Blog reincarnation

 I do like the idea that we are living in a cyclic universe. Cycles repeat, but each new cycle has some memory of the previous cycles. There is a "learning process". Certain features of the previous cycle are being destroyed, new features are being added. We can't really call it a "progress", since "regress" may also happen, for many reasons. It happens all the time: micro cycles, macro cycles, and megacycles.

Here we have an example with my blog. In the past I tried many  options. Now I am going to abandon blogger and test the old-new solution. It took me days to reactivate my old blog, where I had LaTeX installed. Readers can also use inline latex in their comments. That is a big advantage - provided it works. It will take a while to fix all bugs that will certainly appear. But I am optimistic.

So, here is the previous post rendered with latex - to start with: 

Floating in the Einstein Universe

Let me know how it works, and what needs fixing or improving?

Floating in the Einstein Universe

 

This post is the third chapter of the "Primer" saga, based on   "A primer on the (2 + 1) Einstein universe" - a paper is written by five mathematicians, T. Barbot, V. Charette, T. Drumm, M. Goldmann, K. Melnick, while they were visiting the Schrodinger Institute in Vienna and published in print in "Recent Developments in Pseudo-Riemannian Geometry", (ESI Lectures in Mathematics and Physics), Dmitri V. Alekseevsky and Helga Baum eds.,  EMS - Publishing House, Zürich 2008, pp. 179-229. You can download it from arxiv here. The first two chapters: "Perplexed by AI" and "A primer on the Universe" had no math. Now it is time to get serious and to learn how to swim in the Einstein Universe. We will start with learning how to float freely on our backs.

The main arena of "A primer" is the space denoted by Ein^n,1, and we will describe this space now. It is where "geometrical objects" under study live. Then we will start discussing these objects and relate them to the concepts that we have already met in my previous posts, in particular for n=1. What we have done in our sandbox here is a particular case of a universal machine that embeds space-time with n-dimensional space and one-dimensional time. Much of this machine works also with more general, m-dimensional "time", but the case of one-dimensional time is, in a sense, "special". So, here, I will stay with this special case. The paper has "2+1" in the title. This is even a more special case, but a large part of the paper deals with more general n+1 objects. In my recent series of posts we were playing the toy case of 1+1, but we have also given some attention to the 3+1 case relevant to "adult physics". Personally I think that all (n,m,r) cases are pertinent to physics (r is the number of zeros in the signature). But even that will be not enough. Quite probably we will find important uses for ternary, not just quadratic, forms, and complex instead of real structures and geometries. But here let us stay with the synthetic geometry as presented in the "primer". The notion in the paper is sometimes rather original, so we will need a translation between the notation there and the one I was using so far.

Minkowski space E^n,1

We start with the Minkowski space. It is denoted E^n,1. It is defined as an affine space whose underlying vector space is R^n,1 - the space Rⁿ⁺¹ endowed with the quadratic form

q(x) = (x¹)²+....+(xⁿ)² - (xⁿ⁺¹)².

The only difference between E^n,1and R^n,1 is that in  E^n,1 there is no distinguished "origin". Any point cab be selected as the origin, and then  E^n,1can be identified with  R^n,1.

Möbius extension R^n+1,2

The paper does not call it so, but that what it is. We start with R^n,1and add two extra dimensions, one with signature +, and one with signature -.  Thus our quadratic form, written as a scalar product, becomes:

(v,v) = (v¹)² + .... + (vⁿ)² - (vⁿ⁺¹)² + (vⁿ⁺²)² - (vⁿ⁺³)².

The null cone 𝔑^n+1,2.

The paper is using the symbol 𝔑^n+1,2. We have been using just N:

𝔑^n+1,2 = {v∈R^n+1,2: (v,v) = 0}.

The Einstein universe E^n,1 and Ȇ^n,1

These are the same that we have denoted PN and PN₊ in the case of n=1 that we have discussed here. We take the null cone, remove the origin, and identify proportional vectors with proportionality constant λ being non-zero for E^n,1,  and λ>0 for Ȇ^n,1. The authors notice that Ȇ^n,1.is a double covering for E^n,1 ,  and that

Ȇ^n,1. ≈ Sⁿ  × S¹.

For n=1 we recover our torus S¹  × S¹.

The term "Einstein Universe", used by the authors, is not very fortunate. In Einstein Universe time is linear instead of circular. Yes, it is true, the authors consider also the universal covering space with E^~n,1 ≈ Sⁿ  × R¹, but it is not the main subject of the paper. Secondly, the true Einstein Universe comes with a metric - a solution of Einstein field equations for empty space and with a cosmological constant. But E^n,1, carries no natural metric tensor, only a conformal structure. It is true, that we can always endow E^n,1 with a metric compatible with the conformal structure, but such choice depends on which point p we select as the "infinity point". So, let us keep these comments in mind while learning how to float.

The objects

Here is the list of important objects in this synthetic geometry of E^n,1:

• Photons
• Lightcones
• The Minkowski patch

We start with "photons". 

The space of photons Pho^n,1

Here are is an illustration showing two "photons" in Ȇ^n,1 from our previous discussion in Perpendicular light:

These are two light rays forming the infinity p^⟘. 

 In E^n,1 these two light rays intersect in a different way, with just one common point.

For n>1 the infinity p^⟘. consists of a whole family of "photons", so a more general definition is needed. In the paper "photons" are defined as "projectivizations of totally isotropic $2$-planes. But there is another, equivalent, definition.  Namely E^n,1 is automatically endowed with a conformal structure. Therefore the concept of null geodesics is well defined in E^n,1. Photons are just null geodesics. We will return to this subject in the future. The space of photons is denoted Pho^n,1 in the paper.

Next come null cones.

Nullcones L(p)

Next come nullcones, which the authors write as one world.

To be continued....

A primer on the Universe

Galileo Galilei was a key precursor to Isaac Newton and modern science. In Pisa, he famously conducted (or at least proposed) experiments by dropping objects of different masses from the Leaning Tower to demonstrate that their fall speed is independent of mass, directly challenging Aristotle’s longstanding views on gravity. This foundational work paved the way for Newton’s later formulation of the laws of motion and universal gravitation.

Galileo Galilei was a key precursor to Isaac Newton and modern science.

Isaac Newton created the modern view of the universe by conceiving space as a vast, three-dimensional arena and time as a uniform, linear progression—an independent backdrop against which all physical events unfold. He used mathematics to give precise form to the law of universal gravitation, expressing it as a rigorous equation: every mass attracts every other mass with a force proportional to their masses and inversely proportional to the square of the distance between them. In this way, Newton unified terrestrial and celestial mechanics under a single mathematical framework.

Albert Einstein, together with Hermann Minkowski, transformed the Newtonian paradigm by fusing space and time into a single, four-dimensional continuum called spacetime. In this new setting, the geometry of events is governed by Lorentz transformations, revealing that space and time are interdependent and relative to the observer. Their work showed that simultaneity, length, and the passage of time are not absolute but depend on the observer's motion. Physical laws remain invariant under transformations that mix space and time coordinates—a cornerstone of special relativity. For example, two events that appear simultaneous in one reference frame may not be simultaneous in another moving at a different velocity.

Mainstream physics rapidly adopted this paradigm, and today the standard picture still rests on these principles—even though quantum theory and recent astronomical discoveries highlight persistent mismatches with the classical framework. Increasingly, there is a sense that space and time may not be fundamental entities. This shift motivates newer programs such as “process physics” and “relational physics,” in which the basic constituents of reality are not extended in space or time, but are instead conceived as Leibniz-like monads, processes, or relations. In such approaches, our experience of space and time emerges statistically from the collective behavior of countless underlying primary entities, and from the web of connections among them. As a concrete example, some relational models postulate a discrete network of interactions—rather than a smooth spacetime continuum—out of which familiar geometry only appears at large scales.

For further reading on the relational approach, you may refer to the review "Relational Paradigm in Asks and Answers" by M. I. Suslova and A. A. Sidorova-Biryukova here, and "On Explanations of Magnetic Fields of Astrophysical Objects in the Geometric and Relational Approaches" by Y.S. Vladimirov, S.V. Bolokhov, and I.A. Babenko here.

Perhaps physics requires a complete paradigm shift—perhaps it even needs entirely new mathematics, with “circular arrows” as invoked in some relational approaches. Personally, however, I favor a middle path, one based on algebra and geometry. I believe that algebra and geometry have not yet been exploited to their full potential.

Scientists usually believe that there exist fundamental laws or principles, and that their task is to discover and study them. There are at least two possible strategies for this pursuit: we may either collect data, analyze it, and try to deduce the laws by making educated guesses, or we may rely on our intuition or flashes of insight, boldly postulating fundamental principles (such as “duality” or “triality”), and then deducing more complex laws from these "prime principles." In practice, it is usually best to blend these approaches—combining empiricism and intuition—if we are to make significant progress within any reasonable time frame. I witness such a healthy attitude among participants in Yu.S. Vladimirov’s weekly seminar “Foundations of Fundamental Physics” at Peoples’ Friendship University of Russia, which I have been attending regularly for several months. The Vladimirov school is developing a mathematics of complex binary relations and, on this basis, constructing a worldview that incorporates the ideas of Leibniz and Mach, including the key notion of action at a distance.

“Foundations of Fundamental Physics” seminar 2025-11-06.

Y.S. Vladimirov on the far right.

Personally, I am a syncretist, blending elements from different traditions. Building the universe out of primitive yes-no alternatives, even with circular arrows and complex numbers, is a formidable task and may take an extremely long time—even following clever shortcuts. Instead of focusing on elementary monads or "atoms of existence," I prefer the attitude of a chemist, who works with molecules rather than elementary particles. Thus, I take algebras as already given, and geometry—which naturally follows from these algebras—as sufficiently stable to serve as the building blocks of our knowledge. These structures are the prototypes I use for modeling and describing our reality, both material and immaterial.

In my previous post, I mentioned a particular paper, written by five mathematicians, that describes the synthetic geometry of a family of such models. The title of the paper is "A Primer on the (2 + 1) Einstein Universe," and it is intended to introduce the “synthetic geometry of the Einstein universe.” Synthetic geometry is, in principle, based on axioms—like those found in Euclidean geometry. Fortunately, the paper does not pursue this formal path. Personally, I do not care for axiom systems, since they can often be replaced by others that express the opposite or yield incompatible structures. I prefer “constructions.” But constructions, in turn, often come with cumbersome formulas and calculations, making it easy to lose sight of the underlying essence. The paper I refer to, fortunately, takes a balanced approach: it introduces the main constructions, but then focuses on the “objects” involved and the web of relations (mostly incidence relations) between these objects—this is the part of synthetic geometry that I truly appreciate! Let us now delve into the paper’s details.

• The math part will follow in the next post
• A Substack version of this post is here:

Afternotes:

12-11-25 12:02 A passing Reader inquired in a comment to the previous post if I will discuss the Penrose diagram and twistors. Yes, this is related to the content of "A Primer", and we will talk about these concepts. There will be a separate post dealing with the Penrose diagram. BTW Y.S Vladimirov likes to quote the Penrose program and expands it into new areas using new methods,

12-11-25 18:47 Somewhat related new post  on Substack by Laura:
Mind, Matter, and the Epistemic Asymmetry: A Close Reading of Kastrup’s Argument.

Perplexed by AI

 The last post, "Flow of time and flow of space", ended with "In the next post we take a look at what kind of life can reside on the two  infinity circles?".  That was my plan. Trying to bring it to life I started calculating and researching. One thing was leading unavoidably to another, while I was intensively using AI (mainly Perplexity) for digging into "what has been already published on the subject?" My search was mostly unsuccessful. Perplexity could not help itself but to start hallucinating. It was referring me to sources, papers, textbooks, online resources, that did not contain the stuff it was claiming to have. So I was changing my prompts until finally I was led to a paper that made me very very happy. It was like discovering a wonderful meadow with lots of daisies and grazing unicorns. The title of the paper is:   "A primer on the (2 + 1) Einstein universe".

A primer on the (2 + 1) Einstein universe

It was exactly what I needed. The paper is some fifty pages long, and I will be studying it slowly while at the same time describing what I have already learned here. After that we will return to the point where we have left, but with a much broader perspective. The paper is written by five mathematicians, T. Barbot, V. Charette, T. Drumm, M. Goldmann, K. Melnick, while they were visiting the Schrodinger Institute in Vienna. You can download it from arxiv here. It was published in print in"Recent Developments in Pseudo-Riemannian Geometry", (ESI Lectures in Mathematics and Physics), Dmitri V. Alekseevsky and Helga Baum eds.,  EMS - Publishing House, Zürich 2008, pp. 179-229.

Keywords: Minkowski space, spacetime, Lorentzian manifold, conformal structure, Lie algebra, symplectic vector space
"(...) The purpose of this article is to develop the synthetic geometry of the Einstein universe in terms of its homogeneous submanifolds and causal structure..."

I will start with the math in the next post. This post is just an announcement. But I also want to tell you that my view on Artificial Intelligence has evolved. The AI is not "intelligent" at all. It is a useful companion, it has access to huge databases, but occasionally it is simply dumb! It can be also dangerous when you try to rely on it.  Just yesterday I asked Perplexity to sketch a proof of a certain result quoted in the paper I am advertising, but without providing a proof. Perplexity happily "proved" the statement and suggested a reference to look at. I looked and there was nothing even roughly similar there. Then I thought for a while, and decided to ask if another statement, contradicting the first one is true. The answer was: yes, the second statement is true, which means that the first one, the "proven" one, is false. Sure, both statements look "similar", but we are dealing with mathematics. The statements x=y and x≠y look similar! But, while using two-valued logic, only one of them can be true at a time.

Here is another example. I asked Perplexity to analyze the picture below and describe what kind of optical illusion it provides:

Here is the reply:

The attached picture shows a classic impossible object illusion, specifically related to the genre of the Reutersvärd Triangle or Penrose Triangle.

In the image, you see a group of cubes arranged in a triangular configuration on an isometric grid. At first glance, each cube's shape appears normal, but as you mentally connect the arrangement as a 3D object, you realize the structure can't exist in real 3D space: the cubes fit together in a way where the geometry of connections between them is self-contradictory.

This is an example of an impossible figure illusion: the brain tries to interpret the arrangement as a three-dimensional object, but the spatial relationships between the cubes force contradictory perspectives. The drawing leverages local plausibility with global impossibility—meaning each small part looks realistic, but the whole configuration violates Euclidean geometry when viewed as a solid object.

Such illusions are used to illustrate how perception can be tricked by carefully designed two-dimensional images that depict impossible three-dimensional shapes—one of the pioneering styles introduced by Oscar Reutersvärd and later made famous by Penrose and Escher.

Well, I do not see anything "impossible" in the picture!
So I asked the same question Yandex AI, Alice. The answer was much much more satisfactory:

The optical illusion in the attached picture is an example of a 3D isometric illusion (also known as a multistable perception illusion or illusion of depth).

Explanation:

The image shows a group of cubes drawn on a grid using isometric perspective — a technique where parallel lines remain parallel, creating a 3D effect on a 2D surface.

The shading and positioning of the cubes trick the brain into perceiving depth and volume, even though the drawing is flat.

The illusion works by:

Using consistent angles (typically 30° or 45° from horizontal) to represent the three dimensions (length, width, height).

Applying shading to simulate light and shadow, which enhances the 3D effect.

Arranging the cubes in a way that mimics spatial relationships (some appear in front, others behind).

Key characteristics:

Ambiguity: The viewer’s brain struggles to reconcile the 2D lines with the 3D interpretation.

Forced perspective: The grid lines help maintain the illusion by suggesting a consistent spatial grid.

Depth perception: The shading and cube arrangement create a convincing sense of depth.

Similar illusions: This type of illusion is related to:

Necker cube: A classic example of a reversible 3D cube drawing.

Impossible objects: Like the Penrose triangle, which uses similar isometric techniques but creates geometrically impossible shapes.

Conclusion: This is a classic example of how 2D drawings can deceive the visual system into perceiving three-dimensional space, relying on perspective, shading, and spatial arrangement.

For a perplexed Reader here is the link to  Oscar Reutersvärd's "impossible art"

Flow of time and flow of space

Time flows. That much is not surprising. Either time flows through us, or we flow through time—the effect feels the same, as long as the direction stays fixed. We call it the arrow of time. 

The arrow of time mystery

 Here, theory and experience part ways. Physicists and philosophers keep trying to explain—or to explain away—the arrow of time. Is it objective? Or merely subjective? And is its mystery somehow intertwined with that of consciousness? 

 Recently, I attended a seminar on the arrow of time, where one approach drew on the relational theory of Yu. S. Vladimirov. What I noticed was this: while the speaker aimed for an “objective explanation,” he could not avoid repeatedly saying “we.” I resisted the impulse to ask, “Who are these ‘we’ within the framework you propose?” I felt entitled to ask, having already confronted that very question myself—in a paper on the quantum theory of measurement. 

 So we have the arrow of time, we have irreversibility, and we have an unease about time loops—except, perhaps, in our rare dreams.

Here is the new version of my notes, with added Ch. 6.1 "Time-like streams", and 6.2. "Space-like streams," with new Figures 10,11,12.

 

 Link to pdf file.

In the next post we take a look at what kind of life can reside on the two  infinity circles?

Shape of infinity

 Our model toy universe is a torus It is a homogeneous space for the group SO₀(2,2). It does not carry any natural metric. But it carries, as we will see later, a natural conformal structure. Thus there are no ``geodesics'' -- ``shortest'' lines connecting points, but there are natural isotropic lines, or ``null'' lines - they represent ``light rays''. If we select a point p, there are two light rays emanating from p, one ``to the right'', and one ``to the left''. 

Fig. 1 Two Minkowski spacetimes with their boundaries at infinity.

They form two circles. So we have, automatically two circular light rays intersecting at p, and also in the opposite point (-p). These two light rays form p^⟘ .

Shape of infinity

These are red and blue circles on Fig. 1. These two circles of light form the ``infinity'' boundary for the complement, which splits into two disjoint open regions -- red and blue. Each region carries a natural structure of an affine space. It can be identify with Minkowski space-time, endowed with Minkowski space-time quadratic form (or ``metric''). But this Minkowski metric depends on the choice of p. Thus the torus as a whole does not have a natural metric. The stability group of $p$ consists of time and space translations, Lorentz boosts, and dilations. It We can arbitrarily select an ``origin'' in one of these two Minkowski spaces. This further breaks the symmetry to just Lorentz boosts and dilations. Every light ray through a point in one Minkowski space intersects with two light rays at infinity, somewhere in the middle between (p) and (-p). But no such light ray passes through the points (p) or (-p). This opens the following question: which paths in Minkowski space goes to (p) and (-p)?

We will address this question in the forthcoming post.

The whole document containing the details is shown below. You can also download the pdf here. The old notes have been essentially expanded and updated.

Perpendicular light

 Can something be perpendicular to itself? At first glance, this seems impossible. A "thing," in the ordinary sense, cannot stand at right angles to itself. But reality is richer than just "things." Perhaps even the solidity of things is an illusion, a condensation of what might be called "nothing." Take light, for example: it is not a "thing" in the usual sense. In Minkowski space, the four-momentum of a photon has zero length—it is, quite literally, perpendicular to itself.

In this note, we explore a related geometric curiosity. We will look at the set of all points perpendicular to any point lying on the null cone of  R^2,2. The surprising outcome will be a pair of intersecting circles. In the following post, we will uncover their deeper meaning: these circles turn out to be nothing less than two representations of light rays. That, however, lies ahead. For now, let us enjoy the game of pure geometry—elementary trigonometry, sines and cosines leading us into unexpected structures.

Let us first recall some basic definitions pertaining to our subject. We start with more "geometric" coordinate independent way.

Let V be a real 4-dimensional vector space endowed with a quadratic form Q of signature (2,2). Let B: V⨉V → V be the associated bilinear form, so that Q(x) = B(x,x).

Let O(V) denote the orthogonal group of (V,B), that is the set of all linear transformations L of V which leave B invariant: B(Lx,Ly) = B(x,y), for all x,y in V.

Note. Later on we will also consider two subgroups of O(V): special orthogonal  group  SO(V), and its connected component of the identity SO₀(V).

We define N to be the null cone of V:

N = {x∈V: Q(x) =0}

and PN to be the projective null cone:

PN = N⟍{0} / ℝ^×,   

where R^x is the multiplicative group of non-zero real numbers. But we will be mostly interested in PN₊ defined as

PN₊ = N⟍{0} / ℝ^×_>0, 

where ℝ^×_>0 is the multiplicative group of positive (i.e. >0) real numbers.
PN can be obtained from PN₊ by identifying pairs of opposite elements: if we denote by [x] the equivalence class of x, representing a point in PN₊, then we obtain a point in PN by identifying [x] with [-x]. The null cone N⟍{0} is 3-dimensional, therefore PN₊ and PN are 2-dimensional. The group O(V)  acts on PN₊  and on PN in a natural way, by L[x] = [Lx], its action being transitive. So, we have a geometry in the sense of Felix Klein. We will identify the corresponding subgroups later on. Studying geometry is nothing else but studying invariants of this action, or better: studying constructions invariant under this action. Our first construction will be the construction of the space perpendicular to a point.

We first define the relation of orthogonality "⟘" in PN⁺:

Definition 1. For any two points p,q in PN₊, we write p⟘q if p=[x], q=[y], and B(x,y) = 0.

Notice that the definition makes sense, since the condition B(x,y)=0 does not depend on the choice of representatives of equivalence classes. The definition is also invariant in the sense that if p⟘q, the Lp⟘Lq for any L in O(V).

Exercise 1. Verify that, for all p, we have  p⟘p, and p⟘(-p).

Note. Notice that -p is well defined. If p = [x] then -p is defined as [-x]. Moreover L(-p) = -L(p) for any L in O(V).

We  now define:

p^⟘ = {q ∈ PN₊ : q⟘p }                (1)

Now, while PN₊ is 2-dimensional, p^⟘ has one condition more, thus it is 1-dimensional. So it is a curve in PN₊. It contains at least these two points: p and -p. The construction is O(V) invariant: L(p^⟘)=(Lp)^⟘. Thus through each point of PN₊ we have two special curves. They meet again at opposite point -p. What are these curves? How to represent them graphically? We know already that PN₊ can be faithfully represented by a torus. How these two lines will look like on the torus? And what about PN, where p and -p are identified?

Graphical representation of p^⟘.

So far we didn't use coordinates. But to arrive at a graphical representation coordinates are unavoidable. Thus we introduce an orthonormal basis adapted to our point p. So, let e₁,...,e₄ be mutually perpendicular vectors in V, with Q(e₁) = Q(e₂) =1,  Q(e₃) = Q(e₄) = -1, and such that p can be represented by  e₂ + e₄, which is in N.

Any x∈N is of the form x = xⁱe_i, with

(x¹)² + (x²)² - (x³)² - (x⁴)² = 0,                 (2)

or

(x¹)² + (x²)² = (x³)² - (x⁴)² = ρ².                (3)

Since we removed the origin from N, we may assume that ρ = 1. Then x¹, x², x³, x⁴ can be interpreted as cos(α), sin(α), cos(β), sin(β) resp. We will assume that α and β

are in [0,2π]:

x¹ = cos(α),
x² = sin(α),
                    x³ = cos(β),                 (4)
x4 = sin(β).

For graphical representation we draw the torus as a parametric 3D curve

tor(α,β) = { (R + r cos(β)) cos(α), (R + r cos(β)) sin(α), r sin(β)},        (5)

where R and r stand for the major radius and minor radius of the torus. For our graphic representation I choose R = 3, r = 1.

For the point p we have

x¹ = x³ = 0, x² = x⁴ =1,                 (6)

which corresponds to α = β = π/2. Point -p has coordinates x¹ = x³ = 0, x² = x⁴ =-1, which corresponds to α = β = 3π/2. Here is the graphics of our torus with point p depicted in color blue, while -p is depicted in red.

Let us now examine the rest of p^⟘. For x  to be orthogonal to (0,1,0,1) we must have x² - x⁴ = 0, or x² = x⁴. (Why?) But then, from (2), we must have (x¹)² = (x³)² ,   or

x³ =  ∓ x¹.                (7)

1) The case of  x³ = x¹ 

Let us consider first the case x³ = +x¹. Then x = (x¹,x²,x¹,x²), therefore (4) implies cos(β) = cos(α) and sin(β) = sin(α), thus  β = α. The path is depicted in blue on the image below.

2) The case x³ = -x¹.

Now  x = (x¹,x²,-x¹,x²), thus cos(β) = -cos(α), sin(β) = sin(α), which implies β = π - α. The path is depicted in red.

The unoriented case

In this case [x] is identified with [-x]. Thus -p and p represent just one point.

For plotting we will use formulas from "Unoriented and oriented conformal completion of 1+1 dimensional spacetime"

cos(α) = x¹x³ - x²x⁴,
sin(α) = x²x³ + x¹x⁴,
cos(β) = x¹x³ + x²x⁴,
sin(β) = x²x³ - x¹x⁴.

These formulas assume normalization (x¹)² + (x³)² = (x²)² + (x⁴)² = 1. Otherwise we need to divide the right hand side by (x¹)² + (x³)² = (x²)² + (x⁴)² . Assuming, as before, that p is represented by x = (0,1,0,1), we get for this point

cos(α) = - 1,
sin(α) = 0,
cos(β) = 1,
sin(β) = 0.

This corresponds to α = π, β = 0.

Let us now examine the rest of p^⟘.As in the oriented case we must have x³  = x¹ or x³ = -x¹.

1) The case of x³  = x¹

For x³ = x¹ we have the path (x¹,x²,x¹,x²), which leads to

cos(α) = (x¹)² - (x²)²,
sin(α) = x2x1 + x1x2 = 2x¹x²,
cos(β) = (x¹)² + (x²)² = 1,
sin(β) = 0.

We get the coordinate line of α, for β = 0.

2) The case of  x³  = -x¹

For x³ = -x¹ we have the path (x¹,x²,-x¹,x²), which leads to

cos(α) = -1,
sin(α) = 0,
cos(β) =-(x¹)2 + (x²)²,
sin(β) = -2x¹x².

We get the coordinate line of β, for α = π.

The two circles intersect at α = π, β = 0 - the point p.

Note. Notice the important difference: in the oriented case the complement PN₊⟍ p^⟘ of p^⟘ consists of two disconnected regions, each being path-connected and simply connected, while in the unoriented case the complement PN⟍ p^⟘ of p⟘ is path-connected and simply connected.

But what is the meaning of it all?

The physical meaning of so obtained p^⟘ will be discussed in the next post