Sharing Dave Richeson’s tweet explaining the concept of dimension with my younger son

I saw a really neat tweet (and subsequent twitter thread) yesterday:

One way to compute the dimension d of certain shapes:
1) Scale it by a factor of k.
2) This gives you n copies of the original.
3) Then d satisfies n=k^d.
This gives the expected value for a point (0), segment (1), square (2), cube (3). For fractals, the values are not integers. pic.twitter.com/H1iZ0KzT5O

— Dave Richeson (@divbyzero) October 2, 2020

I thought that Dave’s tweet would make for a great project, so we took a close look at it this morning. We started by looking at the tweet and then I asked my son what his definition of dimension was:

Next we worked through a few of the introductory examples in Dave’s tweet – a point, a line, a square, and a cube:

Next we moved on to the fractals. Here we also need to talk about logarithms, so we stuck to two of the examples – the Sierpinski Triangle and the Koch Curve:

Finally, we looked at the most complicated example – the Sierpinski Carpet. At the end of this video we recap and got my son’s thoughts on non-integer dimensions

This was a terrific project. Thanks to Dave for sharing the idea!