Discovery of a new irregular pentagon that can cover the plane
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Re: Discovery of a new irregular pentagon that can cover the plane
#2This 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?
Re: Discovery of a new irregular pentagon that can cover the plane
#3This 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?
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.
Re: Discovery of a new irregular pentagon that can cover the plane
#4math dept is getting shitfaced tonight...
Re: Discovery of a new irregular pentagon that can cover the plane
#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. "
Re: Discovery of a new irregular pentagon that can cover the plane
#6This 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?
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 th…
> 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.
Re: Discovery of a new irregular pentagon that can cover the plane
#7This 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?
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 th…
The angles don't seem so weird though
Re: Discovery of a new irregular pentagon that can cover the plane
#8This 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?
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 th…
I'd be even more curious about the computation used to determine if a given set of sides and angles tessellates, and how long that computation takes. Approaches to enumerate possible expressions don't seem that far-fetched, if the resulting possibilities can be evaluated quickly.
Re: Discovery of a new irregular pentagon that can cover the plane
#9This 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?
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.
Re: Discovery of a new irregular pentagon that can cover the plane
#10I 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.