OP-12046OpenGeometry and topology

Sphere packing density in $\mathbb{R}^4$ ( $C_{36}$ )

Sphere packing density in $\mathbb{R}^4$ : determine $C_{36}$ or improve its bounds; the repository records $\pi^2/16 \approx 0.616850$ (lower) and 0.644421 (upper).

MSC 52C17 · Packing and covering in

n

dimensions (aspects of discrete geometry)

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓

Read original source ↗ 0 Submit progress Lean 4 ↓Discussion (0)

Problem statement

$C_{36}=\Delta_4$ 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]

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Background

References (from the repository page)

[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: “ $4\quad 0.12500\quad 0.13126\quad 0.130587$ .”

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound $\pi^2/16 \approx 0.616850$ ; best known upper bound 0.644421. Values marked with an asterisk have minimal verification.

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]

References

[1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Sphere packing density in $\mathbb{R}^4$ ($C_{36}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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Source & evidence

https://teorth.github.io/optimizationproblems/constants/36a.html

Constant page from github.com/teorth/optimizationproblems at commit 2c1968c, Apache-2.0. The maintainers review submissions for plausibility and replicability but do not certify correctness; asterisked bounds have minimal verification.

Source date: Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differ
Imported: 2026-10-10

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Record

Difficulty: HardResisted sustained effort by specialists.
Posed by: Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers) and contributors
Posed: 2026
Source: Curated problem listOptimization Constants in Mathematics — Davis, Ivanisvili & TaoOptimization Constants in Mathematics — Davis, Ivanisvili & TaoOriginal statement ↗
Citation: Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Sphere packing density in $\mathbb{R}^4$ ($C_{36}$)
Added: 5 Nov 2026
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Fields

Geometry and topology