(March 4, 2022 at 8:51 pm)polymath257 Wrote:(March 3, 2022 at 2:14 pm)GrandizerII Wrote: Is there a way to prove this algebraically without the use of calculus? And without resorting to 3d visual demos, which have caused me more confusion than clarity.
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.
