
Discovery of a new irregular pentagon that can cover the plane - tokenadult
http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/aug/10/attack-on-the-pentagon-results-in-discovery-of-new-mathematical-tile
======
cft
"That same year an unlikely mathematical pioneer entered the fray: Marjorie
Rice, a San Diego housewife in her 50s, who had read about James’ discovery in
Scientific American. An amateur mathematician, Rice developed her own notation
and method and over the next few years discovered another four types of
pentagon that tile the plane. "

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

~~~
1arity
that's awesome. Outsider math and outsider art deserve as much respect as
"outsider" programming ( startups / garage hackers ) garner. Creating semi-
closed communities of insiders, united by what are essentially arcane
handshakes, and dismissive of value emerging outside of themselves, is really
just both a narrow-mindedness that doesn't behoove innovation and also an
admission of an unnecessary insecurity about the lack of substantive reasons
for cohesiveness in that community resulting in the desire to fabricate
arbitrary insubstantial reasons for cohesiveness.

At the same time, you could say that narrow-mindedness was focus and
restraint, things which __do__ work for innovation.

Another way of looking at the cohesiveness issue, is that if your community is
so substantially strong, why do you need to exclude outsiders ? The best
democracies include difference, revealing their strength through diversity.
The worst extremist fascist states put it to death, revealing their dread of
diversity.

Let the ad-hoc groups of intellectual collaborators be like democratic
meritocracies not frightened facist states.

~~~
cousin_it
IMO the analogy doesn't hold up very well. Outsider art is often very good,
and outsider programming seems like a weird idea (do startup hackers have less
training than BigCo employees?) But that's beside the point.

If you're looking at the tiny subset of outsider science that's chosen for
correctness and publication-worthiness, of course you'll be enthusiastic about
it. But these people, like Marjorie Rice, aren't really facing any obstacles
from the establishment! If their work has merit, they usually get a warm
welcome.

The elephant in the room is that >99% of outsider science is extremely bad,
and can't understand why it's bad even after repeated explanations. Pretty
much all the people complaining about the scientific establishment are like
"fermatists", who used to hover around math institutes to have their proofs of
FLT checked. If you're not familiar with such people, I urge you to actually
look at a sample of their work and form your own opinion.

After that I think you'll be less eager to throw accusations around. A
scientist's life is hard enough already, without people telling her she's an
insecure evil fascist for rejecting torsion fields, the EmDrive, or Time Cube.

~~~
3pt14159
> their proofs of FLT checked.

What does FLT stand for? Fermat's Last Theorem?

~~~
jordigh
Yes. Maths cranks are extremely common. If you have an academic maths email,
you're likely to receive crank emails with some regularity.

------
jordigh
What's the point group for the first tiling on that webpage? It's a periodic
tiling (unlike a Penrose tiling, which is only quasiperiodic), so the
crystallographic restriction for two dimensions says the rotation subgroup of
the point group must be one of C_2, C_3, C_4 (not C_5!) or C_6:

[http://mathworld.wolfram.com/CrystallographyRestriction.html](http://mathworld.wolfram.com/CrystallographyRestriction.html)

I can't tell by eye-balling it what the symmetry is for the first one, but its
periodicity says it must be one of those. Quasicrystals with 5-fold symmetry
are not exactly periodic.

There are only 17 wallpaper groups. Since this is a wallpaper, what is its
group?

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

~~~
dooglius
From the table on the wikipedia page, it looks like it's p1

------
ab
It's interesting how your eye naturally groups the pentagons into the larger
primitive unit, like the pinwheels of type 5.

Wolfram Alpha also has some things about tiling:
[http://www.wolframalpha.com/input/?i=pentagon+tiling](http://www.wolframalpha.com/input/?i=pentagon+tiling)
[http://www.wolframalpha.com/input/?i=pentagon+type+5+tiling](http://www.wolframalpha.com/input/?i=pentagon+type+5+tiling)

------
est
Meta question: where can I find list of simple unsolved/undiscovered problems
like these in math?

It does not appear in
[https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_m...](https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics)

~~~
jbrooksuk
I don't think that tiling a plane is exactly a "simple" problem.

~~~
est
I mean something simple enough to understand, and could be played by average
programmer in a IPython Notebook.

I didn't mean simple to solve, just lower barrier of entry.

~~~
leni536
I like this one:

[http://www2.stetson.edu/~efriedma/squinsqu/](http://www2.stetson.edu/~efriedma/squinsqu/)

Can be easily generalized to other shapes and more dimensions too.

~~~
pavel_lishin
Does the lack of n = 16 mean that there's no current proof?

~~~
leni536
> For the n not pictured, the trivial packing (with no tilted squares) is the
> best known packing.

------
sandworm101
Wait a second. Aren't there actually two different pentagon shapes in use
here?

Look at the yellow and blue in the OP. They are actually mirror images of each
other. Maybe a mathematician would say they are the same, but certainly not
someone cutting tile for a bathroom floor. And if these were proteins trying
to form a cell wall, that mirroring would be a serious hurdle.

~~~
cammil
All depends how you define "the same". Mathematicians use "congruency" which
allows mirror images.

~~~
sandworm101
Ah. You learn something every day. I guess the guardian was therefore
incorrect in their definition and should have said "congruent copies" ... but
"copies" is also probably wrong. Congruent congruencies?

>"If you can cover a flat surface using only identical copies of the same
shape leaving neither gaps nor overlaps, then that shape is said to tile the
plane."

~~~
ngneer
"Identical copies" is informal, as is "copies" for that matter. You can think
of the information that is defining the shape as what is being copied, hence
mirroring and other variations are allowed. A possibly related term to look
into would be "isomorphism".

~~~
sandworm101
Is mirroring also allowed when 'filling' a 3d space?

~~~
goldenkey
Mirroring is allowed because in a higher dimension space, it is simply
rotating the lower dimensional object by 180 degrees upon its own axis :-)

Ie. an ant can't flip the hexagon you lay flat, but you can. Similarly, you
cannot "flip" the 3d object you have in terms of chirality, but a 4d being can
:-)

------
unfamiliar
I wish I had a bathroom to tile - I reckon this could be considered "in vogue"
for the next 30 years or so, until they find a newer pentagon. Does anyone
know if this can be coloured with 3 colours? Obviously 4 is possible due to
the 4 colour theorem and 2 will not work due to to three faces sharing a
corner.

~~~
rlpb
> Obviously 4 is possible due to the 4 colour theorem...

Unless your bathroom includes a loop, such as all four walls (even with holes
for windows and doors) or over the ceiling. Then the coloured area is no
longer a plane, and so the 4 colour theorem does not apply.

~~~
T-hawk
Wouldn't all four walls still function as a plane, topologically? Think about
looking into a cube (box) with one side open, so that you see five faces. The
projection of those faces onto your retina or a photograph is a direct mapping
to a plane.

Four walls plus the ceiling are equivalent to that open box. Four walls minus
the ceiling are equivalent to a plane with a hole.

You have to include _both_ the ceiling and the floor to break out of plane
topology. And what you get is a sphere.

~~~
Chinjut
A sphere is also four-colorable, functioning equivalently to a plane for such
purposes; consider stereographically projecting the sphere (minus a point)
onto a plane.

------
Vexs
I think it's kinda funny that most of the tessellations are just using
pentagons to make other shapes that tessellate naturally. I suppose the same
could be said of most tessellations though, but it's still interesting.

~~~
Someone
That's because there are only 17 wallpaper groups
([https://en.m.wikipedia.org/wiki/Wallpaper_group](https://en.m.wikipedia.org/wiki/Wallpaper_group)).
Any _repeating_ pattern must match one of them (non-repeating patterns by
definition do not)

~~~
cLeEOGPw
Can you NOT link to mobile version? Mobile phones switch to mobile
automatically, desktop version does not.

~~~
sp332
The mobile version is very readable on a desktop browser, and it's probably
easier for you to edit the URL on your PC than for the poster to edit it using
their phone.

~~~
snaily
...all effort being equal. And assuming, of course, that comments are read (on
average) once.

~~~
sp332
Assuming that the total effort expended by the few who really dislike the
mobile interface is less than fiddling with a URL on a phone, plus gratitude
toward the person for taking the effort to make the post at all.

------
infinity0
Many of these types are basically combining two pentagons into an octagon (or
even hexagon) then tiling it across the plane. For some reason, intuitively
those seem more easy to me (2 * 3, 2 * 4), so that you could just generate a
bunch of them, and split them in two to create tessellating pentagons?

Even the example in the article can be viewed as a regularly tessellating
nonagon. I don't see what's "irregular" about it? The article doesn't mention
that word, but the HN title does.

~~~
raverbashing
irregular - with sides of different sizes

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

~~~
ColinWright
Sides and/or angles. It's possible to have all the sides the same length, and
yet still not be regular. Similarly, it's possible to have all the angles the
same, and yet still not be regular. It is instructive to construct examples of
both types of failure.

~~~
raverbashing
Ah true. A lozenge has all sides with the same size but not the same angles

------
laverick
"Attack on the pentagon" ...aaaand the guardian has lowered themselves to
buzzfeed's standards.

~~~
matt4077
Not at all. Witty headline have a long history which was unfortunately cut
short by google which, at least for a while, only rewarded the most boring,
literal drivel. It's good to see the tradition making a comeback.

This headline is also completely unlike the buzzfeed-clickbait headlines.
Buzzfeed tries to exploit psychological weaknesses (Mathematicians attack the
pentagon – you won't believe what happened next).

Headlines like the Guardian's are much better in that they're still
entertaining after you've read the story. Sometimes I think they're written
more for the amusement of the editor than anything else.

Other guardian headline I saved: "Blight in Italy leaves pine nut nuts pining
for more"

------
jagermo
very well explained article. even I (with my little grasp on math) can see why
its a big thing. Cudos to the author

------
cammil
"Every triangle can tile the plane. Every four-sided shape can also tile the
plane."

Can someone point me to a proof of this?

~~~
thyrsus
The triangle is pretty easy: take two of the (same size) triangles, with
vertices ABC and A'B'C'. Rotate and translate the second triangle to fit the
matching side of the first triangle, e.g. AB to B'A'. You now have a
quadrilateral with two pairs of equal sides (sides (AB')C and (A'B)C'). The
angle on the corners comprised of the two triangles (e.g., C(AB')C') will add
together to be 180 degrees minus the angle of the adjacent corners, due to the
three interior angles of every triangle summing to 180 degrees. Duplicate that
quadrangle and fit the second to a matching side. The angles put together will
form 180 degrees, i.e., a straight line. Now you have indefinitely extensible
strip. Place the strips next to each other and you've tiled the plane.

------
huuu
Besides tiling your bathroom is there any use case for this? Or is this pure
for fun and gaining knowledge?

Edit: The article is talking about building structures but isn't a triangle
the most rigid form? And triangles are already used in building.

~~~
fractalb
Should it be discounted if it were done for pure fun?

~~~
donkeyd
It's a question, not an opinion. He/she might just be interested in the
practicalities.

------
ben174
With only five points and < 180 degrees at each intersection, couldn't this
problem have been solved by a computer via brute force in seconds?

~~~
tribe
From the article:

“We discovered the tile using using a computer to exhaustively search through
a large but finite set of possibilities,” said Casey. “We were of course very
excited and a bit surprised to find the new type of pentagon.

I would expect it to take longer than seconds, since there are many ways that
these shapes can fit together, and there are many possible edge lengths.

------
devindotcom
This is cool, but I'm not sure why it's a hard problem to solve for a computer
running through a ton of 5-sided polygons with some basic rules and attempting
to tessellate them? Is there a reason that approach doesn't work, other than
being rather un-romantic as far as discovering cool new things like this goes?

~~~
pavpanchekha
Yeah, great reasons why it doesn't work—what 5-sided polygons are you going to
run through? The hard bit is the side lengths; there are good reasons to think
that the angles cannot be too weird. But for example, this pentagon has one
side of side length 1 / (sqrt(2) (sqrt(3) - 1)). How long is your exhaustive
enumeration going to go until it finds that one?

Anyway, it sounds like clever enumeration is _exactly_ what these authors
did—but you do have to be clever to find something at all.

~~~
Pyxl101
> 1 / (sqrt(2) (sqrt(3) - 1))

I don't mean in any way to demean their research, but I think your exhaustive
search could build up formulas in exactly that form and iterate on them. It
starts with a valid pentagon. Then it permutes that pentagon with an
evolutionary method by modifying the formula.

For example, maybe the lengths it tries are:

1 / 2

1 / sqrt(2)

1 / sqrt(2) * 3

1 / sqrt(2) - 1

1 / sqrt(2) * (3)

1 / sqrt(2) * sqrt(3)

It would try both these and many others along the way.

1 / exp(2) ...

2 / sqrt(2) ...

Again, not to trivialize, but there are only so formulas made up of a fixed
number of terms and operators, and as long as it's easy to check whether a
shape is a valid pentagon, and whether it tiles, then I think you could check
a considerable number of them. It looks like a number of the pentagon formulas
have a few sides with complex lengths, while the others are simple or equal to
each other, so you could bias the algorithm to search for those.

I'm sure there's way more complexity I'm overlooking, but that's how one might
get started.

~~~
ntoronto
"I'm sure there's way more complexity I'm overlooking, but that's how one
might get started."

That's how I'd start. There's probably a way to make the search a lot smarter.
Here's the part you're overlooking, though:

"... there are only so [many] formulas made up of a fixed number of terms and
operators..."

"So many" = "countably infinite." Paring it down to finitely many would
require understanding tantamount to having solved the problem in the first
place.

[Edit: I can words.]

------
brunnsbe
For those interested in the subject here is an interesting video about using
penrose tiling for street tiling in Helsinki, Finland:
[https://www.youtube.com/watch?v=yxlEojkVJ0c](https://www.youtube.com/watch?v=yxlEojkVJ0c)

~~~
zimpenfish
There's a lovely building at North Greenwich which is Penrose'd.

[http://wordpress.mrreid.org/2011/02/26/penrose-
tiling/](http://wordpress.mrreid.org/2011/02/26/penrose-tiling/)
[http://www.bdonline.co.uk/foa%E2%80%99s-peninsula-
patterns-f...](http://www.bdonline.co.uk/foa%E2%80%99s-peninsula-patterns-for-
ravensbourne-college/3144928.article)

~~~
jessaustin
I'd never heard that Penrose sued Kleenex. Not his finest hour.

------
hinkley
I don't know why, but this was the first question that popped into my head
after seeing the diagrams:

Does anyone make bricks in these shapes? Those would make an awesome paver
pattern.

------
infogulch
The first of the image of other tilings at the end of the article is almost
obvious: it's just a hexagonal tiling where the hexagons are bisected.

------
Sniffnoy
Does anyone have a link to the actual paper?

~~~
XaspR8d
I'm not certain there is an actual paper yet; I suspect from some of the
phrasings that it was merely an announcement that their program succeeded, but
a formal paper will take a while.

I _did_ find this reddit post by Dr. Mann [1] where he says:

> We were just in the process of debugging and optimizing the code when our
> new example was found. Because we are in the early stages of the
> computational experiments, we were surprised to find this example so
> quickly. We are hopeful of finding more new examples as we proceed.

[1]
[https://www.reddit.com/r/math/comments/3fe347/15th_pentagon/...](https://www.reddit.com/r/math/comments/3fe347/15th_pentagon/cto75m0)

------
ai_ja_nai
we can shape building blocks like those, now

------
zkhalique
Take that, Penrose! LOL

------
1arity
Perhaps the more impressive number is that they found 7 quintillion new
irregular pentagons that can't tile the plane.

~~~
oldstrangers
I've read your comment a few different ways. I'm concluding that you find it
impressive that it took 7 quintillion tries to find a new pentagon that can
tile the plane.

~~~
wingerlang
I think he just meant that the number 7 quintillion is an impressive number,
more than 15.

------
nether
math dept is getting shitfaced tonight...

------
nickysielicki
Kudos for not using the same clickbait title The Guardian did.

Journalistic integrity is dead.

~~~
matthewmacleod
It's a pun, not clickbait.

This is not the death of "journalistic integrity".

~~~
teddyh
No, the original title is clearly clickbait – meant to be misinterpreted as
something other than what the article is about. Here is what a _real_ pun in a
newspaper headline looks like:

[http://threepanelsoul.com/2010/03/22/on-astronomy-
minors/](http://threepanelsoul.com/2010/03/22/on-astronomy-minors/)

~~~
bbcbasic
It's a pun

PUN: Attack on the pentagon results in discovery of new mathematical tile

CB: Breaking News: Pentagon attack

~~~
nickysielicki
Yeah, it's a pun. So what? The title being a pun doesn't make it any less
misleading. Puns and clickbait are not mutually exclusive.

My real gripe is that there's a time and place for everything and a newspaper
ought to know and respect that line. You don't mislead someone skimming your
headlines into thinking the pentagon has been attacked for the sake of a joke.
I like clever headlines as much as the next guy, and I understand that for the
sake of a joke you might leave some ambiguity in the air and mislead your
reader slightly, but this goes far past that.

I think his editor should have caught this.

~~~
bbcbasic
I think using the word 'mathematical tile' indicates that it is a maths/geek
article rather than serious news, and takes this into the safe zone. Just.
It's a fine line though.

