Patrick Honner’s Pi day exercise in 4d – Part 3

We are spending the week working through Patrick Honner’s Pi day exercise in 4 dimensions. The first two parts of our project are

Playing with 4 dimensional shapes using Zometool

Introducing Patrick HOnner’s Pi day idea in 4 dimensions

Also, since I didn’t want to really dive into the “surface volume” and “hyper volume” calculations, this website was critical for today’s project:

Regular Convex Four-Dimensional Polytopes

The main idea today is to calculate “$\pi$” for the first three 4-dimensional platonic solids – the 5-cell, the 8-cell (aka the hypercube), and the 16-cell. A fun twist is that the 5-cell and the 16-cell have some 3d projections that are quite similar, but give quite different values for “$\pi$

So, we started with a super quick review of the 4d formula for $\pi^2$ and then took a look at the 5-cell. Although we didn’t go through the calculation, I liked my son’s guess that the hyper-volume of a 4-dimensional pyramid would be given by (1/4) * volume of base * height.

Next we looked at the 8-cell, or hypercube. Luckily this shape has really easy “surface volumes” and “hyper-volumes.” That allowed us to calculate “$\pi^2$ exactly without too much difficulty – plus we got a little bit of exponent review π

The last shape we looked at today was the 16-cell. This is the most difficult shape to understand, and understanding it is made even more confusing because we have a couple different 3-dimensional projects and they don’t look anything like each other! Also, as noted above, one of them looks a lot like the 5-cell.

It was fun to think about the “spherical-ness” of this shape prior to doing the calculation.

We are really having a lot of fun with this project. Tomorrow we’ll probably focus on the hyperdiamond because it is such a cool shape. Then we’ll talk about the 120-cell and the 600-cell for the grand finale π

A challenge for professional mathematicians

[March 24th, 2016 update – I’m going to link some articles at the end of the blog as I see them. There are two from today. I’m really happy that people are writing about this!]

I saw this article on gravity waves via a Steven Strogatz tweet this morning:

Seeing the article reminded me of the interview that Numberphile did with Ed Frenkel a while back – in particular, the part from roughly 5:00 to 7:00 when Frenkel discuses the need for mathematicians to do better at sharing their ideas with the public:

Frenkel’s point is that even though the ideas in fields such as biology and physics are just as complicated as the ideas in math, these other areas of science are much better at communicating with the public than mathematics is.

I was reminded of Frenkel’s point again this morning when I learned that earlier this month Maryna S. Viazovska solved the 8-dimensional sphere packing problem. Viazovska’s paper on arxiv.org is here:

The sphere packing problem in dimension 8

Maybe I’m a little biased – especially right now because I’ve been spending this week playing around with 4-dimensional shapes with my kids . . .

but I think that the sphere packing problem (i) is something that can be explained to the public (it certainly seems less complicated than gravity waves) and (ii) is something that the public would find to be interesting. There’s not been much of any coverage of Viazovska’s result, though. Here’s what I found doing a simple Google news search:

So, it sure seems this new result is something that would be great to share with the general public. There are, of course, many different directions an article could go – just off the top of my head:

(A) Jordan Ellenberg does a great job explaining the sphere packing problem and the connection to things like the Leech lattice and Hamming codes in How not to be Wrong,

(B) John Cook and Keith Devlin both have recent blog post with connections to higher dimensional spheres / cubes:

The empty middle: why no one is average by John Cook

Theorem: You are Exceptional by Keith Devlin

(C) Two years ago, Steven Strogatz shared this wonderful paper on N-dimensional spheres:

(D) The 2-dimensional problem of circle packing is something anyone can understand and is pretty fun to play with – here’s an old project I did with the boys using disc golf discs, for example:

Sphere packing (well . . . circle packing)

Also, a version of the circle packing problem was in Jim Propp’s most recent blog post about mathematical thinking:

Believe it, then don’t: Toward a Pedagogy of Discomfort

So – come on professional mathematicians!! – here’s a great opportunity to promote a neat result and bring some really cool math to the public’s attention. Don’t let the physics crowd have all the fun!

A few articles that I’ve seen:

On Gil Kalaiβs blog:

A Breakthrough by Maryna Viazovska lead to the long awaited solutions for the densest packing problem in dimensions 8 and 24

Kalai’s blog post also led to a question on Quora:

Why is the solution in dimension 8 such a breakthrough?

Introducing Patrick Honner’s Pi Day idea in 4 dimensions

This will be the 2nd of probably 4 blog post in a series about exploring Patrick Honner’s Pi Day activity in 4 dimensions.

The first project (which includes the background) is here:

Playing with 4 dimensional shapes using Zometool

and Honner’s original post came to my attention via this tweet:

and the main motivation for this 4th dimensional exploration was how my son reacted to working through Honner’s activity:

The point of today’s exercise was to remind my son about Honner’s interesting approach to calculating “$\pi$” for various shapes. The main idea is that the radius of a shape is difficult to determine, but for simple 2-dimensional figures we should always be able to determine the area and circumference. If we want to use this idea we’ll need to find a way to define $\pi$ in terms of area and circumference only:

Having found a new way of defining $\pi$ for circles, we now try to find a similar approach for spheres:

Now we are nearly to 4 dimensions – we just need to find the right way to define $\pi$ for a 4-dimensional sphere. It seems like this task shouldn’t be so hard, but there is a little surprise:

We actually talked about 4-dimensional spheres a few years ago:

Showing the kids about the area of a circle

4-Dimensional Spheres

I really doubt that either of the kids remembers these talks, but it is kind of fun to look back on them now π Tomorrow we’ll look at what our new formula for $\pi$ tells is about the zome shapes we looked at yesterday – namely the 5-cell, the Hypercube (aka the 8-cell), and the 16-cell: