On planar sections of the dodecahedron

Date

Mar 19, 2026

Time

3:00 PM - 4:00 PM

Speaker

Andreas Thom

Affiliation

TU Dresden

Language

en

Main Topic

Biologie

Host

Local Organisers: Nikola Sadovek, Maximilian Wiesmann, Giulio Zucal

Description

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⊆V of vertices as the distribution of isometry types of inclusions Π∩X⊆Π∩V⊆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 R3, 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.

Last modified: Mar 10, 2026, 7:37:54 AM