Sphere packing density in ℝ^4

Fact	Value	Source
Best lower bound with an explicit value (constant 36a)	0.61685 dLOV2014 — dLOV2014 lists the dimension-4 center density lower bound _4≥ 0.12500 (Table 1).	Tao's optimization constants
Best upper bound with an explicit value (constant 36a)	0.644421 dLOV2014 — dLOV2014 lists the dimension-4 center density upper bound _4≤ 0.130587 (Table 1).	Tao's optimization constants

Description of constant

$C_{36} = Δ_{4}$ is the (optimal) sphere packing density in $R^{4}$ , i.e. the largest fraction of $R^{4}$ that can be covered by congruent balls with disjoint interiors. <a href="#CE2003-pack-problem">[CE2003-pack-problem]</a> <a href="#CE2003-def-density">[CE2003-def-density]</a> <a href="#CE2003-greatest-density">[CE2003-greatest-density]</a>

More precisely, for a packing $P$ in $R^{4}$ , 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 $P$ is

$Δ (P) := R \to \infty lim sup p \in R^{4} sup \frac{vol ( P \cap B ( p , R ))}{vol ( B ( p , R ))} .$

<a href="#CE2003-upper-density">[CE2003-upper-density]</a>

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

$Δ_{4} := P \subset R^{4} sup Δ (P),$

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

It is often convenient to work with the center density $δ_{4}$ , defined (for packings of unit spheres) by

$δ_{4} = \frac{Δ _{4}}{vol ( B )},$

where $B$ is a unit ball in $R^{4}$ . <a href="#dLOV2014-center-density">[dLOV2014-center-density]</a>

Known upper bounds

Bound	Reference	Comments
$1$	Trivial	A packing cannot cover more than all of $R^{4}$ .
$0.647791 \dots$	[CE2003](#CE2003)	[CE2003](#CE2003) lists the dimension- $4$ Rogers bound on center density $δ_{4} \leq 0.13127$ (Appendix C, Table 3). Converting gives $Δ_{4} \leq (π^{2} /2) \cdot 0.13127 \approx 0.647791$ . <a href="#CE2003-appC-table3-d4-row">[CE2003-appC-table3-d4-row]</a> <a href="#dLOV2014-center-density">[dLOV2014-center-density]</a>
$0.647742 \dots$	[CE2003](#CE2003)	[CE2003](#CE2003) lists the dimension- $4$ “New Upper Bound” on center density $δ_{4} \leq 0.13126$ (Appendix C, Table 3). Converting gives $Δ_{4} \leq (π^{2} /2) \cdot 0.13126 \approx 0.647742$ . <a href="#CE2003-appC-table3-d4-row">[CE2003-appC-table3-d4-row]</a> <a href="#dLOV2014-center-density">[dLOV2014-center-density]</a>
$0.644421 \dots$	[dLOV2014](#dLOV2014)	[dLOV2014](#dLOV2014) lists the dimension- $4$ center density upper bound $δ_{4} \leq 0.130587$ (Table 1). Converting via $Δ_{4} = vol (B) δ_{4}$ gives $Δ_{4} \leq vol (B) \cdot 0.130587 = (π^{2} /2) \cdot 0.130587 \approx 0.644421$ . <a href="#dLOV2014-table1-d4-row">[dLOV2014-table1-d4-row]</a> <a href="#dLOV2014-center-density">[dLOV2014-center-density]</a>

Known lower bounds

Bound	Reference	Comments
$0$	Trivial	Since densities are nonnegative.
$π^{2} /16 \approx 0.616850$	[dLOV2014](#dLOV2014)	[dLOV2014](#dLOV2014) lists the dimension- $4$ center density lower bound $δ_{4} \geq 0.12500$ (Table 1). Converting via $Δ_{4} = vol (B) δ_{4}$ gives $Δ_{4} \geq vol (B) \cdot 0.12500 = (π^{2} /2) \cdot (1/8) = π^{2} /16 \approx 0.616850$ . <a href="#dLOV2014-table1-d4-row">[dLOV2014-table1-d4-row]</a> <a href="#dLOV2014-center-density">[dLOV2014-center-density]</a>

Additional comments and links

References

<a id="CE2003"></a>[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.
<a id="CE2003-pack-problem"></a>[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 $R^{n}$ can be covered by congruent balls that do not intersect except along their boundaries?”
<a id="CE2003-def-density"></a>[CE2003-def-density] loc: arXiv PDF p.3, Introduction (density definition). quote: “The density $Δ$ of a packing is defined to be the fraction of space covered by the balls in the packing.”
<a id="CE2003-greatest-density"></a>[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.”
<a id="CE2003-upper-density"></a>[CE2003-upper-density] loc: arXiv PDF p.20, Appendix A (upper density definition). quote: “every packing has an upper density, defined by $Δ = lim sup_{r \to \infty} sup_{p \in R^{n}} vol (B (p, r) \cap P) / vol B (p, r)$ .”
<a id="CE2003-appC-table3-d4-row"></a>[CE2003-appC-table3-d4-row] loc: arXiv PDF p.23, Appendix C (Table 3, row “4”). quote: “ $4 0.125 0.13127 0.13126$ .”

<a id="dLOV2014"></a>[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.
<a id="dLOV2014-center-density"></a>[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 $Δ/ vol B$ , where $Δ$ is the density of the packing, and $B$ is a unit ball.”
<a id="dLOV2014-table1-d4-row"></a>[dLOV2014-table1-d4-row] loc: Journal PDF p.14, Table 1, row “4”. quote: “ $4 0.12500 0.13126 0.130587$ .”

What counts as progress

A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
A formal proof (Lean) of a known bound, or a precise error in a claimed one.
New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 87b4ebbd426f, Apache License 2.0; reformatted for this page. New records should also be reported there.

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Current state

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