Two views of FWH solid circular Fano plane at https://www.dropbox.com/s/iofsjqyln0n929d/hele_solid0.gif?dl=0 https://www.dropbox.com/s/zs7pi91p0f99m5o/hele_solid1.gif?dl=0 the first showing 6-fold symmetry about z-axis, the second with z-axis up page showing one (of 3) adventitous double point below right of centre. No coordinate box this time. For nothing is simple. FWH's circle #1, parallel to xy-plane with centre on z-axis, defeated all attempts to plot it until in desperation I rotated everything thru' an entire radian around all three axes ... Maple solve() moves --- or in this case, fails to go anywhere --- in mysterious ways! WFL On 6/30/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
On 6/29/15, David Wilson <davidwwilson@comcast.net> wrote:
Fred, you really came through. I just knew it had to exist. It's quite beautiful.
Sadly there remain 3 circles adventitiously meeting 4-th points disjoint from the corresponding projective lines! [ I brainstormed earlier over symmetry: 3-fold around one cube diagonal is correct. ]
I am currently wrangling its dual (exchanging points with centres), which for some reason is reluctant to display properly; however I strongly suspect that will turn out to present a similar flaw.
So although pretty, it's even more thoroughly crocked than FWH's -- for which I intend to post a graphic as soon as I can engineer its point coordinates -- despite it apparently lacking algebraic proof of correctness.
WFL
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Fred Lunnon Sent: Monday, June 29, 2015 9:23 AM To: math-fun Subject: Re: [math-fun] Fano Plane puzzle
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
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