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 🙂 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:

  

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