Er, on second thoughts --- 7.3 - 6 - 7.1 = 8 ?! WFL On 6/23/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
DWW asks for a properly 3-space Fano configuration, with circles disjoint (except at Fano points).
Consider some general set of 7 planes in 3-space; define the 7 Fano points as an appropriate subset of 3-plane intersections, and the 7 Fano lines as the circumcircles in each plane. Specifying each unit radius constitutes 1 constraint; so modulo isometry, solid configuration set (complex) freedom dimension = 7.3 - 6 - 7.1 = 11 ; and its real planar (presumed) limit dimension = 4 (probably).
But is a planar configuration in general the limit of properly solid ones? We have to be careful here: for instance modulo isometry, set of (7-fold coincident) unit circles with 7 concyclic points is freedom-6; whence limits of (freedom-4) planar configurations necessarily constitute a proper subset.
In mechanical terms, suppose that Fano points can slide along Fano circles. If a planar configuration is picked up and shaken, will it always disorganise properly into 3-space?
It's a pity my points & centres relaxation fails to converge, since it could be modified trivially to solid space (albeit sans graphics).
Fred Lunnon
On 6/23/15, David Wilson <davidwwilson@comcast.net> wrote:
That's pretty neat, I wouldn't have thought to look for planar solutions.
My original hope was a solution with unit circles in R^3 where the circles are Fano lines and circle intersection points the Fano points. You are saying that's not possible?
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Fred Lunnon Sent: Monday, June 22, 2015 9:15 PM To: math-fun Subject: Re: [math-fun] Fano Plane puzzle
<< Does one circle really contain 4 vertices, or does it just look that way? >> DA
I hadn't noticed that rather ugly feature --- one point is rather close to a circle to which it does not belong. Another example avoiding that infelicity: https://www.dropbox.com/s/yl819xpiy716qwh/fano7pt7rg_1.gif?dl=0
Adam seems to have nailed the whole problem very neatly --- I haven't checked his reference yet, but it presumably gives an explicit construction. Incidentally, I also tried relaxation based on 14 points and centres, but it was a total failure!
WFL
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