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:

Some thoughts on celebrating Pi Day.https://t.co/3O4ktH3Dca#math #mathchat #PiDay #PiDay2016 pic.twitter.com/BBYocoEKCa

— Patrick Honner (@MrHonner) March 13, 2016

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

My son trying to figure out how to extend @MrHonner 's Pi day exercise to 4 dimensions 🙂 pic.twitter.com/VisoDm3UdS

— Mike Lawler (@mikeandallie) March 14, 2016

The point of today’s exercise was to remind my son about Honner’s interesting approach to calculating “” 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 in terms of area and circumference only:

Having found a new way of defining 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 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 🙂 Tomorrow we’ll look at what our new formula for tells is about the zome shapes we looked at yesterday – namely the 5-cell, the Hypercube (aka the 8-cell), and the 16-cell:

  

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