@ArterialBlack716  
Regular polyhedra are also called the Platonic solids!

ah, that’s my waifu platonic favorite character whom i love, spending her free time contemplating things like polyhedrons ^^ geometry’s VERY FAR from my strong suit, but if Starlight’s talking about it, that automatically makes it marginally more interesting!

“All these squares make a circle… All these squares make a circle…”

Vree Softy I am surprised no one has done this yet, so anon to the rescue: “NEEEEEEEEERRRRRRRRRRRDDDDS”

@Softy  
(I made this image)  
Defining perimeter and surface area through analysis can be tricky. Taking the limit informally can be dangerous, but with some good taste, such intuitive understanding of infinities helps a lot (it helped Euler and Cavalieri and Archimedes).
 
And modern physicists still use such sloppy infinities (mostly) without consequences. I was trained in Physics Olympiads and used sloppy infinitesimal calculations all the time (including treating the sphere locally as a flat face, which is pretty much saying the sphere is a regular polyhedron with infinitely many flat faces). Surprisingly I have never encountered a difficulty.
 
Even though analysis is rigorized by such things as (ε, δ) and Riemann integration and Lebesgue integration and other abstract nonsense, there’s still value in being sloppy and intuitive. At the very least it helps me think faster.  
Right now I’m looking into the logical foundations of nonstandard analysis, because it seems to be a dream come true: a mathematically rigorous way to think with infinitesimals!
 
_{(I could have written this comment without using so many dead mathematicians’ names, but I feel insecure by the amount of arguments here, so I summoned their dead spirits for defence.)}

@Softy
 
Hey, you did get it! Unfortunately, *peoples’. That one was not intentional. When you realize you made a mistake but it’s too late to edit. :)

@Tyler_Cunn  
I’m probably being more nasty toward Vree than I intended. I just get in a “zone” when I know without a doubt that someone is wrong.
 
I’d say Vree knows “just enough to be dangerous”. Kinda like whoever came up with this proof that 1=2
 
But on the subject of grammar…
 
@Tyler_Cunn  

 
My pretty smart what?  
:-P

@Softy
 
Honestly people who get complicated mathematics like you (and Vree, I don’t know how right or wrong you both are about this but you at least understand what you’re saying) amaze me. Keeping track of numbers on anything remotely complicated has always been my Achilles heel. The only subjects I’ve been any good at is relatively ‘weak’ subjects, like English and Music classes (which is kind of ironic, considering reading music is 90% math). So good on you both, may you forever be blessed with well paying jobs. Amen.

@Tyler_Cunn  
Correcting grammar is fun, too!
 
Math is pretty cool because it’s so pure and unyielding. You get it right, or you get it wrong, but there is a right and a wrong. And it absolutely does not give a crap about the colour of your skin or what’s in your pants or your political affiliations or agendas. It treats everyone equally. You play by its rules, you win. You don’t, you lose. And it’s everywhere and what’s true in math today will be true a billion years from now when the human race itself will be gone. That’s why I always loved math.
 
Except multivariable calculus. That evil shit can FOAD.

@Softy
 
Your pretty smart. Considering I need a calculator for anything more profound than 2 X 2, I’ll just stick to correcting people’s grammar on YouTube.

@Vree  
Look, I don’t mean to insult you, but it’s hard to avoid when you keep being wrong about things as certain and definite as mathematics.
 
Nothing about any of this has to do with “areas”. And the analogy from 3-D to 2-D can be made – in fact, I made it, while you were just misusing terminology.
 
You did not “adequately explain” your point. Your example where you “fold its edges to be closer to the circle” doesn’t produce a regular polyhedron, or a regular polygon, or anything that’s in any way helpful in analyzing Starlight’s statement. Your shape will always have perimeter 8r, and that proves nothing at all.
 
Taking limits of things as certain values approach infinity is a hugely useful concept - y’know pretty much all of calculus depends on it?
 
Now, I never even said that Starlight’s statement is correct. I just said that yours was in_correct, because it is. At best, it’s very sloppy to say that something has “infinitely many” things; there are classes of infinities and things can get weird very fast. I can’t say for certain whether the statment is true – unlike some people, I don’t go around arguing and claiming things I don’t actually know.
 
@Tyler_Cunn  
Well, _actually…. I have a degree in mathematics. However it would be generous to say that I’m a bit rusty on certain concepts; it’s been years. I do know when someone’s trying to BS through simple geometry, though.

@Vree
 
You mean you guys aren’t career mathers?

@Tyler_Cunn  
One of the best parts of communities is taking stuff seriously like this.
 
Now we only need a career mathematician in here to tell why everybody isn’t right.

Are we seriously arguing about circles

@Softy  
First, a polyhedron is 3-dimensional, so none of your 2-dimensional images apply.  
If you’re unable to even see that the case applies to areas just as much, and that I use circumference to explain it because it is easier to demonstrate with a picture, you really shouldn’t attempt to argue with such arrogance. Yeah.
 
Bad try. Not sorry. No dice.
 
The point is not that it will approach infinity. The point is that as long as it is a polyhedron, it can never reach infinity. THe existence of sides and angles will keep it fixed above it. But I adequately explained this. It is not my fault if you don’t follow it.
 
Sphere as infinite-sided-polyhedron would have problems like sides of 0 length, making many of its parameters incvalculable since dividing with 0 is disallowed under regular rules.

@Softy  
Of course there is a law that says whenever you’re correcting someone, you will make a mistake of your own.
 
The circumference of a circle is, in fact, 2πr ≈ 6.28r. I think the rest of my math is OK.

@Vree  
Oh, boy. There are layers of incorrectness in what you just said.
 
First, a polyhedron is 3-dimensional, so none of your 2-dimensional images apply.
 
But, if you want to make an analogy in two-dimensions, then the analogous statement in dispute would be “A circle is just a regular polygon with infinitely many sides.”
 
Considering that statement, what you showed in you second picture is not a regular polygon, so your example falls apart again.
 
Imagine a square around a circle or radius r (as in your first picture).
 
The square has perimeter 8r, which is obviously greater than the circumference of the circle πr ≈ 3.14r.
 
But imagine a regular polygon with double the number of sides of the square – an octagon. (Think of a stop sign.)
 
The perimeter of the octagon is 16(√2-1)r ≈ 6.63r
 
As you increase the number of sides in the polygon, the closer the perimeter will get to πr. In fact, the limit of the perimeter is πr as the number of sides approaches infinity.

Vaporwave Starlight is now my headcanon

@Softy  
A polyhedron with infinite sides is a polyhedron with infinite sides; it is not a sphere. It is never a sphere.
 
There is actually a simple proof for this.  
Imagine a square drawn around a circle.  
 
You fold its edges to be closer to the circle.  
 
You can keep adding new and new right angled edges until from any reasonable distance, the square is indistinguishable from a circle.  
Yet it is still a square. Why? Because if you calculate its circumference, it is still A^2 (A squared), not A x pi. Regardless of how many additional angles you have added. You can do the same with any polygon and any polyhedron, but the bottom line is as long as it has anything that can concievably called an angle and a side, it will never ever become a circle.

@Vree  
No, your logic is wrong.
 
“A cube is just a regular polyhedron with eight faces.” - True statement.  
“Then every regular polyhedron also counts as a cube.” - False statement.

That is actually not true. If it was then every regular polyhedron would also count as a sphere, even impossible ones with weird sides and such.

k