Dimensions review

I’ve managed to download a full copy of Dimensions. In accordance with the Creative Commons license the authors have attached to the work, I’m posting a mirror of the English-narrated versions, converted to mp4 format, and without subtitles. My webspace has decent bandwidth, but probably not enough to justify a full mirror. So, again in accordance with the license:

Dimensions by Jos Leys – Étienne Ghys – Aurélien Alvarez

Now, that out of the way, let’s take a look at it.

The film as a whole explores a few topics in geometry, focusing on the stereographic projection as a tool. It starts by citing the etymology of “geometry” as the science of measuring the Earth. This introduces the 2-sphere as an object of study, and we can wash our hands of reality for the rest of the discussion.

Next the sphere is cut up to show the standard spherical coordinates of latitude and longitude, exhibiting the two-dimensional nature of the sphere. This leads into the idea of making a map by stereographic projection from the North pole. The film discusses the conformality of this map, saying that it “preserves shape”. Clearly this isn’t true, as any newcomer watching the video will surely cry, but they eventually restrict to pointing out that it preserves circles and lines, as illustrated by the meridians and parallels from the coordinate system.

Episode 2 moves on to the third dimension, and down the road towards a familiar exploration of higher-dimensional object, but this time to be presented in terms of stereographic projections. The usual notion of visualizing higher-dimensional shapes by slicing is introduced. Then the film points out how difficult and unintuitive this method can be. Since the focus is primarily on (regular) polyhedra, the shapes are inflated to lie on spheres, which can then be stereographically projected onto the plane. There’s no mention made, though, of the extremely special properties of these polyhedra that allow this construction to proceed.

Before moving to the fourth dimension, there’s a discussion of the coordinate plane and coordinate space, including odd mentions of the terms “abscissa” and “ordinate”, which are all but obsolete in at least American usage. And then the coordinate approach to 4-d discussions is dropped entirely.

Instead, the film introduces the simplices in dimensions 1, 2, 3, and 4 — the last by continuing the combinatorial patterns from the first three. The 4-simplex is passed through 3-space, like the slicing approach to studying 3-d objects in terms of the plane. This is followed by the hypercube, and then the 120-cell and the 600-cell. Next the shadow-projection technique is discussed briefly with the hypercube, the 24-cell, the 120-cell, and the 600-cell. Like slicing, shadow-projection gets too complicated to really get a good feel for the polytopes under consideration.

Instead, we get stereograhic projection. While the animation recaps stereographic projection of the tetrahedron, the narration goes on ahead to discuss how the analogous procedure will work one dimension up (this disjunct gets a little confusing). Next is a parade of the 1-skeleta of the stereographic projections of the 4-simplex, the hypercube, 24-cell, the 120-cell, and the 600-cell. Adding in the 2-faces and rotating the projection allows for a much clearer understanding of the nature of these shapes.

The exploration into 4-polytopes complete, episode 5 starts in on discussing the complex numbers. First comes the geometric interpretation of addition and multiplication along the number line. Of particular importance is the idea of multiplication by -1, which rotates the line 180 degrees around the origin. This introduces Argand’s idea to consider a rotation by 90 degrees as a square root of -1. Of course, there’s no discussion of the arbitrary choice of which direction to rotate, but that would probably only confuse matters.

Anyhow, this gives the Argand plane, and there are geometric interpretations of addition and multiplication of complex numbers. The latter becomes clearer after introducing the modulus and argument of a complex number.

Now the film throws a sphere up against the Argand plane and reverses the process of stereographic projection, mapping the plane onto the sphere. A discussion of some basic conformal transformations follows, but curiously the film doesn’t delve into Möbius transformations, which would have been the most natural topic to merge with the emphasis on stereographic projection. Instead there’s a shakily-motivated digression into the Mandelbrot set and some of its Julia sets. Really, the Möbius transformations would fit a lot better here, and the Mandelbrot set could fill a 2-hour film in its own right.

Episode 7 starts exploring the Hopf fibration. It starts out with an attempt to explain the complex coordinate plane (which of course would have four real dimensions). What looks like the unit circle here is actually then the unit 3-sphere. It intersects each axis in a unit circle in that complex “line”.

Similarly, any other line through the origin in this complex “plane” meets the 3-sphere in a circle. None of these circles intersect each other, since the lines in question intersect only at the origin, which is not on the 3-sphere. Every point on the 3-sphere lies on some line through the origin, and so the entire 3-sphere can be decomposed into a bunch of circles. Even better, these lines through the origin are parametrized by their “slope”, just like in the real coordinate plane. That is, there is exactly one such line for every complex number, and another for infinity. But that’s just the stereographic projection back again! So the 3-sphere is made up of a bunch of circles, one for each point on the 2-sphere.

The really interesting part is seeing how the circles at different points interact with each other. Any two circles form a Hopf link. Further, if we take a whole parallel of latitude on the sphere parametrizing these circles we get a bunch of loops filling up a torus. As we move towards one pole the torus gets thinner and thinner towards the single circle at its core, corresponding to the single point at the pole. At the other pole the torus expands to a straight line through the “doughnut hole” of the torus, and off through infinity. This is the “circle” corresponding to the single point at the other pole.

The circles making up one of these tori are one of four families of circles one can draw on the torus. Of course there are the meridia and the parallels, but there are also two families obtained by cutting along bitangent planes. These latter two wrap around the torus in opposite directions, giving rise to the two mirror-image Hopf fibrations of the 3-sphere. Rotating these tori in the 3-sphere and stereographically projecting to 3-space we find other surfaces covered by four families of circles. And we know they’re all circles because rotations and stereographic projections are all conformal, and so send circles to circles.

The film concludes in episode 9 with a proof of the conformality of stereographic projection from the 2-sphere onto the plane. It uses nothing more than Euclid’s Elements, so it should be comprehensible to a good high-school geometry class. In fact, it might be an interesting exercise to start by viewing the episode as a class and then trying to fill in all the details with the names of theorems the students know from the year’s studies. When the video says “the two triangles are congruent”, the students should be able to call out “hypotenuse-leg!” in response.

If there’s a weak point, I have to say it’s in the narration. The English narrator in episodes 1, 3, 4, 7, and 8 sounds sullen and bored, which communicates the idea that the material is boring. I also can’t say I’m entirely behind the “I’m [historical figure]” conceit. It just feels extraneous, since they never really go into any depth about how the history interacts with the mathematics. To compare: Outside In and Not Knot just use anonymous narrators and make historical citations where the material calls for them to fill in details the animation doesn’t explore. They also sound actively interested in and excited by the material, which encourages the viewer’s interest.