September 11, 2025 by gene.goodreau@gmail.com
Squaring Numbers as Diagonal Morphisms from θ
When we first learn squaring, it’s just arithmetic:
12 = 1, 22 = 4, 32 = 9
But in the Essencia / Q-system, squaring isn’t simply multiplication. It becomes a geometric and
categorical movement starting from the origin θ.
- θ as Origin and Reference
In the Q-framework, θ is the origin point, the source of reference.
At θ, we have a square representing θ then if θˆ2 we have the origin of (0,0). Moving to 1ˆ2 means θ points to the coordinate (1,1).
So instead of ‘1 squared = 1,’ we see: 1ˆ2 = θ → (1,1). - The Case of 2ˆ2
In arithmetic, 2ˆ2= 4. In the Q-system, it can be read in two ways:
- Absolute View: θ at (0,0) points directly to (2,2).
- Relative View: θ, now shifted to (1,1), points forward to (2,2).
- The Diagonal Ladder
Following this logic, squaring builds a ladder along the diagonal of the Cartesian plane:
θ(0,0) → (1,1), θ(1,1) → (2,2), θ(2,2) → (3,3), θ(3,3) → (4,4).
In general: n2 = θ(n-1,n-1) → (n,n). - Category-Theoretic Frame
Seen through category theory, this is a diagonal functor in action:
∆(θ) = (θ,θ), and a morphism maps θ to (n,n). - Why This Matters
This interpretation opens up a new perspective:
- Squaring becomes movement instead of static number.
- Numbers are not just quantities, but locations along a relational diagonal.
- θ isn’t just a zero — it’s the source of projection that generates dimensional steps.
In essence, each square is an act of becoming, a shift from θ into the diagonal plane of existence.
