Math Conjectures

Optimal Sphere Packing in Dimension Five

OPENMajorConjectureProposed c. 1900 · Canonical special case

Canonical statement

For a packing P\mathcal P of congruent closed balls in R5\mathbb R^5 with disjoint interiors, define its upper asymptotic density by
ρ‾(P)=lim sup⁡R→∞sup⁡x∈R5vol⁡5(B(x,R)∩⋃B∈PB)vol⁡5(B(x,R)), \overline\rho(\mathcal P)=\limsup_{R\to\infty}\sup_{x\in\mathbb R^5} \frac{\operatorname{vol}_5\bigl(B(x,R)\cap \bigcup_{B\in\mathcal P}B\bigr)}{\operatorname{vol}_5(B(x,R))},
where B(x,R)B(x,R) is the radius-RR ball centered at xx. Let Δ5\Delta_5 be the supremum of ρ‾(P)\overline\rho(\mathcal P) over all such packings. Then
Δ5=π2152, \Delta_5=\frac{\pi^2}{15\sqrt2},
the density attained by the D5D_5 root-lattice packing.
View source LaTeX
For a packing \(\mathcal P\) of congruent closed balls in
\(\mathbb R^5\) with disjoint interiors, define its upper asymptotic
density by
\[
  \overline\rho(\mathcal P)=\limsup_{R\to\infty}\sup_{x\in\mathbb R^5}
  \frac{\operatorname{vol}_5\bigl(B(x,R)\cap
  \bigcup_{B\in\mathcal P}B\bigr)}{\operatorname{vol}_5(B(x,R))},
\]
where \(B(x,R)\) is the radius-\(R\) ball centered at \(x\). Let
\(\Delta_5\) be the supremum of \(\overline\rho(\mathcal P)\) over all
such packings. Then
\[
  \Delta_5=\frac{\pi^2}{15\sqrt2},
\]
the density attained by the \(D_5\) root-lattice packing.
DG-013:[ConwaySloane1999Packings][CohnElkies2003Bounds][CohnRajagopal2026Five]
Discrete geometryGeometry and topology

Notes

What fraction of R5\mathbb R^5 can be filled by congruent nonoverlapping balls? The sphere-packing problem in general dimension took shape around 1900, and dimension five is the canonical open case treated here. The conjecture asserts that the maximal density is π2/(152)≈0.4653\pi^2/(15\sqrt2)\approx 0.4653, the density of the packing associated with the root lattice D5D_5, the five-dimensional checkerboard lattice [ConwaySloane1999Packings].

The problem is solved in dimensions one through three, and the linear-programming method of Cohn and Elkies [CohnElkies2003Bounds] led to the celebrated sharp solutions in dimensions eight and twenty-four. In dimension five, however, the known analytic upper bounds fall short of the conjectured value: D5D_5 — together with certain nonlattice packings of exactly the same density — remains the best construction known, while the bounds from above leave a definite gap. Recent work of Cohn and Rajagopal studies variations on five-dimensional sphere packings [CohnRajagopal2026Five].

The conjecture is open; a resolution requires an upper-bound method that is sharp at π2/(152)\pi^2/(15\sqrt2), something no current technique achieves.

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.