“It is of course required of a man that he should benefit his fellow-men — many if he can; if not, a few; if not a few, those who are nearest; if not these, himself. For when he renders himself useful to others, he engages in public affairs.” (Seneca, On Leisure 3.5)
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| ... man that he should benefit his fellow-men |
Seneca was a stoic. Having this in mind let us continue from Lie Sphere Geometry Part 3: oriented circles. We have considered there circles St(m), where m is a point on the unit sphere S2 in R3, and t varies between 0 and 2π. In the animation we have endowed each of this circles with a unit normal vector field n(t,ϕ). For t=0 and t=π, the circle shrinks to a point and the normal vectors n(t,ϕ) point in different directions for different values of ϕ.
n(3.2,ϕ), ϕ = k π/4, k=0,...,7
Thus points have their orientations undefined.
We could think that all oriented circles (including points) are parametrized by m∈S2, and 0 ≤ t < 2π. However, looking at the animation of the previous post we can easily visualize the fact that a circle starting at m=(1,0,0) and t = 3π/2 is exactly the same, including its normal vector field, as the circle that starts at m=(-1,0,0) and t=π/2. The first one collapsed to a point (-1,0,0) after t=π, and starts expanding again, the second one simply starts at (-1,0,0). More generally circles (m,t) and (-m,t+π mod 2π) are exactly the same, including their orientations. (Can you see it?)
It follows that the set of all oriented circles on S2 is nothing else but
(S2⨉S1)/Z2,
where Z2 = (+1,-1) acts by
(-1)(m,t) = (-m,t+π mod 2π).
Taking quotient by Z2 may lead to non-orientable surfaces like Mobius strip or Klein bottle. Such surfaces need higher dimensions to embed them in. And that is our plan for the future posts. We will discuss the manifold of all oriented circles using projective geometry. We will add not just one or two, but three extra dimensions!
