As an undergraduate I had the good fortune of taking a class in which Prof. Gerhard Ringel went through his book "Map Color Theorem" in which the coloring problem is solved for every surface except the one that matters - that of the 2-sphere. This includes proofs of the theorem for non-orientable surfaces (odd Euler characteristic). As I recall the proofs were completely developed with little prerequisites required. Prof. Ringel was a delightful instructor (despite a brief, unsavory period in his youth) covering material which was very important to him. I have been enjoying your lectures - your enthusiasm for the subject is obvious. Nice work!
Two hole torus? Only if you have no arms. I count four holes in a shirt. But to fully embrace the torus property, you would have to extend your map to the inside of the shirt, which defeats the purpose of showing it.
@@Achill101actually a shirt is a two hole torus, topologically a hole is something you can put a stick through... A shirt without arms would be a normal torus, because it has a thickness. It was hard to understand for me eather before my math professor explained it to me😂 Sure, you're right, the inside would be colored as well.
@@SM321_ - yes, a short with no arms would be just a torus. And I didn't know how a two-hole torus was defined - less a math problem, but a language problem. . . . We agree that coloring the invisible inside wouldn't be so exciting. Is there any other part of the wardrobe that would be a torus (with holes?) with all sides visible?
@@Achill101 hard to find one. But I think, you can colour the shirt in a way, that all the interesting stuff happens on the outside, you can stretch and squeeze the different colored shapes in a way, that every colour can be seen from the outside. It's at least an interesting thing to think about... PS sorry for my grammar😂
@@SM321_ BTW a shirt would be a 3 hole torus, not 2 hole. Easier to see if you imagine stretching the bottom of the shirt so you can kinda lay it flat, with 3 holes next to each other - left arm hole, neck hole, right arm hole. Pants would have genus 2 though.
As an undergraduate I had the good fortune of taking a class in which Prof. Gerhard Ringel went through his book "Map Color Theorem" in which the coloring problem is solved for every surface except the one that matters - that of the 2-sphere. This includes proofs of the theorem for non-orientable surfaces (odd Euler characteristic). As I recall the proofs were completely developed with little prerequisites required. Prof. Ringel was a delightful instructor (despite a brief, unsavory period in his youth) covering material which was very important to him.
I have been enjoying your lectures - your enthusiasm for the subject is obvious. Nice work!
Super cool!
Wait, why not to design a shirt with a map for which you need 8 colors? I mean the shirt is topologically a two hole torus
Two hole torus? Only if you have no arms. I count four holes in a shirt. But to fully embrace the torus property, you would have to extend your map to the inside of the shirt, which defeats the purpose of showing it.
@@Achill101actually a shirt is a two hole torus, topologically a hole is something you can put a stick through... A shirt without arms would be a normal torus, because it has a thickness. It was hard to understand for me eather before my math professor explained it to me😂
Sure, you're right, the inside would be colored as well.
@@SM321_ - yes, a short with no arms would be just a torus. And I didn't know how a two-hole torus was defined - less a math problem, but a language problem.
. . . We agree that coloring the invisible inside wouldn't be so exciting. Is there any other part of the wardrobe that would be a torus (with holes?) with all sides visible?
@@Achill101 hard to find one. But I think, you can colour the shirt in a way, that all the interesting stuff happens on the outside, you can stretch and squeeze the different colored shapes in a way, that every colour can be seen from the outside.
It's at least an interesting thing to think about...
PS sorry for my grammar😂
@@SM321_ BTW a shirt would be a 3 hole torus, not 2 hole. Easier to see if you imagine stretching the bottom of the shirt so you can kinda lay it flat, with 3 holes next to each other - left arm hole, neck hole, right arm hole. Pants would have genus 2 though.
that's a really good torus.
Yawn.