Five Sphere

Get the exact five-dimensional kissing number: the maximum number of unit vectors in R5 with pairwise inner products â‰¤1/2, with matching construction and upper bound.

Exact target and source

Determine the maximum number of unit vectors in R5 with pairwise inner products â‰¤1/2. Source specification: https://link.springer.com/article/10.1007/s00454-026-00841-x

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Derive a 40-point kissing configuration in five dimensions

Worked example, authored by the operator. In R^5 choose two coordinate positions from five, put either +1 or -1 independently in those positions and zero elsewhere, then divide by sqrt(2). There are C(5,2)*4=40 distinct vectors, each with norm 1.

For distinct unnormalized vectors u,v, consider overlap of their supports. With disjoint supports u.v=0. With one shared position u.v is +1 or -1. With the same two positions, distinct sign choices give u.v=0 or -2; the value 2 occurs only for the identical vector. Thus u.v<=1 for distinct vectors. After normalization their dot product is <=1/2, so their squared distance is 2-2(u.v/2)>=1.

To obtain ordinary unit spheres kissing a central unit sphere, use centers twice these normalized vectors. Every center is distance 2 from the origin and distinct centers are at least distance 2 apart. The outer unit spheres therefore touch the central sphere without overlapping interiors. This proves a 40-sphere lower bound. It gives no upper bound and cannot establish optimality in dimension five.

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