Counting in 4 dimensions

Yesterday we did a neat project based on problem #12 from the 2015 AMC 8:

Problem12

That project is here:

A great counting problem for kids from the 2015 AMC 8

Then I got a nice comment on the project from Alison Hansel:

So, for today’s project we extended the problem from yesterday to 4 dimensions.

Here’s the introduction and a quick reminder of yesterday’s problem. I had both boys review their solutions and then we began to discuss how to approach the same problem in 4 dimensions.

Next we dove a bit deeper into how to approach the 4 dimensional problem. They boys thought a bit about the symmetry that a 4d cube would have and at the end (after a long and quiet pause) my younger son thought that looking at how a square turns into a cube might help us.

In studying how a square transforms into a cube, the critical idea is how 4 edges turn into 12 edges. This video is a little on the long side, but I think the discussion is really interesting. By the end the boys have found the main idea for how to count edges as you move up in dimension.

Next I brought out Henry Segerman’s 4-d cube model and compared the model to the ideas we’d developed up to this point.

An important idea from earlier in this project was that my older son thought that each edge of the 4d-cube would be part of two 3d cube “faces”. Using the model we were able to see that, in fact, each edge is part of 3 cubes.

Finally – with the 4 pieces of prep work behind us! – we were able to answer the AMC 8 question about a 3d cube in 4 dimensions. So . . . how many pairs of parallel edges does a 4d cube have? The answer is 112 🙂

Thanks to Alison Hansel for the great suggestion for how to extend yesterday’s project. I think her idea makes a great way to introduce kids to some simple ideas in 4d geometry.

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