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A sphere with radius 4/pi = 0.6366 cm will fit the bill - after a fashion.

Four points, equidistant on the equator, along with a point at each pole.

Of the 15 possible pairs, all will be 1 cm apart along the great circle apart from the 3 pairs of diametrically opposite points, which will be 2 cm apart.

There is no way in which 6 points can be marked on any sphere so that each of the 15 pairs is the same distance apart.

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Q: How big is a sphere with 6 surface points all 1 cm apart?
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