Discovery of a new irregular pentagon that can cover the plane

[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

[shoo]

Rice's pentagons are illustrated here [1]: Type 9, Type 11, Type 12 and Type 13.

[1] http://www.mathpuzzle.com/tilepent.html

[cokernel]

She has a website for her tesselations (which that Wikipedia article links to), but unfortunately is no longer able to respond to queries: https://sites.google.com/site/intriguingtessellations/home/

There's a nice tiling of butterflies there.

[dunkelheit]

Wow. I personally cannot do much work without being part of a community to share and refine ideas with. It is amazing that someone can produce important work while working essentially in isolation.

[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.

[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

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

[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+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...

[ColinWright]

Not to answer your question, but here's a simple, unsolved problem:

    Dissect the circle into congruent pieces
    such that the center point is in the
    interior of one of the pieces.

It is widely believed to be impossible, but neither proof of impossibility nor construction of an example have been found.

[robinhouston]

There's a nice collection on Math Overflow: Not especially famous, long-open problems which anyone can understand

http://mathoverflow.net/q/100265/8217

[jbrooksuk]

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

[dexterdog]

It might be easier to read the list of all discovered problems. Then everything else is in play.

[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.

[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.

[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.

[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.

[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"

[unfamiliar]

Its a joke, I found it pretty funny.

[Dylan16807]

Isn't that every newspaper that has ever existed?

[CPLX]

I think it's more that they have raised themselves to The Economist standards.

[mahouse]

As you can see, it's some kind of column.

[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.

[function_seven]

Not a formal proof, but close enough I suspect. See comments as well:

http://www.quora.com/Why-do-all-quadrilaterals-tessellate

[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.

[matt4077]

Sure there are. There are many processes where you want to "tile" a surface with identical smaller parts. I'd assume it's something that happens with solar panels. Possibly also on the molecular level with coatings etc. Having more options for the shape of the constituent parts may be an advantage.

see also https://en.wikipedia.org/wiki/Buckminsterfullerene, a molecule inspired by architecture.

[akent]

Can I actually tile my bathroom with tiles like this? Do any tile manufacturers make them that shape?

EDIT: I found these, at least http://www.ballerhouse.com/2010/08/19/precision-concrete-til...

[fractalb]

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

[madcaptenor]

I am buying a house and the bathrooms need updating, so tiling your bathroom is actually the first thing that came to mind.

[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.

[jjxw]

It kind of was:

>“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.

[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?

[nikanj]

Just going to leave this here https://xkcd.com/793/

[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.

[DanBC]

That approach does work. From the article:

"We discovered the tile using using a computer to exhaustively search through a large but finite set of possibilities,” said Casey.

[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

[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/...