This is the final installment about my correspondence with Magnus Wenninger. I didn’t realize I had so much to say! But I am glad to take the opportunity to share a bit about a friend and colleague who contributed so much to the revitalization of three-dimensional geometry in recent years. Talk to anyone truly interested in polyhedra, and they will know of Magnus.
As I mentioned last week, I’ll begin with Magnus’ memoir on the Symmetry Congress (as you can see in the title of his memoir). His friend Chuck Stevens lived near where the Congress was held, and so met him at the airport and was his tour guide for the duration of his visit. (Note: the Society is still active — just google it!)
In this excerpt, Magnus remarks (start in the middle of line 4) that people who don’t know much about polyhedra always ask the same two questions: how long did it take you to make that model, and what do you do with them? I have had similar questions asked of me over the years as well; you just learn to be patient and hopefully enlighten…. Of course Magnus was always kind and generous with his responses.
You might be surprised by Magnus talking with a 10-year-old boy at the conference. Of course it may have been that Josh just happened to be staying at the same hotel, though that is unlikely since he was visiting relatives. More likely is that his aunt or uncle was a conference participant and brought him to the conference. I should remark that it is a common occurrence for a participant in an international conference to plan a family vacation around the trip, so you regularly see children of all ages at such conferences.
I direct your attention to last seven lines here. Magnus was perfectly happy to have his brandy, building polyhedra in an air-conditioned room rather than braving the summer heat to be “cultural.” For me, this emphasizes the simplicity of Magnus’ life. He did not need much to make him happy — some paper and glue, his building tools, his Bible, and perhaps a few other books on philosophy and theology. The quintessential minimalist life of a Benedictine monk.
Here, the second paragraph is interesting. In rereading it, I think I could imagine the exact expression on Magnus’ face when he heard “I’ll take it.” I know that this was a rare occurrence for Magnus. Perhaps it might be less so now; because of Magnus’ influence, as well as the explosion of computer graphics on the internet, people are generally more informed about polyhedra than they were in 1995.
Moreover, more and more high school geometry textbooks are moving away from exclusively two-column proofs, and some even have chapters devoted to the Platonic solids. I don’t think we’re at the point yet where “dodecahedron” is a household word…but we’re definitely moving, if slowly, closer to that point.
The final excerpt I’d like to share is from December 1995. I include this as another example of my collaboration with Magnus — our discussions of “perfect versions” of polyhedra. I’ll go into this example in more detail since it’s a bit easier to understand, but I note Magnus was not a fan of the adjective “perfect.” (And as a historical note, I had used the term “perfect version” and had also corresponded with Chuck Stevens, so Chuck must subsequently have talked to or corresponded with Magnus and used the term, and so Magnus thought Chuck came up with the term.)
I now agree, but have yet to come up with a better term. The basic idea is that some polyhedron models are very complicated to build. But for many of them, there are ways to make similar-looking polyhedra which are still aesthetically pleasing, but a bit easier to construct.
Let’s look at an example I mentioned a few weeks ago: the stellated truncated hexahedron, shown below.
Notice the blue regular octagrams. Now consider the octagrams shown here.
On the left is a regular octagram. If you draw a square around it, as shown, you divide the edges of the square in the ratio Notice that the octagram is divided into 17 smaller pieces by its edges.
However, if you start with a square and subdivide the edges into equal thirds, an interesting phenomenon occurs — there are four points where three edges intersect, resulting in a subdivision of the octagram into just 13 pieces.
You will note that this variation is not regular — the horizontal and vertical edges are not the same length as the diagonal edges. So any polyhedron with this octagram as a face would not be a uniform polyhedron.
However, it would be what Magnus referred to as a “variation” of a uniform polyhedron. So if we took the stellated truncated hexahedron, kept the planes containing the pink triangles just where they are, but slightly move the planes containing the blue octagrams toward the center, we would end up with the following polyhedron:
Note that the octagrams are now the octagram variations. Also notice how the pentagonal visible pink pieces are now rhombi, and the small blue square pieces are completely absent!
Such simplifications are typical when working with this kind of variation. Of course many polyhedra have such variations — but now isn’t time to go into further details. But these variations were among the polyhedra Magnus and I wrote about.
As I mentioned, there is little left of my correspondence with Magnus, since several years of emails have been lost. But I hope there is enough here to give you a sense of what Magnus was like as an individual, friend, and colleague. He never let his fame or reputation go to his head — all he was ever doing, as he saw it, was taking an idea already in the mind of God, and making it real.
He truly was humble, gentle, and kind — and of course a masterful geometer who significantly influenced the last few generations of polyhedron model builders. He will be missed.