A couple of (considerably less feeble) views of solid Fano posted at https://www.dropbox.com/s/i4l1szci7o0ww7r/fano_solid0.gif?dl=0 https://www.dropbox.com/s/vpgp26kcr1ttsgk/fano_solid1.gif?dl=0 the first showing 3-fold symmetry about cube diagonal. WFL On 6/29/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
On 6/28/15, David Wilson <davidwwilson@comcast.net> wrote:
Really, all I want is a stabile hanging in my office consisting of circular wire rings of equal radius (in rainbow colors) welded together with Fano connectivity and otherwise not touching one another.
It's not like I'm hard to please. You would think I had asked for circular Borromean rings. Sheesh.
Right then --- here is an elegant embedding exactly as requested (on numerous occasions) by DWW for a Fano plane composed of points and circles. It has 3-fold symmetry about axis line x = y = z , without extraneous collisions or near-misses. Four points lie at alternate corners of a scaled unit cube; the other three at equal displacements along axis lines.
Approximate values for the points and centres are as follows; as before, a unit circle with centre index i (mod 7) meets points indexed i, i+1, i+3 . q := 0.5912088662; p := 1.224744871; points := [ [q,-q,-q], [-q,q,-q], [-q,-q,q], [0,p,0], [0,0,p], [q,q,q], [p,0,0] ]; u := 0.3203726673; v := 0.2819082667; w := 0.4082482903; centres := [ [u,u,-v], [-v,u,u], [-u,u,v], [w,w,w], [v,-u,u], [u,v,-u], [u,-v,u] ];
I have a proof of correctness, and exact algebraic equations for the point components, though not the centres: 2*p^2 - 3 = 0 ; 36*q^4 - 4*q^2 - 3 = 0 ;
Available on request is a Maple worksheet showing a (rather feeble) 3-D manipulable graphic, which ate up an order of magnitude more coding effort than finding the configuration in the first place!
Fred Lunnon