It’s amazing how much easier it is to make something clear in a visual proof — using pictures rather than equations. Just for comparison, I wrote down the proof I described yesterday using equations instead of pictures, thinking I would post that today.
I encountered two problems: (1) Most people who read this blog won’t be able to follow the equations. (2) Even if they did, there are too many steps.
The biggest difference is at the end, when I showed how those three faces of the cube (which I colored red, green and blue) magically flatten out into three parallelograms to form the hexagon (after you add that one sphere in the center).
From the picture, it’s immediately obvious how this works. But not from the equations. So a story that is simple and elegant when told in pictures becomes a lot less simple and elegant when told with equations.
I wonder whether there might be some sort of hybrid way of describing such things — something half way between equations and pictures, so that you can get the best of both.
Have you read Oliver Byrne’s edition of Euclid’s Elements?
Can you please post the proof? Just want to see it.
When I saw your visual explanation, my first thought was “wow, I’d love to see an animation.” Maybe animation could help make the connection the the elegant visual explanation and the mathematical proof?
It makes me wish Jim Blinn’s Mechanical Universe animations were up on YouTube (a few, badly edited, are… http://www.youtube.com/watch?v=Otr2gojTNyQ )