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 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 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 dimensions is of the form
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 and calculating a simple ratio reveals that you’ve got of your potato left.
Extend this idea into higher dimensions. If your potato-peeling expertise is as good in higher dimensions, you’ll have of your potato left, where is the number of dimensions of your potato. Now 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!