The Lenz Configuration in Four Dimensions

Two circles of radius $1/\\sqrt2$ lying in orthogonal planes of $\\mathbb{R}^4$ and centred at the origin have a striking property: every point of one is at distance exactly $1$ from every point of the other. The reason is that the two points have disjoint coordinate supports, so the squared distance is the sum of two squared radii. Splitting $n$ points evenly between the two circles therefore realises $\\lfloor n^2/4 \\rfloor$ unit distances, and the unit-distance graph contains a complete bipartite graph. This settles the order of the unit-distance problem from dimension four onward, in sharp contrast with dimensions two and three, where the exponent is still unknown.