Originally posted by Grampy Bobby
y = 6 (on a scale of 1 to 6 largest distance between any two points in the plane).
x = 5 (one less, which is the minimum which could apply to all other five points).
[b]y/x or 6/5 = 1.2 (largest distance/smallest distance between any two points).
But are the 6 and 5 just chosen arbitrarily? If so, why not choose 11 and 10 for a ratio of 1.1. Or 101 and 100 for 1.01 and so on, ad infinitum.
I propose the following for consideration:
In order to obtain the smallest y/x ratio, y and x must be as close to equal as possible. Therefore, we want to have the 6 points as equidistant as possible.
Consider the following shape (not drawn to scale, but a helpful visual):
Where the 6 points form a convex, regular hexagon where all 6 'edges' on the perimeter are the same distance apart. This is x. Also 1-6-4-3 forms a square. As do every set of 4 'vertices' whose 'edges' are parallel.
The largest distance here is a diagonal line along any of the squares (e.g. 1-4). This is y.
So y is the diagonal of a square and x is an edge. Therefore, the angle between them is pi/4 (45 degrees). We are only interested in the ratio y/x, so we can assign an arbitrary value to x without any loss of generality. Let x be 1. Thus y / x = y / 1 = y.
Using some trig, we now have:
cos (pi / 4) = x / y
sqrt(2) / 2 = 1 / y.
Therefore y / x = y = 2 / sqrt(2), which is approx. 1.4.