Assume a 3D universe. Cut a cylindrical column out of a block of solid material. Apply force F pushing the two ends of the column toward each other. According to Euler, the column will become unstable against "buckling" if F > (pi/L)^2 * Y * I where Y is Young's modulus of elasticity, I is the areal moment of inertia of column cross section, and L is column length. (Euler's analysis is valid for "slender" columns and concerns stability against infinitesimal "sideways" perturbations.) If other convex shapes are permitted besides circular cylinders, the same thing happens except that the constant pi^2 changes. Specifically, Keller claimed to have computed the optimum convex shape for a 3D column (we always assume ball joints at both ends; Keller also examined having "clamped" ends on one or both sides but his analyses of those cases were attacked) Joseph B.Keller: The shape of the strongest column, Arch. Rat. Mechanics Anal. 5,1 (1960) 275-285. The volume of the optimum convex column is 0.866 times the volume of the uniform column of the same strength. It turns out that Keller's result had already been found by Thomas Clausen in 1851. I prefer to re-express the Euler/Clausen/Keller results another equivalent way as: the NUMBER OF ATOMS N of column-material needed to prevent Euler's buckling-instability (as a function of F and L) is N > constant * F^(2/3) * L^(7/3) for a convex column, valid in the "slender" limit where L is large with F growing at most sublinearly as a function of L. [Derivation: multiply both sides of original Euler ineq by (L^3)/R where R=column radius to get F*L*L*L/R > const * R*R*L = const*N, now solve original Euler for R to find R^3 = const*F*L^2, substitute that in to get N<F^(2/3)*L^(7/3) for instability.] Note the interesting exponent 7/3. Now consider the following "Sierpinski fractal" nonconvex column. Start with the union of a regular tetrahedron of edge length L, with balls centered at the 4 vertices, each of radius R1. Now "remove the middle octahedron" from this tetrahedron, leaving 4 half-edgelength regular tetrahedra each containing one of the 4 corners of the original tet, and union the result with balls (centered at the edge midpoints of the original tet) of radii R2 each. Now on each of these 4 remaining tets, do the same "remove middle octahedron" operation, unioning with balls of radii R3 at the joints... and so on. The recursion stops once we reach the atomic length scale, i.e. after lg(L/atomlength) recursive levels where lg(x)=log(x)/log(2). Each recursion (ignoring the balls) removes exactly half the material, so that the amount left after all levels of recursing is a fraction of order atomlength/L. This constitutes a number of atoms of order L^2, not (which it originally was) L^3. We proclaim the two "endpoints" of this "column" to be any two among the 4 vertices of the original tetrahedron and the handy R1-radius balls easily allow considering "ball joints" at them. If the two end-balls are pushed toward each other by epsilon*L, then this would be compatible with their edge-midpoint moving sideways by sqrt(epsilon)*L (ignoring higher order terms since epsilon assumed tiny). This in turn shortens the 4 length=L/2 edges at the next recursive level by an amount of order sqrt(epsilon)*L each, which can be accomodated by sideways motion of their midpoints by amounts of order L*epsilon^(1/4), and so on. The motion by epsilon*L causes an energy-gain of F*epsilon*L. But there are elastic energy expenditures caused by those distortions (i.e. "sideways" motions). We get instability if the gain exceeds the expenditures. Considering the elastic energies for distorting those balls, the expenditures ought to be of order R2^3*epsilon + 4*R3^3*epsilon^(1/2) + 16*R4^3*epsilon^(1/4) + 64*R5^3*epsilon^(1/8) +... independently of L and F (depends only on epsilon and the Rk's), which is upper bounded by R2^3 + 4*R3^3 + 16*R4^3 + 64*R5^3 +... [we assume the Rk are chosen so that this series converges, for example Rk^3 proportional to F*L*5^(-k)] and is lower bounded by epsilon times that. This suggests we ought to get stability if the number N of atoms obeys N > constant * [ F*L + L^2 ] for this kind of "Sierpinski fractal column" when L is large. =============================================================== We can also do the same kind of analysis but in a 2D universe. In that case a 2D Euler would have argued that for a convex column of length L, stability requires N > constant * F^(1/2) * L^2 atoms, valid in the "slender" limit where L is large but with F shrinking toward 0 as a function of L. [Keller's analysis also works in 2D by the way...] In constrast if we use a "Sierpinski triangle fractal" based on starting with equilateral triangle of side length L, remove middle triangle each recursion then adjoin balls of radius Rk at edge-midpoint joints (at recursion level k)... where the Rk shrink geometrically as a function of k and R1^2 is proportional to L*F, then * number of recursion levels is lg(L/atomlength). * number of atoms (ignoring balls) at deepest recursive level is L^2 * (3/4)^#levels which is of order L^1.584962501 where lg(3)=1.584962501. This causes stability (presumably) if N > constant * [ F*L + L^1.584962501 ]. ================================================================ This way of doing things focuses attention on the "critical exponents" which in 3D and 2D (we have shown nonrigorously) are <=2 and <=1.585 respectively; improvements versus Euler/Clausen/Keller's 7/3=2.3333 and 2 respectively. Can these exponents be shrunk by using better fractals? Can they be justified rigorously? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)