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Discovery of a new irregular pentagon that can cover the plane

theguardian.com

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Re: Discovery of a new irregular pentagon that can cover the plane

#101

Post by aetherson »

ColinWright wrote: post #72
est wrote: post #26

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...

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.

Huh. Is it possible to dissect a circle into congruent pieces that are not, uh, pie shapes? Is that the question, or are there known other ways to dissect a circle into congruent pieces, it's just those pieces also have the center point as an edge point?

Re: Discovery of a new irregular pentagon that can cover the plane

#102

Post by sklogic »

nickysielicki wrote: post #84
sklogic wrote: post #83

Earlier quoted context omitted.

> most people have is that there has been an attack on The Pentagon Most people on this planet do not give a crap about some tasteless Northern American building. It is unlikely the first association with the word "pentagon" for anyone outside of the US.

I agree that most people don't think of The Pentagon when they hear the word "pentagon" alone. But the title says, "Attack on the pentagon". Furthermore, given that the Guardian is a UK newspaper, I would argue that most people in the UK know what The Pentagon is and associate the phrase with the building. But let's assume I'm wrong and most readers don't know what The Pentagon is or wouldn't have associated the titl…

Just asked a number of people around how do they read this title. Nobody had "The Pentagon" in mind. London, UK, geeky population, so there might be some bias.

Re: Discovery of a new irregular pentagon that can cover the plane

#103

Post by T-hawk »

rlpb wrote: post #96
unfamiliar wrote: post #54

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.

> 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.

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.

Re: Discovery of a new irregular pentagon that can cover the plane

#104

Post by morcheeba »

shoo wrote: post #14
cft wrote: post #5

"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

Rice's pentagons are illustrated here [1]: Type 9, Type 11, Type 12 and Type 13. [1] http://www.mathpuzzle.com/tilepent.html

>Types 6-8 were found by R. B. Kershner in 1968. >Type 10 was found R. James in 1975. >Types 9, 11-13 were found by M. Rice in 1976-1977.

Dammit, R. James! You can't just skip numbers like that and mess up Rice's streak!!

Re: Discovery of a new irregular pentagon that can cover the plane

#105

Post by ColinWright »

aetherson wrote: post #101
ColinWright wrote: post #72

Earlier quoted context omitted.

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.

Huh. Is it possible to dissect a circle into congruent pieces that are not, uh, pie shapes? Is that the question, or are there known other ways to dissect a circle into congruent pieces, it's just those pieces also have the center point as an edge point?

Yes.

Summarising, it is possible to dissect a circle into finitely many congruent pieces that do not all touch the center point.

Slightly longer, there are at least two infinite families of solutions:

(a) For every natural number n>1 there are f(n)>0 solutions, with f growing exponentially quickly.

(b) For every natural number n>1 there is an uncountable infinite family.

Thus we have a countable family of solutions, and a countable family of continuous solutions.

So yes, there are solutions that are not all just "slicing a pizza" type solutions.

Re: Discovery of a new irregular pentagon that can cover the plane

#107

Post by Chinjut »

T-hawk wrote: post #103
rlpb wrote: post #96

Earlier quoted context omitted.

> 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.

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…

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.

Re: Discovery of a new irregular pentagon that can cover the plane

#109

Post by function_seven »

cammil wrote: post #106

"Every triangle can tile the plane. Every four-sided shape can also tile the plane." Can someone point me to a proof of this?

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

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

Re: Discovery of a new irregular pentagon that can cover the plane

#110

Post by 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.

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