<< 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 On 6/23/15, Adam P. Goucher <apgoucher@gmx.com> wrote:
Note that this problem can be formulated in a different way, by having 14 points instead of 7 (namely the seven original points and the seven circumcentres). Then the problem is clearly equivalent to:
"Is the Heawood graph a unit-distance graph?"
Looking at the Wikipedia article for the Heawood graph, it transpires that this is indeed one of its properties. It provided a reference to arxiv:
http://arxiv.org/abs/0912.5395
So yes, it is indeed possible for all circles to have unit radius.
(I can't decide whether this is the acceptable practice of `reduction to a known problem' or the underhanded technique of `corollary-sniping'...)
Sincerely,
Adam P. Goucher
Sent: Tuesday, June 23, 2015 at 12:10 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
<< I'm guessing you can't model the Fano plane with seven unit circles. >> DWW
<< This seems decidedly non-trivial to resolve. >> APG
I initially overlooked the possibility that there might exist nondegenerate Fano configurations of unit circles in which all 7 points and all 7 circles are distinct, notwithstanding the trivial degenerate limit where the circles coincide.
Formulating the problem in terms of a semi-algebraic set, we have 14 variables --- x- and y-components of Fano plane points --- and 7 sextic equations of form (uvw)^2 - (2A)^2 = 0 , where A,u,v,w denote area and sides of some triangle formed by 3 collinear/concylic points; along with 42 quadratic inequalities asserting no points and or centres coincide.
Modulo isometry, the freedom remaining in this system has dimension 14 - 7 - 3 = 4 . In principle it is decidable whether there are real solutions via Tarski's algorithm, say using SolveTools[SemiAlgebraic]() in Maple; while such an attack would be easy to launch, its prospect of success is remote.
So after a couple of failed attempts to construct a symmetric solution manually, I drafted an iterative search program which starts from a random point set, and simply attempts to relax it progressively towards unit radii.
The first half-dozen configurations generated were indeed degenerate; then out popped https://www.dropbox.com/s/ymf2f9k4mcgmhac/fano7pt7rg.gif?dl=0 with evidently well-separated parts, and radii within 2.0E-6 of unity --- at which error bound my program terminates, though that could easily be improved. The data is attached below in the format [1, x, y, r] .
Clearly an approximation of this nature does not constitute a proof of existence, however intuitively convincing. So where to from here?
Fred Lunnon
[[1, 0, 0, 0], [1, -0.5668481233, 0.6075796485, 0], [1, -0.3575661931, -0.7701296409, 0], [1, 1.094644573, 0.2231048189, 0], [1, -0.2475264260, 0.1937463870, 0], [1, 0.2495133582, -0.8697660570, 0], [1, -0.2053263145, -1.296737218, 0]];
[[1, 0.3816724709, 0.9242989585, 1.000001220], [1, -1.171379414, -0.1890023228, -1.000000460], [1, 0.1000783952, 0.1190048358, 0.9999993435], [1, 0.4397688784, -0.5326334070, -1.000001220], [1, -0.7324710115, -0.6807994545, -1.000000840], [1, -0.6281769140, -0.3905378926, 0.9999999230], [1, 0.6424250025, -0.7663464105, -0.9999983530]];
On 6/18/15, Adam P. Goucher <apgoucher@gmx.com> wrote:
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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