11–20 of 131 posts
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.
Earlier quoted context omitted.
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…
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.]
Earlier quoted context omitted.
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…
So your iteration procedure will probably get to a good approximation, but might not find the "core" expression.
"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
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?
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…
Earlier quoted context omitted.
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…
mind elaborating on why the angles can't be too weird? I thought it was odd that the angles given were all integers.
EDIT:
To your question as to why angles can't be to weird, consider vertices, where the corners of the pentagons meet. Each vertex is composed of three angles, one from each of the three neighboring vertices (although it is not a-priori obvious that a vertex contains only three angles). The sum of these angles (in rotations) must be 1. This means that, heuritstically, you would want as many permutations of the angles as possible to add up to one.
Journalistic integrity is dead.
Earlier quoted context omitted.
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…
Kudos for not using the same clickbait title The Guardian did. Journalistic integrity is dead.
This is not the death of "journalistic integrity".