On 10/29/10, Fred lunnon <fred.lunnon@gmail.com> wrote:
However 7 rams in tension are (necessary and) sufficient for a robot in general: for example, the Stewart platform with all 6 legs equally extended is opposed by a single extra ram, mounted vertically upward from the platform centre. This presumably constituted the other half of Coxeter's argument, and serves to demolish Edmondson's.
It's not that simple (we've been here before): for all we know, compressing one leg might also compress its neighbour, leaving only 5 in tension.
My earlier proposal for 4 spokes in opposed pairs spiralling on one side, 3 spokes radial on the other side, still looks convincing to me. To prove it would involve partitioning line space into 7 subsets, corresponding to the 7 bases comprising 6 from 7 spokes (appropriately signed), such that always some component with respect to the corresponding basis is positive (denoting tension).
Well, that's perfectly feasible, I suppose ...
Maybe so; also unnecessarily complicated, and wrong anyway. It's actually nec. & suff. that (1) the spokes together span line-space (2) each spoke (Grassman 6-vector, consistently signed) be expressible as a linear combination of others with non-positive coefficients. A reprieve for Edmondson, perhaps? Not so fast, Batman! Because ... Theorem: 8 spokes in tension suffice to mount rigidly a hub in a rim [demonstration below]. Reducing 8 to 7 (earlier proved necessary) presents a slightly trickier prospect: there is less symmetry available, and subsidiary linear relations between the spokes are verboten. Finally, I had earlier omitted to mention one rather important fact: although Euclidean isometries (hence continuous rigid motions) do not commute, their differentials (impulsive screws) do: this permits them to be mapped onto a vector space, with composition mapping to addition. Can somebody recommend a background reference for this material? There must be plenty about it in Helmut Pottmann's book on Line Geometry, which I don't have to hand; or perhaps Whitehead on Universal Geometry (even older than me). Fred Lunnon %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Analysis of a symmetrical 8-spoke wheel lacing: The rim comprises n = 8 (nipple) points at vertices Q[k] of a regular octagon of circumradius r; the hub flanges comprise 8 (elbow) points at vertices P[k] of pair of squares of circumradius q, lying in parallel planes at distance p on either side of the rim, and rotated by pi/4 with respect to one another. With p = hub semi-axis, q = hub radius, r = rim radius, the flange and rim points have projective coordinates [P^0, P^1, P^2, P^3] etc as follows: P[k] = [1, q sin(k pi/4), q cos(k pi/4), p (-1)^k]; Q[k] = [1, r sin(k pi/4), r cos(k pi/4), 0]. for k = 0,...,7. The spokes L[k] connect points as follows L[1] from P[0] to Q[2] L[2] from P[1] to Q[7] L[3] from P[2] to Q[0] L[4] from P[3] to Q[5] L[5] from P[4] to Q[6] L[6] from P[5] to Q[3] L[7] from P[6] to Q[4] L[8] from P[7] to Q[1] with Pluecker coordinates [L^1, L^2, L^3, L_3, L_2, L_1] L[1] = [r, -q, -p, -q r, +p r, 0] L[2] = [(-r-q)c, (+r-q)c, p, +q r, +p r c, +p r c] L[3] = [-q, r, -p, +q r, 0, -p r] L[4] = [(-r-q)c, (-r+q)c, p, -q r, +p r c, -p r c] L[5] = [-r, q, -p, -q r, -p r, 0] L[6] = [(+r+q)c, (-r+q)c, p, +q r, -p r c, -p r c] L[7] = [q, -r, -p, +q r, 0, +p r] L[8] = [(+r+q)c, (+r-q)c, p, -q r, -p r c, +p r c] where c = 1/sqrt(2). It may be verified that spokes have length t = sqrt(p^2 + q^2 + r^2); also that their coordinates satisfy the Grassmann relation. Before normalisation by 1/t^6, a sample 6x6 determinant is | L[1] L[2] L[3] L[4] L[6] L[7] | = 16*(2*r^2/q^2 - 1)*(p*q*r)^3 which (except in the unlikely event that r/q = c) is nonzero; hence they together span line space, and any screw applied to the configuration would be resisted if the spokes were replaced by rams. The spokes satisfy the additional subspace relations L[3] + L[4] + L[7] + L[8] = 0 L[1] + L[2] + L[5] + L[6] = 0 Immediately from these it follows that a screw which compresses any combination of spokes is equivalent to one which tenses some other combination, and so can be resisted by tension alone. [In computing line vectors, it is important that the points are taken consistently in order (say) hub followed by rim, so that the signs are meaningful!] Fred Lunnon [30/10/10] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%