Inscribing a regular octahedron into polytopes

We prove that any simple polytope (and some non-simple polytopes) in $\mathbb R^3$ admits an inscribed regular octahedron

Knaster's problem for (Z2)k(Z_2)^k-symmetric subsets of the sphere S2k−1S^{2^k-1}

We prove a Knaster-type result for orbits of the group $(Z_2)^k$ in $S^{2^k-1}$, calculating the Euler class obstruction. Among the consequences are: a result about inscribing skew crosspolytopes in hypersurfaces in $\mathbb R^{2^k}$, and a result about equipartition of a measures in $\mathbb R^{2^k}$ by $(Z_2)^{k+1}$-symmetric convex fans

Generating polytopes from Coxeter groups

The main goal of this thesis is to study finite reflection groups (Coxeter groups) W and to use these to generate polytopes in two and three dimensions. Such polytopes will be generated as the convex hull of the W-orbit through an initial point. We prove an efficient recipe for finding the stabilizer of an initial point, and examine several examples of such polytopes and illustrate how many vertices, edges and faces these polytopes have. At last we will illustrate how this information can be pictorially encoded on the marked Coxeter diagram for an initial point

The making of geometry

Geometry has been a source of inspiration in the design of the manmade world for millennia; it also provides representational means enabling development of a concept into a built object. In the past three decades computing methodologies have provided the designer with unprecedented tools to explore highly complex forms, create digital models and fabricate them. This paper describes a computational methodology for the transition of forms from abstract geometric configurations to physical objects: a parametric design process assists from the initial ideation to the final prototyping with 3D printing technologies. The five regular polyhedra are used as a case study; this paper explores how parametric based procedures develop these geometric shapes into digital models of structures to be fabricated in different sizes and materials