Math Conjectures

Five-Dimensional Kissing Number

OPENMajorExact constant problemProposed Unknown · Canonical special case

Canonical statement

Let S4={x∈R5:∥x∥2=1}S^4=\{x\in\mathbb R^5:\lVert x\rVert_2=1\}. Determine
τ5=max⁡{∣X∣:X⊂S4, ⟨x,y⟩≤12 for all distinct x,y∈X}. \tau_5=\max\bigl\{|X|:X\subset S^4,\ \langle x,y\rangle\leq\tfrac12 \text{ for all distinct }x,y\in X\bigr\}.
Equivalently, τ5\tau_5 is the maximum number of nonoverlapping unit 55-balls that can simultaneously touch a fixed unit 55-ball.
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Let
\(S^4=\{x\in\mathbb R^5:\lVert x\rVert_2=1\}\). Determine
\[
  \tau_5=\max\bigl\{|X|:X\subset S^4,\
    \langle x,y\rangle\leq\tfrac12
    \text{ for all distinct }x,y\in X\bigr\}.
\]
Equivalently, \(\tau_5\) is the maximum number of nonoverlapping unit
\(5\)-balls that can simultaneously touch a fixed unit \(5\)-ball.
DG-012:[Musin2008Kissing][deLaatOliveiraVallentin2014][CohnRajagopal2026Five]
Discrete geometryInformation theory

Notes

The kissing number τd\tau_d is the maximum number of nonoverlapping unit balls in Rd\mathbb R^d that can simultaneously touch a fixed central unit ball; equivalently, it is the largest number of points on the unit sphere with pairwise angular distance at least 60∘60^\circ. Dimension five is the smallest dimension in which this classical quantity is unknown; it is a canonical open case of a long-standing problem rather than a conjecture with a single date of origin.

The values τ1=2\tau_1=2, τ2=6\tau_2=6 and τ3=12\tau_3=12 are classical, τ4=24\tau_4=24 was proved by Musin [Musin2008Kissing], and the answers in dimensions 88 and 2424 are also known. In dimension five the D5D_5 root system yields 4040 touching balls, and this configuration is widely expected to be optimal. On the other side, semidefinite-programming bounds, which strengthen the classical linear-programming approach to packing problems [deLaatOliveiraVallentin2014], give τ5≤44\tau_5\le 44, so at present 40≤τ5≤4440\le\tau_5\le44. Recent work of Cohn and Rajagopal studies variations on five-dimensional sphere packings [CohnRajagopal2026Five].

The problem remains open: settling it requires either a configuration of more than 4040 spheres or an upper bound below 4141.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.