5-Sided Square - Numberphile

Key Takeaways

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 um 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 on 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 yeah you know i could sort of shape this down here and here 90 degrees down here but not sure i drew 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 number i'll get 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 is something that's always curving inward can i make something and and and the answer is yes i can make a pseudosphere a fake sphere it's funnel shaped it's horn 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 a little little point here that i have to pick the minimum inward going in the and the maximum outward going or the maximum of 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 pseudosphere 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 pseudosphere everywhere has constant negative curvature let me go back for a minute breathe on 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 let's try on the globe true on a sphere on a pseudosphere 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 origamist bob lang showed me how to do this paper that he wrote and uses all sorts of hyperbolic secants and cosecants and and hyperbolic functions abound on the 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 e here's a five-sided figure one two three four five sides each of the corners is 90 degrees 90 there 90 there 90 let's check keep me honest here let's take a piece of paper that's 90 degrees right there's a here's a 90 degree angle right there right there yep 90 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 90 degrees let's and cut out of paper like this here's say here's the piece of paper folded to fit shaped to fit on top the pseudosphere each of these corners are 90 degrees it's a pentagon it's a pentagon whose corners are all 90 degrees so is it a pentagon or square if i say hey a square is a shape that has equal side equal length edges in all corners are 90 degrees then i could claim that this is a five-sided square so its edges are equal length as well each of those edges is the same length 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 and 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 ah no how about if i push this one and 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 pseudospherical 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 our universe because our universe apparently seems to have something of negative curvature our 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 going to 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 going to have loads on it for example mountaineering is something i'm interested in they've got you covered there or how 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