Everything about The Hairy Ball Theorem totally explained
The
hairy ball theorem of
algebraic topology states that there's no nonvanishing
continuous tangent vector field on the sphere. If
f is a continuous function that assigns a
vector in
R3 to every point
p on a sphere such that
f(
p) is always
tangent to the sphere at
p, then there's at least one
p such that
f(
p) =
0.
This is famously stated as "
you can't comb a hairy ball flat", or sometimes, "
you can't comb the hair on a billiard ball". It was first proved in 1912 by
Brouwer, see
(External Link
).
In fact from a more advanced point of view it can be shown that the sum at the zeros of such a vector field of a certain 'index' must be 2, the
Euler characteristic of the 2-sphere; and that therefore there must be at least some zero. This is a consequence of the
Poincaré–Hopf theorem. In the case of the 2-
torus, the Euler characteristic is 0; and it
is possible to 'comb a hairy donut flat'. In this regard, it follows that for any
compact regular 2-dimensional
manifold with non-zero Euler characteristic, any continuous tangent vector field has at least one zero.
Cyclone consequences
A curious meteorological application of this theorem involves considering the wind as a vector defined at every point continuously over the surface of a planet with an atmosphere. As an idealisation, take wind to be a two-dimensional vector: suppose that relative to the planetary diameter of the Earth, its vertical (for example, non-tangential) motion is negligible.
One scenario, in which there's absolutely no wind (air movement), corresponds to a field of zero-vectors. This scenario is uninteresting from the point of view of this theorem, and physically unrealistic (there will always be wind). In the case where there's at least some wind, the Hairy Ball Theorem dictates that there must be at least one point on a planet at all times with no wind at all. This corresponds to the above statement that there will always be
p such that
f(
p) =
0.
In a physical sense, this zero-wind point will be the eye of a cyclone or anticyclone. (Like the swirled hairs on the tennis ball, the wind will spiral around this zero-wind point - under our assumptions it can't flow into or out of the point.) In brief, then, the Hairy Ball Theorem dictates that, given at least some wind on Earth, there must at all times be a
cyclone somewhere. Note that the eye can be arbitrarily large or small and the magnitude of the wind surrounding it's irrelevant.
Application to computer graphics
A common problem in computer graphics is to generate a non-zero vector that's orthogonal to a given one. There is no continuous function that can do this. If the given vector is considered to be the radius vector of a sphere, then it's clear that this is an immediate corollary of the hairy ball theorem.
Lefschetz connection
There is a closely-related argument from
algebraic topology, using the
Lefschetz fixed point theorem. Since the
Betti numbers of a 2-sphere are 1, 0, 1, 0, 0, ... the
Lefschetz number (total trace on
homology) of the
identity mapping is 2. By integrating a
vector field we get (at least a small part of) a
one-parameter group of
diffeomorphisms on the sphere; and all of the mappings in it are
homotopic to the identity. Therefore they all have Lefschetz number 2, also. Hence they've fixed points (since the Lefschetz number is nonzero). Some more work would be needed to show that this implies there must actually be a zero of the vector field. It does suggest the correct statement of the more general
Poincaré-Hopf index theorem.
Corollary
A consequence of the hairy ball theorem is that any continuous
function that maps a sphere into itself has either a
fixed point or a point that maps onto its own
antipodal point. This can be seen by transforming the function into a tangential vector field as follows.
Let
s be the function mapping the sphere to itself, and let
v be the tangential vector function to be constructed. For each point
p, construct the
stereographic projection of
s(
p) with
p as the point of tangency. Then
v(
p) is the displacement vector of this projected point relative to
p. According to the hairy ball theorem, there's a
p such that
v(
p) =
0, so that
s(
p) =
p.
This argument breaks down only if there exists a point
p for which
s(
p) is the antipodal point of
p, since such a point is the only one that can't be stereographically projected onto the tangent plane of
p.
Higher dimensions
The connection with the
Euler characteristic χ suggests the correct generalisation: the 2
n-
sphere has no non-vanishing vector field for
n ≥ 1. The difference in even and odd dimension is that the
Betti numbers of the
m-sphere are 0 except in dimensions 0 and
m. Therefore their
alternating sum χ is 2 for
m even, and 0 for
m odd.
Further Information
Get more info on 'Hairy Ball Theorem'.
|
External Link Exchanges
Do you know how hard it is to get a link from a large encyclopaedia? Well we're different and will prove it. To get a link from us just add the following HTML to your site on a relevant page:
<a href="http://hairy_ball_theorem.totallyexplained.com">Hairy ball theorem Totally Explained</a>
Then simply click through this link from your web page. Our crawlers will verify your link, extract the title of your web page and instantly add a link back to it. If you like you can remove the words Totally Explained and embed the link in article text.
As long as your link remains in place, we'll keep our link to you right here. Please play fair - our crawlers are watching. Your site must be closely related to this one's topic. Any kind of spamming, dubious practises or removing the link will result in your link from us being dropped and, potentially, your whole site being banned. |
