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Another nice puzzle: A cube looks like a square from three orthogonal directions. A cylinder can look like a square from infinitely many directions, but they a
by cousin_it 9y ago
Another nice puzzle:
A cube looks like a square from three orthogonal directions. A cylinder can look like a square from infinitely many directions, but they are all coplanar. Can you find a convex shape that looks like a square from more than three directions, without all of them being coplanar? In particular, can you find a convex shape that looks like a square from two distinct sets of three orthogonal directions? Can you find all such shapes?
- no_identd 9y agoDo mathematicians consider this an open problem or does there exist a well known solution? And what about a name for this problem, does it have one already?
- cousin_it 9y agoNah, it's nothing as grand as that. I saw it on facebook a few months ago and solved it in about ten minutes, just wanted to share the fun.
- no_identd 9y ago...do you know the story of George Dantzig? https://en.wikipedia.org/wiki/George_Dantzig#Mathematical_statistics https://en.wikipedia.org/wiki/George_Dantzig#Mathematical_st... I don't mean to say that it seems applicable here - but I also don't mean to say that it doesn't seem applicable here.
- Avshalom 9y agoAm I allowed to invoke hypercubes because that would cut this knot nicely.
- TehCorwiz 9y agoOrthographic or perspective projection?
- cousin_it 9y agoI only know the solution for orthographic. But it seems like perspective would be too easy, just make any shape with many square sides, and put the cameras close to the sides.
- whatshisface 9y agoCylinders only look like squares in orthographic projection, so my guess is that that's the intention.
- mhei 9y ago(Rot13 as not to spoil the riddle) Ubj nobhg n grgenurqeba? Begubtencuvp cebwrpgvba nybat gur yvar pbaarpgvat nal rqtr zvqcbvag gb gur bccbfvgr rqtr zvqcbvag tvirf n fdhner, lvryqvat 3 fhpu cebwrpgvbaf (cyhf gurve bccbfvgr pbhagrecnegf). Abg fher vs gurer ner bgure fhpu funcrf (vtabevat gevivny zbqvsvpngvbaf bs gur grgenurqeba) - qvq lbh znantr gb svther guvf cneg bhg?
- cousin_it 9y agoA tetrahedron looks like a square from three orthogonal directions, same as a cube. It's possible to get more.
- darkmighty 9y agoI'm thoroughly impressed by your geometric abilities! I didn't know that, and took me a while to check. Any hints on the puzzle, and as a sidequestion, what tools do you use to figure this kind of question? Just imagination, vector algebra, elementary trigonometry?
- cousin_it 9y agoHuh? There's nothing to be impressed about. You can stick a tetrahedron inside a cube so it creates the same square shadows in all three directions: https://i.stack.imgur.com/oAUnH.gif https://i.stack.imgur.com/oAUnH.gif I know a lot of math, but for this puzzle, drawing stuff on paper is enough. Here's a hint: if you cut off one corner of the cube, all shadows are still square. How much can you cut? Can you cut some corners strategically to make at least one new square shadow while keeping all the old ones? How many square shadows can you get?
- octahedron 9y agoAn octahedron?
- cousin_it 9y agoAn octahedron looks like a square from three orthogonal directions, same as a cube. It's possible to get more.
- octahedron 9y agoIs the question posed only in relation to three dimensional space?
- cousin_it 9y agoYeah, it's the kind of shape you can carve from a block of cheese in a minute.