On planar sections of the dodecahedron
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911403218042880 |
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| author | Thom, Andreas |
| author_facet | Thom, Andreas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_12773 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On planar sections of the dodecahedron Thom, Andreas Metric Geometry 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. |
| title | On planar sections of the dodecahedron |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2511.12773 |