Lie Sphere Geometry Part 2: unoriented circles

 

Process philosophy, rooted in thinkers like Alfred North Whitehead and Charles Sanders Peirce, posits that reality is fundamentally dynamic—composed of processes, events, and relations rather than static objects. In this view, becoming precedes being; change and interaction are primary, while "things" are derivative patterns of stability within flux. Category theory aligns with this perspective by emphasizing arrows (morphisms) over objects, treating objects as mere nodes in a web of transformations. Peirce’s semiotics and process-oriented metaphysics anticipated this, framing reality as a continuum of signs and relations, where meaning and existence arise from dynamic interplay rather than fixed substances. Nature, then, is best understood as a network of interdependent processes.

Becoming precedes being

In Parts 19-23 we have discussed circles and conformal maps. Circles are 1D objects, and they have been discussed in two dimensions. But circles may be static or dynamic. A dynamical circle can be expanding or contracting. The dynamical state of the circle can be symbolized by an arrow showing the direction of its rotation with respect to its center. Then, choosing anticlockwise orientation, we draw a perpendicular to this arrow. It will point either towards the center, or outside of the center. State of contraction or expansion. We call them "oriented circles". Points are circles of zero radius. You can not give two orientations to a point. So points are somewhat special. Straight lines can be considered as circles of infinite radius. Lines can carry arrows much like circles. Again there are two areas, one on one side of an infinite straight line, another area on the other side. It may tell us how our line "moves" in one of the directions perpendicular to the line.

Our aim are, in fact, spheres rather than circles - thus one dimension higher. But we will tart with circles, as they are intuitive for our 3D perception. Only after we are done with oriented circles, we will move to oriented spheres.

We first start with unoriented circles. But we want to treat circles, lines, and points in a unified framework.  The first step will be to move from drawing circles on the flat plane to drawing them on a sphere - which is a plane plus a point at infinity. We know that stereographic projection maps the plane into the sphere, and maps circles and lines into circles, preserving the angles between them. We will skip the stereographic projection step, and start directly with drawing our circles on the unit sphere S² in R³.

Unoriented circles

Unoriented circles are simple to deal with. First we define S²:

S² = {x∈R³:  x² = 1}

Then, for every m∈S², and for every r∈[0,π] define

S_r(m) = {x∈S²:  x·m = cos r}.

Then S_r(m) is a circle on S² with center at m. For r=0 it reduces to one point x = m. For r = π it reduces to the opposite point x = -m. For r = π/2 we have a great circle. The parameter r can be considered as a distance between x and m measured along the shortest path (the great circle) connecting x and m. The maximal distance between two points on S² is π.

Let us discard here points and consider only nonpoint spheres. Let me quote (simplified for our needs) Proposition 5.5, p. 120,  from G.R. Jensen et al., Surfaces in Classical Geometries, Springer 2016:

Proposition 5.5. If Σ denotes the set of all nonpoint circles in S², then

Σ = (S² ⨉ (0,π))/ℤ₂,

where ℤ₂ acts on S² ⨉ (0,π) by

-1(m,r) = (-m, π - r).

G.R. Jensen et al., Surfaces in Classical Geometries, p. 121

Proof: Can you see it?

P.S. 28-03-25 13:22 Yesterday's seminar mentioned in a P.S. of the previous post is now available online. Here is a screenshot with me making a comment on simulation of quantum processes with classical computers:

And here is another screenshot, of the speaker, S.A. Vekshenov,  talking about classical and quantum processes

P.S. 29-03-25 12:35 From my recent exchange with Gennady Shipov

Gennady Shipov

 I have learned about his many papers, in particular "The Problem of Inertia as a Cause of Stagnation in Theoretical Physics".  At the end of this paper we read:

"An objective external review of the work done highlights that the obtained results, in my opinion, are worthy of four Nobel Prizes in Physics, namely:

1. For the generalized formula $E(\omega) = m(\omega)c^2$, where mass is defined through the inertia field according to equation (56).

2. For the generalized Tsiolkovsky equation (65), which utilizes the conservation law (59) and asserts the possibility of motion in outer space without mass ejection.

3. For the theoretical justification and experimental confirmation of the motion of a 4D gyroscope under the influence of inertia forces.

4. For the creation of a deterministic quantum mechanics, in which the wave function $\psi$ in equations (87) (formulas (81), (82)) is determined by the inertia field $T^j_{ki}$.

As a far reaching vision, it sounds very interesting for me, in particular 4. So, I will have to take a closer look and see for myself how much of it I am able to understand. T^i_jk is the torsion tensor, and torsion, as it seems,  can couple directly to spin. Perhaps this will help me understand spin?

P.S. 30-03-25 12:38 Yesterday night I asked "myself in the future" about geometric algebras. In particular about the kind of geometric algebras that we will need in the future. The answer was: "Clifford algebras with enhancements". And, moreover, infinite-dimensional. So it's gonna be a really big elephant.

George Gamow, MR TOMPKINS IN WONDERLAND 

P.S. 01-04-25 12:31 AI can be useful. Here are some of the equations of the aether theory (with "slots" d=1,...,7)  based on Clifford Geometric Algebra (CGA), proposed by AI:

Of course human intelligence and inspiration are needed to make sense of these equations.

P.S. 01-04-25 15:52  I thought I will put a new post yesterday or today, but I am not yet completely happy with the level of my understanding of  "oriented circles", as they are described in the literature. I do have some geometric problems with them. And today evening there is another seminar I want to attend:

"Change of entropy-information balance in the processes of radioactive decay and nuclear fusion. Application of generators of non-electromagnetic interactions as an effective and only possible method of controlling the intensity and probability of the state of nuclear processes" by A.V. Karavaikin. 

Thus delay.

P.S. 02-05-25 Here is the full text of Grok 3 AI idea about hyperdimensional Clifford Algebra physics (as I mentioned in a P.S. yesterday):

And from today's session with Grok:

P.S. 02-04-25 18:32 Still fighting with oriented circles for the next post. Produced an animation with Mathematica, but I am not happy with my understanding of the subject.