4 dimensional shadows

Last night at dinner my kids were having a very interesting conversation about shadows of 4 dimensional objects.   The main question they had was whether or not a 3 dimensional shadow of a 4 dimensional object would itself cast a two dimensional shadow? Hmmm . . .  an interesting question for sure, and one that we talked about for today’s Family Math.

First, introducing the question and getting the opinion of each of the kids.  As luck would have it, they have different opinions on the matter:

Next we talked a little bit about what causes a shadow. Here we began by talking about a three dimensional object casting a two dimensional shadow on a plane. We also wondered about how a two dimensional object would see that shadow. Certainly no attempt at any math or physics rigor here, just talking and wondering about shadows.

Before moving on to the 4 dimensional case, I thought it would make sense to talk through one more point about how a 2 dimensional creature living in a 2 dimensional world would see that world. We also tried to extend this two dimensional idea to three dimensions – namely how do we see things? Finally, would would all of this mean for a 4 dimensional creature looking at a 3 dimensional world?

Next we talked about what might happen with a 4 dimensional world when a 3d shadow was cast on a 3 dimensional world? Who knows really, but this was one way to tell the story:

Finally, we finished up with a different way to think about dimension. Rather than shadows which, I think anyway, is a little tricky to talk about, here we talked about what a 4 dimensional object passing through a 3 dimensional world would look like. The easy case to start with was a 3 dimensional object passing through a 2 dimensional world. Maybe the next step is to have the boys read Flatland!

I have no idea how where the original conversation between the boys last night came from. It was fun to hear them talking by themselves about 4 dimensions, though. Hopefully the ideas we talked about today will help them understand 4 dimensions a little bit better.

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