science

38 posts

The Hairy Ball Theorem

If you have the type of not too short hair that completely succumbs to gravity, you will have a cowlick somewhere on your head.** It’s a direct consequence of a mathematical theorem called the hairy ball theorem. Yes, it’s called that and it’s a bona fide mathematical theorem with a real but difficult proof that requires you to get a fancy-schmancy degree in math to really understand it.

cowlick
Double cowlick, what does it mean

But if we just want to feel like we understand it, we can certainly do that. It’s the Internet, after all.

Let’s comb through some informal ways of describing this theorem. One is

  • You can’t comb a hairy ball flat without creating a cowlick.

or closely related

  • You can’t comb the hair on a coconut.

A perhaps more useful, less frivolous description of the theorem is:

  • A cyclonic system must always exist on a planet with an atmosphere.

In other words, somewhere on Earth, at any given moment, there will be a cyclone (in the most general sense).*** I should note that, in certain ways, this is a supremely useless theorem because it doesn’t tell you where the hell the cyclone is. But it is 100% absolutely sure that there is one.

The reason that people built analogies with hair and wind is that the “hairs” on the ball correspond to what are called vector fields for systems that have stuff that moves like a fluid, like wind or piss streaming over someone’s hairy ball. The hair is mostly a visualization tool for particles that move continuously in a closed environment, like the surface of a sphere.

Imagine you could track some representative particles in a fluid. Say, particularly big balls of dust are floating around. Or fireflies are caught in a dust devil at night. And you snap a picture of this with not too short exposure. The balls of dust or the fireflies would come out as blurry streaks that move in thrall to the air. The blurry streaks would correspond approximately to vector fields. So a fairly accurate way of looking at vector fields is to understand it as a concise description of the motion of particles at every single point at a given moment in whatever system you are looking at.

And what do cowlicks and cyclones have to do with this? Basically, the CENTER of the cowlicks/cyclones are points where there is no motion. It’s the eye of the storm you always hear about where everything is calm. These are called the zeros of the vector fields. So if you were to drop something lightweight at the exact center of where the cyclone occurs, that thing you dropped wouldn’t move because there’s nothing moving around it.****

If you’re looking at a stream or a river, you see eddies, the watery analog of cyclones, all the time. But would you still see these if the water were flowing in a completely smooth, straight, artificial indoor stream? Probably not. But now let’s say there’s a sphere and there’s water coursing over the surface. And imagine that it’s completely self-contained. No water drips from it and no water drops on to it. The kicker in this hypothetical situation is this: there will be at least one eddy (i.e. a “cyclone” in our previous parlance or a “zero” in our more technical one) no matter how smooth we make the surface of the sphere. The mathematics of the hairy ball theorem guarantees this.

If you’re curious how the hairy ball theorem is stated in technical terms, it goes like this:

  • There does not exist an everywhere nonzero tangent vector field on the 2-sphere.

**A technical caveat. This doesn’t apply if you’re wearing your hair slicked back or in a ponytail or in some way that makes it defy gravity. It’s mostly applicable if you’ve let your hair loose so that it looks like it’s cascading down your head.

***This ties in with something called the Poincare-Hopf theorem which is even more general than the hairy ball theorem. Because a cyclone is a “cowlick” (i.e. not just any zero, but a special type of zero called a sink of a vector field) with index one, there must be a second cyclone. This more general theorem predicts not one but two cyclones.

****Not in reality, of course. It’s a thought experiment, so bear with me.

Music is my hot hot sex? (And other neuroscientific hyperbole)

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Music is highly valued across all human societies. The specific sounds vary widely, even within cultures. My mom loves to listen to Gregorian chanting; I’d rather be involved in an automobile accident, but I do love The Strokes. The concept is the same, though: an abstract stimulus invokes a pleasurable response.

So is music a drug? New research published last month in Nature Neuroscience(1) indicates a strong similarity. Drugs, sex, and eating have long been known to produce pleasure by releasing dopamine in the mesolimbic system, commonly thought of as the “reward pathway.” These are tangible stimuli that promote (or, our bodies think they promote) survival. Music, though, is abstract. It feels good to listen, but it doesn’t provide us with safety, nutrition, or reproduction.

To test what neural pathways are engaged during musical enjoyment, researchers first had their test group select their favorite music. A “musical frisson” test was used to identify moments of peak pleasure – if you get “chills” on hearing “DON’T STOP – BELIEVING,” you’re actually experiencing a measurable physiological response.

Brain activity was then measured as subjects listened to their chosen piece. Two types of brain imaging were used: PET scans were used for their precision over time, while fMRIs were used for their neural precision. The results were combined to accurately determine exactly what the brain was doing at what point in the listening experience.

Two different neurochemical responses were discovered. In the anticipatory stage – leading up to the subject’s favorite part of the song – dopamine was released in the dorsal striatum. In previous studies(2), the dorsal striatum has been linked to learning and action selection. Researchers at UPenn have specifically linked it to cocaine cravings in addicts.

During the peak pleasure stages (measured by the musical frission response), dopamine was released in the ventral striatum. A previous study(3) strongly linked the ventral striatum to sensations of euphoria associated with amphetamine use in non-addicted individuals .

This biological parallel between drug use and listening to music makes sense, especially if you’ve ever tried to turn off someone’s favorite song “right at the good part.” Only an addict could respond so violently to a 60-second pleasure delay.

1. Salimpoor, V. N., Benovoy, M., Et. al. (2011, January 9). Anatomically distinct dopamine release during anticipation and experience of peak emotion to music [Electronic version]. Nature Neuroscience.

2. Volkow, N. D., Wang, G., Et. al. (2006, June 14). Cocaine cues and dopamine in dorsal striatum: Mechanism of craving in cocaine addiction [Electronic version]. The Journal of Neuroscience, 26(24), 6583-6588.

3. Drevets, W. C., Gautier, C., & Et. al. (2001). Amphetamine-induced dopamine release in human ventral striatum correlates with euphoria. Biol Psychiatry, 81-96.

Life Science

26 comments

Whenever I tell people I’m interested in biological research, I nearly always hear, “Oh, are you going to cure cancer?” There is a winking intonation in this question. We won’t hold you to it, they say with their friendly smiles. I either respond with my easy refrain – “I’m more interested in function than health” – or a laugh and a shrug that drips with faux-modesty. That mantra ignores, of course, that understanding of function leads to technology.

These days, the vast majority of biological research is designed to create product. A cancer drug, an anti-aging technique, a soybean resistant to disease. Why? Because research is expensive – wildly so. No corporation will waste billions on research without an eye toward application. It reminds me of a creation of Margaret Atwood: Crake, a genius whose pragmatism is beyond the reach of empathy towards individuals. “Grief in the face of inevitable death. The wish to stop time. The human condition.” This is what biotech sells. Only the rich can afford it. The rest of humanity is left praying for a trickle down, decades after initial production – AIDS drugs, vaccinations. Voodoo medicine.

“Oh, are you going to cure cancer?” When I was in high school – long before I had much in the way of scientific predilection – I volunteered quite a bit. I spent some time with an organization that worked with pediatric oncology patients and their families. I played with kids before and during their chemo – puzzles, block games, cards. Exhausted parents would thank me as other volunteers whisked them away to ask how they were holding up.

While with the kids, I usually forgot everything, but every once in a while something would jolt me back. A needle bruise on a tiny arm, tear tracks on a mother’s face, a spot of blood from a bloody nose on a pair of light-up sneakers that could fit in my hand. The phone calls telling me my scheduled visit would no longer be needed. I would leave and sit in my car in the parking lot, slumped with exhaustion. I still remember the numbing tingle on my hands from where the steering wheel stitching dug in.

“Oh, are you going to cure cancer?”

More gamez

7 comments

Why is the sky blue?

The gases in the atmosphere scatter blue light most efficiently. Read more here.

Why is grass green?

Chorophyll absorbs blue, red light, reflects green; light —> energy. Read more here.

OR MAYBE YOU’RE JUST NOT SMART ENOUGH. Fine, black holes.

Star implodes. So much gravity nothing escapes; sucks together space-time. Read more here.

Also, fuck you.

String theory. Go.

Not even string theorists know what they’re talking about.

Submit a science question to me either in the comments, to my Gawker account, or to my email. I will answer it in ten words or less. If you ask me a question that neither I nor anybody else has the answer to, I will pretend I never received the question.

Game time!

10 comments

How was the universe…created? Do we even know that?

Short version:
THEORY: Not created, formed. Space/everything began, grew. Before? *shrug*

Long: Read more here.

I was always terrible at physics. Why is [light] both a wave and particles?

Short version:
Light reflects/refracts like waves, has energy/momentum like particles.

Long: Read more here.

Submit a science question to me either in the comments, to my Gawker account, or to my email. I will answer it in ten words or less. If you ask me a question that neither I nor anybody else has the answer to, I will pretend I never received the question.

Just kidding. If any of you can answer this, there’s a Nobel waiting for you. Is the multiverse real?