Kissing number in five dimensions

Six equal disks can touch a central disk in a flat hexagonal arrangement without overlapping. In five dimensions, forty equal balls can surround one central ball. Can more fit under the same touching and nonoverlap rules?[1] [2]

How many equal balls can touch one central ball in five dimensions while their interiors stay separate? Forty are possible, but the cited bounds leave room for more.

For unit-radius balls, outer centers have distance two from the central center and at least two from each other. The triangle between two outer centers and the central center therefore has an angle of at least sixty degrees: the directions have this minimum angular separation. Rescaling the directions gives a spherical code on $S^4$. One forty-direction construction uses all vectors $(\pm1,\pm1,0,0,0)$ with the two nonzero coordinates in any positions, then rescales them.[1] [2]

The maximum lies between forty and forty-four.[2]

The cited July 2026 treatment states $40\leq\tau_5\leq44$. Its new forty-point configuration changes the contact geometry; it does not prove that forty is maximal or raise the achievable number.[2]

The problem compares local crowding with the ways contact directions can be organized. Equal contact counts need not mean equal geometry.[1] [2]

Contact directions link the touching-ball question to separation of points on a sphere. Distinct forty-point configurations show why a contact count alone loses geometric information; local success also need not determine global packing density.[1] [2]