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This may seem like an odd question, but I'm asking myself if the surface of the body/skin can be made into a torus (and how it would look like, e.g. the positions of the legs, arm asf). My intuitive answer is yes, but since I'm not that deep into topology maybe I'm wrong. What do you people think (and if it is true, what could be an approach for creating a mapping between those spaces?)

holistic
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    Perhaps it can be "made into" a torus, but only with some rather radical surgery... –  Feb 18 '14 at 09:38

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If we simplify the human body to only have the one opening where food goes in and one where food goes out, you can see why this would be homeomorphic to a torus. The "hole" is where the food goes through. If we look a little closer we notice this really is an oversimplification, you find a second hole where liquid food comes out and you might find many places where it depends on the definition of the "surface of the body" whether you want to count things as holes or not.

After all, looking at the surface of a food-processing creature as a torus is always a nice start to explain the difference between a sphere and a torus or even surfaces with higher genus at the topological level, since the geometry of the body doesn't look like a standard $2$-torus at all!

Christoph
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If we didn't have a nose (and wouldn't count the ears as holes), the human body would probably be homeomorphic to a torus. Although it also depends if you see the lung and the bladder as dead ends.

Roland
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  • Thanks for the answer! Well..just like as if one would remove the skin, just the surface without any holes :) – holistic Feb 18 '14 at 09:50
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For a normal human male, discounting pores, we have $9$ holes. So our skin homeomorphic to a disk with 8 holes and has fundamental group the free group on 8 generators.

David P
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    What are the 9 holes? 4 are easily identified: mouth, anus, 2 nostrils. Ears wouldn't be considered holes due to the ear drums. One could consider the foramina in the eye sockets for an additional 4. What about for a normal human female, given that the abdominal ostium open into the abdomen? – wflynny Mar 06 '17 at 19:46
  • the question was about the skin. remove just the skin. now think (dont wanna get too graphic here) – David P Mar 07 '17 at 02:29
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    "Now think," Well, I'd hope that my previous comment has shown I have thought about it some. I'm wondering if you're considering the urethra as another hole, but like the ears, the urinary tract is a deadend (at the kidneys). So what are the 9 holes? Anatomical terms aren't graphic. – wflynny Mar 07 '17 at 06:37