[math-fun] where can I buy a flexing hyperboloid?
This is from Courant and Robbins, "What is Mathematics", pg 214 * * * One of the most remarkable properties of the hyperboloid is that although it contains two families of intersecting straight lines, these lines do not make the surface rigid. If a model of the surface is constructed from wire rods, free to rotate at each intersection, then the whole figure may be continuously deformed into a variety of shapes **** So, I'd like to play with such a model. Can I buy one somewhere? Does anything come to mind? I doubt I have patience to make one. I'm also a bit unclear on what "free to rotate at each intersection" means in the quoted passage. Can you fold it into nonhyperboloids? Google was unhelpful. Here's a photo from the book http://static.flickr.com/37/104048548_2327835121_o.gif -- Thane Plambeck http://www.plambeck.org/ehome.htm
* Thane Plambeck <thane@best.com> [Feb 25. 2006 06:28]:
This is from Courant and Robbins, "What is Mathematics", pg 214
* * * One of the most remarkable properties of the hyperboloid is that although it contains two families of intersecting straight lines, these lines do not make the surface rigid. If a model of the surface is constructed from wire rods, free to rotate at each intersection, then the whole figure may be continuously deformed into a variety of shapes
****
So, I'd like to play with such a model. Can I buy one somewhere? Does anything come to mind? I doubt I have patience to make one. I'm also a bit unclear on what "free to rotate at each intersection" means in the quoted passage. Can you fold it into nonhyperboloids?
[...]
Wild guess (please confirm): degree of freedom is same as if connecting two rings with threads and twist them. 0 degree ==> cylinder 180 degree ==> two facing cones everything in between: hyperboloid. That how I visualized hyperboloids on the good old apple2. Was I correct? -- p=2^q-1 prime <== q>2, cosh(2^(q-2)*log(2+sqrt(3)))%p=0 Life is hard and then you die.
The equation of a quadric surface has 10 coefficients, so the freedom of the one-sheet hyperboloid equals 9; subtracting 6 for spatial isometries leaves freedom 3 for the shape/size of the surface. [This is easier to visualise for the 3 orthogonal axes of an ellipsoid.] Since the joints must also have freedom 3, the rods must not only (1) rotate around one another without changing their relative angles, but also (2) rotate "along" each other orthogonally to the previous; and finally (3) slide along one another. Any doubly-ruled surface is a quadric, so this model generates any and only one-sheet hyperboloids. JA's model generates a freedom-1 subset: in particular, its cross-sections are circular rather than elliptical. I think that joints with at most two of the freedoms (1)-(3) restrict the surface to that in which they were assembled; in which case the C & R construction, as reported by TP, appears to be incorrect. Fred Lunnon
participants (3)
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Fred lunnon -
Joerg Arndt -
Thane Plambeck