There's a rather elegant piece of machinery called "Stewart platform" (apparently invented by a fellow called Gough --- you know how these things work), aka 6-jointed parallel robot, comprising 6 hydraulic rams which over a limited range can propel a platform along an arbitrarily specified isometric path. See http://en.wikipedia.org/wiki/Stewart_platform This device occasionally becomes stalled in a position from which it cannot move further. A rather elegant theorem (mentioned by Helmut Pottmann, though I don't think he provides a reference) asserts that such situations occur just when (the Pluecker coordinates of) the ram axis lines become linearly dependent. I conjecture that a similar result applies to the bicycle spoke problem: 6 spokes locate the hub statically relative to the rim, provided they remain always in tension (that is, no spoke can shorten without causing some other to lengthen); and furthermore, they do so "rigidly" (transmitting all torques under tension) provided they are linearly independent. An example of the geometric meaning of linear dependence is provided by quadric surfaces in 3-space: a 3-dimensional subspace of lines in general traces out a quadric regulus (hyperbolic paraboloid or one-sheet hyperboloid if the lines are real). 5-dimensional subspaces (subject to a single linear constraint) are known as "linear complexes" in the line-geometry literature. Fred Lunnon On 10/26/10, Allan Wechsler <acwacw@gmail.com> wrote:
It is possible that spokes are intended to be only tensile, in which case the equality in the constraint should be replaced by "less than or equal", and the answer is almost certainly different.
On Mon, Oct 25, 2010 at 9:32 PM, Dan Asimov <dasimov@earthlink.net> wrote:
From various comments made, especially by Fred, I think the intended problem is something like this:
Given two finite cylinders C_r, C_R of radii r < R and equal heights in R^3, how many constraints of the form
||x_j - y_j|| = c_j, for x_j in C_r and y_j in C_R
in order to require C_r to be fixed in a uniqconcentric
I wrote:
<< In the biography of H.S.M. Coxeter "King of Infinite Space", one section describes his interaction with Buckminster Fuller.
It states that Fuller erroneously claimed that a bicycle wheel needs 12 spokes "to hold it rigid", whereas the correct number is 7. This seems to be stated as a fact of geometry, rather than one of structural engineering.
Anybody have any insight into this claim?
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