4 ms·
Here's an attempt at making sense of that intuition. Let's compute the volume of a pyramid that's inside a cube. Imagine that you're inside a cube room (this wi
by gniv 3y ago
Here's an attempt at making sense of that intuition. Let's compute the volume of a pyramid that's inside a cube. Imagine that you're inside a cube room (this will make it easier to identify the faces). You are sitting in a corner.
From your corner draw lines to the four ceiling corners -- a cube diagonal, two face diagonals and a cube edge. This is the pyramid. Now draw lines to the corners of one of the walls opposite you -- it's the same pyramid, but on its side. And lastly draw lines to the other wall opposite you -- again the same pyramid. You have now covered the entire cube with three identical pyramids, so the volume of the pyramid is a third of the volume of the cube. The interesting part is that the bases of these pyramids are on the three dimensions, giving the intuition that the /3 is due to the dimension. This proof might even generalize in higher dimensions.
Edit: I think this is also the gist of the top answer in TFA, but it's weirdly formulated imo.
- justinpombrio 3y agoThat's a beautiful proof! And yeah, it generalizes to higher dimensions. If you're in the corner of an N-dimensional hypercube, then the hypercube has 2n faces of dimension n-1. You're touching half of these faces, leaving n faces to form pyramids out of. And just like in the 3D case, they're arranged symmetrically so they must all have the same volume. EDIT: got around to reading TFA and it's also the top answer, so there's a nice visualization. Diving deeper into the proof, say the hypercube is a unit hypercube, spanning from (0, ..n, 0) to (1, ..n, 1) -- where "..n" means "a sequence of length n". Then: Each corner is a point (b_1, ..n, b_n) where each b_i is 0 or 1. You're located at (0, ..n, 0). The hypercube has 2n faces, which can be represented as a pair (i, b) where 0<=i<n and b is 0 or 1. The face (i, b) touches the 2(n-1) corners whose i'th coordinate is equal to b. You touch half of those faces: specifically the faces (i, 0) for each 0<=i<n. You can draw a pyramid to the other half of the faces, since you don't touch them. These pyramids must all have the same volume, by the symmetry of coordinate permutations, which is an operation that preserves volume.
- dadoomer 3y agoVery cool! Thanks for sharing.
- dp-hackernews 3y agoTFA? :-/ TFA - What does TFA stand for? The Free Dictionary https://acronymS.thefreedictionary.com/TFA https://acronymS.thefreedictionary.com/TFA
- gniv 3y agoApologies for the slang: https://www.urbandictionary.com/define.php?term=TFA https://www.urbandictionary.com/define.php?term=TFA