The hairy ball theorem

http://en.wikipedia.org/wiki/Hairy_ball_theorem

by acangiano • 17 years ago

63 36 17 years ago

en.wikipedia.org

neat, this is very important in classical mechanics. Orbits of your system trace out tori in the phase space, the cool thing is how the tori break up as you perturb your system from integrable to non-integrable. Chaos, without dissipation.

davi• 17 years ago

more/links?

Retric2• 17 years ago

I just added this to the comment about the weather:

"This is not strictly true as the air above the earth has multiple layers, but for each layer their must be a point with zero horizontal windspeed."

Feel free to edit it.

graphene• 17 years ago

I think it would be better to delete the eintire paragraph above your addition, as the notion that this theorem guarantees at least one cyclone somewhere on earth is ridiculous.

byrneseyeview3• 17 years ago

Is it wrong because the earth is not a perfect sphere, or for some other reason?

russell• 17 years ago

> Note that the eye can be arbitrarily large or small and the magnitude of the wind surrounding it is irrelevant.

Given that the eye can be arbitrarily small, I would accept the statement that the earth would always have at least one circular wind pattern. The paragraph is not talking about just hurricane size.

pixcavator3• 17 years ago

Considering multiple layers doesn’t provide more insight here. The atmosphere form one "thick" layer around the Earth and the air moves within this layer. Assuming that this layer is closed (does not gradually comes to nothing), what we have is a "thick" sphere. Its Euler characteristic is still 2, thick or not. Now you can think of the wind as a continuous transformation of this solid. This map is homotopic to the identity map and, therefore, has Lefschetz number equal to the Euler characteristic, i.e., 2. Hence there is a fixed point. With some extra work you can show that this implies that the vector field has a stationary point. At that point the wind speed is 0, and not just the horizontal component. That point, certainly, does not have to be a cyclone. However, there is nothing wrong with the conclusion about the weather presented in the article; it’s just more idealized than this.

Retric2• 17 years ago

The point about layers is for each vertical cm around the earth you need to have that point with zero air speed. Same deal for any width you feel like talking about. You also need to have a line that connects all those points from earth to space. However, that line's path need not be strictly vertical it could wrap around the earth 3 times before reaching space.

However, "In meteorology, a cyclone refers to an area of closed, circular fluid motion rotating in the same direction as the Earth[1][2]." which is not nessisarily true.

IsaacSchlueter• 17 years ago

Hahahah...

Hairy balls.

samlittlewood2• 17 years ago

Thus the popularity of unit quaternions in 3D simulation and graphics. They cover the sphere twice, so each rotation can be represented by two possible values.

My hack minds eye view of this is that you take two combed layers with nulls at the poles. The rotation then jumps between layers as it crosses the poles.

zkz2• 17 years ago

Could you explain in more detail? Why do people in 3d simulations and graphics care about the existence of a point where the function value is 0?

samlittlewood2• 17 years ago

They want a rotation representation that:

1) can be accumulated (quaternion multiplication) and interpolated (slerp) 'smoothly',

2) and in a way that does not depend on the absolute orientation.

- Smooth:

If you dive in and start using 3 angles (yaw,pitch,roll) to represent 3d rotation, then the rotations are like the hairy ball - however you order them, there will be values where you can't easily step to another physically very close rotation without a huge change in values - the discontinuities in the hairy ball.

This is classically called 'gimbal lock', where a gimbal mount is a physical manifestation of angle representation. In simulations, it might show itself as strange behaviour when pitched up/down close to 90deg, or an interpolation between apparently close rotations going wild.

By using unit quaternions (4 values), and avoiding the discontinuties (but now having two representations for each physical rotation) it becomes easy to acuumulate the rotations that come from a flight model, animation, user etc. without out ever getting stuck in a corner.

- Independant of absolute rotation:

A related point is that an incremental rotation by multiplying quaternions will always produce the same relative rotation irrespective of absolute attitude. You can see angle based represetnations going wrong when, for example, a camera control system offers buttons that are suposed to be left/right/up/down relative to the camera, but produce varying results as the camera rotates.

It's possible to do the same things with 3x3 matrices - but it is a bit more tricky to keep them normalised over time, and obviously takes more storage. A typical solution to all the above angle problems in a simulation would be to convert (yaw, pitch, roll) to a matrix, accumulate, then convert back. My brief experience of aerodynamics texts is that all the equations are obfuscated with unpicking and rebuilding angles. It's a bit like doing arithmentic with Roman numerals - possible, but painful.

Dilpil• 17 years ago

Any other interesting applications of topology to physical phenomenon? I can only seem to find applications to other areas of math.

beaker• 17 years ago

The theorem lacks a certain aesthetic beauty. Apply Occam's Razor to create a smoother, more palatable package.

The hairy ball theorem

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