Lie Sphere Geometry Part 3: oriented circles and Seneca

 “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)

 ... 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 S_t(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!

"It is worth noting that, beyond affirming the presence of a Creator in the Universe, the article "On the Question of Speed" unwittingly brushes against the enigma of exploring Jung’s elusive flux. For if the soul, in an instant, reunites with a body cast dozens of light-years apart, then the transfer of knowledge, unmarred by time’s delay, becomes conceivable—

A whisper of proof for the flux’s existence."

(Эдуард Веркин, «Сорока на виселице», Москва: «Эксмо», 2025)

P.S. 06-04-25 16:24 Sunday Strip: Jumping Off a Cliff

 P.S. 07-04-25 11:45 Found it interesting:

" It is still unknown whether S⁶ admits an integrable almost complex structure. Many well-known and respected mathematicians have written papers purporting to answer this question one way or the other, but all the proofs have been found to be wrong or incomplete."

John M. Lee, Introduction to Complex Manifolds, AMS 2024, p. 43.

P.S. 07-04-25 13:43 Temporology seminar tomorrow (time travel included). Very interesting subject for me:

What is a Photon

March 11, 2025, Tuesday, 19:00

Host: Koltovoy N.A.

1. Koltovoy Nikolai Alekseevich. Overview of photon models.
2. Klesnikov Alexander Alexandrovich. Photon model based on the deformation model of space-time.
3. Egorov Evgeny Ivanovich. Photon.
4. Afonin Vladimir Viktorovich. Mechanical structure of the universe. Photon.
5. Godarev-Lozovsky Maxim Georgievich. Cosmological ideas of Alexey Georgievich Shlenov (philosophical aspect).
6. Mirkin Vladislav Iosifovich. Photon.