5-Sided Square - Numberphile

90 degrees 90 degrees if I have a shape that has 90-degree angles and all the sides are the same length it's a square yeah but wait you know as well as I do if I go to the earth go on the earth and say oh I'm going to start on the equator go up to the North Pole then move over 90 degrees and take the line of longitude down that goes right through the Gulf of Mexico and Yucatan and hit there oh I've got a 90 degree angle here a 90 degree angle here and a 90 degree angle here not a sphere I can have triangles that have more than 180 degrees in them I can cut a piece of paper on here that would be a three-sided square I could sort of shape this down here and here 90 degrees down here it's not sure I do it correctly but let's try this one each of the angles here are 90 degrees here's a 90 degree angle here's a 90 degree angle and here's almost a 90 degree angle here's a three-sided square all three sides are the same length and all the angles are 90 degrees let's hear not to be confused with a ball but a sphere has constant Gaussian curvature which says that if I pick a point that point there and draw two lines through it and choose the maximum and the minimum curvatures of those things multiply them together positive number times a positive net positive a sphere everywhere has constant positive curvature okay neat can I get something that has constant negative curvature is it possible to make something that is the opposite of a sphere something that instead of always bulging outward if something that's always curving inward can I make something and and and the answer is yes I can make a pseudo sphere a fake sphere it's funnel-shaped it's Horan shaped and it has the delicious property that if I pick any point at all on it say right here in one direction it'll bulge inward in the other and the other direction it bulges outward I have to make little little point here that I have to pick the minimum inward going in the and the maximum outward going or the maximum is either one of them I can't just rotate this 90 degrees I have to or a few degrees this way so it's like an X I have to pick these so that in one direction it's bulging out and the other one it's bulging in turns out that a pseudo sphere has everywhere constant negative curvature that means that if I multiply to get the Gaussian curvature here where the positive going here is not very big but the negative going is huge and over here the positive the negative going curvature is kind of small but the positive curvature along here is quite big a pseudo sphere everywhere has constant negative curvature let me go back for a minute breathe I'm an ordinary Euclidean piece of paper every triangle if I sum the angles I'll get 180 degrees on a sphere if I draw a triangle any triangle at all I draw on it the sum of the three angles will be always greater than 180 degrees it's true on the globe true on a sphere on a pseudo sphere no matter how I draw a triangle its angles will always sum to less than 180 degrees they'll be pointy in an odd way they'll come to little pointy shapes we can take advantage of that to draw a shape that has 90-degree corners in five sides origami Asst Bob Lang showed me how to do this from paper that he wrote and uses all sorts of hyperbolic secant and cosecant sand and hyperbolic functions abound on this surface but the cool thing is you can take a piece of paper get it wet wrap it around here everywhere has negative Gaussian curvature by choosing my points just just right I can find that eat here's a five sided figure one two three four five sides each of the corners is 90 degrees 90 there 90 there I did it okay check keep me honest here let's take a piece of paper that's ninety degrees right there's a here's a 90 degree angle right there right there yep ninety degree angle there over here sure enough 90 degrees 90 degrees 90 90 here's a five sided figure the five catches a five sided figure all corners are ninety degrees let's say and cut out a paper like this here is say here's the piece of paper folded to fit shaped to fit on top the pseudo sphere each of these corners are ninety degrees it's a Pentagon it's a Pentagon whose corners are all ninety degrees if I say hey a square is a shape that has equal side equal length edges in all corners are ninety degrees then I could claim that this is a five sided square so it's edges are equal lengths as well each of those edges is the same what happens if I bash it flat well a nifty thing about topology says it shouldn't be possible if I try bashing this flat it's like taking a section of a sphere taking a section of the globe in bashing it flat there's not a good mapping that will preserve areas that will preserve directions we get into the problem of conformal mapping and non conformal mapping and things like this we can do this but as I push down here that bends out push down here oh I'll just bend over here oh maybe if I push down here oh then it bumps up here bumps up there I know how about if I push this one in this one down this one this one this I've got five fingers one two three one two three four five and up but now it's bumping up here I can sort of put it in a press but it's trying to pop out the bottom of the table you can't perfectly map a spherical or a pseudo spherical surface onto a Euclidean plane it's a problem that map makers have had for a long time and it's a problem that shows up when you start mapping your universe because our universe apparently seems to have something of negative curvature now thanks to the great courses plus for supporting this video if there's anything you want to learn about including mathematics the great courses plus is gonna have it have a look at this one it's called crazy kinds of connectedness but it's not just mathematics pretty much anything you want to learn about they're gonna have loads on it for example mountaineering is something I'm interested in they've got you covered there or how about dog training this is a 24 part course on dog training which I find fascinating and I especially love because have a look Lulu this dog here is called Lulu my dogs called Lulu but it's not all about dog training seriously you name it they've got it and it's going to be taught by world-class experts from institutions and universities all across the globe now if you go to the great courses plus comm slash numberphile there's also a link in the description you can check it out and do a free trial access to everything I think it's something like 10,000 videos you've got it our thanks to the great courses plus for supporting this video

Channel: Numberphile

Views: 1,364,162

Rating: 4.9388847 out of 5

Keywords: numberphile, squares, topology, spheres

Id: n7GYYerlQWs

Length: 9min 15sec (555 seconds)

Published: Mon Aug 13 2018

Reddit Comments

[u/snipers501]

He's so happy about all of this!

I wish I could be excited about math.

[u/typhyr]

does anyone know where he got the negative curvature of the universe thing from? i thought experiments showed it was flat, with a ~1% or less margin of error, and googling led me to the same result.

[u/psdnmstr01]

Cliff Stoll is too pure for this world

[u/IkonikK]

what about the 11-sided square?

[u/iloveciroc]

Cliff Stoll is the Mr. Rogers of math. I love him!

[u/Ahmadthenewb]

Cool video but isnt a square defined as specifically a 4 sided figure with equal sides and 4 right angles? So saying a 5 sided 'square' is wrong isnt it? Am i missing something?

[u/EdPeggJr]

I'll raise with a square on 5 sides, otherwise known as the d10.

[u/[deleted]]

I love Cliff so much <3 He is one of my heroes!

[u/Ay-Shaaaan030]

I am loving his enthusiasm! 😁