Tuesday, October 5, 2010

Water in a Cube

I don't often come across visualization problems that I can't see the answer to in a few seconds. (I mention this since I too frequently come across people who assume that because I'm female I must be bad at this. Remember, I do geometry.) So I was very excited when I was recently given the following tricky visualization problem:

Suppose you are given a cube. Stand it on one vertex so that the diagonally opposite vertex (the one with which it shares no faces) and your vertex create a line perpendicular to the ground. Now fill the cube halfway with water. Looking straight down, what shape is the surface of the water?
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John Moeller said...

It's a hexagon.

Anonymous said...

For me the reason the answer is unintuitive is because cubes are so often associated with powers of 2, and not their euclidean dimension.

John Moeller said...

Do the same with a regular dodecagon. It took me a few seconds longer because I had to count.

Boris said...

Here's another visualization problem with a cube. Hopefully you and your readers will enjoy it.

Take a cube and fix two opposite vertices. Then spin the cube. Looking from a direction perpendicular to the axis of rotation, what shape do you see?

That is, what is the union of all of the rotations of the cube that fix two opposite vertices?

Anonymous said...


Aren't cubes normally associated with powers of three?


Anonymous said...

I'm late to the party, but I want to say that the best I can come up with is that it's obviously isomorphic to a disc. Oh, and convex even.

I've always been pretty hopeless in 3d or higher.