Last week I ended with a remembrance of my wonderful visits to see Magnus at St. John’s Abbey in Collegeville, MN. These were certainly unique among my visits with other friends and mathematicians.
I left off with an excerpt from 10 June 1994; this next one is dated 18 June 1994.
This is another excerpt describing elements of our collaboration together; I point your attention to the first few sentences, where “stellations of two cores” are mentioned. Again, since the focus is on my work with Magnus, I will elaborate a bit on this idea without giving all the necessary background. (If you’re really curious, the internet has all the answers to your questions about stellations.)
Consider the following uniform polyhedron, called the stellated truncated hexahedron.
This name is perhaps a misnomer, since careful observation reveals that the octahedron bounded by the eight pink triangles in fact lies entirely inside the cube (hexahedron) bounded by the eight octagrams. So the octahedron does not actually truncate the cube.
Here is a partial figure from a paper I’m currently writing based on the notes referred to in the above excerpt.
Notice in d) how the red square truncates the black square to create a regular octagon. So this figure shows how the lines containing the edges of an octagon (analogous to faces of polyhedra in three dimensions) divide the plane into different regions.
But in h), we have a figure bounded by lines containing edges of two squares, but one lies entirely within the other. So we cannot choose a single polygon as generating this diagram. Therefore, we say this figure is generated by two cores — the two squares which do not intersect each other.
Of course there are many other polyhedra which may be thought of in this way, but hopefully these two examples illustrate this polyhedral thread in my collaboration with Magnus.
This excerpt is from November 1994. I include it as it illustrates how industrious Magnus was in building polyhedra. He would build polyhedron models and sell them at craft fairs to raise money for the Abbey or other related causes.
“Sales average only about a dollar an hour for my work.” Now 1994 was not all that long ago…a dollar an hour hardly seems fair for the work of a world-renowned model builder! But most people just see a few brightly colored bits of paper held together with a few drops of glue, and have no real idea about the mathematics behind the models or what it takes to build them.
So building polyhedra really was a labor of love for Magnus; he would have built them just to give them away. I recall one trip where I brought back three large trash bags filled with models which Magnus built but had no room to store. I kept some for myself, but then freely distributed them to mathematics teachers and students of all levels so they could take pleasure in holding and studying them. I hope some were inspired to build a few models on their own….
Again, Magnus stayed rather busy! Note that in 1995, he had already been a priest for 50 years. Also notice the reference to the ISIS Congress in Washington, D.C. Magnus wrote a fairly detailed “memoir” about his attendance at that conference. I’ll show you several excerpts from that memoir in next week’s post, as it comes rather later in the year (and so there isn’t room for those excerpts this week).
This is another example of a figure Magnus sent me as part of our ongoing collaboration. At the beginning of our correspondence, I had begun a textbook on polyhedra based on spherical trigonometry, which I used for a course I taught at the college level.
In particular, I was investigating the mathematics in Spherical Models. Recall that I wasn’t satisfied with numerical approximations — I wanted exact mathematical expressions for the angles used to build geodesic models based on polyhedra.
The above figure is based on one I used for designing spherical models where some of the faces of the polyhedron are pentagons. When projected onto a sphere, you needed to draw geodesics (great circles) connecting the vertices and calculate the individual segments in the spherical pentagram just created.
Now in the plane, when you join the vertices of a pentagon with diagonals, the diagonals are subdivided into the ratio φ : 1 : φ, where φ is the golden ratio. But the angles into which the diagonals of the spherical pentagram are subdivided isn’t a matter of calculating a simple ratio — there is a lot of trigonometry involved. Moreover, the ratio varies with the size of the original spherical pentagon.
In any case, here it not the place to discuss all the mathematical details. I just wanted to illustrate again one aspect of my correspondence with Magnus. I feel certain I must have sent him all my draft chapters, but likely he gave me his commentary when I visited, or perhaps somewhere in those emails that have been lost forever in some virtual black hole….
This last excerpt shows Magnus’ gentle nature. Perhaps you are not aware of the difference between the geometrical terms trapezoid and trapezium (see this article for a brief history) — but the meanings in the UK and US are swapped. What is a trapezoid here is a trapezium in the UK, and vice versa.
In any case, I made a reference to “Those crazy Brits!” when discussing this difference in nomenclature, and Magnus was sympathetic to their cause, as shown in the last few sentences of the excerpt. I can honestly say I cannot recall him ever saying a disparaging word about anyone (try counting the number of people you can say that about). He was always keen to understand all sides of an issue — and this always made philosophical and theological discussions so interesting.
I’ll stop with this comment today. Next week, we’ll begin by looking at Magnus’ memoir on the ISIS conference in Washington, D.C.