4 Dimensional Spheres

I hope that everyone has seen the nice video that Numberphile and Ed Frenkel did about why people hate math:

As I write this blog post the video has about 130,000 hits, so I guess there’s still a few views to go before everyone has seen it.  Too bad.

The statement that really struck me in the video comes near the end (around 8:00).  Talking about the beauty of math, Frenkel says – “What if I told you there is this beautiful world out there and you don’t even have to travel anywhere to find it.  It’s right at your fingertips . . . . this is the coolest stuff in the world.”
The day before Numberphile posted the video I had done a fun project with my kids about circles and spheres.  We found how to calculate that the area of a circle was $\pi * R^2$ (two ways, actually!) and also how to find that the volume of a sphere was $\frac{4}{3} \pi * R^3.$    That talk with my kids had been inspired by a paper that Stephen Strogatz had posted earlier in the week that discussed the volume of a sphere in any dimension.  Pretty neat stuff for sure, but I stopped with just the circle and sphere with my kids.

After hearing Frenkel’s message at the end of the video, I changed my mind and plan to spend the next week talking with them about the 4 dimensional sphere.  They seemed pretty excited about the idea, and I’m super excited to try it so we started tonight.

The first thing that I wanted to talk about was some basics about circles.  They remembered some of the discussion from the weekend  ( it was sort of neat to see that they remembered the methods more than they remembered the results) and we quickly reviewed the rest.

After the review I backed up – what is a circle in the first place?  How can you extend that definition to a sphere?  What is a one dimensional version of a circle?    All fun things to discuss.

Finally I showed how we could create a 2 dimensional circle from a bunch of 1 dimensional circles and then build a sphere out of a bunch of 2 dimensional circles.   Of course, this was what we’d done over the weekend, but it probably didn’t hurt for them to see it again since we’ll use the same method to build up our 4 dimensional sphere.

All in all, it seemed like a nice start to our little project.

to be continued  (Jan 20, 2014)

Jan 21, 2014

Thought I’d be shoveling snow tonight so we moved on to our next talk about 4D spheres this morning.  My goal today was to help them get a slightly better understanding of how we move from zero dimensions to 4 dimensions.    To help with the geometry we built some models using our Zometool set.  The models were of squares and cubes rather than circles and spheres, but that just gave us two examples in the higher dimensions.  I took advantage of the two different shapes in each dimension to remind them of how we calculated the area and volume for the circle and sphere and to show that we could use the same method for the square and cube.    Seemed like the had a good time and they spent about 30 minutes after me made the movie building other stuff with the Zometool set.    Haven’t decided what we’ll talk about tomorrow, but I’m happy with how these two talks have gone.

to be continued (Jan 21, 2014)

Jan 22, 2014

I decided that the next step would be reviewing two methods that we’d talked about previously for finding the area of a circle and the volume of a sphere ( see this blog post:  https://mikesmathpage.wordpress.com/2014/01/18/showing-the-kids-about-the-area-of-a-circle/) .  I settled on this approach because I wanted the kids to see how you could extend the concepts in these methods even though you didn’t necessarily know exactly what a 4d Sphere looked like.  Unfortunately this talk didn’t go nearly as well as I was hoping and I feel that I could have tied the points together much better if we went through it again.  Oh well, teaching lesson learned (hopefully!).

The main idea I’m trying to emphasize is breaking up a problem into smaller problems that you already know how to solve.  The specific example in the problems at hand is chopping a circle up into rectangles and chopping a sphere up into discs.  We extend that idea into chopping a 4 dimensional sphere up into spherical discs.  As we found with the circle, this method for finding the volume of a 4D sphere leaves us with a complicated sum involving square roots, and we’ll use Wolfram Alpha to help us understand that sum a little better tomorrow.

So, although my explanations in this talk were a little clumsy, I hope that the main ideas did come across and I’m excited to move on to understanding the new sum that will tell us the volume of the 4D sphere.  There’s a big surprise waiting for them in that sum!

to be continued (Jan. 22, 2014)

We took a break from shoveling out this morning to finish off the problem of finding the area of a 4 dimensional sphere.  After a quick review of what we’d done the last few days, we took a final look at the equation for the volume of one of the slices of the 4d sphere and jumped over to Wolfram Alpha to help us with the sum.  I didn’t want to dwell too much on how to evaluate the sum anyway, but using Wolfram Alpha had also had a special purpose in the lesson.  Since the sum we were looking at has a value of $\pi / 2$, I was wondering if either of the kids would be able to recognize the number.  Turns out that my oldest son actually did guess the right number after I asked him a few questions.  My youngest son thought it was pretty neat that we found a value for the sum, and also really cool that the volume of the 4d sphere with radius 1 was $\pi^2 / 2.$

Anyway, that’s the end of the main journey.  This exercise turned out to be one of the most enjoyable math talks that I’ve had with the boys.  I’m really happy that we were able to make it all the way to this neat result.