Theorem: _________________________________________________________________ 7 spokes in tension are necessary and sufficient to mount rigidly a hub in a rim. _________________________________________________________________ Necessity of 7 is essentially a consequence of the sufficiency method, there being no linear relation possible between 6 members of a basis. Sufficiency follows the same strategy as earlier for 8 spokes: the production of linear relations, involving all the spokes, whose coefficients are all non-negative. This configuration involves synchronised regular 12-gons of holes P[k], R[k], Q[k], on parallel planes for left, right flange, rim, with radii q, q, r, flanges at distance +p, -p along the common central hub (z-axis). The lacing connects on the left side (radially) L[1] from P[0] to Q[0] L[2] from P[4] to Q[4] L[3] from P[8] to Q[8] and on the right side (spirally in opposing pairs) L[4] from R[1] to Q[5] L[5] from R[5] to Q[1] L[6] from R[7] to Q[11] L[7] from R[11] to Q[7]; the squared lengths of the spokes are respectively s^2, t^2 = p^2 + (r-q)^2, p^2 + (r+q)^2. The determinant of the first 6 spokes (before normalisation by s^3 t^4) is 9 sqrt3 (r - q) p^3 q r^4, establishing spanning of line-space; and the single relation between the spokes is 4*(L[1] + L[2] + L[3]) + 3*(L[4] + L[5] + L[6] + L[7]), establishing reaction under tension. QED It would be interesting to know how much of this Coxeter actually knew (off the top of his head?), or whether he just guessed lucky (I reckon I'm scoring around 50% here --- about as bad as could be!) Since Fuller & co. were apparently unaware of this technique (er --- the screw theory, that is), it presumably is not common knowledge amongst structural engineers --- which I find surprising. There's obviously more questions that could be asked at this juncture --- such as, what hole layout and spoke lacing minimises the maximum spoke tension in response to a unit applied screw? [Note that the relevant metric appears to be the purely directional sqrt( (L^1)^2 + (L^2)^2 + (L^1)^2 + (L^1)^3 ), being also the length of a line segment between normalised points.] But perhaps I'm going to be able to put this beauty and myself to bed now (doubtless I'm not alone). Nice problem, Dan! Dan? Are you still there? Fred Lunnon [30/10/10] On 10/30/10, Fred lunnon <fred.lunnon@gmail.com> wrote:
... 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^8, 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]