Human body and topology

1. Aug 14, 2008

DaveC426913

I understand that topo-physiologically, the human body is a donut. i.e. not only are we a donut physically, but we are a donut as an organism.

Our skin is an interface between the outside world and the inside of our bodies. Bacteria and other microbes have to get through our skin defense before they can infect us. But this barrier persists right though our GI tract as well, which is why we can eat dirt without getting infected. Physiologically, our intestines are considered outside the body.

So, to my question: if we count the nostrils, we've got 4 orifices. (I wouldn't count orifices that lead to the interior of the body, such as urethra, or dead ends such as the ears or Eustacian tubes).

So, are all blobs with 4 openings topologically equivalent? Or are there several flavours such as holes branching off into multiple holes and such?

2. Aug 14, 2008

DaveC426913

Upon reflection, I answered my own question. The attached diagram contains two distinct flavours of "4-holed blob". They are not topologically equivalent.

Attached Files:

• donuts.gif
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3. Aug 14, 2008

Defennder

I believe we have a lot more holes than you posited. Every single one of our skin pores is also a hole.

4. Aug 15, 2008

maze

Although this is a bit of a tangent, there is quite a bit of interesting topological effects that arise when 3D model makers make polygonal models of the human body.

The basic principles of subdivision modeling are that quadrilaterals are prefferred over triangles or other n-gons, and that each vertex have 4 edges leaving it. The number of holes in the mesh topologically require that you can't always achieve both these objectives, and so your mesh is going to have vertices connected to more or less than 3 edges, and these vertices are called "poles". eg: 3-poles, 5-poles, etc. The terminology comes from the standard quadrilateral mesh of a sphere, where all vertices have 4 edges except at the north or south pole.

Where you place these poles has dramatic effect of the flow of edgeloops along the mesh, and so there is a considerable art to placing them so that the resulting edge flow corresponds with the way the structure of the body flows. By twiddling edges, you can more poles around, and interesting things happen to edgeloops when poles collide or cross.

more in this epic thread with lots of pictures:

Last edited: Aug 15, 2008
5. Aug 15, 2008

LukeD

The pores don't count because they're just indentations (they stop at some point). Similarly, neither do ears canals, urethra, vagina, etc. However, the OP forgot about the eye sockets, so we have 6 (unless I'm forgetting something).

6. Aug 15, 2008

HallsofIvy

But the eye sockets also are "just indentations", aren't they?

7. Aug 15, 2008

Defennder

How can this be so when the sweat from our pores originate from the water consumed through our oral cavities?

8. Aug 15, 2008

HallsofIvy

Because the liquid passes through cell membranes. If you want to argue that there must be "holes" in the membranes that the liquid passes through, you will have to go down to the molecular level at which point the human body is just a collection of separate molecules.

9. Aug 15, 2008

DaveC426913

In the OP, I was explicit to point out that I'm defining the "surface" of this blob as the barrier through which microbes cannot normally penetrate. This is why I was careful to name it as a topo-physiological issue, not simply a topological issue. (Maybe I should have said topo-bio-physiological.)

While there's no question that this blob is porous to some things at some times, it does not simply let everything pass. Skin pores do not normally let microbes pass. They are topo-physiologically culs-de-sac - dead-ends.

Similar with the GI tract. A microbe that finds itself in our GI tract is still outside our body, as it will, for the most part, still have to fight its way through our defenses and through the GI wall.

Last edited: Aug 15, 2008
10. Aug 15, 2008

LukeD

No, the eye sockets connect to the oral cavity. Just like you'll squirt milk out your nose if you laugh while drinking, you can squirt it out your eyes as well (though I've never done it and I have no idea how). There's actually a record in the Guiness Book for the furthest that someone has shot liquid (after drinking it) out of their eye sockets (you'll see some crazy things on their tv show)

11. Aug 16, 2008

HallsofIvy

Thanks for the information and confirmation of why I have never watched that tv show!

12. Aug 16, 2008

DaveC426913

Yes, that's through the tear ducts. I'm not sure whether the tear ducts are outside the body biologically. You can get infections in there.

13. Jul 24, 2009

ghoshg

Yes, you are right. The process water molecules are transported through cell membranes in our body is called 'osmosis'; the same process water molecules get transported in a tree from its root to its leaves.

14. Jul 24, 2009

Tibarn

If we go down to the atomic level, the human body doesn't really have an inside. Anything sufficiently small can pass between the atomic nuclei. We are over 99.9% empty space after all.

15. Jul 24, 2009

ghoshg

What you are talking about is similar to quantum tunneling. Do you have any knowledge about tunneling effect ? In our body there is no tunneling as such.

16. Jul 24, 2009

DaveC426913

No, he's expressing himself incorrectly. He doesn't mean 'between nuclei' as if things could pass through the space inside an atom, he means simply between atoms, or at least, between molecules.

17. Jul 25, 2009

YellowPeril

I dont believe that one can make a distinction about the ears not being an opening, afterall if one considers that the body is comprised of tubes and boilers essentially at low pressure with valves all over the place then the eustacian tube should not be treated any differently from the valve which closes off the stomach for example. Another point is that some people topologically different from others e.g. women with earings or might be different at different poses e.g. if you put your thumb and forfinger together does that affect the topology? i.e. it is necessary to be specific about the rules that apply to the body before describing it's topology.

18. Jul 25, 2009

DaveC426913

Fair enough. OK, Eustooshun toobs* count as holes.
We will assume the default human state.
Definitely no. Two parts of an amorphous blob touching definitely does not turn it not a hole.
I am fairly certain this goes without saying in topology, unless explicitly specified otherwise.

*obscure M*A*S*H reference

19. Jul 30, 2009

YellowPeril

Sorry DaveC426913,

You were originally correct about the ears, the eustachian tube sits behind the ear drum to equalise pressure therefor making it a blind opening. Saw a diagram today which reminded me.

20. May 11, 2011

barrylyndon

My differential geometry book says that according to the classification theorem of surfaces, every [compact?] orientable [2-] surface [in 3-space] is homeomorphic to a sphere or a sphere with a finite number of holes in it.
So with just three facial orifices and one down below, we're a three-holed torus, or, a donut with two handles glued onto it.
In Dave's thumbnail .gif, we are the second blob, since all of those true orifices are connected. Stretch out hole that runs from the top to the bottom (don't think about it too clearly) wide enough, relative to the rest of it, and what you have is a donut with two more holes punched through its sides. Each of those holes can be re-shaped to be like a "handle" on the main body of the torus.
I guess with the eyes, that's adding two more handles, or two more ways to tunnel through, making us a 5-holed torus.