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.