(March 5, 2022 at 8:03 am)GrandizerII Wrote:(March 4, 2022 at 8:51 pm)polymath257 Wrote: Imagine a cube. Consider the top left, front vertex. Opposite that is the bottom, right, back vertex. There is a *three* fold rotation around the line joining those opposite vertices.
If you look at the pyramid consisting of the top, left, front vertex and the bottom square, that three fold rotation will take that pyramid to two other copies that, together, fill the cube.
Hence, the volume of that pyramid is 1/3 the volume of the cube.
Thanks, polymath. But I think the issue with me is that I can't visualize that three-fold rotation well. Even if I were to watch a video/animation of that occurring, I still would probably struggle to have that intuition once I look away from the screen. Thanks for coming up with something simpler than other stuff I've checked online, though.
I'll get to Fire's article soon. If still struggling, maybe it's easier for me to just go with the calculus route then.
Imagine holding the cube between your thumb and forefinger by opposite corners of the cube. if you have a pair of dice, take one and do it. You will find that you can rotate around the axis between your fingers and the resulting rotation is three fold.
There are three edges coming out of each vertex. The three fold rotation cycles those edges out of the opposite corners. it also cycles the faces that come together at those corners.
It *is* a difficult rotation to imagine. But it is well worth it.
If you take a (planar) cross section perpendicular to the line between opposite corners and half way between them, the cross section on the cube is a hexagon (!).
