Sunday, July 06, 2014

You’re not a donut. And not a mug either.

A topologist, as the joke goes, is somebody who can’t tell a mug from a donut.

Topology is a field of mathematics concerned with the properties of spaces and their invariants. One of these invariants is the number of ways you can cut out slices of an object without it falling apart, known as the “genus”. You can cut a donut and it becomes an open ring, yet it is still one piece, and you can cut the handle of a mug and it won’t fall off. Thus, they’re topologically the same.

The genus counts essentially the number of holes though that can be slightly misleading. A representative survey among our household members for example revealed that the majority of people count four holes in a T-shirt, while its genus is actually 3. (Make it a tank top, cut open shoulders and down the front. If you cut any more, it will fall apart.)

Every now and then I read that humans are topologically donuts, with anus, excuse me, genus one. Yes, that is obviously wrong, and I know you’ve all been waiting for me to count the holes in your body.

To begin with the surface of the human body, as any other non-mathematical surface, is not impenetrable, and how many holes it has is a matter of resolution. For a neutrino for example you’re pretty much all holes.

Leaving aside subatomic physics and marching on to the molecular level, the human body possesses an intricate network of transport routes for essential nutrients, proteins, bacteria and cells, and what went in one location can leave pretty much anywhere else. You can for example absorb some things through your lungs and get rid of them in your sweat, and you can absorb some medications through your skin. Not to mention that the fluid you ingest passes through some cellular layers and eventually leaves though yet another hole.

But even above the molecular level, the human body has more than one hole. One of the most unfortunate evolutionary heritage we have is that our airways are conjoined with the foodways. As you might have figured out when you were 4 years old, you can drink through your nose, and since you have two nostrils that brings you up to genus three.

Next, the human eyes sit pretty loosely in their sockets and the nasal cavities are connected to the eye sockets in various ways. I can blow out air through my eyes, so I count up to genus 5 then. Alas, people tend to find this a rather strange skill, so I’ll leave it to you whether you want to count your eyes to uppen your holyness. And while we are speaking of personal oddities, should you have any body piercings, these will pimp up your genus further. I have piercings in my ears, so that brings my counting to genus 7.

Finally, for the ladies, the fallopian tubes are not sealed off by the ovaries. The egg that is released during ovulation has to first make it to the tube. It is known to happen occasionally that an egg travels to the fallopian tube on the other side, meaning the tubes are connected through the abdominal cavity, forming a loop that adds one to the genus.

This brings my counting to 5 for the guys, 6 for the ladies, plus any piercings that you may have.

And if you have trouble imagining a genus 6 surface, below some visual aid.

Genus 0.

Genus 1.

Genus 2.

Genus 3.

Genus 4.

Genus 5.

Genus 6.



  1. what's next?!

  2. Euler's bridge to nowhere?

    Paradox can be a useful tool even to an experimentalist. It may even present a thought experiment in the form of your post above? What then can we use to describe something that rolls down a hillside and come to rest in the valley, a stone?


  3. It's all string worldsheets to me. ;)

  4. "Tanquam ex ungue leonem," Johann Bernoulli. "I can blow out air through my eyes," Bee. The historic footnote is set. Drop opposite shoes in vacuum. Torsion balances are rare,so DOI:10.1063/1.1147553 and

    4-oxa-D_3-trishomocubane (one methylene replaced by an ether). Helium-entrained racemate vapor is de Laval nozzle vacuum supersonic expanded, reducing its rotation temperature to ~1 kelvin. The cryogenic molecular beam passes through a chirped-pulse FT microwave spectrometer to high resolution map its ground state rotation spectrum. A doubled spectrum explains quantum gravitation.

    Then, homework. "8^>)

  5. The overall quality of topological illustrations has improved a lot at this blog recently.

  6. Zephir,
    On the contrary. This time her theory is full of holes.
    With a little aether one can take a tee shirt off and leave the outer garnet invariant.
    So, if two Mobieus firewalls strip the cold isotropic reKleined bottle super imposed can hold water.
    But it is uncertian if the starry polka dot pink shift dress is Minnie Mouse or the T-Mobile model.
    (Maybe I should lay off those Monster Chaos Energy drinks when reading Barrow) :-)

  7. Uncle Al,
    What if in the distant noise resulting in replication ringing bells, a third shoe drops? One super partner at a time please for our vague dreams of inflation.

  8. Sabine, let me ask a silly question: How many holes has the event horizont of a black hole?
    May I say it has no holes at all?
    Or we may suppose, for instance in the case of Kerr black holes, they're like torus shape?

  9. The homework assignment is incomplete - the backside view is missing.

  10. You have to sum over all possible completions and evaluate the path integral ;)

  11. Nemo:

    Depends on the number of dimensions. Black hole horizons can have many different topologies, see eg this paper.

  12. Oh Yes, Sabine that's right. Thank you

  13. LOL! You might mark such posts "NSFW" - not safe for work - viewing such topologies on corporate equipment is against corporate policies (at least here in the US).

  14. You left out the illustrations for the string-theory approach.

  15. "The genus counts essentially the number of holes though that can be slightly misleading. A representative survey among our household members for example revealed that the majority of people count four holes in a T-shirt, while its genus is actually 3. (Make it a tank top, cut open shoulders and down the front. If you cut any more, it will fall apart.)"

    This is because the holes in the T-shirt are connected, right? A normal mug has genus 1, a double-handled mug genus 2 (since the holes are no connected, i.e. you can't go in one and out the other), right?

  16. One might imagine topology dealing with Sierpinski gasket, Apollonian network, and Menger sponge lingerie. What happens with Kazimierz Kuratowski's K_5 and K_(3,3) string bikinis?

    (Bolt-ons? Feh. Silicone belongs in a crystallizing dish with a nichrome coil. Saline jobs are remarkable for Serratia marcescens contamination, admittedly colorful. You do not want to be at the intersection of Heyer-Schulte and "coconuts.")

  17. Hi Phillip,

    Saying that the holes are connected is equally misleading. The problem is that the word "hole" in colloquial terms doesn't mean exactly the same as it means in topology. If you take a balloon and you pinch one hole in it, it's still simply connected - it has the topology of a mug without a handle. Best,


  18. In a phone conversation with J.Stashift asked to consult in topology for string theorist in 1985 or so at UNC, I tried to convey my idea of my Quasic space. Along the way past the initial lecture for a non student of initial branching of. Two or one sided topolopy I said "So there are no holes. " then after our long conversation he replied, "That is correct. "

  19. A torus has genus one, as does the surface of a coffee mug with a handle. This is the source of the joke that "a topologist is someone who can't tell his donut from his coffee mug."

    So a teacher prepares his student as Stanley Mandelstam did for his student Polchinski?

    ...okay, so its all math?

    In an ordinary 2-sphere, any loop can be continuously tightened to a point on the surface. Does this condition characterize the 2-sphere? The answer is yes, and it has been known for a long time. The Poincaré conjecture asks the same question for the 3-sphere, which is more difficult to visualize.

    So we might call it a topography of energy yes?

  20. The Poincaré conjecture, proved in 2003 by Grigori Perelman, provides that the 3-sphere is the only three-dimensional manifold (up to homeomorphism) with these properties.

    The 3-sphere is homeomorphic to the one-point compactification of \mathbb{R}^3. In general, any topological space that is homeomorphic to the 3-sphere is called a topological 3-sphere.

    Now when I say you may have gone beyond I was being specific, with regard to the joke.(?)

    Or, it is another one of these issues you have with the Platonists?:)

  21. So we stand quite firmly with, "A sphere with three handles (and three holes), i.e., a genus-3 torus.*" :)-*Weisstein, Eric W. "Triple Torus." From MathWorld--A Wolfram Web Resource.

    The sphere, can be a useful object?

  22. Plato - Hegel,

    The general sphere group is at once the most and least interesting.
    Topology is only skin deep and Platonism can be seen as ultimate perfection. Putting aside the asymmetry, especially bilateral and thoughts of dialectical hierarchies perturbative or not, a perfect model can be photoshoped by finer tuning. Or by fixed or erasible artistic tattoos paint on her a garnet that phenomenonally fools readers of the Sports Illustrated swimsuit editions.
    So while I understand Pontclaire's conjectures are now proved it still does not cover the depth and span of physics problems in today's cosmological speculations.
    Three space with its unique properties stands alone to be understood among the other higher and lower dimensions.
    Beauty as the guiding good for our expanding growth inflation's allows variations in perception of which is ugly duck or emerging swan, and conversely.
    On which side of mirrors does Nature constrain processes screen thru one way?

  23. Bee,

    You have human anatomy sufficiently covered. ;)

    Are there equations of physical dynamics in which the mathematics of topology plays a significant role?


  24. As with any compact Riemann surface, the sphere may also be viewed as a projective algebraic curve, making it a fundamental example in algebraic geometry. It also finds utility in other disciplines that depend on analysis and geometry, such as quantum mechanics and other branches of physics

    Also seen in applications....

    The Riemann sphere has many uses in physics. In quantum mechanics, points on the complex projective line are natural values for photon polarization states, spin states of massive particles of spin 1/2, and 2-state particles in general (see also Quantum bit). The Riemann sphere has been suggested as a relativistic model for the celestial sphere. In string theory, the worldsheets of strings are Riemann surfaces, and the Riemann sphere, being the simplest Riemann surface, plays a significant role. It is also important in twistor theory.

  25. "You have human anatomy sufficiently covered. ;)"

    Better to have it sufficiently uncovered.

  26. Phillip,
    From the comsmological constant to Casmir effects on the microscale, perhaps the same place yet where we have to unify the algebraic and geometric interpretations as in QM and the old e i π 0 1, on what side of the debate the other sees no back reaction.
    No matter what our simulations say we still wonder and all understand the search for naked singularity. :-)

  27. Curious that e to the i pi plus one equals zero. Or is it minus one? (This being an almost remembered oddment)

    In any case, I am wondering if the notion of slope is conveyed in the mathematics of topology. Is there a metric?

    Perhaps this question is too remedial.

  28. Topology in breaking news! It appears that all Brazucas are chiral left-handed.

    Any chiral fullerene with T (C_44, C_52, C_92, C_100), O, or I (C_140, C_260,) symmetry is geometrically perfectly chiral re Petitjohn (J. Math. Phys. 40, 4587 (1999)). They must lack all S_n axes, thus not T_h, T_d, O_h, or I_h point groups.

  29. This comment has been removed by the author.

  30. In hyperbolic geometry, pairs of pants are sewn together, leg to leg, or leg to waist (there is no distinction between the legs and the waist), to create Riemann surfaces of arbitrary genus; conversely, Riemann surfaces can be cut into pairs of pants by cutting along closed geodesics. Because the "legs" can be twisted before being sewn together, there is a large amount of freedom in how the pants can be assembled. This ambiguity gives the Fenchel–Nielsen coordinates for the moduli space of the Riemann surface, which has complex dimension 3(g − 1) = 3g − 3 for g > 1.

    How can you ever choose to enter such a world as an abstraction, if you did not have Riemann at your side, and the world of Einstein leading, by having merged and being as an example, of the way?

    Isometries of Riemann surfaces

    Layman, working his way throu

  31. So, how does one retain that example by what was previously mentioned? Just guessing:)

    In general relativity, conformal maps are the simplest and thus most common type of causal transformations. Physically, these describe different universes in which all the same events and interactions are still (causally) possible, but a new additional force is necessary to effect this (that is, replication of all the same trajectories would necessitate departures from geodesic motion because the metric is different). It is often used to try to make models amenable to extension beyond curvature singularities, for example to permit description of the universe even before the big bang.

    Imagine, a cosmologist might be persuaded to see more then a square box?:)

  32. Plato,

    Topology and algebra could meet in some models.

    What if nature does not know if it exists on a torus or a cylinder? Or that what we see of such phenomena of space shows something in between for example degrees of negative probabilities or some excess or lack of light or matter in a given model.

    Our systems and assertions are at once too simple and too complex (well complicated as to how well complex numbers apply can be a matter of degree too) Is there some degree of causation? At least in a wider environment of possibilities?

    The effects or stress from the uncertainty of indistinguishable torus genus or cylinder geometries like ink dropped into water making closed and descending inflation like rings at some point of division.

    Sheets, sheaves of jumps that suggest Riemann models, locally these fade in or out not necessarily cyclic. But can the effect of interference not precipitate out of the manifold of perfection to paint a purely quantum picture? Might not gravity waves be such interference and superposition? Thus in a sense there is evidence?

    I am not sure what Sabine is thinking but it is important and keyed this thought this morning from a dream.

    It is not always the issue of casual feminine comfort and causal masculine position that finds new geometries when in weightlessness there is an equivalence principle. Or as you cite some idea of gluing twisted pants together where there are none or dividing a torus into thirteen solid parts. or the crossing over of three things connected to three other things dividing the surface of a torus into seven areas...

    Sometimes in the naked truth and it is simple really we stumble upon surprising new configurations.

    What is the genus of the universe?

  33. In 1919, Kaluza sent Albert Einstein a preprint --- later published in 1921 --- that considered the extension of general relativity to five dimensions. He assumed that the 5-dimensional field equations were simply the higher-dimensional version of the vacuum Einstein equation, and that all the metric components were independent of the fifth coordinate. The later assumption came to be known as the cylinder condition. This resulted in something remarkable: the fifteen higher-dimension field equations naturally broke into a set of ten formulae governing a tensor field representing gravity, four describing a vector field representing electromagnetism, and one wave equation for a scalar field. Furthermore, if the scalar field was constant, the vector field equations were just Maxwell's equations in vacuo, and the tensor field equations were the 4-dimensional Einstein field equations sourced by an EM field. In one fell swoop, Kaluza had written down a single covariant field theory in five dimensions that yielded the four dimensional theories of general relativity and electromagnetism. Naturally, Einstein was very interested in this preprint

  34. While towards the end of May 1919 Einstein had not yet fully supported the publication of Kaluza’s manuscript, on 14 October 1921 he thought differently:

    “I am having second thoughts about having kept you from the publication of your idea on the unification of gravitation and electricity two years ago. I value your approach more than the one followed by H. Weyl. If you wish, I will present your paper to the Academy after all.”86 (letter from Einstein to Kaluza reprinted in [49], p. 454)

    History of Unified Field Theories

  35. Sabine, I like the pictures, but you got the math wrong. "One of these invariants is the number of ways you can cut out slices of an
    object without it falling apart, known as the genus". True if all your cuts are circular: you must end at the same point where you started. If fact, punching or sewing a circular hole does not change the genus of a surface. So, the genus of a T-shirt is zero, because after sewing sleeves and top you have a pillow-case.

  36. Plato,
    That hundred year old account is interesting ( I think Weyl-Eddington is vastly superior to Kaluza-Klein ) but it really gets down to the arithmetic, five partitioning and so on. Does not mean 5 dimensions necessarily.
    I commented to Michel J. on visualizing gravity this:
    While the analogy suggests gravity waves vaguely gravitational waves are harder to see.this shaped boat is stable as the force of waves is self cancelling at the torus surface. One does not get sea sick in the center of such a craft.

    So, Alexy Gavrilov, what math are you talking about and in what context? If we cut a circular path to a torus does that make a difference in genus?

  37. If you cut a disk off a torus, making a hole, then the genus does not change (it remains one). Of course, if you cut it along a meridian, it becomes a cylinder, with genus zero.

    Naturally, math I am talking about here is topology.

  38. Alexy,

    Groups, invariants, dimensions. I find it remarkable that to do arithmetic in our heads we need to memorize 15 operations. But we are talking about 16 things here in a matrix as if two of them are somehow superimposed.
    So the question is, at least algebraically, just what can we perceive of so many abstract dimensions? Klein certainly has had an influence and hidden curved dimensions is half the picture where we naturally see a third of things but the other part is useful too. If there are 9 natural dimensions we can see only three viable ones at a time. If we allow contiguous forces to merge rather than remain separate we only see one of three axes as continuous or none of them.
    I am suggesting when you say the genus is one or zero, that in the bigger picture the question is one of treating this as information theory, perhaps in a more relaxed way as other equivalent views of abstract intrinsic curvature can be more direct for our quantum like or Einstein-Mach like computations.
    This would apply to the remarkable seeming coincidence that (by Klein) all the Euclidean and non-Euclidean geometries stand logically or fall together. But our thoughts on fluid analogies also have to sort out this picture

  39. Great its a new dimension of beauty & black holes. So pretty. Like this post

  40. This reminds me of the old proof that there could only be seven heavenly bodies in the solar system, because humans have only seven holes in their heads. (The seven holes were two eyes, two nostrils, two ears and one mouth. The seven heavenly bodies were the Moon, Earth, Mercury, Venus, Mars, Jupiter and Saturn. Knowing this I've always found the bad jokes about the name Uranus rather ironic as it was the 8th planet discovered.)

    P.S. The bathing suit genus thing was pretty clever.

  41. Everything is about scale; if we considers only the atomic nuclei, there are only holes, so many holes that it is easy to project himself in 2D without any nuclei intersection.

  42. For the homework problem, do we need to consider the holes in the lace, or just the large holes?



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