Over a decade ago, it was shown that every edge unfolding of the Platonic solids was without self-overlap, yielding a valid net. We consider this property for regular polytopes in arbitrary dimensions, notably the simplex, cube, and orthoplex. It was recently proven that all unfoldings of the $n$-cube yield nets. We show this is also true for the $n$-simplex and the $4$-orthoplex but demonstrate its surprising failure for any orthoplex of higher dimension.

Given a convex polyhedron, the star unfolding of its surface is obtained by cutting along the shortest paths from a fixed source point to each of its vertices. We present an interactive application that visualizes the star unfolding of a box, such that its dimensions and source point locations can be continuously toggled by the user.

The associahedron is a convex polytope whose 1-skeleton is isomorphic to the flip graph of a convex polygon. There exists an elegant geometric realization of the associahedron, using the remarkable theory of secondary polytopes, based on the geometry of the underlying polygon. We present an interactive application that visualizes this correspondence in the 3D case.

Consider two simple polygons with equal area. The Wallace-Bolyai-Gerwien theorem states that these polygons are scissors congruent, that is, they can be dissected into finitely many congruent polygonal pieces. We present an interactive application that visualizes this constructive proof.

The phylogenetic tree space, introduced by Billera, Holmes, and Vogtmann, is a cone over a simplicial complex. In this short article, we construct this complex from local gluings of classical polytopes, the associahedron and the permutohedron. Its homotopy is also reinterpreted and calculated based on polytope data.