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Project

The Discovery of a New Series of Uniform Polyhedra

The main topic of my research is my discovery of a new series of uniform polyhedra, which I have named ‘uniform helical star columns’, or for short UHSCs. Uniform helical star columns are polyhedra built from equilateral triangles, in which the triangular faces intersect. There is an infinity of UHSCs and yet we can distinguish two different classes. One of the reasons to do this research was the aesthetic quality I found in most of the UHSCs. It became the basis for my sculptural work.

The history of the regular polyhedra starts around 400 B.C., but important developments are still taking place, as we can see in Branko Grünbaum’s paper ‘”New” uniform polyhedra’, published in 2003. In fact, the question as to what constitutes a polyhedron is one that has been actively discussed over many centuries, and can still be a controversial topic today.  All definitions seem to agree that a polyhedron consists of vertices, edges, and faces, in which the faces are (planar) polygons, the vertices are corners and edges are sides of those polygons, and each edge is incident with exactly two vertices and exactly two faces.  Also, any two faces (any two vertices) of a polyhedron are connected by a chain of mutually incident faces (mutually incident edges).  All polyhedra discussed in this thesis satisfy these requirements, and additional conditions are described for special polyhedra. In order to get a good understanding of the new uniform polyhedra, and especially of the UHSCs it is important to have adequate tools for visualizing these polyhedra. Therefore, the first part of this thesis is devoted to the visualization of polyhedra. Visualization of polyhedra also has a long history, but for these new polyhedra I had to develop new tools, for which I made use of the now available computer techniques. The helical shape and the helical movement are essential for the new UHSCs. It turns out that helical movement can be used in different ways, not only to construct, but also to explain the structure of polyhedra with intersecting faces. The helical movement called ‘helixation’ is defined as a transformation and is compared to another helical transformation, the Jitterbug transformation. This led to the discovery of yet another infinite uniform polyhedron, an addition to the three Petrie-Coxeter polyhedra. The construction of the UHSCs is investigated in various ways (both virtual and real). Traditional folding techniques, but now transformed to virtual reality, are used to show how a UHSC can be constructed.

Date:10 Feb 2016 →  7 Sep 2020
Keywords:Polyhedra
Disciplines:Architecture, Art studies and sciences
Project type:PhD project