Mathematical physicist and mentor to mathematically talented youth. Talent is that which bridges the gap between what can be taught and what must be learned.
Joined June 2012
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This is incredibly moving. A great master, performing a timeless masterwork. Unforgettable.
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For almost ten years now, I've been on Twitter. I've tweeted more than 7,000 times. And I'm profoundly demoralized by this experience; I think it's been a waste of time. I've tried hard, but I've communicated very little to very few, and inadvertently engendered much ill will.
Any circle in the plane may be identified with the set of all (x, y) satisfying
(x - a)^2 + (y - b)^2 = c^2
for some a, b, c. But this equation may be put in the form
A(x^2 + y^2) + 2Bx + 2Cy + D = 0,
where A is non-zero, and
a = -B/A
b = -C/A
c^2 = (B^2 + C^2 - AD)/A^2.
That point will be external to the unit sphere, and so will determine a fourth circle on the unit sphere, which will be tangent to the three pairwise-tangent circles on the unit sphere.
And that fourth circle will correspond to a circle in the plane tangent to the initial three.