§ 2.15 Chopp octahedral shrinker candidateEAAEF7E5907F

(Numerical genus-seven octahedron) Chopp computes a closed genus-seven shrinker candidate in $\mathbb R^3$ by starting with a sphere bored in the eight directions $(\pm 1,\pm 1,\pm 1)$. There are four drilling axes and eight openings into the central cavity. The candidate has octahedral symmetry, dimension two and codimension one; its intended topology is a closed orientable surface of genus seven.

The construction is numerical level-set evolution with dilation and level selection. It supplies neither a closed formula nor a proof of an exact solution. In normalized coordinates the target is $\vec H=-X^\perp /2$, with the proposed flow $M_ t=\sqrt{-t}\, \Sigma $ for $t<0$. The conversion from the paper's scale is described in Example 2.14. The intended surface is compact, embedded and without boundary, hence would be complete and proper with no ends if realized as a smooth surface. These attributes describe the numerical candidate, not a proved shrinker.

Section 4.2 reports $E=0.0197$ for the self-similarity diagnostic and approximate shape preservation under further MCF. Figure 13 uses an $80\times 80\times 80$ grid. No certified convergence theorem, uniqueness, curvature classification, stability spectrum, Gaussian area or entropy value is given for this computed surface. See [Chopp1994], Section 4.2, pp. 7–9, Figures 13–15. The suggestion there that other regular polyhedra should yield shrinkers is a conjectural extension, not additional computed examples. The schematic shows the eight passages and genus seven. Later Platonic constructions are recorded separately in Examples 2.36 and 2.27; sharing a symmetry does not identify those surfaces with Chopp's candidate.