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You or someone you know may have struggled to get a cowlick to just stay down already, but you can take solace in the fact that these inconvenient hair tufts have a lot to teach us about the world around us.

Hosted by: Michael Aranda

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Sources:
https://www.math.hmc.edu/funfacts/ffiles/20005.7.shtml
https://people.ucsc.edu/~lewis/Math208/hairyball.pdf
“Concepts of Modern Mathematics” textbook https://books.google.com/books?id=4WPDAgAAQBAJ&lpg=PA156&ots=7o3UdW_Mtl&dq=%22now%20topologically%20a%20dog%20is%20a%20sphere%22&pg=PP1#v=onepage&q=a%20dog%20is%20a%20sphere&f=false
https://www.sciencedirect.com/science/article/pii/S2211379715000716
https://blogs.scientificamerican.com/roots-of-unity/my-head-is-not-a-hairy-ball/
https://settlement.arc.nasa.gov/70sArtHiRes/70sArt/art.html
http://assets.press.princeton.edu/chapters/i8722.pdf
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Images:
https://commons.wikimedia.org/wiki/File:Menschliche_doppelte_Haarwirbel.JPG
https://settlement.arc.nasa.gov/70sArtHiRes/70sArt/Torus_Model_AC76-0492.1_5696.jpg
[INTRO ♪].

Cowlicks can feel like some of the most inconvenient things in the world of hair, and they always seem to pop up at the worst times, too. Whether it’s picture day or prom night, there’s a good chance somebody out there is freaking out about that one lock that just…won’t… stay… down.

But, hey, it’s okay. If that’s you, you’re going to be fine, because mathematically speaking, your cowlick is totally fixable. See, there’s a classic theorem in topology called—no joke—Hairy Ball Theorem.

It says that if a ball is covered in hair, the hair can’t be combed smoothly without at least one bald spot or cowlick. That’s good news for you, because your head isn’t totally covered in hair. So, according to math, you should be able to smooth everything down somehow.

But… why does this matter? Well, this theorem might sound weird, but it’s something mathematicians have spent time investigating. Because here’s the thing: Hairy Ball Theorem might have a strange name, but it can help scientists in every field from meteorology to space engineering.

You can think about this theorem in terms of bald spots and cowlicks, but mathematicians usually phrase it differently. According to Harvey Mudd College, one definition is this: “Any continuous tangent vector field on a sphere must have a point where the vector is zero.” To unpack exactly what this means, let’s use one pretty weird example of this theorem: a creepy spherical dog from the textbook Concepts of Modern Mathematics. When you look at this dog, you might just see a ball of hair.

But mathematicians see that so-called “continuous tangent vector field”. A vector is a value with direction and magnitude, like an arrow, or one piece of dog hair. And since it’s tangent, that means the hair touches the dog at only one point: the one it grew out of.

Theoretically, you could have a tangent vector field where all of the hairs stick out at super weird angles. But since this field is continuous, that means each hair has a direction that’s barely different from the hairs surrounding it, as if they’d all been brushed in one smooth motion. Except, that is, for one vector.

According to Hairy Ball Theorem, the dog has to have one point that’s either a bald spot with no vector at all, or a cowlick—a point where the vector doesn’t fit in the continuous field. The reason why is pretty technical, but it means that, no matter how hard you try, you can’t get a perfectly groomed spherical dog. Knowing this isn’t just some strange trivia fact, either.

Once you move past the weird dog example, the idea is important to all kinds of fields from nanotechnology to computer graphics. Like, instead of hair, the vectors could be the light bouncing off a reflective sphere. Somewhere on that sphere is a point where the light reflects directly back to its source, so there’s one point where your reflection isn’t distorted.

Wind is also a vector, so by Hairy Ball Theorem, as long as there’s wind on the Earth, there must also be somewhere without wind. That could be a calm day, or the center of a cyclone. Hairy Ball Theorem tells us about shapes other than spheres, too.

For example, donut-shaped toroids are one shape that can have smooth ‘dos. A furry donut could have its fur combed continuously, or—maybe more importantly—a toroidal space station potentially could have a steady air flow. At the end of the day, Hairy Ball Theorem is a classic example of how math influences things we deal with every day.

It might not be able to help you tame your cowlick right this second, but, hey… maybe that’s what hats are for. Thanks for watching this episode of SciShow, and special thanks to our channel members for your help in making it happen! Becoming a channel member is one way you can support the show and help us make more content like this.

As our way of saying thank-you, you’ll get a sweet badge by your name that gets fancier the longer you stay a member, along with things like members-only posts in the community tab. This shoutout is actually brought to you by one of our channel members, Shadow Marie Weatherholt. So thanks for your support!

If you want to join Shadow Marie, you can become a channel member by clicking on the button that says “Join” below this video. [OUTRO ♪].