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 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
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 using 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 Zometool! And that's exactly what I did: I built Zometool models
of all regular convex and star
polychora!
In addition, 4-dimensional space admits a total of 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 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 constrain the number of
models. Despite these, 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 show you my collection, providing brief
descriptions of the objects represented by the models and their basic geometric properties.
Second, to describe the many geometric relations between the objects.
More detailed descriptions - and in particular the associated mathematical concepts - are provided
by the links and especially the references.
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, 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 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. 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:
In this picture, taken in 2019 by Aris Noutsos, I am holding a model of the 600-cell.
News:
- 2021 August - present: re-writing the whole site, learning from the models.
- 2021 July: Completed the set of paper models of the regular compounds of regular polyhedra.
- 2020 December 13: Completed the compound of 120 5-cells. As far as I know this is the first physical model of this compound. This completes my collection
of Zometool models.
- 2020 May 10: Using vZome, Scott Vorthmann finds that the compound of 120 5-cells can be built with Zometool.
- 2020 April 28: Completed frame of the Compound of 75 16-cells. On May 10, I finished the blue 3-intersections.
- 2019 January 26: Built model of Gosset's 8-dimensional figure. This was later disassembled.
- 2018 December 19: Completed the Compound of 15 16-cells.
- 2018 August 4: Completed the Compound of 5 24-cells.
- 2017 November 11: Completed the Great grand stellated 120-cell. This completes the list of models of all 16 regular polychora.
- 2017 November 2: Some of my models are being displayed at the Gutenberg store
in Bad Godesberg. See poster here by Ⓒ Zometool Europa / Lyn Taylor,
with photos by Aris Noutsos and Jason Wu.
- 2016 November 12: Completed larger scale version of the rectified 120-cell. Later disassembled.
- 2016 February 13: Completed rectified 600-cell. Later disassembled.
- 2015: Moved to new office.
- 2014 August 21: Completed model of the Grand 600-cell.
- 2014 August 6: Completed model of the stellated 120-cell.
Next: Polyhedra