## My Polyhedra Textbook, II

In my last post, I talked about a textbook I had written to use in a course which, among other things, introduced students to a non-Euclidean geometry — spherical geometry.  And while the primary purpose of the text was for teaching a college-level course, the actual content of the text had a different origin.

As I have mentioned previously, I got interested in polyhedra during graduate school, and was very fond of taking books about polyhedron models out of our mathematics library at Carnegie Mellon.  It was easy enough to photocopy the nets provided, or use the numerical data to construct nets on my own.

But what was missing for me, a mathematician-in-training, was a more rigorous discussion of these intriguing models.  For example, Wenninger’s Polyhedron Models contains many nets with some idea of how to construct them from a more general geometric point of view, but few details.  And his Spherical Models contains many tables of data of angular measures to three decimal places, but many derivations are missing.  Other books included metrical data such as the measures of dihedral and edge angles and circumradii, but again, often only numerically.

I should point out that this is not necessarily a criticism of these wonderful books, since each book had its own purpose.  What I found, though, was that there was really no book out there which took a more mathematically sophisticated viewpoint at an intermediate level.

So I decided that if I wanted to know more about polyhedra — that is, know the exact metrical data associated with polyhedra, not just numerical approximations — I needed to do some work on my own.

I used two approaches — coordinates and linear algebra, and spherical trigonometry.  In both cases, I wanted precise results — so typically, this meant knowing exact values for the cosines of dihedral and edges angles, for example.  When you know the cosine of an angle, you know the angle.

Why the cosine?  The sine has the disadvantage of being ambiguous, since many dihedral angles of polyhedra are obtuse — so you need more information than just the sine of the angle.  Using the tangent involved the troublesome case of $\dfrac\pi2$ in formulas — which occurred rarely, but was still a nuisance.  Some sources use the tangent of the half-angle, but this often involves additional calculations.

For my geometry course, I focused on spherical trigonometry — recall that I did not want linear algebra as a prerequisite for the course so that it would be accessible to a broader student demographic.  But I still wanted to keep the mathematical rigor — use spherical trigonometry to calculate exact values of the cosines of relevant angles.

To give you an idea of what’s involved, let’s calculate the dihedral angle of a dodecahedron — that is, the angle between any two pentagonal faces of a regular dodecahedron. To use spherical trigonometry (see a previous post on spherical geometry for a refresher), we imagine a small sphere centered at a vertex of the dodecahedron — it will carve out a spherical triangle, as shown below. Here, a is a vertex of the dodecahedron, and points b, c, and d are the points where the small sphere intersects the edges of the dodecahedron meeting at a.  This creates a spherical triangle whose sides all have measure 108°, since the interior angles of a regular pentagon all have measure 108°.

The angles between the sides of the spherical triangle with vertices a, b, and c are the dihedral angles of the dodecahedron, whose measure we call D.  We may apply the law of cosines for spherical triangles in order to find (see the relevant Wikipedia page; in the textbook, spherical trigonometric formulas are discussed in detail): $\cos108^\circ=\cos108^\circ\cos108^\circ+\sin108^\circ\sin108^\circ\cos D.$

To find $\cos D,$ we need to know the following (also explained in the text): $\cos108^\circ=\dfrac12(1-\tau),\quad\sin108^\circ=\dfrac12\sqrt{\tau+2},$

where $\tau$ is the golden ratio.  (I use $\tau$ for the golden ratio as this is the notation Coxeter used in his well-known Regular Polytopes, which inspired generations of polyhedral enthusiasts.)

Solving yields $\cos D=-\dfrac1{\sqrt5},$

so that D is approximately 116.6°.

Now this is not the only way to find D; it is possible to find Cartesian coordinates for the vertices of the dodecahedron and use some linear algebra, for example.  But using spherical trigonometry is straightforward and elegant — and is surprisingly versatile.

When I teach my polyhedra course, we use spherical trigonometry to find edge and dihedral angles of all the Platonic and Archimedean solids.  Further, we use spherical trigonometry to design geodesic models, like the 4-frequency icosahedron shown here. So the emphasis is on applying spherical trigonometry to the construction of physical models.  In a typical two-lecture-per-week course, one lecture is always a hands-on laboratory, where we build a polyhedron or geodesic model we studied previously.  We never build a model without knowing precisely how it is designed — we first understand the mathematics of the model, and then we build it.  Included in the text is a week-by-week outline of how I’ve used the text in the classroom.

Now the classroom textbook is just Part I of the book.  Part II is a bit more technical, and intended for the true polyhedral enthusiast.  It turns out that spherical trigonometry is a powerful tool for studying polyhedra.  In fact, the dihedral angles of all the uniform polyhedra may be calculated using spherical trigonometry — even the most complex snub polyhedra.  However, it is sometimes necessary to solve sixth-degree polynomials in order to do so!

I thought to add Part II to the volume as I had already done all the calculations some years ago.  And, to my knowledge, the calculation of the dihedral angles of all the uniform polyhedra using spherical trigonometry has not been published before.  So I hope to contribute something to the literature of classical polyhedral geometry by publishing this book, in a way that someone with a modest mathematical background can understand.

As I continue with the book project, I’ll post updates as I discover interesting geometrical tidbits along the way!

## My Polyhedra Textbook, I

Now that I’ve moved on to semi-retirement, there is time to take on a 25-year-old project:  my textbook on polyhedra!

I became interested in polyhedra during graduate school, when I was fascinated by the trio of books by Magnus Wenninger:  Polyhedron Models, Dual Models, and Spherical Models.  I don’t know how many times I checked out Polyhedron Models from the mathematics library at Carnegie Mellon University, but I have clear memories of flipping pages back and forth over and over again, trying to understand the various and subtle relationships among the 100+ models shown in the book.

Of course this prompted me to build my own models — but at a time when there weren’t nets available online.  So I designed my own nets using Postscript.  Since I loved coding, this was no problem at all.  I essentially wrote my own turtle graphics package in Postscript, and used this to create any net I wanted.

Having moved around several times in recent years, I have very few models that I’ve built.  And like many model builders, I’ve given most of them away, anyway.  But here are a few I made for my friend Sandy (whom I’m visiting as I write this post).

Eventually I finished graduate school and went on to my first university teaching position, where I stayed for fourteen years.  I was at a small, liberal arts school, where many of the mathematics majors were destined to be middle school or high school mathematics teachers.  Moreover, I was to replace a retiring faculty member who had taught a course entitled, “Higher Geometry.”

I eagerly agreed to take on this mantle, but was interested in shifting the focus.  In particular, I wanted to make the course about polyhedra rather than the usual content of a Higher Geometry course, which often included a lengthy discussion of hyperbolic geometry.

Allow me a moment to step on my pedagogical soapbox here.  Yes, I understand the importance of introducing students to a non-Euclidean geometry.  But as many of my students were prospective teachers, I knew there was really no way they would be able to introduce hyperbolic geometry to their students.

But spherical geometry is also an example of a non-Euclidean geometry, and further, you can actually build physical models of non-Euclidean objects by building geodesic models.  So while you can’t really see that a triangle in hyperbolic geometry has an angle sum less than 180°, you can actually see that a spherical triangle has an angle sum greater than 180°.

You can also look at axiomatics in spherical geometry, with the added bonus that you expose students to the important concept of duality.  Finally, you can ramp up the mathematical content of such a study by introducing students to spherical trigonometry.  I should remark that, very likely, fewer than 1 in 10 (or perhaps even 1 in 100) mathematicians can rattle off the cosine law for spherical triangles — so exposing students to spherical trigonometry is significant.  It’s practical as well — think of flight paths — but I never went into this application as there just wasn’t enough time.

Stepping off my soapbox now…suffice it to say that I was given free reign to retool the Higher Geometry course.

I decided to have the course be centered on spherical trigonometry.  Why?  First, the course needed some substantial mathematical content; spherical trigonometry can be quite challenging, especially some of the more involved derivations.  This also allowed for a fairly detailed study of polyhedra, as the edge and dihedral angles of all the uniform polyhedra can be found using spherical trigonometry.

Now it is possible to find edge and dihedral angles of polyhedra in other ways, but these usually involve linear algebra applied to Cartesian coordinates in three dimensions.  And in the typical undergraduate curriculum, linear algebra follows the calculus sequence.

So if I wanted the course to be accessible to other students — such as those needing a mathematics elective but were too advanced for, say, college algebra — I couldn’t have linear algebra as a prerequisite.

And so a new “Higher Geometry” was born.  I did eventually rename the course to “Polyhedra and Geodesic Structures,” as it was more apt — one main application of spherical trigonometry I introduced was building spherical models, like those described extensively in Wenninger’s Spherical Models.  It was a highly successful course, which I taught off and on at various institutions for about twenty years.  I also conducted any number of workshops for both teachers and students of all ages over the same time span.

Essentially, students and teachers of all ages just loved the hands-on aspect of building polyhedra and spherical models.  They often commented on how building their own models made mathematics “real.”  There was always the added bonus that they got to take their work home with them!

Yes, model building is a fun activity.  But I always made sure to balance content with the hands-on laboratory experience.  We never built any models without understanding some aspect of the geometry underlying the models.

Naturally, that geometry varied with the students involved.  For middle school students, working with spherical trigonometry was far too advanced.  But we could always see how Euler’s formula applies to convex polyhedra.

In my university-level course, we actually proved Euler’s formula using spherical geometry with the method attribute to Legendre; despite others’ claims to the contrary, it is in fact the most elegant proof of Euler’s formula….

And this is just the first part of the book!  In my next post, I’ll say a little more about the genesis of the first part, and then go on to describe the second part of the book.  Expect a long thread about polyhedra and three-dimensional geometry in the upcoming months….

## Enumerating the Platonic Solids

The past few weeks, I outlined my approach to a series of lectures on polyhedra.  One of my constraints is that students will not have seen a lot of trigonometry yet, and will not have been exposed to three-dimensional Cartesian coordinates.  But there is Euler’s Formula!  I just finished a pair of lectures on the algebraic enumeration of the Platonic solids using Euler’s Formula, and I thought others might be interested as well.

As a reminder, Euler’s Formula states that if $V,$ $E,$ and $F$ are the number of vertices, edges, and faces, respectively, on a convex polyhedron, then $V-E+F=2.$

How might we use this formula to enumerate the Platonic Solids?  We need to make sure we agree on what a Platonic Solid is:  a convex polyhedron with all the same regular polygon for faces, and with the same number meeting at each vertex.

To use this definition, we will define a few more variables:  let $p$ denote the number of sides on the regular polygons, and let $q$ denote the number of polygons meeting at each vertex of the Platonic solid.  (Those familiar with polyhedra will recognize these as the usual variables.)

The trick is to count the number of sides and vertices on all the polygons in two different ways.  For example, since there are $F$ polygons on the Platonic solid, each having $p$ sides, there are a total of $pF$ sides on all of the polygons. But notice that when we build a cube from six squares, two sides of the squares meet at each edge of the cube.  This implies that $2E$ also counts all of the sides on the polygons.  Since we are counting the same thing in two different ways, we have $pF=2E.$

We may similarly count all the vertices on the polygons as well.  Of course since a regular polygon with $p$ sides also has $p$ vertices, there are $pF$ vertices on all of the polygons.

But notice that when we put the squares together, three vertices from the squares meet at a vertex of the cube.  Thus, if there are $V$ vertices on a Platonic Solid, and if $q$ vertices of the polygons come together at each one, then it must be that $qV$ is the total number of vertices on all of the polygons.  Again, having counted the same thing in two different ways, we have $pF=qV.$

Thus, so far we have $V-E+F=2,\quad pF=2E,\quad pF=qV.$

Note that we have three equations in five variables here; in general, such a system has infinitely many solutions.  But we have additional constraints here — note that all variables are counting some feature of a Platonic Solid, and so all must be positive integers.

Also, since a regular polygon has at least three sides, we must have $p\ge3,$ and since at least three polygons must come together at the vertex of a convex polyhedron, we must also have $q\ge3.$

These additional constraints will guarantee a finite (as we know!) number of solutions.  So let’s go about solving this system.  The simplest approach is to solve the last two equations above for $E$ and $V$ and substitute into Euler’s Formula, yielding $\dfrac{pF}q-\dfrac{pF}2+F=2.$

Now divide through by $F$ and observe that $F>0,$ so that $\dfrac pq-\dfrac p2+1>0.$

Multiply through by $2q$ and rearrange terms, giving $pq-2p-2q<0.$

How should we go about solving this inequality?  There’s a nice trick here:  add $4$ to both sides so that the left-hand side factors nicely: $(p-2)(q-2)<4.$

Now we are almost done!  Since $p,q\ge3,$ then $p-2$ and $q-2$ must both be integers at least $1;$ but since their product must be less than $4,$ they can be at most $3.$

This directly implies that $p$ and $q$ must be $3, 4,$ or $5.$

This leaves only nine possibilities — but of course, not all options need be considered.  For example, if $p=q=5,$ then $(p-2)(q-2)=9>4,$

and so does not represent a valid solution.  But when $p=3$ and $q=4,$ we have the octahedron, since $p=3$ means that the polygons on the Platonic Solid are equilateral triangles, and $q=4$ means that four triangles meet at each vertex.

So out of these nine possibilities to consider, there are just five options for $p$ and $q$ which satisfy the inequality $(p-2)(q-2)<4.$  And since each pair corresponds to a Platonic Solid, this implies that there are just five of them, as enumerated in the following table: Actually, this implies that there are at most five Platonic Solids.  How do we know that twelve pentagons actually fit together exactly to form a regular dodecahedron?  A further argument is necessary here to be complete.  But for the purposes of my lectures, I just show images of these Platonic Solids, with the presumption that they do, in fact, exist.

Now keep in mind that in an earlier lecture, I enumerated the Platonic Solids using a geometrical approach; that is, by looking at those with triangular faces, square faces, etc.  I like the problem of enumerating the Platonic Solids since the geometric and algebraic methods are so different, and emphasize different aspects of the problem.  Further, both methods are fairly accessible to good algebra students.  The question of when to take an algebraic approach rather than a geometric approach to a geometry problem is frequently difficult for students to answer; hopefully, looking at this problem from both perspectives will give students more insight into this question.

## Teaching Three-Dimensional Geometry, III

This is the last of a three-part series on teaching three-dimensional geometry.  A few weeks ago, I had begun describing how I would go about putting together a series of about 20 online videos on 3D geometry, each lasting 5–7 minutes.  I just finished a discussion of buckyballs, and why regardless of the number of hexagonal faces on a buckyball, there are always exactly 12 pentagonal faces.

Euler’s Formula was key.  We’ll look at another application of Euler’s Formula, but before doing so, I’d like to point out that students at this level have not encountered Cartesian coordinates in three dimensions, and so I need to find things to talk about at an accessible level.

On to the truncation of polyhedra!  Again, we can apply Euler’s Formula, but it helps to think about the process systematically.  You can count the number of vertices, edges, and faces on a truncated cube, for example, one at a time — but little is gained from a brute force approach.  By thinking more geometrically, we would notice that each edge of the original cube contributes two vertices to the truncated cube, giving a total of 24 vertices.

We can continue on in this fashion, counting as efficiently as possible.  This sets the stage for a discussion of Archimedean solids in general.  A proof of the enumeration of the Archimedean solids is beyond the scope of a single lecture, but the important geometrical ideas can still be addressed.

This concludes the set of lectures on polyhedra in three dimensions.  Of course there is a lot more that can be said, but I need to make sure I get to some other topics.

Like spherical geometry, for instance, next on the slate.  There are two approaches one typically takes, depending how you define a point in spherical geometry.  There is a nice duality of theorems if you define a Point in this new geometry as a pair of antipodal points on a sphere, and a Line as a great circle on a sphere.  Thus two distinct Lines uniquely determine a Point, and two distinct Points uniquely determine a line.

This is a bit abstract for a first go at spherical geometry, so I plan to define a Point as just an ordinary point on a sphere, and a Line as a great circle.  Two points no longer uniquely determine a Line, since there are infinitely many Lines through two antipodal Points.

But still, there are lots of interesting things to discuss.  For example, there is no such thing as a pair parallel lines on a sphere:  two distinct Lines always intersect.

Triangles are also intriguing.  On the sphere, the sides are also angles, measured by the angle subtended at the center of the sphere.  So all together, there are six angular measures in any triangle.

Since students will not have had a lot of exposure to trigonometry at this point, I won’t discuss many of the neat spherical trigonometric formulas.  But still, there is the fact the angle sum of a spherical triangle is always greater than $180^\circ.$  And the fact that similarity and congruence on the sphere are the same concept, unlike in Euclidean geometry.  For example, if the angles in a Euclidean triangle are the same in pairs, the triangles are similar.  But on a sphere, if the angles of two spherical triangles measured the same in pairs, they would necessarily have to be congruent.

In other words, students are getting further exposure to non-Euclidean geometries.  (I did a lecture on inversive geometry in a previous section.)  One nice and accessible proof in spherical geometry is the proof that the area of a spherical triangle is proportional to its spherical excess — that is, how much the angle sum is greater than $180^\circ.$  So there will be something  I can talk about without needing to say the proof is too complicated to include….

The final topic I plan to address is higher-dimensional geometry.  The first natural go-to here is the hypercube.  Students are always intrigued by a fourth spatial dimension.  Ask a typical student who hasn’t been exposed to these ideas what the fourth dimension is, and the answer you invariably get is “time.”  So you have to do some work getting them to think outside of that box they’ve lived in for so long.

One thing I like about hypercubes is the different ways you can visualize them in two dimensions. Viewed this way, you can see the black cube being moved along a direction perpendicular to itself to obtain the blue cube.  Of course the process is necessarily distorted since we’re looking at a static image. This perspective highlights a pair of opposite cubes — the green one in the middle, and the outer shell — and the six cubes adjacent to both. And this perspective is just aesthetically very pleasing, and also has the nice property that every one of the eight cubes looks exactly the same, except for a rotation.  Again, there won’t be any four-dimensional Cartesian coordinates, but still, there will be plenty to talk about.

I plan to wrap up the series with a discussion of volumes in higher dimensions.  As I mentioned last week, I’d like to discuss why you should avoid peeling a 100-dimensional potato….

Thinking by analogy, it is not difficult to motivate the fact that the volume of a sphere $n$ dimensions is of the form $Kr^n.$

Now let’s look at peeling a potato in three dimensions, assuming it’s roughly spherical.  If you were a practiced potato peeler, maybe you could get away with the thickness of your potato peels being, say, just 1% of the radius of your potato.  This leaves the radius of your peeled potato as $0.99r,$ and calculating a simple ratio reveals that you’ve got $0.99^3\approx0.97$ of your potato left.

Extend this idea into higher dimensions.  If your potato-peeling expertise is as good in higher dimensions, you’ll have $0.99^n$ of your potato left, where $n$ is the number of dimensions of your potato.  Now $0.99^{100}\approx 0.366,$ so after you’ve peeled your potato, you’ve only got about one-third of it left!

What’s happening here is that as you go up in dimension, there is more volume near the surface of objects than there is near the center.  This is difficult to intuit from two and three dimensions, where it seems the opposite is the case.  Nonetheless, this discussion gives at least some intuition about volumes in higher dimensions.

And that’s it!  I’m looking forward to making these videos; I actually made my first set of slides today.  As usual, if I come across anything startling or unusual during the process, I’ll be sure to post about it!

## Teaching Three-Dimensional Geometry, II

A few weeks ago, I began a discussion of what I’d be presenting in a series of twenty (or so) 5—7 minute videos on three-dimensional geometry. I didn’t get very far then, so it’s time to continue….

So to recap a bit, I’ll begin with the usual cones/cylinders/spheres, looking at surface areas and contrasting flat surfaces with the surface of a sphere. Then on to a prelude to calculus by looking at the volume of a cone as a limiting case of a stack of circular disks.

Next, it’s on to polyhedra! A favorite topic of mine, certainly. Polyhedra are interesting, even from the very beginning, since there is still no accepted definition of what a polyhedron actually is. The exception is for convex polyhedra; a perfectly good definition of a convex polyhedron is the convex hull of a finite set of points not all lying in a single plane. Easy enough.

But once you move on to nonconvexity, uncertainties abound. For example, from a historical perspective, sometimes the object below was a polyhedron, and sometimes it wasn’t. Sounds odd, but whether or not you consider this object a polyhedron depends on how you look at the top “face,” which is a square with a smaller square removed from the center. Now is this “face” a polygon, or not? Many definitions of a polygon would exclude this geometrical object – which is problematic if you want to say that a polyhedron has polygons as faces. So this brings us to a definition of a polygon, which is problematic in its own way – to see why, you can look at a previous post of mine on the definition of a polygon.  Now the point here is not to resolve the issue in an elementary lecture, but rather point out that mathematics is not “black-and-white,” as students tend to believe. Also, it provides a nice example of the importance of definitions in mathematics.

Now this would be discussed briefly in just one video. Next would be the (obligatory) Platonic solids – where else is there to begin? The simplest starting point is the geometric enumeration by looking at what types of polygons – and how many – can appear at any given vertex of a Platonic solid. This enumeration is straightforward enough.

Next, I plan on computing the volume of a regular tetrahedron using the usual $Bh/3$ formula. This is not really exciting in and of itself, but in the next lecture, I plan to find the volume of a regular tetrahedron by inscribing it in the usual way in the cube by joining alternate vertices.

Of course you get the same result. But for those of us who work a lot in three-dimensional space, we deeply understand the simple algebraic equation, $2 \times 4=8.$ What I’m referring to, specifically, is that the number of vertices on a three-dimensional simplex is half the number of vertices of a three-dimensional hypercube.

This simple fact is at the heart of any number of intriguing geometrical relationships between polyhedra in three dimensions. In particular, and quite importantly, the simplex and the cross-polytope together fill space. This relationship is at the heart of many architectural constructions in additional to generating other tilings of space with Archimedean solids. But most students have never seen this illustrated before, so I think it is important to include.

Then on to a geometry/algebra relationship: having enumerated the Platonic solids geometrically, how do we proceed to take an algebraic approach? A fairly direct way is to use Euler’s formula to find an algebraic enumeration.

No, I don’t intend to prove Euler’s formula; by far my favorite (and best!) is Legendre’s proof which involves projecting a polyhedron onto a sphere and looking at the areas of the spherical polygons created. This is a bit beyond the scope of this series of videos; there simply isn’t time for everything. But it is important to note the role that convexity plays here; yes, there are other formulas for polyhedra which are not essentially “spheres,” but this is not the place to discuss them.

Next, I want to talk about “buckyballs.” I still have somewhat of a pet peeve about the nomenclature – Buckminster Fuller did not invent the truncated icosahedron – and so the physicists who named this molecule were, in my opinion, polyhedrally rather naïve. But, sadly (as is the case so many times), they did not come to me first before making such a decision…

The polyhedrally interesting fact about buckyballs is this: if a polyhedron has just pentagonal and hexagonal faces, three meeting at every vertex, then there must be exactly twelve pentagons. Always.

Now I know that the polyhedrally savvy among you are well aware of this – but for those who aren’t, I’ll show you the beautiful and very short proof. Once you’ve seen the idea, I don’t think you’ll ever be able to forget it. It’s just remarkable – even with 123,456,789 hexagons, just 12 pentagons.

So let $P$ represent the number of pentagons on the buckyball, and $H$ represent the number of hexagons. Then the number of vertices $V$ is given by $V=\dfrac{5P+6H}{3},$

since each pentagon contributes five vertices, each hexagon contributes six, and three vertices of the polygons meet at each vertex of the buckyball.

Moreover, the number of edges is given by $E=\dfrac{5P+6H}{2},$

since the polygons on the buckyball meet edge-to-edge. Of course, $F=P+H,$ since the faces are just the pentagons and hexagons. Substitute these expressions into Euler’s formula $V-E+F=2,$

and what happens? It turns out that $H$ cancels out, leaving $P=12!$

Amazes me every time. But what I like about this fact is that it is accessible just knowing Euler’s formula – no more advanced concepts are necessary.

And yes, there’s more! This is now Lecture #12 of my series, so I have a few more to describe to you. Until next time, when I caution you (rather strongly) against peeling a 100-dimensional potato….

## Bay Area Mathematical Artists Seminars, XI

This past weekend marked the eleventh meeting of the Bay Area Mathematical Artists Seminars.  Our host this month was Scott Vorthmann, the mastermind behind vZome.  Scott lives in Saratoga, and so those participants who live in the San Jose area were glad of the short commute.

It seems that the content of our seminars is limited only by the creativity of the artists involved, meaning fairly limitless….  Scott invited anyone interested to come early — 1:00 instead of our usual 3:00 — and be involved in a Zome “build;” that is, the construction of a large and intricate model using Zome tools.  Today’s model?  The omnitruncated 24-cell!

This is not the place to have a lengthy discussion of polytopes in four dimensions.  In a nutshell, the 24-cell is a polytope in four dimensions with 24 octahedral facets.  This polytope is truncated in a particular way (called omintruncation), and then projected into three-dimensional space.

But there is just one problem with the projection Scott wanted to build.  You can’t build it with the standard Zome kit!  No matter.  Scott designed and 3D-printed his own struts — olive, maroon, and lavender.  If you’ve ever played around with ZomeTools, you’ll understand what a remarkable feat of design and engineering this is.

The building process is a modular one — six pieces like the one shown below needed to be built and painstakingly assembled together. Scott built two of the modules before anyone arrived, so we had something to work from.  That left just four more to complete….

The modules were almost done, but we needed to move on.  In addition to the Zome build, we had two other short presentations.   Andrea and Andy were planning to present a workshop at Bridges 2018 in Stockholm, but at the last minute, were unable to attend.  So they brought their ideas to present to us.

The basic idea is to encode a two-dimensional image using two overlays, as shown here. Your friend has an apparently random grid (pad) of black and white squares.  You want to send him a secret message; only you and he have the pad.  So you send him a second grid of black and white squares so that when correctly overlaid on the pad, an image is produced.

This is a great activity for younger students, too, since it can be done with premade templates and graph paper.  And even though Andrea and Andy were not able to attend Bridges, their workshop paper was accepted, and so it is in the Bridges archives.  So if you want to learn more about this method of encryption, you can read all the details about the process in their paper in the Bridges archives.

Our next short presentation was by pianist Hans Boepple, a colleague of Frank Farris at Santa Clara University.  Frank happened to have a very stimulating conversation with Hans about a mathematics/music phenomenon, and thought he might like to present his idea at our meeting.

The idea came from a time when Hans happened to look down a metal cylinder of tubing, like you would find at a hardware store.  It seemed that there was an interesting pattern of reflections along the sides of the tubing, and knowing about music and the overtone series, he wondered if there was any connection with music.

Here is part of a computer-generated image of what Hans produced using paper and pencil many years ago: How was this picture generated?  Below is how you’d start making the image. You can see that the red lines take two zigzags to move from one corner of the rectangle to another, the blue lines take three zigzags, the green four, and the gold lines take five.  If you keep adding more and more lines, you get rather complex and beautiful patterns like the one shown above.  Those familiar with the overtone series will see an immediate connection.

Of course, the mathematical question is about proving various properties of this pattern.  It turns out that the patterns are related to the Ford circles; BAMAS participant Jacob Rus has created an interactive version of this diagram.  Feel free to explore!

In any case, we were delighted that Hans could join us and share his fascination with the relationship between mathematics and music.  You can  learn more about Hans in this interview in The Santa Clara, which is Santa Clara University’s school newspaper.

When Hans finished his presentation, it was time to finish building the omnitruncated 24-cell.  It was quite amazing, as Scott is certainly one of the foremost experts on ZomeTools in the world.  Here is the finished sculpture, suspended from the ceiling in his home.  Just getting the model up there was an engineering feat in its own right! It is difficult to describe the intricacy of this model from just a few pictures. Here is an intriguing perspective of the model, highlighting the parallelism of the blue Zome struts.  It seems there is no end to the geometrical relationships you can find hidden within this model.

And, as usual, the afternoon didn’t end there.  Scott arranged to have Thai food — one of our favorites! — catered in, and we all chipped in our fair share.  We all were having such a great time, the last of us didn’t leave until about 8:30 in the evening.  Another successful seminar!

It is quite heartwarming to see so many so willing to take on hosting our Bay Area Mathematical Artists Seminars.  We have all enjoyed these meetings so much, and we are so glad they continue to happen.  I am confident there will be many, many more delightful Saturday afternoons to experience….

## Teaching Three-Dimensional Geometry, I

I have recently had a rather unusual opportunity.  I’ve talked a bit over the last few months about my consulting work producing online videos for a flipped classroom; I’ve been working busily on the Geometry unit.

Now the last section of this unit is on three-dimensional geometry, and I’ve been given pretty free reign as to what to cover in this 20-lectures series of 5-7 minute videos.  And given my interest in polyhedra (which I could focus on exclusively with no shortage of things to discuss!), I felt I had a good start.

But the challenge was also to cover some traditional topics (cones, cylinders, spheres, etc.) — as well as more advanced topics — while not using mathematics beyond what I’ve used in the first several sections of the Geometry unit.

There is, of course, no “correct” answer to this problem.  But I thought I’d share how I’d approach this series of lectures, since geometry is such a passion of mine — and I know it is for many readers as well.  The process of reforming high school geometry courses is now well underway; I hope to contribute to this discussion with today’s post.

Where to start?  Cones and cylinders — a very traditional beginning.  But I thought I’d start with surface areas.  Now for cylinders, this is pretty straightforward.  It’s not much more difficult for cones, but the approach is less obvious than for cylinders.

Earlier in the unit, we derived the formula for the area of a sector of a circle, so finding the lateral surface area of a cone is a nice opportunity to revisit this topic.  And of course, finding the lateral surface area of a cylinder involves just finding the area of a rectangle.

Now what do both of these problems have in common?  Their solution implies that cones and cylinders are flat.  In other words, we reduce what is apparently a three-dimensional problem (the surface area of a three-dimensional object) to a two-dimensional problem.

This is in sharp contrast to finding the surface area of a sphere — you can’t flatten out a sphere.  In fact, the entire science of cartography has evolved specifically in response to this inability.

So this is a nice chance to introduce a little differential geometry!  And no, I don’t really intend to go into differential geometry in any detail — but why not take just a minute in a lecture involving spheres to comment on why the formulas for the surface areas of cones and cylinders are fairly easy to derive, and why — at this level — we’re just given the formula for the surface area of a sphere.

I try to mention such ideas as frequently as I can — pointing out contrasts and connections which go beyond the usual presentation.  Sure, it may be lost on many or most students, but it just may provide that small spark for another.

I think such comments also get at the idea that mathematics is not a series of problems with answers at the back of the book…on the face of it, there is no apparent reason for a student to think that finding the surface area of a cone would be simpler than finding the surface area of a sphere.  This discussion gets them thinking.

Next, I’m planning to discuss Archimedes’ inscription of a sphere in a cylinder (which involves the relative volumes).  This is a bit more straightforward, and it’s a nice way to bring in a little history.

I also plan to look at inscribing a sphere in a right circular cone whose slant height is the same as the diameter of the base, so that we can look at a two-dimensional cross-section to solve the problem.  In particular, this revisits the topic of incircles of triangles in a natural way — I find it more difficult to motivate why you’d want to find an incircle when looking at a strictly two-dimensional problem.

Now on to calculus!  Yes, calculus.  One great mystery for students is the presence of “1/3” in so many volume formulas.  There is always the glib response — the “3” is for “three” dimensions, like the “2” in “1/2 bh” is for “two” dimensions.

When deriving these formulas using integration, this is actually exactly a fairly solid explanation.  But for high school students who have yet to take calculus?

It is easy to approximate the volume of a right circular cone by stacking thin circular disks on top of each other.  If we let the disks get thinner and take more and more of them, we find the volume of the cone as limit of these approximations.  All you need is the sum $\displaystyle\sum_{k=1}^n k^2=\dfrac{n(n+1)(2n+1)}6.$

I plan to prove that $\displaystyle\sum_{k=1}^nk=\dfrac{n(n+1)}2,$

and then prove (or perhaps just suggest — I’m not sure yet) the formula for the sum of squares.

I think a fairly informal approach could be successful here.  But I do think such discussions are necessary — in calculus, I’ve routinely asked students why certain formulas they remember are true, and they struggle.  As a simple example, students can rarely tell me why the hypotenuse of a 30-60-90 triangle is twice as long as the shorter leg.

When teachers just give students formulas and ask them to plug numbers in to get answers to oversimplified word problems, of course there is a sense of mystery/confusion — where did these formulas come from?  I’m hoping that this discussion suggests that there is a lot more to mathematics than just a bunch of formulas to memorize.

As usual, I realize I have much more to say on this topic than I had originally supposed…I’ve only discussed up to the fifth lecture so far!  Since I have not had extensive experience teaching more traditional topics at the high school, it has been an interesting challenge to tackle the usual geometry topics in a way that grabs students’ attention.  It’s a challenge I enjoy, and of course I’ll have much more to say about it next week….