Sphere packing density in $\mathbf{R}^4$

Description of constant

$C_{36}=\Delta_4$ is the (optimal) sphere packing density in $\mathbb{R}^4$, i.e. the largest fraction of $\mathbb{R}^4$ that can be covered by congruent balls with disjoint interiors. [CE2003-pack-problem] [CE2003-def-density] [CE2003-greatest-density]

More precisely, for a packing $\mathcal{P}$ in $\mathbb{R}^4$, let $P$ denote the union of all balls in the packing, and let $B(p,R)$ denote a (Euclidean) ball of radius $R$ centered at $p$. The (upper) density of $\mathcal{P}$ is

\[\Delta(\mathcal{P}) := \limsup_{R\to\infty}\sup_{p\in\mathbb{R}^4}\frac{\operatorname{vol}(P\cap B(p,R))}{\operatorname{vol}(B(p,R))}.\]

[CE2003-upper-density]

Then the sphere packing density in $\mathbb{R}^4$ is

\[\Delta_4:=\sup_{\mathcal{P}\subset\mathbb{R}^4}\Delta(\mathcal{P}),\]

the greatest packing density in $\mathbb{R}^4$.

It is often convenient to work with the center density $\delta_4$, defined (for packings of unit spheres) by

\[\delta_4=\frac{\Delta_4}{\operatorname{vol}(B)},\]

where $B$ is a unit ball in $\mathbb{R}^4$. [dLOV2014-center-density]

Known upper bounds

Bound	Reference	Comments
$1$	Trivial	A packing cannot cover more than all of $\mathbb{R}^4$.
$0.647791\ldots$	CE2003	CE2003 lists the dimension-$4$ Rogers bound on center density $\delta_4\le 0.13127$ (Appendix C, Table 3). Converting gives $\Delta_4\le (\pi^2/2)\cdot 0.13127\approx 0.647791$. [CE2003-appC-table3-d4-row] [dLOV2014-center-density]
$0.647742\ldots$	CE2003	CE2003 lists the dimension-$4$ “New Upper Bound” on center density $\delta_4\le 0.13126$ (Appendix C, Table 3). Converting gives $\Delta_4\le (\pi^2/2)\cdot 0.13126\approx 0.647742$. [CE2003-appC-table3-d4-row] [dLOV2014-center-density]
$0.644421\ldots$	dLOV2014	dLOV2014 lists the dimension-$4$ center density upper bound $\delta_4\le 0.130587$ (Table 1). Converting via $\Delta_4=\operatorname{vol}(B)\,\delta_4$ gives $\Delta_4\le \operatorname{vol}(B)\cdot 0.130587 = (\pi^2/2)\cdot 0.130587\approx 0.644421$. [dLOV2014-table1-d4-row] [dLOV2014-center-density]

Known lower bounds

Bound	Reference	Comments
$0$	Trivial	Since densities are nonnegative.
$\pi^2/16 \approx 0.616850$	dLOV2014	dLOV2014 lists the dimension-$4$ center density lower bound $\delta_4\ge 0.12500$ (Table 1). Converting via $\Delta_4=\operatorname{vol}(B)\,\delta_4$ gives $\Delta_4\ge \operatorname{vol}(B)\cdot 0.12500 = (\pi^2/2)\cdot (1/8)=\pi^2/16\approx 0.616850$. [dLOV2014-table1-d4-row] [dLOV2014-center-density]

Additional comments and links

References

[CE2003] Cohn, Henry; Elkies, Noam. New upper bounds on sphere packings I. Annals of Mathematics 157 (2003), 689–714. DOI: 10.4007/annals.2003.157.689. arXiv PDF: https://arxiv.org/pdf/math/0110009.pdf. Google Scholar.
- [CE2003-pack-problem]
  loc: arXiv PDF p.1, Introduction (opening paragraph).
  quote: “The sphere packing problem asks for the densest packing of spheres into Euclidean space. More precisely, what fraction of $\mathbb{R}^n$ can be covered by congruent balls that do not intersect except along their boundaries?”
- [CE2003-def-density]
  loc: arXiv PDF p.3, Introduction (density definition).
  quote: “The density $\Delta$ of a packing is defined to be the fraction of space covered by the balls in the packing.”
- [CE2003-greatest-density]
  loc: arXiv PDF p.3, Introduction (greatest packing density).
  quote: “One can prove that periodic packings come arbitrarily close to the greatest packing density.”
- [CE2003-upper-density]
  loc: arXiv PDF p.20, Appendix A (upper density definition).
  quote: “every packing has an upper density, defined by $\Delta=\limsup_{r\to\infty}\sup_{p\in\mathbb{R}^n}\ \mathrm{vol}(B(p,r)\cap P)/\mathrm{vol}\,B(p,r)$.”
- [CE2003-appC-table3-d4-row]
  loc: arXiv PDF p.23, Appendix C (Table 3, row “4”).
  quote: “$4\quad 0.125\quad 0.13127\quad 0.13126$.”
[dLOV2014] de Laat, David; de Oliveira Filho, Fernando Mário; Vallentin, Frank. Upper bounds for packings of spheres of several radii. Forum of Mathematics, Sigma 2 (2014). DOI: 10.1017/fms.2014.24. arXiv PDF. Google Scholar.
- [dLOV2014-center-density]
  loc: Journal PDF p.13, Section 1.4 (paragraph before Table 1).
  quote: “the center density of a packing of unit spheres being equal to $\Delta/\mathrm{vol}\,B$, where $\Delta$ is the density of the packing, and $B$ is a unit ball.”
- [dLOV2014-table1-d4-row]
  loc: Journal PDF p.14, Table 1, row “4”.
  quote: “$4\quad 0.12500\quad 0.13126\quad 0.130587$.”

Contribution notes

Prepared with assistance from ChatGPT 5.2 Pro.