Uncategorized – Creativity in Mathematics

The Puzzle Cabaret

I decided to undertake a new venture while staying home during the pandemic: write a puzzle book!

I’m not sure how the idea ultimately condensed from a gaseous nebula of random ideas, but one thought was that with so many people at home now, they needed something fun to do. Why not puzzles? With several print-on-demand platforms available, I could even self-publish!

If you’ve been following my blog from the beginning, you know I’ve written many posts about creating new and interesting puzzles. (You can search the index to easily find them.) And as a consumer of puzzle magazines for decades, I’m pretty familiar with what’s out there. Some of the current magazines include exactly the same type of puzzles they had when I was growing up.

Well, I could do better, I thought. (Although it will ultimately be up to my readers to decide that!) Since I love word, logic, and math puzzles, I’d make sure there was a nice variety. There was a particular challenge — if you look at puzzle magazines, you’ll notice that particular authors contribute the same type of puzzle to one or more different magazines. But there is just one of me.

Further, if you go to amazon.com, you might be (like I was) underwhelmed by the quality of many of the puzzle books. So I needed to create a wide variety of challenging puzzles which were not currently in circulation.

And so The Puzzle Cabaret was born! I put some sample puzzles on The Puzzle Cabaret’s website and my Twitter feed, @cre8math, so feel free to go there and check them out.

What made this such a fun project for me was that I got to use so many different tools in the toolbox I’ve outfitted over the years. Of course there is the creative task of creating new and innovative puzzles. I relied on Mathematica for some help, such as a puzzle involving words made up of three-letter words, like lithographer — LIT + HOG + RAP + HER. Rather than look through a dictionary myself, I wrote a program which searched Mathematica‘s large dictionary for 6-, 9-, and 12-letter words like this.

Or what about letters that can come before “Q” in a word? Easy to search a dictionary. I did learn a lot of new words this way — such as the only word (aside from its plural) which contains the letters “LQ” in order. Hats off to you if you know it!

So having some programming experience was helpful. My experience with mathematical art — a main thread in my blog — was helpful. I was able to do all the layout myself in LaTeX with the TikZ plugin. It’s nice because each page is essentially a sheet of graph paper — you can give exact coordinates of any page element you want so everything lines up perfectly.

I especially enjoyed creating borders around special pages — like the Table of Contents or separator pages between sections of the book. The challenge was designing something intriguing in black-and-white which would look good even if the print quality was not what you’d expect in a publisher-printed book.

Experience writing papers, books, book chapters, and teaching materials meant that I’ve become a pretty decent editor. So I could do that on my own, too. It’s tricky, though — looking at something you write over and over again. I undertook several rounds of proofreading, and kept findings things to fix each time. But at some point, you just have to stop and accept the fact that yes, there will still be a few typos in the finished book….

My design experience also meant I was able to create my own cover. It was a fun project. I went for “less is more,” and was driven by one overriding thought — my cover should shout “CLICK ME!!!” Well, I’d definitely click on it….

So having a diverse skill set was definitely helpful. But there are really so many other things which go into self-publishing a book. Learning terminology — trim size, gutter margins, embeddable fonts. Getting an ISBN. Ordering proof copies. Etc., etc. Volume 2 will be so much simpler now that I’ve finished Volume 1!

It was definitely a challenge — but one that I hope will be worth it. Because there’s the one overarching question: will anybody actually buy The Puzzle Cabaret?

I chose to sell it on amazon.com since it was the simplest marketing strategy. (Click here to see it on Amazon!) Maybe I’ll try something different for Volume 2, but my goal was to keep the monetary investment to a minimum until I knew this type of book would be popular. I really only had to pay for the ISBN, ordering proof copies, and of course paper and toner for all of the printing….

So we must let time tell! And while I was able to do a lot on my own, I must acknowledge that I have some pretty awesome friends who also have some truly exceptional skills. Stacy Speyer (see what she’s up to) was a huge help with layout as a result of her vast experience with art and graphic design. Phil Webster (see what he’s up to) play-tested many, many puzzles and gave me invaluable feedback. And Stephen Campbell (see what he’s up to) gave good advice about looking forward to future volumes.

Advice for would-be self-publishers? Patience! I’ll just give one example. Everything was going smoothly until I tried to upload my manuscript and order a proof copy to see what the final product would look like. No luck. I got some errors, and the (not-so-) helpful people at amazon.com essentially said that well, since I was using LaTeX, too bad.

Stacy came to the rescue — she’s published many books before. She just took my pdf file, uploaded it into her software, and exported it as — you guessed it — a pdf. Bingo! Of course this violates a basic mathematical principle: while 2 = 2, it is absolutely not the case that pdf = pdf. Oh, well. You get the point. With so many different variables present in the self-publishing process, something(s) will inevitably go awry. So the second piece of advice? Have smart friends!

Largest Number Found!

Late last month, Professor Warren Porterdunk of Berryville University announced that he had, in fact, found the largest number. I was lucky enough to arrange a virtual interview with him, which is summarized here.

Of course, there is no largest number. “That is the conventional mathematical wisdom, yes,” said Porterdunk. “But when you include trans-transcriptional numbers — that is, numbers which cannot actually be written down — well, that’s a game-changer.” Porterdunk calls these numbers Titans, derived from the acronym TTN, which he finds apt.

So what is the largest number? Just a single “1” followed by infinitely many zeroes. “That’s only one representation,” said Porterdunk. Some real numbers have more than one representation, such as $1=0.\overline9$ . The “1” followed by infinitely many zeroes — which Porterdunk calls “die Uberzahl” as an homage to Nietzsche — also has multiple presentations. “What makes the analysis of Titans a bit tricky is that they may have uncountably infinitely many representations,” said Porterdunk.

Of course I had to ask: What about a single “2” followed by infinitely many zeroes? “That’s always the first question I get,” chuckled Porterdunk. He says that the intuition which causes us to ask that question comes from the usual right-to-left algorithm for multiplying numbers. “But the Uberzahl has no rightmost digit, so we need to throw many intuitions out the window and develop a Titan arithmetic.”

How is this done? First, you need something called the Extendability Axiom, which postulates that (1) for a given prime number $p,$ the limit $\lim_{n\to\infty}p^n$ exists, and (2) if $p_1<p_2,$ then $\lim_{n\to\infty}p_1^n<\lim_{n\to\infty}p_2^n.$ Porterdunk calls numbers of the form $\lim_{n\to\infty}p^n,$ where $p$ is prime, uberprimes.

Then, enumerate the primes, and define a Titan to be a product $\prod_{n=1}^Mp_n^{k_n},$ where either $M=\infty$ or at least one of the $k_n$ is infinite. “The trick is to find a well-ordering on the Titans,” said Porterdunk. “Then show that the set of Titans has a unique maximal element. Done!”

And the trick? Consider the uberprime $P=\lim_{n\to\infty}2^n,$ and the Titan $Q=3\cdot5\cdot7\cdot11\cdots;$ that is, the product of all the odd primes. Which is larger?

You might think $Q$ is, since each odd prime is larger than 2. On the other hand, if you look at the first two powers of 2 multiplied together, the result is larger than 3. Then the next three together are larger than 5, the next three larger than 7, and so on. So $P$ must be larger. “Consequently, you must be very, very careful when well-ordering the Titans. Pitfalls are everywhere,” remarked Porterdunk.

A staggering 263 pages of formal proof are necessary to demonstrate that the Titans do indeed have a unique maximal element. “Yes, the devil is in the details,” commented Porterdunk. “One sticky point is that Titans usually have a unique factorization into finite powers of primes and uberprimes. But quantifying what “usually” means is a very delicate task.”

Another sticky point revolves around a representation theory for Titans. Consider the Uberzahl. At first glance, it may seem that a “1” followed by infinitely many zeros is just the product of the uberprimes $\lim_{n\to\infty}2^n$ and $\lim_{n\to\infty}5^n.$ “If it helps you to sleep at night, go ahead and think that,” said Porterdunk. “But I can assure you, there is a bit more than that to a representation theory for Titans.” Rather an understatement, I should say!

The next steps for Porterdunk? “I’m working on topological induction right now,” he said. Our discussion of this was quite involved, but I’ll try to briefly summarize. First, create a topology on the Titans such that the set of finite products of uberprimes is dense. Then use the well-ordering to induce a Titan metric on the space. Next, invoke the metric to define convergence in the usual way. Then the principle of topological induction states that if $Q=\lim_{n\to\infty}q_n$ is the limit of a sequence of Titans $q_n,$ and if some property holds for each of the $q_n,$ then the property also holds for $Q.$

“When you’ve been immersed in Titan arithmetic as I have,” said Porterdunk, “topological induction just seemed like the natural next step to take. I think there’s a lot of potential there.”

What applications might the principle of topological induction have? “Well, it’s very premature to make any definitive statement. Very. But let’s just say it might be quite interesting to several folks to know that after developing a little bit of analytic Titan theory, it may be possible to show that the Riemann hypothesis is actually equivalent to the P versus NP problem.”

Well, if that isn’t a mathematical cliffhanger, I don’t know what is! Porterdunk said it will be a few years before he can tidy up all those details. It definitely would be a coup for Titan arithmetic. I’ll be sure to keep you up-to-date on this and other intriguing applications of topological induction. Look out for the interview with Porterdunk in the New York Times next week — and don’t hesitate to visit http://www.dieuberzahl.org, where you’ll find a complete set of references and several papers to download. I hope you’ll enjoy learning about trans-transcriptional number theory as much as I have!

The Fourth Dimension, IV

In the past few weeks, we looked at the hypercube in four-dimensional space. We approached this intriguing geometrical object using thinking by analogy; that is, we looked at one-, two-, and three-dimensional hypercubes (segments, squares, and cubes), and imagined potential properties — such as the number of vertices, edges, faces, and cells — of a four-dimensional analogue.

Moreover, we made these counts in two different (although related) ways, and obtained the same results. We ended last week’s discussion with the following question: yes, we obtained these counts in two different ways, but exactly what were we counting? In other words, what was this hypercube we were analyzing? Did we make the extension into four dimensions in a mathematically sound way?

We won’t be able to go into all the details in this post, but we’ll outline the main ideas which suggest that our reasoning so far in the series to this point could be made rigorous.

So what we need to do is actually say just what a hypercube in four dimensions actually is. There are different ways to make this precise, but perhaps the easiest way is to use Cartesian coordinates. Most readers are likely fairly familiar with representing a point in the plane by the Cartesian coordinates $(x,y),$ and many have encountered $(x,y,z)$ as a representation of a point in three-dimensional space.

What about four-dimensional space? It is natural to just “add” another coordinate, and represent a point in four-dimensional space by $(x,y,z,w)$ (yes, the “ $w$ ” comes last, so the other coordinates continue to make sense in the usual way).

You might ask, how can we just add another coordinate? In other words, how can we just add a dimension? To do this rigorously, we would need to study some linear algebra — in particular, the geometry of $n$ -dimensional real space, usually denoted by the symbol “ ${\mathbb R}^n.$ ” In the general case, we have lists of $n$ real numbers representing points in an $n$ -dimensional real space.

So let’s take this point for granted, and do a little more thinking by analogy. Or perhaps it might be better to say “defining” by analogy. For example, we might define a square, together with its interior, as

$\{(x,y)\,:\,|x|\le1,|y|\le1\}.$

Now this certainly seems a bit more complicated than how a square is typically defined in elementary geometry. But this set of points is easy to sketch in the plane. Moreover, given this algebraic description of the square, it is easy to enumerate the vertices: $(1,1),$ $(-1,1),$ $(-1,-1),$ and $(1,-1).$

Alternatively, the set containing these four vertices is given by

$\{(x,y)\,:\,|x|=1,|y|=1\}.$

Moreover, we can algebraically describe the top edge of square:

$\{(x,1)\,:\,|x|\le1\}.$

But do you see the catch? Suppose we are given an algebraic description of a set of points in the plane, such as the square above. How do we algebraically define its vertices? edges?

Ask yourself the following question: can you find a point in the set described above — the square — which has the property that there exists a line which intersects the square in only that point? The answer is yes: but such a point must be a vertex of the square. So we might use this property to define the vertices of a set.

But we must be careful. With this definition, every point on this pair of line segments would then be a vertex!

Day173Square2

This is certainly not what we want. To see how to address this issue, we might introduce another concept: convexity. You might remember this idea coming up in my post about polygons, but here is a brief refresher. A convex set has the property that the segment joining any two points of the set lies entirely within the set.

Day106Convexity

Using polygons as examples, the two polygons on the left would be convex, but the two on the right would not be convex.

This idea generalizes into any number of dimensions; but we’ll look a little more at two dimensions right now. Looking back at the picture of the square, and the picture of the two line segments, we observe that the square is a convex set, while the pair of segments is not.

So we might be tempted to say that if a set is convex, then any point which is the intersection of that set and some line is a vertex of the set. This works for the square, but…consider the circle shown below.

Day173circle

Are there points which are the intersection of this convex set and a line? Yes there are — any point on the boundary of the circle, since a tangent line to a circle always intersects the circle in a single point.

So does a circle have infinitely many vertices? In this case, we would actually call points on the circle extreme points of the convex set. So a vertex would be just one example of an extreme point of a convex set.

Now what about the convex set here?

Day173Square3

The dashed lines indicate that part of the boundary is missing. This set has just two extreme points, not four. The problem here is that this set is not closed in a topological sense. I won’t say more about topology here — it’s quite a huge topic! — but you can see that the number of vertices on a “square” depends critically upon whether the boundary is included.

I hope you can see that there are several issues involved in gleaning geometric properties from a purely algebraic description. But the point is this: we may define a hypercube as the set

$\{(x,y,z,w)\,:\,|x|\le1,|y|\le1,|z|\le1,|w|\le1\}.$

This is a completely unambiguous definition, but the difficulty lies in describing geometrical properties of a set using only rigorous mathematical definitions and not relying on what it looks like.

So this is why, mathematically, we can say just what a four-dimensional hypercube really is. It takes a bit of work to tease out all the properties of a hypercube, but it can be done.

And no, this isn’t the only way to define a hypercube either….concepts such as convex hull or Cartesian product may be used to give other descriptions as well. Feel free to look these ideas up if you’re interested in learning more about them….

Fr. Magnus Wenninger, O.S.B., V

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.

14-8-95-1 — Excerpt from 14 August 1995.

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!)

14-8-95-2 — Excerpt from 14 August 1995.

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.

14-8-95-3 — Excerpt from 14 August 1995.

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.

14-8-95-4 — Excerpt from 14 August 1995.

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.

W92

Notice the blue regular octagrams. Now consider the octagrams shown here.

Day130octagrams

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 $1:\sqrt2:1.$ 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:

W92pv

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.

Fr. Magnus Wenninger, O.S.B., IV

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.

W92

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.

twocores2d

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.

Fr. Magnus Wenninger, O.S.B., III

I just returned yesterday from the Joint Mathematics Meetings in San Diego, where I gave my talk commemorating Fr. Magnus Wenninger. Last week, I posted several excerpts from my correspondence with Magnus which I included in that talk. Today, I’ll continue that discussion, again adding commentary as appropriate.

I included this quote because of Magnus’ translation of Kepler: “Beyond doubt there exists in the mind of God the eternal form of all these shapes, which we call its truth.” Magnus had a very philosophical approach to mathematics, and always considered that his work was some manifestation of the Divine.

Is mathematics invented or discovered? This age-old and unresolved question has sparked much debate over the centuries. But for Magnus, the answer is clear: mathematics is discovered. The polyhedra he created all existed in the mind of God long before he reimagined them with a few pieces of paper and a little glue.

So Magnus’ vocation as a Benedictine monk strongly influenced how he thought about his life’s work. Having personally held and studied some of Magnus’ models, it is not a stretch to say that perhaps they were in fact touched by the Divine….

27-2-94 — Excerpt from 27 February 1994.

This excerpt again shows how very active Magnus was during the 1990s. But this particular letter is part of a thread in our correspondence where I was planning my first visit to see Magnus.

What made the trip a little more challenging is that I also wanted to visit Peter Messer in Wisconsin as well. Magnus introduced me to Peter as someone very interested in studying stellations of polyhedra, both from concrete and abstract viewpoints. (I also have a folder of correspondence from Peter, but that for another time!) Also, Peter was using Mathematica, which I was quite familiar with. So it seemed appropriate that we should meet.

Peter just recently retired, but was a dermatologist at the time, and hence had a busy professional schedule. So I was trying to juggle three schedules to arrange a series of visits — all through written correspondence! Now we would consider this approach somewhat anachronistic, but at the time, it was perfectly reasonable.

The title of my talk at the Joint Mathematics Meetings was Working with Magnus Wenninger. So in addition to excerpts which documented our getting to know each other, I also included excerpts which illustrated various aspects of our collaboration.

If you look back at the very first excerpt from last week’s post, you’ll notice the second paragraph begins “I’m in no hurry to get information about barycentric coordinates.” In my introductory letter to Magnus, I mentioned that I had begun work on finding coordinates of polyhedra using a barycentric coordinate system.

But over the next several months, I did send Magnus notes on barycentric coordinates, and he did study them. Now is not the time to go into a detailed discussion of barycentric coordinates — the point is that Magnus occasionally included diagrams in his correspondence to present his perspective on different aspects of our collaboration.

The first figure shows barycentric coordinates relative to a six-frequency triangle. The second figure shows how he would use this abstract idea to create a template which he would use to make what would be called a six-frequency geodesic icosahedron. Again, now is not the time to go into details, but I did want to include a brief description for those who are a little more familiar with polyhedra and geodesic structures.

Although, as I mentioned just a few moments ago, I began arranging my first visit with Magnus through letters, I must have finalized arrangements with him and Peter over the telephone. This excerpt from 10 June 1994 was the first letter since February which mentioned my visit; I must have realized that calling both Magnus and Peter was ultimately more efficient than writing them. I should mention that Magnus had difficulty hearing over the phone, and so we never had any substantial conversations on the telephone; our chats were confined to arranging logistics of visits or other such details.

I always enjoyed my visits with Magnus. I would stay in a guest room in the Abbey at St. John’s University in Collegeville, Minnesota — this was a spartan room with a simple bed, a small desk and chair, and I think maybe another chair to sit in…it’s been too many years to remember the exact details.

I’ll recall a typical day during one of our visits — although the individual visits tend to blur together. I’d say I made a half-dozen visits while I lived in the Midwest, including a time when Magnus visited me and I drove him to a conference at the University of Illinois Urbana-Champaign.

In any case, there was a common room for guests in the abbey for breakfast. Nothing too elaborate, but enough to get you going. I’d be on my own in the morning — working on polyhedral pursuits, or perhaps just walking around the beautiful campus. My visits were usually during the summer, so I was spared the cold of Minnesota winters.

Magnus would then find me around lunchtime. He had permission for me to have lunch and dinner with the other monks — a real treat, since this was not the usual protocol. But Magnus and I often had philosophical and theological discussions as well as conversations about polyhedra, and so I was welcome at meals. I can’t remember details, but the conversations were always quite stimulating. The Benedictines at St. John’s were rather liberal.

In the afternoon, we’d meet to discuss various aspects of polyhedra or perhaps work on some models. Often Magnus would be building while we talked.

Then I’d be on my own for a bit. Often I would go to one of the Offices in the Abbey church before dinner, and I’d meet Magnus after the service (the monks sat in their own section) and we’d walk down to dinner and more interesting conversation.

Our evening sessions were a bit more informal, and our discussions would often be more philosophical rather than polyhedral in nature. Magnus would have his nightly brandy — or perhaps a little Grand Marnier, since I would usually bring him a bottle when I visited.

Then it was early to bed for Magnus, although I was up a little later. I am not sure when the first Office was in the morning, but I am fairly certain I was always in bed at the time….

We must have made arrangements for future visits either by phone or email, since this is the only correspondence I have which includes any details of visits with Magnus. But they were wonderful times which I shall always remember fondly.

I’ll continue with excerpts from my correspondence with Magnus next week…so stay tuned!

Fr. Magnus Wenninger, O.S.B., II

In this next installment of my commemoration of Fr. Magnus Wenninger, I’ll excerpt various snippets from our correspondence together which illustrate Magnus’ unique perspective as well as highlight our collaboration together.

This is by no means meant to be a complete biography of Magnus’ life; rather, I hope you get to know Magnus a little through our particular correspondence. Of course everyone who knew Magnus had a slightly different experience.

I have about 65 letters in my folder of correspondence, which includes both letters from Magnus and my replies. I had to be selective in preparing my talk for the Joint Mathematics Meetings, and what follows are the excerpts I chose (in chronological order), along with some personal commentary.

I chose this excerpt for a few reasons. First, Tom Banchoff will also be giving a talk at the JMM in our session commemorating Magnus. But I also wanted to highlight the fact that Magnus, although already in his early seventies, was still very active. He was routinely invited to give talks and presentations, and also published papers occasionally.

And, as he indicated, when not otherwise occupied, he was always building models of one sort or another. And I really do mean always — even at conferences, he would often have a table set up where he would build models and discuss polyhedra with anyone interested enough to stop by and see what he was up to. Which was usually very many people….

This excerpt is from a letter dated 30 August 1993. It would take too long to set the stage for what this letter discusses, but I want to draw attention to the words “eye-balling” and “guessing” here.

Magnus’ focus was on building models, not on studying their precise mathematical properties. So if he didn’t actually know exact coordinates for a model, he would approximate, and build what he called a variation of the abstract mathematical model.

But here, Magnus is being somewhat humble — for his eye-balling and guessing is scaffolded on decades of model-building experience. It’s like giving a chef who graduated from Le Cordon Bleu a bowl of soup, say, and asking him or her to “guess” what spices were used. You’d get a lot more than just a guess….

This intuition about three-dimensional geometry was absolutely key to Magnus’ model-building success. If a spike on a model was too “pointy,” Magnus would simply design a variation with the features he wanted. And somehow, almost magically, they always turned out wonderfully.

13-9-93 — Excerpt dated 13 September 1993.

This next excerpt is dated 13 September 1993. The important quote here is the last complete sentence. As I mentioned last week, Magnus is best known for his trio of books Polyhedron Models, Spherical Models, and Dual Models.

Magnus is not the first to have enumerated all the uniform polyhedra; this had been done by a few others earlier — Coxeter (et al.) published a complete enumeration in 1954, Sopov proved completeness of the list in 1970, and Magnus’ Polyhedron Models was published in 1971.

And even though Polyhedron Models was always the most popular of the three books, Magnus’ felt that Dual Models was more significant since it was the first time all the duals of the uniform polyhedra were published.

I also note his use of words: “more significant.” I don’t think Magnus would never had said “I am more proud of Dual Models.” He was not a prideful man, always giving credit to his God for his insights and abilities.

But I think significance here is all relative. In terms of popularizing polyhedra and model building, Polyhedron Models was certainly the most significant of the three books. But in terms of a contribution to the mathematics of polyhedral geometry, I would agree with Magnus that Dual Models was the most significant of the three.

Of course Spherical Models is quite wonderful, too…. But in this book, Magnus indicated how to build (or approximately build) a wide variety of spherical models. However, moving in this direction is much more difficult mathematically and in some sense broader than just considering the uniform polyhedra or their duals, since there are many ways to build a spherical model based on a given polyhedron. So there would be little hope of being so comprehensive when discussing spherical models.

I include this next excerpt since it is one of the few times Magnus has mentioned his own abilities in his letters — it also echoes the excerpt from 30 August 1993. So there is little more to say here, except to note that he mentions his intuitive abilities are derived from building over a thousand models, not to any inherent ability. Again, an instance of his humility.

In this excerpt, I point to the phrase, “…holding a real model in my hands is still my greatest thrill.” Those of us who knew Magnus absolutely knew he felt this way about model building.

Because of his extensive polyhedral network, he had connections with mathematicians who helped him with precise, computer-generated diagrams to help him make models for his books. But he was less interested in the details of how the diagrams were made than he was in diving right into building the models themselves.

He was also well aware that computer graphics were an excellent tool which could be used to discover new and wonderful polyhedra — changing a parameter here or there would result in a different polyhedron, and it is easy to look at parameterized series of polyhedra with a computer.

But, this was all in aid of model building, not an end in itself for Magnus. There are plenty of others who focus on the computer graphics — as it should be, since any polyhedral enthusiast will have their own particular areas of interest and expertise.

Only five excerpts, and it’s time to finish! I didn’t realize I had so much to say about these snippets of correspondence. But I hope you are beginning to have some insight into who Magnus was and what he accomplished. I’ll have still more to say as in the next installment of excerpts from my correspondence with Magnus.

Fr. Magnus Wenninger, O.S.B., I

Fr. Magnus Wenninger was an astoundingly prolific polyhedron model builder, having built many thousands of models during his life. He is best known for his trio of model-building books: Polyhedron Models, Spherical Models, and Dual Models. Ask anyone who is an enthusiastic model builder, and I will bet that they own at least one, if not all three.

These books were important because they opened up the world of polyhedron model building to a much wider audience, giving detailed instructions on how to build dozens of models, as well as discussing the mathematics underlying polyhedra (although to a limited extent).

But perhaps equal to Fr. Wenninger’s model-building capabilities was his ability to connect polyhedron enthusiasts with one another. Because of his books, other model builders would correspond with him describing their particular interests, and he would connect them with others who had written to him with similar interests. In 1996, he created a mailing list where those interested could exchange ideas about all aspects of polyhedra. It was very active for several years, but because of the proliferation of sites about polyhedra in recent years, it is somewhat less so now. It is now maintained by Dr. Roman Mäder.

I was first introduced to Fr. Wenninger’s books by finding them among the stacks at the mathematics and science library at Carnegie Mellon. I don’t recall how many times I checked out Polyhedron Models; I was completely engrossed. I would page back and forth, over and over, looking for similarities between the various models and underlying geometrical patterns.

I began corresponding with Magnus in the summer of 1993. I still have all of our correspondence — I made sure to photocopy any letter I sent him so I would have a continuous record of our polyhedral conversations.

Sadly, Magnus Wenninger passed away last February at the age of 97. When I heard about this, I thought it would be fitting to organize an Invited Paper Session at the Joint Mathematics Meetings (on 12 January 2018 in San Diego) in his honor. I invited some of his colleagues from the polyhedron mailing list, and others as well.

This morning, I began reading through the letters exchanged with Magnus in preparation for a talk I will be giving in the session. Yes, this was in the day when people wrote letters, and moreover, wrote them by hand. Amazing! Such an interesting treasure trove of ideas and thoughts.

For my talk, I am excerpting text from Magnus’ letters to me, taking pictures of the text, and using these excerpts as the main body of slides. I am not sure how many others had an extensive correspondence with Magnus, but I thought this would provide a unique glimpse into Magnus’ life.

And so begins this series of posts commemorating Fr. Magnus Wenninger. I’ll continue this post by giving an overview of our relationship. Then later, I’ll share with you some excerpts from his letters to me, as well as provide commentary when appropriate. It is fitting that he should be remembered; organizing the Invited Paper Session in his memory as well as writing about him in my blog will serve as my contribution.

I was simply fascinated by the beauty, intricacy — and to an extent, simplicity — of three-dimensional polyhedron models. But I was also a graduate student at Carnegie Mellon in mathematics, which was not insignificant.

And just what was the significance of being a graduate student? Well, I was being trained to think rigorously, mathematically. At the time, however, all the accessible books on polyhedral geometry were at a relatively elementary level.

What I mean is this. When you study a polyhedron, there are many metrical features evident, such as edge lengths and various angles; for example, the angles between two faces of a polyhedron (like the 90° angles between faces of a cube). And in many of these books, these lengths and angles were given — but in most cases, only in tables with approximations to enough significant figures necessary to build reasonably accurate models.

I started to wonder how all those numbers were calculated. I wasn’t satisfied with approximate results; I wanted exact results. Thus began my polyhedral self-education, some thirty years ago.

So I began playing around, and when I finished graduate school, mustered up the courage to write Magnus. I was rather intimidated at the time — I had just started learning about polyhedra, and he had published at least three books on the subject! But it turns out Magnus was unusually generous with his time and talent, and replied within a few weeks.

My introductory letter was dated 5 July 1993, and the last letter I have from Magnus bears the date 22 October 1997. I am quite sure that this is because we continued our correspondence online. However, virtually all of that correspondence is lost, since most of it was conducted through a university email address I no longer have access to.

There are a few lingering emails in my gmail account, since my correspondence with Magnus waned after I left my first university position. But in addition to maintaining an active correspondence with Magnus, I did visit him a number of times at St. John’s Abbey in Collegeville, MN. I would stay in a guest room at the Abbey, and join Magnus for lunch and dinner. Throughout the day, I would visit Magnus’ room where we would talk about polyhedra, or perhaps I would work on my own, or just take a walk around the campus.

My interactions with Magnus were absolutely inspirational. Looking back at notes from those years, I am amazed at how much I accomplished — and this at a time when computer graphics were much less sophisticated than they are now. I would often share results with Magnus in my letters, and he would provide his unique perspective on my current work. Even when he was critical, he was unfailingly kind.

So in my next post, I’ll begin sharing excerpts from Magnus’ letters to me. That is, after all, how you get to know a person — one interaction at a time. Hopefully you will be able to get a sense of the humble, brilliant, generous man Magnus was and continues to be for those who knew him well. His legacy lives on through us, as we strive to be for others who Magnus was for us.

Bay Area Mathematical Artists, III

Another successful meeting of the Bay Area Mathematical Artists took place yesterday at the University of San Francisco! It was our largest group yet — seventeen participants, include three new faces. We’re gathering momentum….

Like last time, we began with a social half hour from 3:00–3:30. This gave people plenty of time to make their way to campus. I didn’t have the pleasure of participating, since the campus buildings require a card swipe on the weekends; I waited by the front door to let people in. But I did get to chat with everyone as they arrived.

We had a full agenda — four presenters took us right up to 5:00. The first speaker was Frank A. Farris of Santa Clara University, who gave a talk entitled Fibonacci Wallpaper Spirals.

He took inspiration from John Edmark’s talk on spirals at Bridges 2017 in Waterloo, which I wrote about in my blog last August (click here to read more). But Frank’s approach is rather different, since he works with functions in the complex plane.

He didn’t dive deeply into the mathematics in his talk, but he did want to let us know that he worked with students at Bowdoin College to create open-source software which will allow anyone to create amazing wallpaper patterns. You can download the software here.

Where do the Fibonacci numbers come in? Frank used the usual definition for the Fibonacci numbers, but used initial values which involved complex numbers instead of integers. This allowed him to create some unusually striking images. For more details, feel free to contact him at ffarris@scu.edu.

Next was our first student talk of the series, My Experience of Learning Math & Digital Art, given by Sepid Ebrahimi. Sepid is a student in my Mathematics and Digital Art course; she is a computer science major and is really enjoying learning to code in Processing.

First, Sepid mentioned wanting to incorporate elements into her work beyond simple points, circles, and rectangles. Her first project was to recreate an image of Rick and Morty, the two main characters in the eponymous cartoon series. She talked about moving from simple blocks to bezier curves in order to create smooth outlines.

Sepid then discussed her second project, which she is using for her Final Project in Mathematics and Digital Art. In order to incorporate sound into her work, she learned to program in Java to take advantage of already-existing libraries. She is creating a “live audio” program which takes sound input in real time, and based on the frequencies of the sound, changes the features of various geometrical objects in the video. Her demo was very fascinating, and all the more remarkable since she just started learning Processing a few months ago. For more information, you can contact Sepid at sepiiid.ebra@gmail.com.

The third talk, Conics from Polygons: the Chord Ratio Construction, was given by Scott Vorthmann. He is spreading the word about vZome, an open-source virtual environment where you can play with Zometools.

The basis of Scott’s talk was a simple chord ratio construction, which he is working on with David Hall. (Here is the GeoGebra worksheet if you would like to play with it.) The essential idea is illustrated below.

Begin with two segments, the red and green ones along the coordinate axes. Choose a ratio r. Now add a chord parallel to the second segment and r times as long — this gives the thick green segment at y = 1. Connect the dots to create the third segment, the thin green segment sloping up to the right at x = 1. Now iterate — take the second and third segments, draw a chord parallel to the second segment and r times as long (which is not shown in the figure), and connect the dots to form the fourth segment (the thin green segment sloping to the left).

Scott then proceeded to show us how this very simple construction, when iterated over and over with multiple starting segments, can produce some remarkable images.

Even though this is created using a two-dimensional algorithm, it really does look three-dimensional! Conic sections play a fundamental role in the geometry of the points generated at various iterations. Quadric surfaces in three dimensions also come into play as the two-dimensional images look like projections of quadric surfaces on the plane. Here is the GeoGebra worksheet which produced the graphic above. For more information, you can contact Scott at scott@vorthmann.org.

The final presenter was Stacy Speyer, who is currently an artist-in-residence at Planet Labs. (Click here to read more about art at Planet Labs.) She didn’t give a slideshow presentation, but rather brought with her several models she was working on as examples of Infinite Polyhedra Experiments with Planet’s Satellite Imagery.

One ongoing project at Planet Labs is planetary imaging. So Stacy is taking high-resolution topographical images and using them to create nets for polyhedra. She is particularly interested in “infinite polyhedra” (just google it!). As you can see in the image above, six squares meet at each vertex, and the polyhedron can be extended arbitrarily far in all directions.

One interesting feature of infinite polyhedra (as you will notice above) is that since you cannot actually create the entire polyhedron, you’ve got to stop somewhere. This means that you can actually see both sides of all the faces in this particular model. This adds a further dimension to artistic creativity. Feel free to contact Stacy at cubesandthings@gmail.com for more information!

We’ll have one more meeting this year. I am excited to see that we’re making so much progress in relatively little time. Presentations next time will include talks being prepared for the Joint Mathematics Meetings in San Diego this coming January, so stay tuned!

Bridges 2017 in Waterloo, Canada!

Bridges 2017 was in full swing last weekend, so now it’s time to share some of the highlights of the conference. Seems like they keep getting better each year!

The artwork was, as usual, quite spectacular. I’ll share a few favorites here, but you can go to the Bridges 2017 Gallery to see all the pieces in the exhibitions, along with descriptions by the artists.

My favorite painting was Prime numbers and cylinders by Stephen Campbell.

campbell

What is even more amazing than the piece itself is that Stephen makes his own paint! So each piece involves an incredible amount of work – and the results are worth it. Visit Stephen’s website to see more of his work and learn a bit more about his artistic process.

I also liked these open tilings of space by Frank Gould.

IMAG6969

Although they are simple in design, the overall effect is quite appealing. I often find that the fewer elements in a piece, the more difficult it is to coax them into interacting in a meaningful way. It is challenging to be a minimalist.

This lantern, Variations on Colourwave 17 — Mod 2, designed by Eva Knoll, shows a pattern created using modular arithmetic.

IMAG6988

The plenary talks this year were almost certainly the best I have ever experienced at any conference. Opening the conference was a talk by Damian Kulash of the band OK Go. He described his creative process and gave some insight to how he makes his unbelievable videos.

They really cannot be described in words – one example he talked about was this video filmed in zero gravity in an airplane! There are no tricks here — just unbridled creativity and cleverness.

Another favorite was the talk by John Edmark. Many of his ideas had a spiral theme, like the piece you see below.

edmark

But what is fascinating about his work is how it moves. When this structure is folded inward, it actually stretches out. This is difficult to describe in words, but you can see a video of this phenomenon on John’s website. What made his talk really interesting is that he discussed the mathematics behind the design of his work.

Stephen Orlando talked about his motion exposure photography.

orlando2

Although seemingly impossible, this image is just one long-exposure photograph — the colors you see were not added later. Stephen’s technique is to put programmable LED lights on a paddle and photograph a kayaker paddling across the water.

But where is the kayaker? Notice the background — it’s almost dark. It turns out that if the kayaker moves at a fast enough rate, the darkness does not allow enough time for an image of the kayaker to be captured. Simply amazing. Visit Stephen’s website for more examples of his motion exposure photographs.

There were also many other interesting talks given by Bridges participants, but there is not enough room to talk about them all. You can always go to the Bridges 2017 website and download any of the papers you’re interested in reading.

But Bridges is also more than just art and talks. Bridges participants are truly a community of like-minded people, so social and cultural events are also an important aspect of any Bridges conference.

I shared an AirBnB with Nick and his parents a short walk from campus — the house was spacious and comfortable, and really enhanced the Bridges experience. Sandy (Nick’s mother) wanted to host a gathering, so I invited several friends and colleagues over the Friday night of the conference for an informal get-together.

It was truly an inspiring evening! There were about fifteen of us, mostly from around the Bay area. Everyone was talking about mathematics and art — we were so engrossed, it turns out that no one even remembered to take any pictures! One recurring topic of discussion was the possibility of having some informal gatherings throughout the year where we could share our current thoughts and ideas. I think it may be possible to use space at the University of San Francisco — I’ve already begun looking into it.

Lunch each day was always from 12:00–2:00, so there was never any need to rush back. This left plenty of time for conversation, and often allowed time to admire the art exhibitions. It seems that you always noticed something new every time you walked through the displays.

On Saturday evening of the conference was a choir concert featuring a cappella voices singing a wide range of pieces spanning from the 15^th century to the present day. It was a very enjoyable performance; you can read more about it here.

The evening of Sunday, July 30^th, was the last evening of the conference for us. Many participants were going to Niagara Falls the next day, but we were all flying out on Monday. We decided to find a group and go out for dinner — and about a dozen of us ended up at a wonderful place for pizza called Famoso.

We chatted for quite some time, and then split up — some wanted to attend an informal music night/talent show on campus, but others (including me and the Mendlers) went to shoot some pool at a local pool hall. Again, a good time was had by all.

While the first day of the conference seemed a little slow, the rest just flew by. Another successful Bridges conference! Nick and I both had artwork exhibited and gave talks, caught up with friends and colleagues from all around the globe, enjoyed many good meals, and got our fill of mathematical art.

Although the location for the next Bridges conference is usually announced at the end of the last plenary talk, these is still some uncertainty about who will be hosting the conference next year. But regardless of where it is, you can expect that Nick and I will certainly be there!