Guest Blogger: Percival Q. Plumtwiddle, II

As promised, here is the second installment of my friend Percy’s essay on the significance of our Earthly existence being three-dimensional.  I hope these few words give food for much geometrical thought….

On Threeness (continued…)

Popping over a few dimensions in the meanwhile, the knowledgeable geometer is certainly aware that in dimension four and, indeed, in all dimensions exceeding this critical dimension, the situation is somewhat overwhelming, to say the least. So many thousands of uniform polyhedra abound that the prospect of their enumeration appears beyond the scope of even the most ambitious geometer. (I might recommend such a project, however, to an advanced practitioner of one of the Eastern religions who might, as a result of sufficient spiritual development, continue the undertaking in subsequent reincarnations.)

But in dimension three, how manageable our delight!  Seventy-odd uniform polyhedra clog the pores, multitudinous compounds and stellations frolic with the celestial spheres, etc.,  such as to render the average mortal awestruck.  Details are, of course, voluminous (though within reason), and more eminent geometers than yours truly have authored copious memoranda on said frolicking.  And there is yet much left unsaid, or perhaps, unmemorandized.  To presume that our friendly Supreme B. was unaware of such phantasmagoria would be heresy first class.  Fie!, I say, to such presumption!

As convincing as I have striven to be, I shall endeavor, with difficulty, not to be so narcissistic as to suppose that I have left no room for doubt in the reader’s mind. And so I entice with a few non-Euclidean morsels…

I shall not doubt the reader’s familiarity with the method for calculating the length of a circular arc, given the appropriate parameters. Perhaps less well-known is the method for calculating the area of a spherical triangle, with the attendant formulae relating the various parts of such a triangle only slightly more obscure. I purposefully omit their statements, secretly hoping that the reader will rediscover their simplicity and elegance afresh… In any case, the knowledgeable reader is well acquainted with the vast number of beautiful yet stable geodesic structures which may be constructed, the data for which constructions being directly calculated by way of the aforementioned simple formulae. Let not the skeptical reader be amused by such hyperbole — for one who has indeed fabricated such a structure or two with a few sheets of stiff paper and a bit of rubber cement will certainly view my descriptions as grave understatement.

In any event, as the informed reader is well aware, the formula for finding the area of the spherical triangles mentioned above may be successfully implemented by a child of ten, or perhaps a precocious child of eight, with a minimum of instruction, a pocket calculator, and, if necessary, the promise of an ice cream in the event of their successful completion. The informed reader is equally well aware that analogous calculations in four dimensions may be successfully implemented by no less that a ranking member of the cranially elite given the necessary data (and perhaps a few extraneous ones for good luck), a personal computer, and, if necessary, the promise of tenure in the event of their successful calculation. The superiority of the three-dimensional scenario, both fiscally and otherwise, is evident.

I pause, as a prelude to a conclusion, to ask the reader to reflect upon his or her stance vis a vis the proposed heightened status of Threeness. I imagine the reader to be of one of four minds. First, the reader may be insufferably bored at the current exegesis, in which case he or she will have undoubtedly ceased its perusal long ago. In this case, there is no reason for a continuation of discourse. Secondly, the reader may, to my ineffable delight, be wholly in agreement with the propositions contained herein. Should this be the case, I am not compelled to offer further justification for them. Thirdly, the reader may, to my profound disappointment, be immovably opposed to my thesis. (As an apology to those readers who fall in the second category, I must, rather that be labelled as a righteous fanatic, admit this frankly incomprehensible possibility.) I would not prolong the displeasure of such readers, those that there may be. Fourthly, the reader may be genuinely undecided, perhaps having never mused upon such matters previously, or perhaps still wrestling with deep foundational issues and at an impasse with regard to their resolution. In the former case, I urge an immediate excursion to that local library known for its excellent collection of volumes on the subject, followed by a thorough study of those treatises, and then a rereading of the current manuscript. In the latter case, I believe the matter rather more philosophical than mathematical, and I might suggest several worthwhile Buddhist sources (and caution the reader to avoid the twentieth-century European existentialists). In neither case would I find a need to argue further in this matter of Threeness.

And so, dear reader, I take my leave. It is my fondest hope that these few words have, at the very least, given cause for a leisurely intellectual frolic amidst sunny geometrical meadows, and at the very best, profoundly elevated the spirit. In either case, my work will not have been in vain.

Guest Blogger: Percival Q. Plumtwiddle, I

As I’m out of town for a few weeks, I thought I’d invite my good friend Percy Plumtwiddle to be a guest blogger.  He is an ardent enthusiast of geometry, and we have had many animated discussions on the geometry of three and higher dimensions.

But he has his stubborn side (don’t we all?).  And yes, his language is flowery and over the top, but well, we all have our eccentricities, too.  Nonetheless, he makes some interesting points — just how interesting, I’ll have to let you decide.

Also, his essay, while not too long, is a bit long for just one post — so I’ll break it into two installments.  I’ll return to my keyboard in a few weeks.  In the meantime, enjoy Percy’s unique perspective on geometry, and more broadly, life in general….

On Threeness

Were I God — and I assure you, dear reader, that such is not remotely an aspiration; and were it, even my dear friends and those who hold me in highest esteem (those that there may be) would chortle audibly at the consideration of such a prospect — and were it incumbent upon me to create a sentient being or two (admittedly, I would be somewhat embarrassed, were I God, to admit to the creation of many such beings which presently spoil an otherwise hospitable planet) — and had I the necessity of establishing a place of residence for the aforementioned sentient beings (not that it would be required, matter being at times inconvenient, especially when taking the form of a piece of chocolate cake hidden under one’s outer garments at the precise moment when a head-on collision with a rotund nephew is imminent) — I would not hesitate for a moment in making such a domicile, in dimension, three.

Lest the frivolous reader imagine such a dimensionality to be merely the whim of a somewhat eccentric, although omniscient, Supreme Being, or lest the pious reader imagine that such a dimensionality foreshadows the fairly recent (according to even the most conservative of cosmological estimates) doctrine of the Trinity popular in some Western religions, and lest the banal reader imagine that this third dimesion would be, as goes the cliche, a “charm,” and lest the reader of belles lettres be tempted to make a thoroughly trite reference to MacBeth — apply the cerebral brakes forthwith!  I intend to provide sound reason to dispel such imaginations, should the reader find it bearable to forage among slightly less green pastures.

I appeal not only to the reader’s intellect — being, unfortunately, acquainted much more intimately with self-proclaimed members of the cranially elite than I presently desire — but also to the reader’s aesthetic sensibilities, as potentially volatile as such an appeal may be.  But I have profound faith in this universal Threeness, so I proceed, fully cognizant of the perilous quagmire in which I may soon find myself immersed, should the fates allow.

Indeed, my argument is entirely geometrical, being of the school of thought that such argumentation is lacking in no essentials.  Should the reader be otherwise disposed, I urge an immediate cessation of the perusal of this document, the administration of a soothing tonic, and perhaps the leisurely reading of a light novella.

Several remarks of the Euclidean persuasion present themselves.  Oh, indeed, we are all familiar with the usual formulae for distance, angle, etc., etc.  These pose little difficulty, even in thirty-seven dimensions, once we are introduced to our good friend the Greek capital sigma.

But let us turn our eyes to the imaginitive world of polyhedra, a subject quite dear to my heart.  Another discussion entirely!  It is within this discussion that the force of my words comes to bear.  For indeed, I should be greatly surprised if when, upon strolling through the Pearly Gates and bidding good-day to St. Pete, I find that God did not have amongst his cabinet (a gaggle of geese, perhaps, but the precise nomenclature for an approbriation of Archangels momentarily eludes me) one for whom polyhedra are a consuming passion.

And what would give cause for raised eyebrows?  Follow along.  Examining your regular polyhedra of dimension two, we find squares, triangles, etc., etc., and, while these are no doubt pleasing to the eye, one might find oneself a trifle bored after pondering a polygon of forty-six sides.  For next comes the all-too-exciting forty-seven sider, and while one might appear to the casual observer nonplussed, the more astute attendant of human psychology would undoubtedly sense a consuming ennui.  Hardly worthy of God, or even a lesser member of his cabinet.  Of course, one may counter that I have neglected the likes of five-pointed stars, nineteen-pointed stars, and eighty-five-pointed stars, and one would be correct.  But these, too, soon become tiresome.

This ends the first installment of Percy Plumtwiddle’s essay, On Threeness.  Be sure to catch the second (and final) installment next week!

International Dodecahedron Day 2016!

I have always been fascinated by geometry in three dimensions.


I can still remember an eighth-grade project in my algebra class where we built polyhedra — mine was a white icosahedron with smaller, orange equilateral triangles connecting the midpoints of the edges.  This is what an icosahedron looks like:


I even naively tried to build an icosahedron by trying to glue twenty regular tetrahedra (triangular pyramids) together, thinking that if you took a face of the icosahedron and connected its vertices to the center of the polyhedron, the result would be a regular tetrahedron. It’s pretty close — but not quite there. I wondered why my model didn’t close up.

Next, I remember walking down the aisles of the mathematics and science library at Carnegie Mellon, looking at mathematical “picture books.” I don’t know how many times I checked out Polyhedron Models by Magnus Wenninger, just looking at all the photos of the paper polyhedron models, flipping back and forth between them, trying to see how they were related to each other.

With each model, there was a net — a set of connected polygons you could use to create the model. The net you see here is for a polyhedron called the rhombic dodecahedron.  I wondered how you could make each net.  Day028rhdodecaLater, I found out that there were lots of data published about various lengths and angles in different polyhedra — but those data were often numerical approximations, not exact values. What were the exact values?

These questions stimulated me to study more deeply — using tools such as coordinate geometry, linear algebra, and spherical trigonometry. I eventually answered many of the questions I asked so far, but of course generated many more questions, which were usually more difficult to answer.

Once I finished graduate school, I started writing a book on the mathematics of polyhedra, and eventually used it in a university-level geometry course. As I gained experience, I was asked to help design a senior capstone course for a local high school which used my text. A few years into this, my colleagues Todd Klauser and Sandy Spalt-Fulte helped organize a project where students in this senior course — as well as Todd’s other geometry students — went to a local middle school and taught the younger students how to make three-dimensional models of dodecahedra. And so Dodecahedron Day was born.

Dodecahedron Day is celebrated on December 5 of each year (for the 12 pentagons on a dodecahedron), and was first celebrated in 2005. Perhaps it’s a bit early in the year to talk about it — but just yesterday, I ran a booth at a fundraiser for the San Francisco Math Circle where we had students build three-dimensional models of different types of dodecahedra.


What I love about this type of activity is how much the students love it as well.  The dodecahedron you see here was created by a young girl of about eight or nine years old — she worked painstakingly with her glue stick for close to an hour getting it together.  Her focus was intense.  No, it wasn’t perfect.  But it was hers.

Interestingly, the parents of several students also built models — and took home nets to build more!  And one of my student assistants, Simon, got very creative with decorating a net for a small stellated dodecahedron.


Students have fun building polyhedra. Frankly, I think mathematical activities which are just fun are very useful activities — improving students’ attitudes about math is really critical to their success. Students perform better in subjects they like.

But in addition to being fun, building models requires focus and attention to detail, and also develops spatial abilities. In fact, an undergraduate in my 3D geometry course who later went on to get a Ph.D. in chemistry told me that my geometry course helped her in graduate school more than any of her chemistry courses! Just think about the geometry inherent in studying orbitals.

Over the years, I’ve developed many activities for Dodecahedron Day, and include some on the Day’s website For younger students, I’ve created activities involving pentominoes, since there are 12 pentominoes, each made up of five connected unit squares (keeping with the 12/5 theme). I’m sure others have and will continue to develop different activities — the important thing is that students take a day to truly enjoy doing geometry.

One thing I do insist upon, though, is that teachers don’t create a contest out of who makes the “best” dodecahedron. There’s too much competition in schools anyway, and it defeats the purpose of Dodecahedron Day to have a student who is genuinely proud of his model to leave the day thinking, “Mine was really nice, but hers was better. The teacher said so.”

The great thing is that once students learn basic model-making skills, they can search the internet for printable nets of almost any polyhedron they can think of. Pieces can be made from different colors, and particular color arrangmements of pieces can create really beautiful models.

Another reason for writing about Dodecahedron Day a bit early is that it really takes some planning. If a school or school district wants to set aside December 5 for activities, it’s best to make that decision before the school year starts — otherwise there may be time pressure to coordinate with course syllabi, school leadership and other teachers. Further, if teachers want to take their students to other schools so that their students can teach younger pupils how to build models, it helps to develop relationships with local schools if they don’t already exist. I can tell you from experience, the earlier you start, the better.

So if you’re a student reading this, ask your teacher to celebrate Dodecahedron Day this year! If you’re a teacher or school principal, think about it — and feel free to comment with questions, concerns, or ideas. I’m happy to help in any way I can.

And if you’re not from the US, consider introducing Dodecahedron Day into schools in your country! You’ll notice the title of today’s post — I’m hoping to make Dodecahedron Day 2016 an international event, and I can’t do it alone. Perhaps materials need to be translated, or simply reformatted for A4 paper…. By starting early, it’s possible to create an enjoyable experience for everyone involved.

So in anticipation, Happy Dodecahedron Day 2016! Let’s make this a day to introduce a passion for geometry to students all around the world!