Sunday Special - From Fairy Tales to Math: The Power of Threes

 Ever noticed how many things come in threes? Like in a story, three chances always seem just right, or in comedy, the third punchline really lands. There's even a term for it: "the rule of three," or in Latin, "omne trium perfectum"—meaning all things in threes are perfect. According to an English language forum I stumbled upon, “the rule of three” shows up everywhere from fairy tales to Hollywood blockbusters because things in threes are inherently more humorous, satisfying, and impactful.

Think about it. In storytelling, the protagonist’s first attempt fails, the second is almost there, and the third? Bingo! Success! But there’s more to this number than just fairy-tale magic. It even shapes the foundations of mathematical structures in Clifford algebras.

omne trium perfectum

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Clifford Algebras: A Trio of Transformations

In any dimension, Clifford algebras feature three key transformations—three "involutions," if we’re being formal. These are not just fancy operations; they’re structural quirks that make Clifford algebras unique.

1. Main Automorphism (Π) - It flips the sign of odd products of vectors.
2. Main Anti-automorphism (τ) - Changes the order of vector products (also known as "reversion").
3. Composition (Π∘τ = τ∘Π) - A combined transformation, also called "conjugation."

Together, these involutions create a complete set that operates consistently within every Clifford algebra. And here’s the kicker: they’re a part of our world’s fundamental structure. Our very own 3D Euclidean space obeys these algebraic principles. It's as if nature decided three dimensions were "just right"—Goldilocks style. The aim? To extract all we can from these three dimensions before venturing off to other realms.

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Three as the First Prime: A Mathematical Treasure

But there's even more to three! Mathematically, three is the first truly prime number (2, which comes before 3,  is truly exceptional among prime numbers, as it is the only even prime number - which is odd!). The revered mathematician Leonard Euler had plenty to say about prime numbers and divisors. 

Here is the reasoning of Leonard Euler (G. Polya, Mathematics and Plausible Reasoning, Vol. 1, Induction and Analogy in Mathematics, Princeton University Press 1990, p. 91):

"[...] 2. A prime number has no divisors except unity and itself,"and this distinguishes the primes from the other numbers. Thus 7 is a prime, for it is divisible only by 1 and itself. Any other number which has, besides unity and itself, further divisors, is called composite, as for instance, the number 15, which has, besides 1 and 15, the divisors 3 and 5. Therefore, generally, if the number p is prime, it will be divisible only by 1 and p; but if p was composite, it would have, besides 1 and p, further divisors. Therefore, in the first case, the sum of its divisors will be 1+p, but in the latter it would exceed 1+p. As I shall have to consider the sum of divisors of various numbers, I shall use the sign σ(n) to denote the sum of the divisors of the number n. Thus, σ(12) means the sum of all the divisors of 12, which are 1, 2, 3, 4, 6, and 12; therefore, σ(12) = 28. In the same way, one can see that σ(60) = 168 and σ(100) = 217. Yet, since unity is only divisible by itself, σ(l) = 1. Now, 0 (zero) is divisible by all numbers. Therefore, σ(0) should be properly infinite. (However, I shall assign to it later a finite value, different in different cases, and this will turn out serviceable.)

3. Having defined the meaning of the symbol σ(n), as above, we see clearly that if p is a prime σ(p) = 1+p. Yet σ(1) = 1 (and not 1+1); hence we see that 1 should be excluded from the sequence of the primes; 1 is the beginning of the integers, neither prime nor composite. If, however, n is composite, σ(n) is greater than 1+n."

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The Law of Three: Good, Evil, and the Uncharted Middle

Stepping away from math, let’s explore something a bit more cosmic: "The Law of Three." While perusing academia.edu, I came across a work titled ARCHONS HIDDEN RULERS THROUGH THE AGES by Anbr Cama, which discusses ethical and cosmic dualities. Here's the essence of the Law of Three:

“There is good, there is evil, and there is the specific situation that determines which is which.”

The cosmic perspective suggests that both good (often linked with Service to Others or STO) and evil (Service to Self, STS) are necessary. It’s not a question of resolving this tension; rather, it’s about choosing a path. Human ethics can feel binary, but the Law of Three reminds us of the "third factor"—the context. Just as in storytelling and mathematics, three isn’t simply two opposites with a bridge; it’s a unique, cohesive whole.

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The Universe in a Droplet: The Holographic Principle and 3D Space

And now, the grand finale: the Holographic Principle. Imagine if all of the universe’s information were contained in the tiniest droplet. If the universe’s vastness could be distilled, it might exist encoded within the Clifford algebra of our 3D space.

This implies something wild—time itself might be woven into this 3D algebraic fabric! If we could untangle the layers within our spatial dimensions, we might glimpse time’s secrets. It’s as though the universe has handed us a compact manual for existence, neatly packed into three dimensions.

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Embracing the Power of Three

So there you have it: from fairy-tale patterns to mathematical beauty, from cosmic ethics to the secrets of time—three, it turns out, is a number that holds endless layers of depth. The Law of Three isn’t just a principle; it’s a worldview. Whether in jokes, mathematical theorems, or cosmic reflections, threes show us that life, like a good story, often unfolds in a satisfying trilogy. And maybe, just maybe, there’s some cosmic rule nudging us toward threes for good reason.

P.S. 03-11-24 19:42 In reply to a comment below by Bjab, concerning  the use of AI:

"AI has blown this world open. Almost all pressure to write has dissipated. You can have AI do it for you, both in school and at work.

The result will be a world divided into writes and write-nots. There will still be some people who can write. Some of us like it. But the middle ground between those who are good at writing and those who can't write at all will disappear. Instead of good writers, ok writers, and people who can't write, there will just be good writers and people who can't write."

https://www.paulgraham.com/writes.html

The Spin Chronicles (Part 8): Clifford Algebra Universal Property

 In The Spin Chronicles (Part 7): Whispers of the Cosmos Beneath the Fig Tree we have met quaternions and biquaternions living peacefully within the Clifford algebra of our 3D space. Today we will widen a little bit our horizons.

The Secret of Light

"God's imaginings extend from rest to rest in His three-dimensional radial universe of length, breadth and thickness--to become the stage of space for His imagined radial universe of matter, time, change and motion." (Walter Russell,  The Secret of Light, University of Science and Philosophy 1947). (Fig. 4)

Once we have space, it gets curved, but since All is One, the structure of the whole Cosmos is reflected in the structure of its smallest cubes, and what we are studying now is the algebra of such a primitive cube. We already know it's eight-dimensional. Three dimensions of space generated 2³ dimensions of its, first Grassmann Λ(V), then Clifford, algebra Cl(V). The Clifford algebra structure allows us to measure its substructure. And now we are ready to delve deeper into this issue. We will start with the natural automorphisms  and anti-automorphism of Cl(V). Recall that we have realized Cl(V) on the canvas of Λ(V):

Λ(V) = Λ⁰(V) ⊕ Λ¹(V) ⊕ Λ²(V) ⊕ Λ³(V)

by deforming its Grassmann product u∧v into the Clifford product uv

uv = u∧v+(u⋅v)1,

for u,v in V. Elements of each Λⁱ(V) are called homogeneous of degree i. We then write: i = deg(a).  Grassmann product of homogeneous elements is again homogeneous. Clifford product, on the other hand, does not preserve homogeneity.  We will return to this point soon, but first let us make some side observations

Observation 1. Our construction of the Grassmann and Clifford algebras can be repeated for any n-dimensional (real or complex) vector space V (there is nothing special about n=3 in this respect) and for any symmetric bilinear form B(u,v) (rather than the Euclidean product (u⋅v)). We just write

uv = u∧v+B(u,v)1,

istead of uv = u∧v+(u⋅v)1.

Sometimes certain properties of the Clifford algebra are even easier to prove for a general Cl(V,B) than for just our particular case.

Observation 2. Every Clifford algebra Cl(V,B) has a very important universal property. I will now state this property, without proving it. Some authors are even using this property to define Clifford algebra. 

Universal Property: Let φ be a linear map from V into an associative algebra A (with unit 1), such that φ(x)² = B(x,x)1 for all x in V. Then φ can be extended to a unique algebra homomorphism  φ^~ from Cl(V,B) to A.

We will now show two simple applications of this universal property. Since it is a nice exercise in a logical and precise thinking, we will do it in details. First take for A the algebra Cl(V,B) itself. We can do it since Cl(V,B) is an associative algebra with unit 1. For φ we take the map φ: x ⟶ -x. We consider it as a map from V to Cl(V,B). The x on the left hand side is considered as a vector in V, the -x on the right hand side is considered as an element of Cl(V,B). We can do it, since V is (identified with)  a subspace of Cl(V,B) - the subspace of elements of degree 1. We take φ(x) = -x. This is -x in Cl(V). We take its square in Cl(V). We get: (-x)(-x) = x²  = xx = B(x,x)1. Therefore φ satisfies the assumptions in the Universal Property. Thus it extends to a unique homomorphism φ^~ from Cl(V,B) to Cl(V,B). The fact that  φ^~  extends φ means  that φ^~ (x) = φ (x) = - x for x in V. But  it is a homomorphism of Cl(V,B). A composition of two homomorphisms is a homomorphism. Thus φ^~ ∘ φ^~ is also a homomorphism. But (φ^~ ∘ φ^~) (x) = x for x in V, and V generates Cl(V). Therefore (φ^~ ∘ φ^~) is the identity on Cl(V,B). Therefore φ^~  is not just a homomorphism, it has an inverse, thus it is an automorphism of Cl(V,B). It is called the canonical (or principal ) automorphism, and we denote it by Π. It is clear that for homogeneous elements of Cl(V) we have

Π(u) = (-1)^deg(u) u.

The elements u of Cl(V,B) on which Π(u) = u form the subalgebra of Cl(V,B) - it is called the even subalgebra, and denoted Cl(V,B)⁺.  The elements u of Cl(V,B) for which Π(u) = -u form a vector subspace of Cl(V,B), denoted Cl(V,B)^-. We have

Cl(V) = Cl(V,B)⁺ ⊕ Cl(V,B)^-.

We now move to the canonical (or principal ) anti-automorphism. This is another instructive exercise in semi-precise logico-algebraic thinking. This time we will take for A the algebra Cl(V,B)^T opposite to Cl(V,B). That means Cl(V,B)^T is the same as Cl(V,B) as a vector space, but multiplication in Cl(V,B)^T, which we denote as u*v is defined as opposite to the product uv in CL(V,B):

u*v = vu.

The algebra Cl(V,B)^T is also associative algebra with the same unit 1. The identity map id: Cl(V,B) → Cl(V,B)T has the property:

id(a)*id(b) = id(ba).

  Let now φ stand for the identity map φ: x ⟶ x, from V to Cl(V,B)^T. Then φ (x)*φ (x) = x*x = xx = B(x,x)1, and again we can apply the universal property to deduce that   φ extends to a unique algebra homomorphism  φ^~ from Cl(V,B) to Cl(V,B)^T . Since 1 and V generate (by sums of products) all CL(V,B) and all Cl(V,B)^T, and since φ^~  is the identity both on 1 and on vectors of V, it follows that φ^~  is a vector space isomorphism, therefore (φ^~ )^-1 exists. We define  τ: Cl(V,B)⟶Cl(V,B) by

τ = id^-1∘φ^~.

Then , for any u,v in Cl(V,B) we have

τ(uv) = vu,

and for any x in V we have τ(x)=x. We also have τ(1)=1. Thus τ is an anti-automorphism of Cl(V,B) - the canonical anti-authomorphism. It is, in fact, a unique anti-automorphism with these properties. Sometimes we will write u^T instead of τ(u), and call it a transposition.

Exercise 1. Show that Π∘τ = τ∘Π.

In the following chronicles we will use Π and τ rather often. They are the main characters of the Clifford algebras and spinors story.