I realized that after I posted it. You could choose your points in R^3 slightly perturbed from seven points equally spaced on a unit circle. I think that way the circles could be made arbitrarily close to unit circles, with the smallest radius any value < 1. I'm guessing you can't model the Fano plane with seven unit circles.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Fred Lunnon Sent: Wednesday, June 17, 2015 7:56 PM To: math-fun Subject: Re: [math-fun] Fano Plane puzzle
The minimum radius has no maximum. WFL
On 6/18/15, David Wilson <davidwwilson@comcast.net> wrote:
Let F be the Fano Plane.
Let FL be the set of 7 lines in F.
Let FP be the set of 7 points in F.
For L in FL, let P(L) be the set of 3 points on L.
Now let SP be a set of 7 points in general position in R^3.
Let m : FP รณ SP be a bijection.
For each line of L of F, let C(L) be the circle in R^3 through the 3 points m(P(L)).
Let SL = C(FL).
Let S = (SL, SP).
S is then a model of the Fano Plane in R^3 with circles for Fano lines and points for Fano points.
If we scale S so that the largest circle has radius 1, how large can we make the radius of the smallest circle of S by judicious choice of SP?
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