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

Upper bound, last-listed row

| $0.644421\ldots$ | [dLOV2014](#dLOV2014) | [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] |

exact pinned row

| $0.644421\ldots$ | [dLOV2014](#dLOV2014) | [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 $\Delta_4\le \operatorname{vol}(B)\cdot 0.130587 = (\pi^2/2)\cdot 0.130587\approx 0.644421$. <a href="#dLOV2014-table1-d4-row">[dLOV2014-table1-d4-row]</a> <a href="#dLOV2014-center-density">[dLOV2014-center-density]</a> |

tracked since 2026-07-24 — no movement seen yet

Lower bound, last-listed row

| $\pi^2/16 \approx 0.616850$ | [dLOV2014](#dLOV2014) | [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] |

exact pinned row

| $\pi^2/16 \approx 0.616850$ | [dLOV2014](#dLOV2014) | [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 $\Delta_4\ge \operatorname{vol}(B)\cdot 0.12500 = (\pi^2/2)\cdot (1/8)=\pi^2/16\approx 0.616850$. <a href="#dLOV2014-table1-d4-row">[dLOV2014-table1-d4-row]</a> <a href="#dLOV2014-center-density">[dLOV2014-center-density]</a> |

tracked since 2026-07-24 — no movement seen yet