Lie Sphere Geometry Part 6: The Gospel of Q

I am kind of a Platonist. But I'm not the only one!

We continue from Part 5. Let us first recall the definitions.

We consider a 6-dimensional real vector space V with a scalar product of signature (4,2). We select an orthonormal basis in this space, e_i (i=0,...,5) , so that

(e₀,e₀)=(e₁,e₁,)=(e₂,e₂)=(e₃,e₃)=1, (e₄,e₄)=(e₅,e₅)= -1.

with vectors ei being orthogonal to each other:

(e_i,e_j) = 0 for i≠j.

We call this space, with a selected basis,  R^4,2. A general vector x of V  is then written as

x = x⁰e₀+...+x⁵e₅.

We define Q is the set of all  isotropic lines through the origin in V. Alternatively Q can be written as (see Part 5):

Q = {[u]∈P(V): (u,u) = 0},

where P(V) is the real projective space of V: P(V)=(V∖{0})/∼, with x∼y if and only if there exists  a real λ such that y=λx. We now want to prove the following proposition:

Proposition. The following formula defines an explicit isomorphism between the space of oriented spheres (discussed in Lie Sphere Geometry Part 4: oriented spheres) and Q:

(m,t) ⟼ x(m,t) = [m+cos(t)e₄+sin(t)e₅].                    (1)

Proof. Let us start with analyzing the meaning of the formula (1). On the left hand side we have m and t. Together they define an oriented sphere as explained in Part 4.  But in Part 4 we had coordinates x¹,...,x⁴, while here we have coordinates x⁰,...,x⁵. How should we understand (1)? On the right hand side we see m+cos(t)e₄+sin(t)e₅. This suggests that m in (1) should be understood as m=m⁰e₀+m¹e₁+m²e₂+m³e₃.  So our S³ in Part 4 is now in the space of coordinates x⁰,...,x³ instead of x¹,...,x⁴ as it was previously.  Since m is assumed to be on the unit sphere S³, we have (m,m)=1. We also have (m,e₄)=(m,e₅)=0.

Then we find that

(x(m,t), x(m,t)) = (m,m)-sin²(t)-cos²(t)=1-1=0,

so that x(m,t) is an isotropic vector in V. It is certainly a non-zero vector (Why?), so it defines a point of Q.

Next, we remember from Part 4 that (m, t) and (-m, t+π mod 2π) define the same oriented sphere? Do they define the same point of Q? We easily see, from the properties of sin and cos functions, that x(-m,t+π mod 2π) = -x(m,t). Therefore indeed (1) defines a well defined map from the set of oriented spheres  (S³⨉S¹)/Z₂ into Q.

It remains to be shown that the map is one-to one and onto. Let us first check that the map is one-to-one. Suppose [x(m,t)]=[x{m',t')]. That means x(m',t')=λx(m,t), or, explicitly

λ(m+cos(t)e₄+sin(t)e₅)=(m'+cos(t')e₄+sin(t')e₅) .

Since 1=cos²(t)+sin²(t) = cos²(t')++sin²(t')=1, it follows that λ=±1. If λ=1, then t'=t and m'=m. If λ=-1, then t'=t+π mod 2π, and m'=-m. Thus (m,t) and (m',t) define the same oriented circle. So our map is injective.

We will show now that it is also surjective. The simplest way of doing this is by explicitly calculating the inverse map from Q to (S³⨉S¹)/Z₂. Let's do it. We take any point [u] in Q. Then (u,u)=0 and u≠0. That means

(u⁰)²+(u¹)²+(u²)²-(u³)²-(u⁴)²-(u⁵)² = 0,

or

(u⁰)²+(u¹)²+(u²)²-(u³)² = (u⁴)²+(u⁵)² .

But u≠0, therefore  (u⁴)²+(u⁵)² > 0. Set u'=λu, where λ = ( (u⁴)²+(u⁵)² )^-½ . Then [u']=[u]. But now (u'⁴)²+(u^'5)²= 1, therefore there exists a unique t in [0,2π) such that u'⁴=cos(t), u'⁵=sin(t). Set m=u'⁰e₀+u'¹e₁+u'²e₂.+u'³e₃. Then x(m,t)=u', therefore [x(m,t)]=[u]. QED.

The mystery of zero-radius spheres

The space (S³⨉S¹) looks like a homogeneous space - no of its points is better than any other. And (S³⨉S¹)/Z₂ or isomorphic to it Q doesn't look any better. In fact, we will see later on that both are homogeneous spaces under the action of SO(4,2). And yet among the oriented spheres in S³ there are those with zero radius and no orientation. The same applies to our experiences in the 3D world we live in. How can these two different views  be reconciled? That is a bugging question. Here is my attempt to answer it, and details I hope to give in future posts.

What we see, what we experience, is a projection, like a shadow of a higher-dimensional structure on a lower-dimensional wall. A sphere becomes a point, but the sphere itself is just a shadow. Change the position of the lamp, and the sphere will become a point and the point will become a sphere.

Now, formally it looks like a reasonable answer, but after some thinking the next question comes to mind: how can we do it in practice? Can we? Or is it just mathematics that has nothing to do with the physical world?

I think we can, and, I think, it has to do a lot with physics. Special Relativity deals with inertial frames. But there are no truly inertial frames in nature. Lorentz transformations are transformations between inertial frames. It is an idealization valid as long as accelerations can be neglected. This is our overwhelming experience. We do not have much experience with accelerations. Well, fighter jet pilots have some, but still their experiences are local and short, not more than second's long 10 Gs. That's next to nothing on the Universe scale. What happens inside elementary particles? We have no idea. Their internal structure is still a pure guess. We have models that answer some questions, but leave other questions open. I will approach this subject from the side of mathematics. The physical interpretation and possible application of the mathematical formulas is a hard and demanding work. And, anyway, what we do here is just playing with one particular model, a model that, perhaps, catches some of the properties of Reality, but leaves aside other important properties.

So, in the next post we study group action on Q. And after that I will propose a way how to get rid of taking quotient by Z₂, which bothers me aesthetically.  Why must we take this quotient? To take another shadow for reality? As you see I am kind of a Platonist. But I'm not the only one!

Lie Sphere Geometry Part 5: Lie Quadric

 The symbol of space, in its most elemental and crystalline form, is the cube—a solemn and silent sentinel of three dimensions. With its six square walls, it encloses a finite void, a miniature cosmos balanced on the symmetry of eight vertices. Each edge traces the logic of extension, and each face confronts its opposite in silent equilibrium.

But nature, as always, plays in dualities. Opposite the cube stands its geometric twin, the octahedron—a figure of eight triangular faces and six converging vertices (square bipyramid). Where the cube stands firm, the octahedron spins, airy and precise, each point tapering like a thought reaching outward. And it is among these six vertices that we find a deeper enigma.

But nature, as always, plays in dualities.

Two of these six are not like the others.

In the esoteric grammar of Lie sphere geometry—a geometry that sees through appearances and speaks the language of contact and curvature—these two points take on a special role. They break symmetry, not by defect, but by signifying something beyond the solid form. They whisper of "time"—not time as a mere parameter, but as a two-dimensional entity, a subtle twin-threaded fabric that cuts across the frozen lattice of space.

Where the cube represents the fixed scaffold of extension, these special vertices in the dual figure suggest movement, directionality, and the possibility of becoming. Thus, in this quiet interplay between cube and octahedron, between solid and point, space and time touch—not as opposites, but as intimate correspondents in a deeper, unseen order.

According to Sophus Lie the Universe is 6-dimensional

6 = 3 + 1 + 2.

But who was Sophus Lie? Here is a relevant part from the history:

"In 1894 in the Russian city of Kazan, an international prize was instituted to commemorate  the  mathematician  Lobachevsky.  The  prize  was  to  be  awarded  to  a mathematician who  had made prominent contributions  to  geometric  research, particularly in the development of non-Euclidean geometry. In 1897  Klein  was asked by the prize committee to provide a description of Lie's work, and Klein's evaluation led to Lie receiving the award in 1897 - the very first recipient of this esteemed prize. In his argument, over and above everything else, Klein pointed to the third volume of Theorie  der Transformationsgruppen, where the theory was applied to  the principle axioms  of geometry. Klein's  evaluation was  printed in Mathematische Annalen a year later. "

Arild Stubhaug, The Mathematician Sophus Lie", Springer 2002

And here is the relevant part from Kazan's University site:

"At the time of the establishment of the prize in 1895, the remaining principal capital amounted to 6,000 rubles in gold, and a prize of 500 rubles was paid out of the interest on it every 3 years. At the first three awards, the person who wrote a critical review of the nominee’s work was awarded the N. I. Lobachevsky gold medal.

1897 — Lee Sophus , for work on the theory of transformation groups; the gold medal was awarded to the referee Felix Klein.

Now that we have a clue about the person, we can move to his six-dimensional Universe - the home of spheres. We shall do it in slow steps, carefully,  to avoid lurking dangers.

The three main monographs discussing the subject are:

[1] Benz Walter, Classical Geometries in Modern Contexts: Geometry of Real Inner Product Spaces, chapter 3: Sphere geometries of Möbius and Lie,  Birkhäuser 2007.

[2] Cecil, Thomas E. Lie sphere geometry, Springer 2008.

[3] Jensen G.R. et al., Surfaces in Classical Geometries, Springer 2016.

Wikipedia article "Lie sphere geometry" contains additional references. In my exposition I am following mainly Ref. [3]. To my surprise I did not find anything on this subject written by Russian mathematicians. If there is something that I am not aware of, I will be thankful for an advice. I have consulted AI on this issue and received the following answer:

"Lie sphere geometry isn’t as mainstream a term in Russian mathematical literature as, say, hyperbolic geometry or Lie group theory. It’s often subsumed under broader topics like conformal geometry or contact geometry. English-language works, such as Thomas E. Cecil’s Lie Sphere Geometry: With Applications to Submanifolds (translated or referenced globally), dominate specific treatments, and Russian equivalents might not have been as distinctly branded. Soviet mathematicians tended to embed such ideas within larger frameworks rather than isolating them in monographs."

Following this advice I did another search to find this:

Код УДК    Описание
5    Математика и естественные науки
51    Математика
514    Геометрия
514.1    Общая геометрия. Геометрия в пространствах с фундаментальными группами
514.15    Геометрия в пространствах с другими фундаментальными группами
514.152    Конформная геометрия и ее аналоги
514.152.6    Геометрия сфер Ли

And then, just minutes ago,  I was able to  find this:

А. И. Бобенко, Ю. Б. Сурис, О принципах дискретизации дифференциальной геометрии. Геометрия сфер, УСПЕХИ МАТЕМАТИЧЕСКИХ НАУК, 2007 г. январь — февраль т. 62, вып. 1 (373).

A 48 pages long survey. I will have to study it yet!

The 6D octahedral universe of Sophus Lie

We take six-dimensional real vector space R⁶ with coordinates x⁰, x¹, x², x³, x⁴, x⁵. There we introduce the indefinite scalar product

(x,y) = x⁰y⁰ + x¹y¹ + x²y² + x³y³ - x⁴y⁴ - x⁴y⁵.

We denote by R^4,2 the resulting inner-product space. We endow R^4,2 with orientation and denote by e₀,...,e₅ the corresponding orthonormal basis in R^4,2. Thus 

(e₀,e₀)=(e₁,e₁,)=(e₂,e₂)=(e₃,e₃)=1, (e₄,e₄)=(e₅,e₅)= -1.

Now we go to projective space by introducing in R^4,2 the equivalence relation

x∼y if and only if there exists  a real λ≠0 such that y=λx.

The equivalence classes [x] form the projective space P(R^4,2). It is a compact 5-dimensional manifold.

Definition. The Lie quadric Q⊂P(R^4,2) is the smooth quadric hypersurface

Q = {[u]∈P(R^4,2): (u,u) = 0}.

We notice that Q is well defined: if λ≠0 the (u,u)=0 if and only if (λu,λu) = 0. The condition defining Q takes away one dimension from the five dimensions of P(R^4,2). Thus Q is a four dimensional and compact.

Proposition. The following formula defines an explicit isomorphism between the space of oriented spheres (discussed in Lie Sphere Geometry Part 4: oriented spheres) and Q:

(m,t) ⟼ [m+cos(t)e₄+sin(t)e₅].

The analysis, the proof, and the formula for the inverse map will be discussed in the next post.

Lie Sphere Geometry Part 4: oriented spheres

 Circles are one-dimensional. They live in a two-dimensional space. If you throw a stone on a water surface - circular waves appear and move away from the source of the disturbance.

Circular waves appear and move away from the source of the disturbance

Their radii measure time. But we live in a three-dimensional space. If, from outside of space,  we throw a stone on the ether of space, spherical waves appear and move away from the source point. Their radii measure time. So, let us move from circles to spheres. This is harder to imagine, but more realistic.

But our space is spherical itself. Thus we are going to discuss the space of spheres S² on the sphere S³.

To draw S³ we need four-dimensional space R⁴. The scalar product in R⁴ is

x·y = x¹y¹+x²y²+x³y³+x⁴y⁴.

and the norm

||x||² = x·x

There the equation of S³  is

S³ = {x∈R⁴: x·x = 1} .                        (1)

For each m∈ S³ we will create now the family spheres S_t(m):

S_t(m) = {x∈S³: x·m = cos(t)} .           (2)

Remark 1. In (2) both x and m are in S³. From the Cauchy inequality we have |x·m| ≤ ||x|| ||m|| = 1. Therefore -1 ≤  x·m  ≤ 1. Thus setting x·m = cos(t) for some t in [0,2π] makes sense. The set S_t(m) is invariant under all rotations R of R⁴ that fix the rotation axis m. Indeed, if Rm=m, from invariance of the scalar product under rotations Rx·Ry =x·y we get Rx·m =x·m. Thus  S_t(m) has a spherical symmetry. I will not go into why and in which sense it is a "sphere". It is clearly a "point" for t=0 and t=π. Can you see it?

But cos(t) = cos(2π-t). To remove the redundancy we introduce oriented spheres. Or, in other words, we introduce also sin(t) into the description. Every nonpoint sphere S_t(m) is a two-dimensional surface in a three-dimensional manifold S³. Given any point x∈S³, the tangent space to S_t(m) at x is two-dimensional. It is in the three-dimensional space T_x(S³) tangent to S³ at that point. Thus there is a one-dimensional subspace of T_x(S³) perpendicular to T_x(S_t(m)). Therefore there are two opposite unit vectors perpendicular (or "normal") to S_t(m) at x. Each of them define an orientation, it defines what is "inside" the sphere (the opposite is then considered as being "outside").

We choose the following formula to define the normal unit vector n(x) for each x in a nonpoint S_t(m):

n(x) = ( m-cos(t)x )/sin(t).        (3)

Making sense of Eq. (3). Making sense of Eq. (3) requires some work - a dissection of the formula. For that we need some tools, and to take appropriate safety measures.

Safety measures should come first.

Safety measures should come first. Are we safe? There is a dangerous denominator sin(t). What if it is zero? That would be a catastrophe! But we are discussing nonpoint circles, thus t is different from 0 and  π. In the interval [0,2π), and for t different from these two dangerous points, sin(t) is nonzero. We are safe!

Now comes the dissection work. The vector n(x) should be perpendicular to any vector tangent to S_t(m) at x. Is it? How can we see it? To answer this question we need an algebraic characterization of the tangent space T_x(S_t(m)). Let u be a vector tangent to S_t(m) at x. Let x(s)∈S_t(m), s∈R, be a smooth curve through x, with x(0)=x, (dx(s)/ds)|_s=0 = u.  Now x(s) is on the unit sphere S³, thus

x(s)·x(s) =1 for all s.

Differentiating with respect to s at s=0 we get

u·x = 0.                   (4)

On the other hand x(s) is on the same sphere S_t(m) for all s. Thus

x(s)·m = cos(t)   for all s.

Differentiating with respect to s at s=0 we get

u·m = 0.                   (5)

Equations (4) and (5) characterize vectors tangent to S_t(m) at x. Vector u has four components. These four components satisfy two equations. 4-2 =2, thus we have two-dimensional space of vectors tangent to the two-dimensional sphere. It looks good. Now we need to check that our vector n(x) given by (3) is orthogonal to all such u. Since n(x) is a linear combination of m and x, both being orthogonal to u, thus n(x) is also orthogonal to u. But n(x) should be tangent to S³. This condition gives us an additional requirement:

n(x)·x = 0.

This we need to check. So we check, suppressing the inessential denominator:

(m-cos(t)x)·x = m·x -cos(t)x·x = cos(t) - cos(t) = 0.

This can be also understood by writing m-cos(t)x as m - (m·x)x. We subtract from m its orthogonal projection on x, therefore we obtain vector perpendicular to x.

Exercise 1. Verify that n(x) given by Eq. (3) is of unit norm.

Exercise 2. Verify that (m,t) and (-m, t+π) define the same oriented sphere - the same sphere as a set, and the same normal vector field n(x).

Conclusion: The set of all oriented spheres on S3 is isomorphic to

(S³⨉S¹)/Z₂,

where Z₂ = (+1,-1) acts by

(-1)(m,t) = (-m,t+π mod 2π).

We are now ready to add dimensions and to move to the world of conformal transformations.

 The world of conformal transformations.

P.S. 09-04-25 14:44 From yesterday's seminar "What is photon?"

Nikolai Magnitskii

Federal Research Center “Computer Science and Control”, Moscow State University, Ltd “New Inflow”, Russian Federation
"Theory of compressible oscillating ether Results in Physics 12 (2019) 1436–1445"

The paper considers ether as a dense inviscid compressible oscillating medium in the Euclidean three-dimensional space, given at each instant of time by the velocity vector of propagation of the ether density perturbations and satisfying the continuity equation and the ether momentum conservation law. It is shown that a generalized nonlinear system of Maxwell-Lorentz equations that is invariant with respect to Galileo transformations, the linearization of which leads to the classical system of Maxwell-Lorentz equations; laws of Biot-Savart-Laplace, Ampere, Coulomb; representations for Planck’s and fine structure constants are obtained from the system of the two ether equations as well as formulas for electron, proton and neutron, for which the calculated by formulas values of their internal energies, masses and magnetic moments coincide with an accuracy to fractions of a percent with their experimental values which are anomalous from the point of view of modern science. A concept of an ethereal theory of atom and atomic nucleus is presented, which makes it possible to answer many questions about the structure of atom, on which modern science is unable to answer.

Keywords: Ether, Maxwell-Lorentz equations, Ampère and Coulomb laws, Planck’s and fine structure constants, Proton, Electron, Neutron, Atomic nucleus.

Slide from the presentation