Visualizing the higher dimensions: the regular polytopes in four dimensions and higherToeplitz Colloquium

by Paulo Freire (Max-Planck-Institut für Radioastronomie Bonn)

Monday Apr 27, 2026, 4:00 PM → 6:00 PM Europe/Berlin

Endenicher Allee 60/1-016 - Lipschitzsaal (Mathezentrum)

Abstract

In this talk, I will present some projections of higher-dimensional polytopes to our 3-D space. This includes especially the four-dimensional regular polytopes, but I will also present some additional six-dimensional and one eight-dimensional polytope. I start the talk by making a brief presentation of the Zometool system. This will be followed by a discussion of some of the different types of projections, with an emphasis on orthographic projections, highlighting the fact that they are affine transformations, and what this implies for the projections. I then continue discussing the properties of the Platonic solids and their projections to two dimensions, especially their symmetries and how the symmetry of the polyhedron is related to the symmetry of the projection. I will also discuss how the symmetries affect directly the geometric properties of polyhedra and their projection. I then introduce the four-dimension analogues of the Platonic solids, the regular polychora, with a discussion of some of their projections to three-dimensional space represented by Zometool models, with a detailed discussion of their symmetries and the symmetries of their projections. At this stage, the audience will be able to participate in the construction of a model of one of the regular polychora, the Icosahedral projection of the 600-cell. Finally, I will mention some interesting objects in higher dimensions, and some general principles for constructing their projections.