Welcome!

Hi, welcome to my office at the Max Planck Institute for Radio Astronomy, in Bonn!

First, let's have a look around!


Fig. 1a: This is me in my polytope garden, holding a model of an orthographic projection of the 600-cell to 3 dimensional space. Photo by Aris Noutsos.


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. Yet, 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 visualize. 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 system, 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 building models of the 4-dimensional polytopes, the polychora. The reason for this is that something wonderful happens in 4-dimensional Euclidean space (4D): while higher dimensional spaces have only three regular polytopes each and no regular star polytopes, in 4D there are 16 regular polytopes, with 10 of them being star polytopes! Interestingly, using Zometool, we can build models of the most symmetric projections of all of them! And that's exactly what I did! All my models are shown in the following pages.

There are, in addition, 52 regular polychoron compounds. With Zometool, we can build projections of 11 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. 1b: A wide view of the office. 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 recognize immediately when you see the models. Of course, they are also mathematically more interesting. Finally, limitations of space constrain the size of any collection. Despite these limitations, my collection is now complete in the sense that it includes models of all regular polyhedra, all regular polychora, all regular polyhedron compounds and all regular polychoron compounds that can be built with the Zometool system, plus a few closely associated forms.

The objective of this site is twofold. First, to provide brief descriptions of the objects represented by the models. More detailed descriptions - and in particular the associated mathematical concepts - are provided by the links and especially the references. Second, and more importantly,to describe the many geometric relations between those objects - duality, rectification, stellations and facetings of different kinds - which provide, by themselves, a good geometrical understanding of these objects. Many of these relations only became clear to me while contemplating the models themselves.

In this page, I named a "new" stellation operation that occurs among some regular star polychora, "dual aggrandization", where a cell is expanded into a larger dual in the same 3-D "plane". Furthermore, I describe explicitly and name the faceting transformations that are the "duals" of the stellation operations between star polytopes. These operations can be used to derive, in a simple manner, some of the intriguing properties of the regular star polytopes, especially why so many share vertex, edge and even face arrangements. As far as I know, these simple concepts have not been discussed explicitely in the existing literature.

I hope these models will inspire some of you to start your own journey into the world of polytopes and mathematics. 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. 1c: A narrower view towards some of the most complex models of 4-dimensional polytopes, taken in 2018. Photo by Aris Noutsos.

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