Welcome!

Although I am an astronomer (I study binary pulsars), I also have other interests and hobbies. One of those hobbies has resulted in the unusual decoration of my office at the Max Planck Institute for Radio Astronomy, in Bonn!

First, let's have a look around!



Fig. 1.1: This is me in my polytope garden, holding a model of an orthographic projection of the 24-cell to 3-dimensional space. Picture taken in 2022 by Alexander Kappes.

Why polytopes?

A few decades ago, I started noticing polyhedra, partly because of Carl Sagan's Cosmos, but also because of the art of M. C. Escher. After making a few simple white paper models of the Platonic solids, I was struck, as Bertrand Russell put it, by

" ... a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show."

Mathematical objects have this beauty because they are finished - they can only be the way they are and cannot logically be anyway else; nothing is missing, and nothing can be taken away. Being logically necessary, they were not invented, they were discovered.

Among mathematical ideas, regular polytopes - the generalization of the concept of a regular polyhedron to any number of dimensions - are easy to understand and amongst the easiest to visualise. This makes them a gateway to that world of sublimely pure beauty beyond our own.

Because of this, I continued making polyhedron models. In the late 1990's - early 2000's I built 60 models of uniform polyhedra. When I moved to Bonn, in 2009, I started making a few paper models of polyhedra for my office. In late 2013, I started playing with the Zometool, making some models of stellations and facetings of polyhedra (a particularly large sub-class of these is the stellations of the Rhombic triacontahedron).

Since 2014 I have been using the Zometool to build models of shadows (technically, orthographic projections to 3-D space) of 4-dimensional (4D) polytopes, the polychora. The reason for this is that something wonderful happens in 4D space: while higher dimensional spaces have only three regular polytopes each and no regular star polytopes, in 4D there are 16 regular polychora, with 10 of them being star polytopes! Interestingly, the most symmetric projections of all of them can be built with the Zometool! And that's exactly what I did: I built models of all regular convex and star polychora!

In addition, 4-dimensional space admits a total of 36 fully regular polychoron compounds! With the Zometool, we can build projections of 9 of them, and of course I built them all. These objects are especially interesting because, apart from being stellations and/or facetings of the regular convex polychora, their projections are, like those of the regular polychora, beautiful, complex, and challenging to build, and because of that a lot of fun. Furthermore, they nicely illustrate some important relations between different types of geometric symmetry and thus relations between the associated symmetry groups.



Fig. 1.2: A narrower view towards some of the most complex models of 4-dimensional polytopes, taken in 2018. Photo by Aris Noutsos.


With time, my collection has been focusing more and more on the regular forms; the main reason is their unique beauty, which you can recognise immediately when you see the models. Of course, they are also mathematically more interesting. Finally, limitations of space and my inventory of the Zometool parts constrain the number of models: thus almost all models of Archimedan polychora (some of which appear in Fig. 1.2) have been replaced by models of regular compounds (Fig. 1.1). Despite these, the permanent collection in my office is now complete in the sense that it includes: The partially and quasi-regular polyhedra are important for understanding the regular and partially regular polychora. All other the Zometool models in this site were disassembled, all other paper models are somewhere else.

The objective of this site is a presentation of my polytope models, providing simple descriptions and a basic geometric understanding of the polytopes, their symmetries and their geometric relations. I generally try to explain the observed geometric properties, but some facts are presented as given - detailed explanations would require a more rigorous mathematical treatment; such as those found in the references. The metric properties of the polygons and polyhedra are covered briefly, and only to the extent thet they help us to understand the properties of some of the the 4-dimensional polytopes. Things like Euler's polyhedral formula and Descartes' theorem for angular deficiencies are not discussed in detail, as they don't apply to the non-convex polyhedra we discuss, or apply only with complex modifications, the same happens for higher-dimensional polytopes. I don't use Schläfli and Wythoff symbols, because these are partially redundant with the more informative Coxeter-Dynkin graphs, which I use to describe symmetries and polytopes derived from them using the Wythoff construction.

One important aspect is that many of these geometric properties and relations have only became clear to me while contemplating the models themselves! This highlights their usefulness as learning aids. Therefore, this site is also an evolving notebook of what I have been learning from these models. In particular, the Zometool is a great tool for displaying geometric relations. One of the features that makes it especially useful is that, in orthographic projections of uniform polychora, it highlights clearly which elements are being projected without distortion - these always appear in the same color (blue or green, depending on the model). This is great for the study of the regular and equatorial edge sections of the regular polytopes with Triangular faces, which has been most illuminating.

I hope these models will inspire some of you to start your own journey into the world of polytopes and mathematics. Follow the references! If you have any comments or suggestions for improvements, please let me know. Special thanks go to Scott Vorthmann, Nan Ma and David Richter.



Fig. 1.3: In this picture, taken in 2019 by Aris Noutsos, I am holding a model of the 600-cell.

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