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**gurthbruins****Member**- Registered: 2010-05-09
- Posts: 157

A perfectly spherical egg is resting in a perfectly hemispherical egg cup of the same radius.

So that every point on the surface of the cup is touching (mated to) a point on the egg.

The egg is now removed and can be rolled around at random or by design.

The egg is now replaced in the cup. Regardless of how this is done, there will be at least one point on the cup that is touching the same point on the egg as originally.

*It's the activity of the intelligence above all that gives charm to existence.*

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**namealreadychosen****Member**- Registered: 2011-07-23
- Posts: 16

What if the egg is exactly turned upside down so that every point which was touching the egg before is not touching it any more? It is necessary to include the upper boundary of the egg cup as part of the egg cup for Gurth's theorem to always hold.

This is basically a case of the theorem that every mapping leaves at least one point unchanged.

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**gurthbruins****Member**- Registered: 2010-05-09
- Posts: 157

namealreadychosen wrote:

What if the egg is exactly turned upside down so that every point which was touching the egg before is not touching it any more? It is necessary to include the upper boundary of the egg cup as part of the egg cup for Gurth's theorem to always hold.

This is basically a case of the theorem that every mapping leaves at least one point unchanged.

- Agreed.

*It's the activity of the intelligence above all that gives charm to existence.*

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