21 examples
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- In other words, whenever one attempts to comb a hairy ball flat, there will always be at least one tuft of hair at one point on the ball.
- However, the hairy ball theorem says there exists no continuous function that can do this for every point on the sphere (i.e. every given vector).
- Then, turning his face towards the land, he scurried away over the ice like a hunted partridge, or a hairy ball driven before an Arctic breeze. Cited from Red Rooney, by R.M. Ballantyne
- It handles vertices at tangent-space discontinuities by making duplicates, thus solving the hairy ball problem.
- A consequence of the hairy ball theorem is that any continuous function that maps an even-dimensional sphere into itself has either a fixed point or a point that maps onto its own antipodal point.
- In 1912 he proved the hairy ball theorem for the two-dimensional sphere, as well as the fact that every continuous map from the two-dimensional ball to itself has a fixed point.
- The hairy ball theorem of algebraic topology states that there is no nonvanishing continuous tangent vector field on even-dimensional n-spheres.
- In the case where there is at least some wind, the Hairy Ball Theorem dictates that at all times there must be at least one point on a planet with no wind at all and therefore a tuft.
- Similarly, the hairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a cowlick."
- He came up again at the edge of a shallow riffle over which the water ran like the rapids at Niagara in miniature, and for fifty or sixty yards he was flung along like a hairy ball. Cited from Baree, Son of Kazan, James Oliver Curwood
- Attached to the gray worsted stocking which covered his fleshless calf was a fluffy black hairy ball, with one little red eye glancing up, and the gleam of two white teeth where it held its grip. Cited from The Refugees, by Arthur Conan Doyle
- For an ordinary sphere in three dimension space it can be shown that the index of any vector field on the sphere must be two, this leads to the hairy ball theorem which shows that every such vector field must have a zero.
- This is a corollary of the hairy ball theorem.
- In mathematics, the discussion of vector fields on spheres was a classical problem of differential topology, beginning with the hairy ball theorem, and early work on the classification of division algebras.
- The most well-known example is the hairy ball theorem, where the Euler class is the obstruction to the tangent bundle of the 2-sphere having a nowhere vanishing section.
- If I could only dine off that fox I saw a fortnight ago, curled up into a delicious hairy ball, I should ask nothing better; I would have eaten her then, but unluckily her husband was lying beside her, and one knows that foxes, great and small, run like the wind. Cited from The Orange Fairy Book, Andrew Lang, Ed.
- Tangent bundles are not, in general, trivial bundles: for example, the tangent bundle of the sphere is non-trivial by the hairy ball theorem.
- However the hairy ball theorem shows that a continuous vector field tangent to the surface of a sphere must fall to zero at one or more points on the sphere, which is inconsistent with the assumption of an isotropic radiator with linear polarization.
- Regardless of the particular paradigm, the intuitive meaning of the Chern class concerns 'required zeroes' of a section of a vector bundle: for example the theorem saying one can't comb a hairy ball flat (hairy ball theorem).
- In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem and the Borsuk-Ulam theorem.