a different O() class, or something like that.

~~~
dekhn
As for Conjectures: I tried one of them
([http://norvig.com/beal.html](http://norvig.com/beal.html)) but didn't have
any luck (see the part just before the Conclusion). Like Peter says, better
spend your brainpower on a proof than your computerpower on finding
counterexamples.

It didn't need a subject domain expert- just computer engineers who understood
the problem and how to test counterexamples efficiently.

------
inetknght
> _attack on the pentagon_

Wow that definitely sounds like a clickbait title.

~~~
jcutrell
On top of it, “plane” is all over the article.

~~~
fit2rule
Well.. I think its fair to acknowledge this as a perfectly cromulant "next
generation Peace Symbol" as a consequence.

------
JasonFruit
Sometimes I feel like there's not much left people haven't figured out, and
then I see something like this: pentagons that can tile a surface aren't fully
worked out mathematically. You'd think that would be something we'd have
covered, but no — there's wonder left in the world!

~~~
thekingofh
You might be interested in these
[https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_m...](https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics)

Notably some of them like the Millennium Prize offer a reward of 1M dollars if
you solve it. There's definitely still some absolute mysteries out there!

------
Grue3
Wikipedia [1] says it has been proven in 2017 that no convex pentagons that
can tile the plane other than these 15 types exist. I've been wondering if
there's a single convex tile that can only tile the plane irregularly, guess
not.

[1]
[https://en.wikipedia.org/wiki/Pentagonal_tiling](https://en.wikipedia.org/wiki/Pentagonal_tiling)

~~~
pgreenwood
It's an open problem to find a single aperiodic tile. It's called the Einstein
Problem.

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

~~~
ljcn
The _einstein_ problem (lower-case E).

Not to be confused with the physicist.

~~~
whatsstolat
TIL ein stein means one tile.

------
quickthrower2
Previous discussion:

[https://news.ycombinator.com/item?id=10045297](https://news.ycombinator.com/item?id=10045297)

Man those 3 years went quick!

------
dandare
I wish I know how to cut these shapes out of regular square tiles in large
quantities. I would love to have Penrose tiling on other exotic patterns in my
bathroom.

~~~
matte_black
Do a polished concrete floor but stamped with the tiling pattern cut out
through a CNC.

Done. Super cheap.

~~~
ISL
Penrose tilings are aperiodic. The stamp (or combined area of all CNC-ed
stamps) would be as large as the floor.

~~~
tropo
Within a fixed distance of a given type of tile, the possibilities are not
unlimited. Stamp with occasional overlap, and you could get the job done with
only a few different stamps.

------
lousyd
The article says that "tiling the plane" is:

"If you can cover a flat surface using only identical copies of the same shape
leaving neither gaps nor overlaps"

So the trick, if I'm understanding this, it's that the shape must have sides
that can push up against other of its sides, leaving no gaps. Obviously a
circle isn't gonna do it.

Does the plane have to be a plane in the mathematical sense, i.e. infinite?
Because then you're obviously not going to cover it with anything. It goes on
forever.

If just a section, does the plane have to be a square or other rectangle?

~~~
joppy
Making the plane infinite makes the problem easier, since you don’t need to
worry about the boundary of any finite section of the plane. For example, if
you choose a triangular section, you won’t be able to tile that section by
squares. But it is still clear the infinite plane can be tiled by squares;
just keep laying the squares edge-to-edge.

------
nautilus12
Very dishonest title

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darkstar999
> “We discovered the tile using using a computer to exhaustively search
> through a large but finite set of possibilities,”

Brute force. That's my kind of math.

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saagarjha
> Every triangle can tile the plane. Every four-sided shape can also tile the
> plane.

The triangle case makes sense to me, intuitively, but the four-sided one makes
much less sense (maybe I need a sheet of paper?). Does this also have a
“trivial” proof?

~~~
jawarner
Hint: The interior angles of any four sided shape add up to 360°.

------
apo
An example of when Title Case would be most misleading.

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jlebrech
I want to build boats that are shaped like this, then you can tile them
together and create a new country.

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raverbashing
Yeah the headline is a bit misleading, it sounded like a cyberattack on a US
government institution

But interesting nonetheless

~~~
make3
it's not misleading, it's a joke

~~~
mulnz
It's prime clickbait. It's published on a major news site, and they know what
many people's first reaction will be.

Definitely funny, definitely dishonest.

------
tzahola
Startup idea: ceramic tiles for geeks.

~~~
Razengan
Indeed. Some of the examples [0] are very aesthetically pleasing. I imagined
them as season-themed stained-glass windows, representing leaves, nuts,
icicles etc., maybe in a house with indoor/outdoor rooms for each season. :)

[0] [https://i.guim.co.uk/img/static/sys-
images/Guardian/Pix/pict...](https://i.guim.co.uk/img/static/sys-
images/Guardian/Pix/pictures/2015/8/6/1438878897109/cb0eea28-8fe7-4cc3-a0ce-896c6f8a6fb4-bestSizeAvailable.png?w=620&q=20&auto=format&usm=12&fit=max&dpr=2&s=4f52798b5c17e781e33274833682d36b)

