Wikipedia:Reference desk/Archives/Mathematics/2015 February 7
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February 7
[edit]Animation of anchored rotating cube
[edit].gif)
I recently saw an animation of a cube rotating around an axis with each of the six faces anchored externally via its own flexible chord (three colours, one for the pair of chords in each direction), I think in Wikimedia Commons. Can anyone provide the link? —Quondum 17:10, 7 February 2015 (UTC)
- Cancel that. I found it. —Quondum 21:23, 7 February 2015 (UTC)
- Wow, that's very cool! Do you know if anyone has made a working physical model (kinetic_sculpture) of this? -- ToE 14:18, 8 February 2015 (UTC)
- Not that I know but it would not be that hard to make a model of it. The key is it rotates around the up axis, i.e. about the green faces. Further the green chords though they may be flexible don't change much but just rotate. So make a cube with the green chords stiff and rigidly attached. Make the red and blue ones out of something flexible. Then fix the top and bottom of the green chords and rotate about the line between them. The hardest part is probably choosing materials for the flexible chords, as I can't think of a material that behave's exactly like that. Perhaps something like flexible springs, i.e. coils of wire which could be covered in blue and red material.--JohnBlackburnewordsdeeds 17:03, 8 February 2015 (UTC)
- I would construct it to show that the chords do not twist for added effect (note that the green chords cannot be both rigid and firmly attached to the cube): attach 6 flexible tubes firmly to the cube and to the environment i.e. housing walls (no rotation at points of attachment). The green chords could contain rigid bent rods that are lubricated to rotate inside the green tubes, and could be driven externally (via the end) to produce the rotation. The tubes could be flexible plastic or as described, covered springs. The others are more problematic: they could contain springs, but would flop around a bit as the needed length changes. I've also wondered whether one could construct something like this with the rotation axis at an angle so that the sequence of deformations each chord is identical to that of the others, though out of phase. A challenge for someone to animate! —Quondum 18:42, 8 February 2015 (UTC)
- Ah yes, looking closely the cube is going around twice for every rotation of the green chords. This perhaps could be solved using motors. Have motors that rotate the green chords then additional motors in the cube that rotate the cube at the same rate about them (or it as it could be just one long bent wire) so twice as fast. To get it to start and stop you could just switch power on and off. It would probably drift over time but that might be improved by tuning. For the other colours I was thinking springs as they both stretch and flex, as well as returning to their original position. Given enough freedom but anchored well they should in theory keep trying to adopt a position with minimum energy which I think will be close to what's in the animation. I don't know where you'd get springs of the desired size, they might need to be made specially.--JohnBlackburnewordsdeeds 04:23, 9 February 2015 (UTC)
- I would construct it to show that the chords do not twist for added effect (note that the green chords cannot be both rigid and firmly attached to the cube): attach 6 flexible tubes firmly to the cube and to the environment i.e. housing walls (no rotation at points of attachment). The green chords could contain rigid bent rods that are lubricated to rotate inside the green tubes, and could be driven externally (via the end) to produce the rotation. The tubes could be flexible plastic or as described, covered springs. The others are more problematic: they could contain springs, but would flop around a bit as the needed length changes. I've also wondered whether one could construct something like this with the rotation axis at an angle so that the sequence of deformations each chord is identical to that of the others, though out of phase. A challenge for someone to animate! —Quondum 18:42, 8 February 2015 (UTC)
- Not that I know but it would not be that hard to make a model of it. The key is it rotates around the up axis, i.e. about the green faces. Further the green chords though they may be flexible don't change much but just rotate. So make a cube with the green chords stiff and rigidly attached. Make the red and blue ones out of something flexible. Then fix the top and bottom of the green chords and rotate about the line between them. The hardest part is probably choosing materials for the flexible chords, as I can't think of a material that behave's exactly like that. Perhaps something like flexible springs, i.e. coils of wire which could be covered in blue and red material.--JohnBlackburnewordsdeeds 17:03, 8 February 2015 (UTC)
- Wow, that's very cool! Do you know if anyone has made a working physical model (kinetic_sculpture) of this? -- ToE 14:18, 8 February 2015 (UTC)
zeros of zeta function
[edit]how to calculate zeros of zeta function and graph of zeta(1/2+it)
I've read several recent books on the RH for the lay audience. However, none show how to actually calculate a zero of the Riemann zeta function. For example, how does one calculate:
14.134725142
21.022039639
25.010857580
30.424876126
32.935061588
I would really like to know an algebraic expression to do this? Possible? My level of math is that of a typical engineer (including 1 semester of complex analysis). My understanding of infinite series is shakey at best. — Preceding unsigned comment added by 151.236.166.33 (talk) 21:36, 7 February 2015 (UTC)
- Calculating zeros of the zeta function is very hard (there is no formula for it, algebraic or otherwise). I believe it is usually done by constructing an approximation to the zeta function that is valid in the critical strip, where the error in the approximation can be controlled (e.g., the Riemann-Siegel formula). This is how Titchmarsh computes the first few dozen (in 1935, by hand!), basically plugging numbers into an approximation until a change in sign is observed. He does have a heuristic that bounds the number of zeros in suitable intervals on the characteristic strip, so he is able to rule out zeros other than the ones he finds on in this way. I don't know what the current state of the art is, but it's not likely to be much easier to understand than the Titchmarsh paper (which is referenced in the article Riemann hypothesis.) Sławomir Biały (talk) 22:54, 7 February 2015 (UTC)
