We can get arbitrarily close to 1 by beginning with SP being the vertices of a regular 7-gon of unit circumradius, and perturbing the points by epsilon so that they're in general position. So the question is: can we get equality to hold? That is to say, is it possible for all circles to have unit radius? This seems decidedly non-trivial to resolve. Sincerely, Adam P. Goucher
Sent: Thursday, June 18, 2015 at 12:56 AM From: "Fred Lunnon" <fred.lunnon@gmail.com> To: math-fun <math-fun@mailman.xmission.com> 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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