# Some Thoughts on “A Breakthrough in Higher Dimensional Spheres”

Since I have an interest in the popularization of math, I was excited to hear about a new PBS series called “Infinite Series” and their first YouTube video “A Breakthrough in Higher Dimensional Spheres.” So I decided to write down some thoughts I’ve had about this video and some general trends in pop math.

My general impression of this video is that it’s good. It strikes a good balance between being correct and understandable for the layperson. Many articles about math tend to fail at this, using inappropriate analogies to try to illustrate the point. I am slightly disappointed that some details weren’t provided (more on this later), but it’s extremely hard to balance the level of detail with being entertaining.

To get to the details, the video starts with an introduction the show and its premise, which seems to be current progress in mathematics. It’s not a bad idea, but since most mathematical ideas are built from older ones, this will either limit the topics covered or require each a portion of each episode to go over some background material. Based on how this episode went, I’m going to guess both factors will come into play. I don’t expect to see an episode on the Kadison-Singer problem or the classification of nuclear C*-algebras, for example.

The introduction of spheres and Euclidean space of arbitrary dimensions was good. The sphere packing problem was stated well, but I thought the examples and explanations lacked some important details. The 2d sphere packing is based on square grids and hexagonal grids. Even outlining this fact in the image would have gone a long way to illustrating how that works. It would also help in understanding the higher dimensional generalizations. The 3d case also does little to explain how the sphere packing works and the shapes involved. Also, it’s unclear if, as the picture seems to indicate, the 8d case is a “duplication” of the hexagonal sphere packing. Also when discussing the general case, I think it would be helpful to know where the difficulty lies. Is it the case that we have candidates for other dimensions, but can’t prove that it’s the best as it was for the 3d case? I felt that the sphere packing problem and its solutions were inadequately explored.

The counterintuitive nature of higher dimensional spheres, on the other hand, I thought was explained well. The ratio of the volume of the sphere to the volume of the circumscribing cube going to zero was interesting. (This might be an interesting worksheet in a calculus class, I should make a note of it.) The analogy with the basketball court and the grain of sand was good, due to the fact that such an object with all those properties is impossible to conceive of in our limited space.

Despite some flaws, I think the video was pretty good. Its discussion of the sphere packing problem itself, I think was of moderate success. But the discussion of higher-dimensional spheres worked well to highlight its counterintuitiveness. I am looking forward to seeing more videos from this program, and I hope you’ll join me.