Pytkeev space

String girdling Earth is a mathematical puzzle with a counterintuitive solution. String is wrapped around the equator of a perfectly spherical Earth. This string is cut and a piece 1 metre in length is added in. The string is now rearranged so that it is at an even height above the equator. The question that is then posed is whether the gap between string and Earth will allow the passage of a car, a cat or a thin knifeblade.

Solution

Template:Hidden begin Considering that 1 metre is almost negligible compared with the 40 000 km circumference, the first response will be that the new position of the string will be no different from the original surface-hugging position. Surprisingly, the answer is that a cat will easily pass through the gap, the size of which will be $\frac{1}{2\pi }$ metres or about 16cm. Even more surprising is that the size of the sphere or circle around which the string is spanned, is irrelevant, and may be anything from the size of an atom to the Milky Way - the result will remain about 16cm.

A corollary is that an athletics track, whether the circumference of the stadium is the standard 400m or the size of the Milky Way, would have the same offset between starting lines on each lane, equal to $2\pi$ the width of the lane. Template:Hidden end

References

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