4D Equable Shapes – can you help?
In year 8 (ages 12 and 13), we have a project called Equable Shapes which starts off with trying to find a rectangle where the area and perimeter have the same value (ignoring units). Students often stumble across some through trying various shapes and are more successful if they are systematic in their approach. Usually, someone eventual points out that just trying rectangles isn’t very efficient or, more bluntly, calls it tedious. This is the perfect cue for “You’re right, there is a quicker way…” and demonstrating an algebraic approach. It’s really nice to get to the point where students are actually asking if they can use algebra to solve something.
Then, it can go in various ways:
- Compound rectangles,
- Triangles (with pythagoras),
- Some other interesting 2d shapes,
- 3D – surface area = volume.
Now, I have a couple of students who noticed that in 2D, a 4×4 shape is equable and in 3D, a 6x6x6 shape is equable and so, they suggested that in 4D, an 8x8x8x8 shape should be equable with volume = to hypervolume?
This seems a reasonable prediction. What I’d like help with is how to determine if that’s true. I was able to describe that the 4D shape in question would be bounded (somehow) by 8 cubes of size 8x8x8, which I hope is actually true. What is 4D space called? It’s fair to say that my 4D shape work has been somewhat lacking in the past and I feel that I’m now at a stage in my life where I’m happy to delve into 4D. Can anyone help? Oh, and if you could do it in such a way that a 12 year could understand that’d be great.