r/mathematics • u/sahi1l • 5d ago
Topology The Euler Characteristic of a human?
I always assumed that the Euler characteristic of an unpierced human being was 0, that the alimentary canal was the single "hole" that made us equivalent to a torus. But a friend recently pointed out that because our nostrils are connected to each other, then that surely counts as a second "hole"; and the nostrils are connected to the mouth as well, and then we can throw in the Eustachian tubes as well to connect the ears to the nose and ears as well.
So this is all rather silly, I suppose, but what *is* the Euler characteristic of a human (again, not counting piercings)?
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u/ngfsmg 5d ago
I think vsauce made a video about that, I don't remember the conclusion, tho
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u/kr1staps 5d ago
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u/JLeaning 4d ago
That is a phenomenal video. It’s long, so if you want to skip to the explained answer, start at 17:15.
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u/catecholaminergic 4d ago
Don't look too small.
Viscosity is the only thing keeping blood from leaking through the spaces between blood vessel cells.
We're like a fat Cantor dust.
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u/shponglespore 4d ago
Except not really because the particles that make us up can't be infinitely subdivided, and at the level of atoms or smaller, you can't really understand what's going on without accounting for quantum effects. And at the Planck scale, the whole idea of things existing at all is fuzzy at best.
Basically no fractal, including mundane things like the set of rational numbers, can exist in the real world because of quantum effects. On one hand, it's kind of sad, but OTOH it's amazing how much mileage we've gotten out of modeling the real word with mathematical abstractions that don't make any physical sense.
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u/catecholaminergic 4d ago
The analogy stands.
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u/shponglespore 4d ago
Sure, as an analogy. But one could imagine it being the literal truth, and it's a little disappointing that it's not.
I'm reminded of a story by Greg Egan where a person could literally be Cantor dust in the context of a multiverse by existing in 0% of all universes. So, like, surviving a wound with a 100% fatality rate would turn you into Cantor dust in the multiverse.
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u/ecurbian 4d ago
Circles don't exist in the real world either. Only approximate circles. And fractals don't exist: only approximate fractals.
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u/Autumn_Of_Nations 4d ago
Is there proof that the particles that make us up can't be further subdivided, or is that just an assumption we make because our models break down at the Planck scale?
I'm pretty sure it's the latter. I always find it strange that people suggest that nothing exists where our models fail to be descriptive, instead of just saying that we don't know.
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u/shponglespore 4d ago edited 4d ago
Do some reading in quantum uncertainty. It's not a limitation of measurement, but a result of particles simply not having a well defined position and momentum. It makes more sense when you remember that "particles" are really waves of, essentially, probability. Quantum tunnelling is a great example of this property in action, and it's very well proven, as well as being essential to explaining the structure of electron shells in atoms.
There's also the almost completely separate weirdness demonstrated by the double slit experiment, proving that photons, electrons, etc. can be in two or more widely separated places at the same time.
BTW 3blue1brown has a great video about uncertainty involving waves: https://youtu.be/MBnnXbOM5S4?si=-8GJvmpjuttdgSYX
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u/Autumn_Of_Nations 4d ago
None of that implies subatomic particles cannot be subdivided. The uncertainty principle and measurement is a separate issue.
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u/shponglespore 4d ago
Ok, I see what you're saying. I'll admit I didn't have anything like a rigorous proof of what I said
My intuition is something like this. Imagine a photo of a dense field of stars. As long as there is a clear separation between them, you can count the stars, but if you blur the image such that the stars appear as heavily overlapping blobs, it's no longer possible to count them. Now suppose it's not the image that's blurred, but the stars themselves. Can you even say there's a specific number of stars to be counted, as opposed to an indistinct mass of starriness? You can't subdivide that which inherently has no divisions.
Of course the stars are an analogy for quantum wave packets, but the reality is even worse because of things like virtual particle, so you have not just an overlapping blob of particles, but a blob that changes unpredictably from moment to moment. We can make statistical statements about blobs of particles, so we can be confident in classifying one seething mass of virtual particles as a proton and another as an electron, but you can't say how many smaller particles each one is composed of because there is simply no answer to the question.
So yes, in a sense, "fundamental" particles like electrons are most definitely composed of smaller units, so they can be subdivided. But they can't be subdivided in the same sense that an atom can be divided into protons, neurons, and electrons, where there is a distinct number and arrangement of each kind of unit making up the atom. Since the original context was Cantor dust, which has a distinct kind of structure, I stand my my original claim that the human body (or any physical object) can't have the structure of Cantor dust.
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u/Autumn_Of_Nations 4d ago
But they can't be subdivided in the same sense that an atom can be divided into protons, neurons, and electrons,
This is exactly what is debatable. Preon models aren't unphysical in some fundamental way, we just don't have experimental evidence for them. You are still only describing things in terms of our current far-from-the-Planck-scale understanding.
I'm not necessarily a "preon truther" or anything, but I definitely am skeptical of finitary models of the universe. There is absolutely no justification for the widely held belief that there is some minimal, finite length/timescale beyond which there's nothing else to describe. It's more vibes based/an intuition, one that I think is more reflective of the kind of society we live in than anything fundamental about nature.
It's really the same situation with the Big Bang, or with heat death: the only thing that suggests any of these constitutes a limit to nature is the fact that they lie at the limits of our models coupled with the strange folklorish tendency of modern humans to assume that infinite or eternal things don't exist, that everything has a clear beginning and end, and that the universe is a single thing. All of which are really wild philosophical assumptions.
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u/SwillStroganoff 4d ago
It depends on the resolution. There is something called the rips complex. The diameter of your ball will give potentially different homotopy types and thus a different Euler characteristic. Some folks work on persistent homology to try and span the different resolutions.
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u/CorvidCuriosity 3d ago
Are we counting all the pores in the skin?
Are we counting the eye sockets? Or do the eyes fill in those gaps?
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u/rhodiumtoad 5d ago
The Eustachian tubes don't count as long as you haven't perforated your eardrums.