On planar sections of the dodecahedron

TL;DR

The paper studies whether random planar sections determine a 3D structure in a discrete vertex-model on the regular dodecahedron. It defines the planar statistic $\mathrm{PS}(X)$ as the multiset of planar congruence types of inclusions $\Pi\cap X \subseteq \Pi\cap V \subseteq V$ over vertex-planes $\Pi$, and proves non-identifiability by exhibiting two non-congruent 7-vertex subsets $S$ and $T$ with $\mathrm{PS}(S)=\mathrm{PS}(T)$, explicitly $S=\{0,1,2,3,4,11,17\}$ and $T=\{0,1,3,4,5,11,17\}$. This combinatorial obstruction is then encoded geometrically by constructing polytopes $K_S$ and $K_T$ obtained from the dodecahedron via small vertex truncations; these have identical distributions of planar sections, despite $K_S\not\cong K_T$. The argument combines a careful orbit-type analysis of vertex-plane incidences with a decomposition of the affine Grassmannian, yielding a 3D analogue of Mallows–Clark non-uniqueness results in planar geometry. The findings demonstrate fundamental non-uniqueness in geometric tomography even in a single polytope setting and highlight how high-level shape information can be lost under random planar sampling.

Abstract

In the analysis of three-dimensional biological microstructures such as organoids, microscopy frequently yields two-dimensional optical sections without access to their orientation. Motivated by the question of whether such random planar sections determine the underlying three-dimensional structure, we investigate a discrete analogue in which the ambient structure is the vertex set of a Platonic solid and the observed data are congruence classes of planar intersections. For the regular dodecahedron with vertex set $V$, we define the planar statistic of a subset $X\subseteq V$ of vertices as the distribution of isometry types of inclusions $Π\cap X \subseteq Π\cap V \subseteq V$, and ask whether this statistic determines $X$ up to isometry. We show that this is not the case: there exist two non-isometric $7$-element subsets with identical planar statistics. As a consequence, there exist two polytopes in $\mathbb R^3$, whose distribution of isometry classes of two-dimensional intersections is identical, while the polytopes are not themselves isometric. This result is an analogue of classical non-uniqueness phenomena in geometric tomography.

Figures (2)

• Figure 1: Schlegel diagram with all 20 vertices labeled.
• Figure 2: 7-element subsets $S$ (left) and $T$ (right)

Theorems & Definitions (6)

• Definition 1
• Remark 2
• Theorem 3
• proof
• Theorem 4
• proof