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Discovery of a new irregular pentagon that can cover the plane

theguardian.com

121–130 of 131 posts

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

#121
post #120

Earlier quoted context omitted.

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."

"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".

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

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

#123
post #122

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?

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.

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

#124
post #122

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?

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.

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

#125

Earlier quoted context omitted.

It's a pun PUN: Attack on the pentagon results in discovery of new mathematical tile CB: Breaking News: Pentagon attack

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 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.

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

#127
post #36

Earlier quoted context omitted.

I like this one: http://www2.stetson.edu/~efriedma/squinsqu/ Can be easily generalized to other shapes and more dimensions too.

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

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

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

#128
post #120

Earlier quoted context omitted.

"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".

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

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 :-)

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

#129
post #23
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

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…

Science works when it embraces the unconventional and the unknown, anything that steps away from this limits the power of science to shine the light of a theory and an experiment onto something mysterious.

It is positive, constructive and it works for outsiders to be included in science and art. it is also positive and correct to critically look at thing when you hold the highest potential of those things as important. Nothing is sacred beyond critical inquiry, that way lies dogmatic extremism. Being an apologist for a broken culture doesn’t help that culture, nor the context in which it operates.

If you choose to pretend that outsider art is all good and outsider science is all bad, and that the culture of art, science and math ( and tech ) are beyond reproach, and that the points raised have no value as opportunities to create improvements, and that scientists ( or mathematicians, or the establishment ) are poor fake victims who are suffering hardship for considering whether their cultures possess opportunities for improvement, if this is the case then I can understand how pretending that seems to make it easier for you, since you can hold to an unchallenged perspective and not experience the disruption that comes with questioning the lay of the land under your feet. Tho that choice isn't as likely to lead to reflection, as being open to questioning. And without reflection, no improvement.

And without questioning, no science.

Science is inspiring. Science isn't about protecting yourself from reflection, science is brave, science embraces reflection and self critical thinking, and science embraces belief in the face of the unknown, science embraces acknowledging the unknown, and running the experiment. Science embraces the question. Science is the opposite of dogmatic. Anything that claims to be science and shies away from openness, is the mere color of science.

Science is that very ideal : that strength is open vulnerability, and not dogmatic inflexibility. Science is an inspiration.

I believe the very sensitivity of these comments ( and the fact that they created the accusation in their interpreting, fabricating a connection to an accusation that was not stated ) reveals clearly that this issue is on people’s minds, and that in expectation of it being raised, and in expectation of its raising being troubling, they are not only defending against it, they are prematurely defending against it, and creating themselves the accusation that was not present and which they have hallucinated in their fear of it.

To be afraid of something is to suspect it.

So the very ideas I have raised have triggered this apparition to be conjured in these minds giving life and proof to the very questions I raised, and expanding the scope of relevance of the critique. It's not just about maths, clearly. It's about science as a whole. The argument was never "could this occur?" for we all know it could, and we have seen it, as have we all seen the kind of reflexive apologist rhetoric get triggered that was triggered here in denial of anything wrong with these culture’s held beyond question. The argument instead was a warning against the very defensiveness seen triggered in response to it, and a warning put now in clearer relief against the background of comments that display that very inflexibility and refusal of critical questioning that can lead to the exclusionist frightened culture warned about.

These comments are then nothing except my loyal supporters, they stand beside my warning as additional support for its importance, and in them we can see how the beginnings of dogmatic inflexibility take hold : through defensive denial of divergent ideas and refusal to question the conventional.

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

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

#130

Earlier quoted context omitted.

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

Why not? It seems simple to me. Simple doesn't mean easy. Factoring a 2048 bit number is very simple and very hard.

Because then anything that can be proven in a given axiom system is considered simple because it can be done by enumerating each possible sequence of transformations.
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