Counting a Figurate Property in Cayley–Dickson Algebras

Cayley-Dickson Algebras

Define something once, observe what follows, then redefine it using what came before. In simple terms, that formidable game of redefining based on prior objects. Is there anything more to it when we write recursive algorithms?

Figurate numbers, as I mentioned in earlier posts, overflow what I would call their combinatorial side $\binom{n}{k}$, the one that defines them, and irrigate neighboring or even subterranean places. In this very brief post, I want to offer a faint taste of possibility (more fragile than anything logically immediate). Many of these inquiries come from questions closer to daydreaming:

given an object, this object also comes from somewhere else.

Yes, a simple question in the spirit of what I imagine the inverse Galois problem to be, more precisely, a realization problem: given a structure, which ambient object contains it or generates it? Figurate numbers also encode the dimensions of algebras (you can read an introduction to the topic here).

When I said repeat, I was referring to the Cayley-Dickson construction, which generates a sequence of algebras over $\mathbb{R}$, the numbers we usually use to plot functions, starting with the reals and doubling the dimension at each step:

where $A_{n+1}$ is the set of ordered pairs from $A_n$, equipped with a new multiplication. Notice that the product defined for

as

is similar to the complex case, and yes, it coincides exactly when $n = 1$. Here $\bar{a}$ denotes conjugation in $A_n$ (analogous to the conjugation of $i$ in the complex numbers). Conjugation in $A_{n+1}$ is then defined by

You can take a pen and repeat it. In fact, things get progressively less natural. The construction successively loses commutativity (at $\mathbb{H}$), associativity (at $\mathbb{O}$), and alternativity beyond that (as we will see again below).

In a post from last months, in this post on modules over a ring, I showed how to define figurate numbers in a type system. Let me quote exactly what I said there:

Let $R$ be a ring and $M_R$ a free module over $R$, and let $m \ge 3$. Then we can define the polygonal figurate numbers (extensible by recursion to many others) in a type system as

I also copy the simplest example with triangular numbers (you can also see the type-level difficulties in that post):

type VectorSpace = A["length"] extends dim
  ? A
  : VectorSpace;

type DirectSum = VectorSpace;

type V1 = VectorSpace<200>;
type V2 = VectorSpace<335>;
type V3 = DirectSum;

And to compute, you simply run:

type VecTriangular
  n extends number,
  T_1 extends number[] = VectorSpace<1>,
  IterCount extends number[] = [1]
> = IterCount["length"] extends n
  ? T_1
  : VecTriangular
      n,
      DirectSum>,
      [1, ...IterCount]
    >;

type T_3 = VecTriangular<3>;
type T_5 = VecTriangular<5>;
type T_20 = VecTriangular<20>;
type T_50 = VecTriangular<50>;

With this we have described numbers as dimensions. Why did I bring all of this back? It has to do with the next section.

Natural inclusions, two by two

Let us enter symbolically into the idea of a river from that post mentioned above. Each algebra embeds into the next via natural injection morphisms

Landing on the form in which we would study them numerically, and observe how each step sheds algebraic properties:

From this it is immediate that each algebra $A_n$ has dimension $2^n$. And since figurate numbers also go to infinity, we are interested in the unbounded growth pattern of the sequence of dimensions, which satisfies

This coincides exactly with the impolite numbers, i.e., numbers that cannot be expressed as a sum of consecutive positive integers, which turn out to be precisely the powers of $2$, computable with my library figuratenum:

from figuratenum import PlaneFigurateNum as fgn
# Generate impolite numbers
gen = fgn().impolite()
impolite_num = [next(gen) for _ in range(32)]
# [2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, ...]

The initial question appears when playing with Legos and assembling ad hoc algebras that respect the monotone growth of figurate numbers. In fact, figurate numbers can naturally capture a dimensional particularity of other algebraic spaces. But how many interesting algebras can I build this way? A second question, arising from a somewhat more discerning perspective, is:

Given a not necessarily injective sequence of algebras, does there exist a figurate number that counts some really interesting properties of it?

Like music, each note, or each algebra, is part of a single composition, with each stage filtering and chaining the harmony of the whole. We can think of ourselves as that function which computes all possible interpretations of subalgebras, units, idempotents, musical phrases, and dynamics.

It is time to play Debussy algebra.