Counting geometric properties in 4 and 6 dimensions

During yesterday’s project, my younger son asked me about tesseracts a couple of times. I’m not sure what caused his interest in 4-dimesional cubes yesterday, but I decided to explore tesseracts starting with Cifford Pickover’s Math Book as a way of catering to that interest. Pickover’s section on tesseracts has an interesting chart that served as a nice way to get the project going.

To start off I asked the boys about what patterns they saw in Pickover’s chart:

 

Next we moved to the living room to explore the beginning of the chart with our Zometool set. The goal was for the boys to see a connection between the corners and line segments as you moved up in dimension. It took a little extra time for them to see this connection, but eventually they were able to see the pattern.

 

Next we made a cube and counted up the corners, edges, and faces. Here the boys saw that there was a relationship between the number of faces in dimension n and the faces and edges in dimension (n-1). In fact, the relationship is exactly the same as the relationship we saw in the last video!

At the end of this video, my younger son describes how he thinks we can make a Zometool hypercube.

 

We made our hypercube off camera. I asked the boys to describe the new shape – it was interesting to hear their descriptions since I did not expect them to describe this one in terms of sliding. Their descriptions involved scaling rather than sliding.

After their descriptions, we tried to count the shapes that were in the chart in the Pickover book. They had a tough time seeing the 3-dimensional shapes at first, but eventually were able to count the 8 cubes (or “pyramids” as they called them).

The real trick was counting the 2-dimensional faces – this part was hard!

 

I took a break in the last video after about 5 minutes. The problem of counting the faces of the hyper cube was giving the boys a tough time. They knew from the chart that there were 24 faces, but their initial count was 36, so they knew they were over counting. But what and by how much were they over counting?

We worked through two different ideas in the counting process – one where you count each face twice, and one where you only count the “pyramid” sides. This two part process seemed to help them see how to count the faces.

Interestingly, counting the edges and corners was easy.

 

To wrap up the project, we returned to the kitchen table and the Pickover chart. We used the patterns that we saw from our Zometool set to fill in the numbers for a 6 dimensional cube. It was neat to hear them talking about the various geometric pieces of a 6 dimensional cube. Showing how to count all of these shapes without being able to visualize the complete 6-dimensional shape is a nice example of the power of math!

 

So, this project was a little longer than normal. Counting the properties of the hypercube was hard, but I’m glad we were able to work all the way through it. The neat thing is that the patterns themselves are not too difficult – multiple one thing by 2 and add another thing –

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