The problem is yet another variation on the recursion which produces the Koch snowflake. I discussed the Koch snowflake in one my first posts, so visit my Day007 post on this fascinating fractal for a refresher.

So what is the variation here? Consider the general recursive scheme

where represents a forward move and the indicate counterclockwise turns, which would be common instructions in a turtle graphics setting.

The Koch curve is generated by choosing

Previously I’ve studied cases where and the resulting image possesses *central* symmetry, unlike the Koch curve (see my Day008 post for examples). But this is my first venture into exploring instances where Things work a bit differently here….

So let’s investigate the example illustrated above in more detail. The recursive scheme which generates this spiral is given by

How does this scheme draw a spiral arm? Let’s look at the figure below.

We begin at the origin, and then draw a segment at labeled “1” in the figure. Next, we turn counterclockwise (since ) and draw the segment labeled “2”.

The next turn is so we turn completely around and begin retracing the arm in the opposite direction. Then turn counterclockwise and move forward. Note that since this has the effect of taking us right back to the origin.

What next? It is at this point that we invoke a recursive call — and so the next angle tells us what direction the next arm will be drawn in. Of course, like before, the three angles after that will continue drawing the arm and then bring us back to the origin, awaiting the next recursive call.

This behavior can be illustrated in the following chart.

Here, the angles turned are listed in rows of four, one after another. The first three angles in each row are the same, and their job is to complete the arm once a direction is chosen. But the direction chosen is determined by the fourth column (separated by the divider), whose behavior is not periodic and highly recursive.

So where do we go from here? The next chart we’ll look it is a chart of the directions the arms are drawn in.

We read this chart in rows as follows: the first arm is drawn at an angle of directly to the right. The second arm is drawn at an angle of as can be seen in the figure above.

But the third arm *exactly* retraces the second arm drawn, since it is pointed in the direction of as well. The fourth arm drawn — at an angle of — exactly retraces the first arm drawn.

So you can see what’s going on. One important consequence of the recursive algorithm is that the arms keep being retraced over and over again.

When will the complete spiral be drawn? Well, we need to see *every* multiple of in the chart above. Of course eight rows isn’t enough. But what is really surprising is that it takes over 1,000,000 rows before every multiple of 30 is encountered!

How would you show this? We can’t go into all the details here. The important observation is that if you read across the chart one row at a time, you get the *same* sequence of angles by reading down the first column of the chart.

But it isn’t enough to just *observe* this — it has to be *proved.* This is where some elementary number theory comes in. No more than what might be seen in an undergraduate number theory course, but beyond the scope of this post.

Slight step up on my soapbox — while I usually deplore the definition of mathematics as the “science of finding patterns” (as this only scratches the surface), in this case, finding patterns is critically important. With some trial and error, you hit upon making charts in rows of four, and then putting the directions the arms are drawn in rows of four, and then stare at the numbers until you notice patterns in the charts. The trick is knowing *what* to prove — once stated properly, the results almost prove themselves.

You’ll have to wait for the paper to come out for all the details. But here is a brief summary. Suppose you divide the circle into *n* parts, where *n* is even. Then devise a recursive scheme using the turning angles

Now define

When *n* is divisible by 4, the spiral has *n* arms, and you need to draw exactly segments to render the entire spiral. In our case, with 12 arms, it takes segments to render the spiral.

But when *n* is even but *not* divisible by four, the spiral only has *n*/2 arms, and it takes segments to draw a complete image of the spiral.

Absolutely amazing, in my opinion. I formulated these conjectures over three years ago, but got stuck. A few weeks ago I had the house to myself for a while, and I just sat down and said to myself, “Look, you’ve already written one paper on these recursions. You can *do* this.”

And within two days, I worked out the patterns. Then the proofs, and within a week, a draft of the paper.

I am always humbled by such a seemingly innocuous problem — generating a simple spiral . But there are so many levels to this problem, and so much interesting mathematics to be discovered. I’ll continue exploring recursively drawn images, and share the amazing results with you when I find them!

]]>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 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 *D *(see the relevant Wikipedia page; in the textbook, spherical trigonometric formulas are discussed in detail):

To find we need to know the following (also explained in the text):

where is the golden ratio. (I use 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

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!

]]>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….

]]>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 and many have encountered 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 (yes, the “” 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 -dimensional real space, usually denoted by the symbol “” In the general case, we have lists of real numbers representing points in an -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

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: and

Alternatively, the set containing these four vertices is given by

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

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!

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.

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.

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?

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

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….

]]>

So we begin by looking at the square and cube in a different way, and thinking (again) by analogy from there.

Let’s start with the number of edges on a square this time. We’ll start with one dimension less than the object we’re examining — later we’ll use the squares on a cube, and the cubes on a hypercube. Now to count vertices, we note that each edge has two vertices — for 4 x 2 = 8 vertices. But when we join the edges at vertices, two vertices merge, so we’ve overcounted by a factor of 2. Thus, there are 8 / 2 = 4 vertices on a square (as we know).

What happens when we look at the cube in the same way? Let’s start with the six squares. To count edges, we see that each square on the cube has 4 edges, for a total of 6 x 4 = 24 edges. But as with the vertices on the square, when we join two squares at their edges, two edges merge, so we’ve overcounted by a factor of 2. This means there are 24 / 2 = 12 edges on the cube.

For vertices, we note that three squares meet at each vertex. So there are 6 x 4 = 24 vertices on the 6 squares — but when we join the squares together, *three* vertices merge to one. This means we’ve overcounted by a factor of 3, so there are 24 / 3 = 8 vertices on a cube.

Now begin with the eight cubes on a hypercube. We note that the “8” comes from the sequence 2, 4, 6, 8,… for 2 vertices on a segment, 4 edges on a square, 6 squares on a cube, etc. On the square, we saw that 2 vertices merged. On the cube, we observed that 2 edges merged, then 3 vertices. By analogy, on the hypercube, we should have 2 squares merging, 3 edges, and 4 vertices. This pattern continues into higher dimensions as well.

Let’s use this analogy to check that we’ve counted correctly. First, we count squares. With 8 cubes, we have 8 x 6 = 48 squares. But since the cubes meet square-to-square, we’ve overcounted by a factor of 2, so that there are just 48 / 2 = 24 squares on a hypercube.

Now on to the edges. Since 8 cubes contribute 12 edges each, there are 96 edges in total. But three cubes meet at each edge, so we’ve overcounted by a factor of 3. This implies that there are in fact 96 / 3 = 32 edges on a hypercube.

Finally, we count the vertices. With 8 vertices on each of 8 cubes, we have 64 vertices all together. But four cubes meet at each vertex, so we have in fact overcounted by a factor of 4. This results in just 64 / 4 = 16 vertices on a hypercube.

So we counted correctly! Well, almost…. You might have noticed that we haven’t exactly *defined* what a hypercube is in a rigorous mathematical way. Until we do, the best we can say is that we’ve counted *something,* although we don’t know precisely what they something is at the moment. That will be a subject for a later post.

As you might expect, there are other ways to represent a hypercube besides the figure shown in last week’s post. We chose that particular representation because of the way we were thinking by analogy.

We’ll look at two additional ways. The first is analogous to looking at a cube face on, as shown below.

Of course the inner and outer squares are the same size on a cube — but we can’t help distorting faces of the cube when we make a two-dimensional sketch. Four of the squares are distorted into trapezoids here.

Here is the analogous representation of the hypercube.

In this figure, the inner cube (in green with black edges) is in fact the same size as the outer cube (with blue edges); these cubes are directly opposite each other on the hypercube. This is directly analogous to the inner and outer squares we saw in the earlier figure.

Further, note how the other six cubes are distorted into frustums of square pyramids — you can easily see the trapezoidal faces, which we also saw in the earlier figure. If you look carefully, you can count the 16 vertices and 32 edges. The 24 squares are a bit trickier — but begin with the 12 squares on the inner and outer cubes. Each of the other 12 squares contains exactly one black edge and exactly one blue edge — they are going “out” from the black edges to the blue edges. The perspective is different, but all the squares are there.

And of course there are the eight cubes — the inner cube, the outer cube, and the six frustums of square pyramids. All the elements of the hypercube are indeed present, but again, in a different perspective.

I did save the best for last — my favorite representation of a hypercube, shown below.

I just love the symmetry of this image — the octagram inside the octagon. If you look at the left image, you’ll see one of the eight cubes highlighted in blue. When each of the eight cubes is transparently colored in the right image, you’ll see an interesting overlap of colors.

Now let’s count the vertices, edges, faces, and cells in this figure. The 16 vertices are readily apparent in the inner octagram and the outer octagon. The 32 edges can be seen by counting eight edges from both the octagram and octagon, and two additional edges connecting each vertex of the octagon to two vertices of the octagram.

The squares are a bit trickier here as well — but eight are easily visible as undistorted squares sharing one edge of the octagon. But if you look carefully, you’ll also see 16 additional squares as rhombi in the figure. Eight of these rhombi each share two edges with the octagon, and the eight others each share two edges with the octagram. It might take staring for a minute, but they are all there.

And finally, there are the eight cubes. As seen in the left image, three consecutive edges of the octagon are enough to determine one of the cubes. Since we can take three consecutive edges of an octagon in exactly eight ways, we have found the eight cells on the hypercube.

So again, all the elements of a hypercube are present — it just takes looking from the right perspective to see them all.

Still remaining is to obtain the same counts by considering a rigorous definition of a hypercube. That for a later post….

]]>So the fourth spatial dimension must be, for us, an abstraction. To think about it, we must somehow relate it to things we already know — in particular, geometry in dimensions zero, one, two, and three. Therefore, to think about the tesseract — the four-dimensional analogue of the cube — we must somehow think by analogy.

In particular, we can think about how we go from a point (zero dimensions) to a line segment (one dimension), and then from a line segment to a square, a square to a cube, and then make the analogous leap from a three-dimensional cube to a four-dimensional tesseract, or hypercube. We’ll look one way to do this in today’s post, and next week, we’ll see a different way. It’s a good idea to look at this “new” fourth dimensional from different perspectives to make sure our thinking by analogy is accurate.

To create a line segment from a point, we think of it moving one unit (we need to be specific) along a “new” first dimension to create another point — these two points are vertices of the line segment, and one edge is created.

Now move this segment one unit along a second dimension which is *perpendicular* to the line segment, as shown below.

Now let’s count. We have two vertices and one edge for each of the two segments (the one at the bottom in black, and the one at the top in blue), and each vertex creates another line segment (shown in red above) as it moves from the bottom segment to the top segment. This gives a total of four vertices and four edges on a square — as expected.

Thinking by analogy, we now imagine a square moving *up* along a perpendicular dimension, as shown below.

Counting vertices, edges, and faces, we have 8 vertices, 8 edges, and 2 faces from the bottom and top squares, shown in black and blue, respectively. Now each vertex of the bottom square creates an edge as it moves up (creating 4 more edges, shown in red), and each edge of the bottom square creates a new face (creating 4 new faces, shown in pale yellow). This gives a total of 8 vertices, 12 edges, and 6 faces. Notice the strategy: count the bottom and top figures, and then notice what is created by vertices and edges as they move along a perpendicular dimension.

Now it’s time to extend this strategy into the fourth dimension. To do so, we need to imagine a cube moving *out* along a fourth spatial dimension — and of course, it is difficult to imagine because we are so used to our three-dimensional world.

Look at the above figure. Let’s think of the cube outlined in black as the base cube. Move this cube *out* along a fourth spatial dimension — so that each vertex creates an edge (shown in red) as it moves to the top cube (shown in blue).

Before we start counting, we need to introduce a little terminology. The four-dimensional analogue of a polyhedron is called a *polytope,* and in addition to vertices, edges, and faces, we have three-dimensional *cells* on a polytope.

Now let’s count the vertices, edges, faces, and cells on a hypercube in just the same way as in the previous two examples. For vertices, we have 16 total — 8 from the base cube, and 8 from the top cube. For the edges, we have 12 each for the base and top cubes, and each vertex of the base cube creates a new edge (shown in red). This gives a total of 12 + 12 + 8 = 32 edges on the hypercube.

What about faces? Well, we have 6 faces each for the base and top cubes, and each edge of the base cube creates another square face (as it did when we created the cube). This gives us 6 + 6 + 12 = 24 faces. Finally, to count the cells, we have the base and top cubes, and each square face of the base cube moves *out* to create another cube (such as the one shown in yellow, with the square from the base cube shown in darker yellow). This gives 2 + 6 = 8 cells (cubes) on the hypercube.

How can we be sure this is correct? Next week, we’ll look at a different (though certainly related) way to count the number of vertices, edges, face, and cells on a hypercube. We will see that we do in fact obtain the same counts.

As a final remark, we briefly look at Euler’s Formula in four dimensions. Recall that where and represent the numbers of vertices, edges, and faces on a convex polyhedron. Now if represents the number of cells on a polytope, the four-dimensional analogue of Euler’s Formula is

In our case, we have the true statement

We won’t go into more depth here, as that would take us quite a bit further afield. But if you’re interested to know more, you can always look up Euler’s Formula in higher dimensions, and extensions which have to do with the *topology* of geometrical objects, which is fairly straightforward when the objects are convex. When they’re nonconvex, the situation is decidedly more difficult.

Stay tuned for next week’s post, where we look at yet another way to count the numbers of vertices, edges, faces, and cells on our friendly tesseract!

]]>One of my projects has been writing an introductory book on polyhedra, and I’m in the middle of a proposal to have the book published. A draft chapter was on the fourth spatial dimension, so I thought I’d share it here. As much as I love the topic, I rather surprised myself by looking at my blog index and finding I’d never talked about it before….

What follows is the draft chapter, slightly edited as a stand-alone series. Enjoy!

“What is the fourth dimension?

No, it’s not time.

Well, maybe it is if you’re studying physics. Even then, we have certain intuitive ideas about how time works.

For example, if I imagine that it’s 9:17 a.m. in San Francisco, then I know *what time it is* in any other city in the world. In New York City, it must be 12:17 p.m., since I add three hours to convert to Eastern Standard Time.

This assumption is just fine for getting along in daily life — and as far as most people are concerned, this way of thinking about time is *right.* But in fact, it works only because in our daily lives, we move around fairly slowly — at least compared to the speed of light.

To understand what happens when particles do move close to the speed of light, you need to study *special relativity* — and here, our ordinary intuitions about time are no longer valid.

But we’re interested in a fourth *spatial* dimension. How is this possible? Where is it? We are so used to living in a three-dimensional world, the idea of a fourth spatial dimension seems rather fantastic.

In 1884, Edwin Abbott’s delightful novella *Flatland* was published (Abbott, Edwin A. *Flatland: A Romance in Many Dimensions.* New York: Dover Thrift Edition. 1992). The protagonist was none other than A Square, an inquisitive four-sided being living in a purely two-dimensional world called *Flatland.*

He was chosen as the Flatlander to receive a visit from a Sphere on the eve of the Third Millenium. This was an unnerving visit for A Square, since the Sphere kept suggesting he consider the direction upward, but there was no upward for A Square. There was North, South, East, and West, but A Square just couldn’t fathom this direction, “upward.”

Finally, the Sphere lifted A Square out of his two-dimensional world to show him the glory of Space. Quite a revelation!

We are in the same predicament as A Square when it comes to contemplating a fourth spatial dimension. Yes, we can look forward, backward, to our left and right, up and down, but nowhere else. It doesn’t seem that there is enough room for a fourth dimension. Where would it be?

In a typical high school geometry class, you would likely have been introduced to points, lines, and planes — and perhaps were told that points are 0-dimensional, lines are 1-dimensional, planes are 2-dimensional, and that all objects of these types live in a 3-dimensional space. But while there were infinitely many points, lines, and planes, there was only *one* 3-dimensional space.

And while we do not encounter a fourth spatial dimension on a daily basis, we don’t actually encounter points, lines, or planes, either. Can we actually see a point if it has no length? How could we possibly see a line if it has no width? It would be invisible. These geometrical ideas are in fact mathematical abstractions — and once we enter the world of mathematical abstraction, our universe becomes almost unimaginably vast. A. R. Forsyth wrote almost one hundred years ago (A. R. Forsyth, *Geometry of Four Dimensions, *Cambridge University Press, New York, 1930, p. vii.):

Mathematically, there is no impassable bar against adventure into spaces of more than three dimensions of experience.

To get a handle on how to imagine the fourth dimension, we’ll look at one of the most popular and well-know denizens of that rarefied world — the hypercube, also known as a tesseract.”

What follows is an extended example of “thinking by analogy,” which would make this post much longer than I usually allow myself. So that’s where we’ll start next week!

]]>As mathematicians are well aware, mathematics is both a historical and cultural phenomenon. I know relatively little about the history of mathematics, but I did do some research into the relationship between Kepler and Tycho in preparation for a talk on Kepler’s Laws and the Music of the Spheres. The facts that Tycho was not only interested in astronomy, but was also a member of the Danish nobility (and hence had the money to design and create instruments which allowed for more accurate measurements than previously possible) were crucial in providing the data necessary for Kepler to deduce the laws which we know by his name today. The story is of course much more involved; the point is that providing some context for statements such as Kepler’s Laws makes them more accessible to students rather than just stating them as if they were abstractions devoid of cultural and historical context.

My contribution was running a workshop on polyhedra (the Platonic solids in particular). Participants were familiar with the Platonic solids at various levels, so I made sure there were nets available at varying levels of difficulty. For example, we were going to discuss the geometry of fullerene molecules, so I also had some truncated icosahedron nets printed out as well.

The weather was particularly warm this weekend; plans were altered so we could be outside for much of the workshop. So I moved everything I wanted to discuss using the computer to the first hour. For example, to put our work into some context, I discussed my correspondence with Magnus Wenninger (which I excerpted in a series of blog posts earlier this year).

I also wanted to make a reference to Kepler’s Harmonices Mundi, as Snezana had mentioned Kepler earlier in the conference. Among many other things, Kepler included a discussion of the Platonic solids.

Further, I wanted to discuss the fairly recent (certainly compared to Kepler) discovery of fullerenes, as this discovery led to an interesting, if relatively simple, result: in a fullerene molecule, where each carbon is adjacent to exactly three others, and the carbon atoms come *only* in rings of five and six, then there must be exactly twelve rings of five carbons, regardless of how many rings of six there are.

This is actually not very difficult to prove using Euler’s formula (and I have done so in a previous post), but it is somewhat surprising the first time you hear it. More difficult is to prove that there can be any number of hexagons *except* one; I didn’t go into this in any more detail, however. But I did want to go to the Wikipedia article and put the question in some historical context.

After the lunch break, we proceeded to an outdoor classroom which we populated with two movable whiteboards. First, I had participants build at least one of the Platonic solids using the double-tab method I learned from Magnus Wenninger. We then moved on to an algebraic enumeration of the Platonic solids as an application of Euler’s formula (which was also mentioned earlier). As some of the participants were involved with teaching younger students, I wanted to avoid spherical trigonometry.

Then, as mentioned above, we looked at fullerenes. Since the technique involved here was similar to that used to enumerate the Platonic solids, after we defined the problem, I gave the participants time to solve it on their own. Relatively quickly, one of the educators did prove that there were exactly twelve rings of pentagons, and presented her solution to the others. This was a great way to end the afternoon; afterwards, we took a walking tour of the busier section of Jagodina.

Of course one interesting aspect of attending any international conference is learning about the similarities and differences between education in different countries. In Serbia as well as the United States, teachers are not paid as well as other professions. But mathematicians have fewer opportunities for employment than they do in the US, so they are more likely to go into teaching as a career.

One striking difference revolves around school subjects and the corresponding teacher training. As early as fifth grade, the sciences are separated as subjects. All students take biology and informatics (computer science) beginning in the fifth grade, physics beginning in the sixth grade, and chemistry beginning in the seventh grade. Moreover, teachers of mathematics, biology, physics, and chemistry have to *major* in that subject to teach in grade five and beyond. (Up to the fourth grade, there is just one teacher for all subjects except foreign languages.)

A suggestion was once made to change this in mathematics: just allow mathematics teachers in middle school grades to take a few courses, similar to what we could call a *mathematics endorsement* here. This was strongly rejected by those responsible for training mathematics teachers. We can learn a lesson from this, as many teachers of middle school mathematics in the US are not adequately prepared for their classrooms. However, as the teaching of informatics is a fairly recent addition to the curriculum, a major in computer science is not necessary to teach informatics in the middle school grades.

Another significant difference is how mathematics is taught at university. Each major in the sciences has their own sequence of mathematics courses, taught by faculty in that department; physics typically has a three-semester sequence, and chemistry a two-semester sequence. So there is no “one size fits all” calculus sequence, as is commonplace in American universities.

One consequence of this is that there is no gateway proof-writing course in the mathematics major. Theory and proof are integrated into mathematics courses from the very beginning. Thus, to earn the highest marks in an introductory calculus course, a student must show an ability to think abstractly and write mathematical arguments. This would seem like such a luxury for the typical American mathematics professor – a Calculus I course consisting entirely of math majors! But this is just standard operating procedure in universities in Serbia.

So my first experience in Serbia was an enjoyable one, and I learned a lot about the culture and education. I look forward to a few days in Belgrade before returning to the US later on in the week. I leave you with this interesting piece of found art I saw walking down the main street of Jagodina — you truly can never tell what surprises await when visiting somewhere new!

]]>Again, the assumption here is that we are working in a more traditional system, where students must be assigned grades, and these grades must in large part be based on performance on exams.

Given IMSA’s statements about the advanced nature of their curriculum, I had concerns about the fairly traditional exams we gave in mathematics. In my mind, there was little to distinguish our exams from those given in any other rigorous calculus course.

The reason given to me by other mathematics faculty was that there just wasn’t time in a roughly hour-long exam to assess conceptual understanding. I wasn’t convinced, and I started thinking of an alternative. I did agree, however, that it wouldn’t work to have a conceptual question on an exam which might take half the exam period for a significant fraction of the students to complete.

What I finally settled upon was including a range of conceptual problems for which students only needed to provide a reasonable approach to solving. If you chose a conceptual problem which happened to be centered on a student’s weakness, you wouldn’t be able to assess a broader conceptual understanding. And if you insisted the problem be worked completely through, you encountered significant time constraints.

I’d like to share one last anecdote. I recall a parent visitation day one Saturday, which happened to be the day after I gave a calculus exam. Two of the parents approached me after the session and told me how much their son or daughter enjoyed my exam. This indicated to me that for the student who *can* perform the routine procedures easily, they *want* to be challenged to think outside the box, and indeed they *thrive* on such challenge. Shouldn’t we, as educators, find ways to stimulate *all* of our students, rather than be content with having students in the middle earn their B’s, making sure the struggling students earn their C’s, and relegating the very capable students to a sustained boredom?

And now for the last installment….

“Where does this bring us? Here are some key points as I see them.

- We should move away from assigning grades punitively.
- We should reconsider the “point'”system of evaluating student performance. Referring to the TIMSS (Third International Mathematics and Science Study): “In our study, teachers were asked what ‘main thing’ they wanted students to learn from the lesson. Sixty-one percent of U.S. teachers described
*skills*they wanted their students to learn. They wanted students to be able to perform a procedure, solve a particular kind of problem, and so on….On the same questionnaire, 73 percent of Japanese teachers said that the main thing they wanted their students to learn from the lesson was to think about things in a new way, such as to see new relationships between mathematical ideas.” (Stigler and Hiebert,*The Teaching Gap,*ISBN 0-684-85274-8, pp. 89-90.) A point system reflects the assessment of procedural knowledge. - “We can think of all assessment uses as falling into one of two general categories — assessments
*FOR*learning and assessments*OF*learning.” (From an internal document distributed to mathematics teachers at IMSA.) But why? The distinction is artificial. There are many other ways to compartmentalize assessments, such as timed/untimed, individual/group, skill/conceptual, procedural/relational, short-term/long-term, etc. The main argument for focusing on the “for/or” distinction is its relationship to student motivation — but we are given no context for it. I suggest that our typical IMSA student*is*highly motivated — certainly in relation to the average student in a typical high school classroom. - We should consider the assignment of letter grades in general. Right now, it would be impractical to suggest that we have formal written evaluations of each student in each class. But is it desirable? And if so, what resources are necessary to support such a system?
- We should discuss the assessment of problem-solving.

Will any of these suggestions help to illuminate the power of ideas? I’m not sure. With the current need to assign grades, and their current cultural meaning and importance — especially when it comes to applying to college — there will be the necessary compromises in the classroom. I realize that many suggestions are of the “move away” rather than the “move toward” type. But I suppose that if there is something I am moving toward, it’s giving students at all levels more of a BC Fast-Track experience regardless of the depth of content.

This means actively moving toward a classroom environment where earning good grades is subordinate to learning complex concepts. Of course the two are not mutually exclusive — but I’d rather have students earn good grades because they learned, rather than learn in order to get good grades.

Of course many issues brought up in these remarks have been left hanging or only tentatively developed. These brief comments are meant to suggest questions for discussion, not definitive answers.

I can’t resist ending with the following challenge from Maslow: “In order to be able to choose in accord with his own nature and develop it, the child must be permitted to retain the subjective experiences of delight and boredom, as *the* criteria of the correct choice for him. The alternative criterion is making the choice in terms of the wish of another person. The Self is lost when this happens. (Maslow, source cited earlier, p. 58.) Is it possible to create a mathematics curriculum which can survive this test of course selection?”

Thanks for staying with this series! No, there is no simple resolution to any of the issues described in this essay. But that doesn’t mean we shouldn’t be involved in a conversation about them….

]]>I remind you that this scale is not ideal; the purpose was to come up with some system of assigning grades which wasn’t punitive, but rather which motivated students to learn concepts rather than to avoid losing points on exams.

We continue with a discussion of how to use such a system in practice.

“Now let’s consider this in the context of an exam. The first part of an exam is a skills portion, with, say, ten short problems of roughly equal length. Expectations for this part of the exam [meaning a grade of B] are seven problems “essentially” correct, *and* four problems completely correct. These expectations are written on the exam for students to see.

Are these expectations too low? Perhaps. But then an A [refer to the chart below] means eight problems “essentially” correct, with at least five completely correct. Of course we must ask what it means for a problem to be “essentially” correct — but when in doubt, err on behalf of the student. (Students rarely suggest that their scores be lowered.)

Then grading is actually somewhat easier, and grades can be assigned as follows, with the abbreviations EC and CC meaning essentially correct and completely correct, respectively (for simplicity, a grade of C is assigned for all other cases not accounted for):

Now this eliminates the need for partial credit — but does require a judgment as to what “essentially correct” means.

This also makes grading much easier. I would suggest that each problem be marked as “EC,” “CC,” or left blank. Few comments, if any are necessary. This is the approach I have taken in BC Fast-Track, and it encourages *further* learning as it leaves the student in the position of needing to work through their mistakes.

I would have students keep a section of their notebooks for exams and revisions, and there they can keep their reworked problems, should they choose to do so. Then — as I did in BC Fast-Track — students could visit me periodically with their notebooks and I can take a look at their ongoing progress. This “additional” work, if sufficiently well done, could boost their grade at the end of the semester.

I think this could have the same effect it did in BC Fast-Track — exams were easier and more enjoyable to grade. But there were *more* discussions in my office about reworked exams and sources of error that were initiated by the students themselves, and these discussions were *not* about points, but about *concepts.*

Now what about the part of the exam which is intended to be more conceptual? Let us suppose that there are three problems, roughly comparable in length, and of various difficulties. Then grades might be assigned as follows:

More details about how this would fit in a classroom environment may be found in a later document [I cannot recall which document is being referred to here]. But this system allows for a more qualitative approach to grading. Performance expectations are also clearer, but such expectations depend critically upon the nature of the problems given. Moreover, grades are not assigned punitively, but the emphasis is on doing problems completely *and* correctly.

For an example, below could be a set of ten skills problems and three conceptual questions for a basic assessment on the rules of differentiation. This would a 70-minute assessment. Given expectations for completely correct problems, I think this is reasonable.

Skills questions:

1. Evaluate

2. If find

3. Find

4. Using the quotient rule, find

simplifying as much as possible.

5. Find the derivative of

6. Find the equation of the tangent line to at

7. Using a definition of the derivative, find the derivative of

8. Assume that and are differentiable functions. Find

9. Find the derivative of

10. Let be the greatest integer function. Using the definition of the derivative, determine whether or not the derivative exists at

Conceptual questions:

1. Using the product rule, find

Explain your result.

2. Suppose that the line is tangent to both and Find and

3. Suppose that is a differentiable function. Discuss the following limit:

Stay tuned for next week’s final installment of this series on assessment….

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