A Structural–Geometric Reframing of the “Squaring the Circle” Problem
For centuries, squaring the circle has been treated as a symbol of impossibility.
An abstract limit.
An asymptotic pursuit.
A thought experiment rather than a testable geometry.
Most approaches drift toward:
- infinite limits
- heuristic packing
- visual approximation
- probabilistic optimisation
This work asks a simpler — and more precise — question:
What if squaring the circle is treated as a finite geometric problem, not an idealised one?
🧠 Geometry Before Approximation
Classical packing studies often relax constraints early:
- corners are approximated
- boundaries are softened
- acceptance becomes statistical
The Finite Structural Area Experiment (FSAE) takes the opposite path.
It does not ask how densely squares can fill a circle in theory.
It asks:
How many squares can be placed such that every square is exactly inside the circle — with no tolerance, no approximation, and no visual inference?
📐 What FSAE Actually Does (Without Guesswork)
FSAE is built using principles from Shunyaya Structural Universal Mathematics (SSUM), but its rule is deliberately simple and classical:
For every square, all four corners must satisfy:
x_corner^2 + y_corner^2 <= R^2
There are:
- no softened boundaries
- no probabilistic acceptance
- no simulation loops
Every square is either certified or rejected.
🔢 Finite Enumeration, Not Infinite Limits
This study works with:
- a fixed circle radius
R - a fixed square side length
s - explicit lattice configurations
Both axis-aligned and rotated square lattices are evaluated under identical rules.
Rotation is treated as a bounded geometric parameter, not a free optimisation trick.
Translation fairness is enforced explicitly.
The result is not a curve, a trend, or an estimate —
but a finite, integer count.
🧪 The Method (High Level, Deterministic)
FSAE proceeds by:
- enumerating square lattice centers
- rotating and translating them deterministically
- analytically checking all four corners of every square
- certifying results strictly as PASS / FAIL
No learning.
No solvers.
No Monte Carlo sampling.
The same inputs always produce the same outcome.
🌟 What the Geometry Reveals
For the reference case:
R = 10s = 1
The results are unambiguous:
- Axis-aligned lattice:
277 → 279 - Rotated lattice:
277 → 279
Improvements occur only at specific geometric alignments.
Small parameter changes often do nothing.
This reveals a key insight:
Geometric improvement is discrete, not smooth.
✨ The Beauty of Structural Plateaus
What makes this result important is not the number itself —
but how the number changes.
There is no gradual optimisation.
No continuous drift.
Instead:
- geometry advances in plateaus
- alignment unlocks capacity suddenly
- translation matters as much as rotation
This behaviour is invisible in approximate or asymptotic methods —
but obvious under exact certification.
🔍 The Key Insight
“Squaring the circle” does not need to be infinite to be meaningful.
When treated as a finite structural problem:
- geometry becomes testable
- results become verifiable
- disagreement becomes explicit
In short:
Exact geometry reveals structure that approximation hides.
🌍 Why This Matters Beyond a Puzzle
FSAE is not about squares and circles alone.
The same certification discipline applies to:
- finite packing problems
- layout constraints
- bounded spatial design
- geometry-first diagnostics
Before optimisation, simulation, or material assumptions,
geometry itself can be certified.
🧭 Observability, Not Optimisation
FSAE does not:
- claim optimality
- predict physical performance
- replace engineering judgement
It provides geometric observability only.
All decisions live above the result —
while the geometry remains exact and untouched.
🔗 Where the Work Lives
🔬 Executable Verification — SSUM Observatory (Case 08)
https://ompshunyaya.github.io/ssum-observatory/08_finite_structural_area_experiment/
https://github.com/OMPSHUNYAYA/ssum-observatory
📘 Finite Structural Area Experiment (FSAE)
https://github.com/OMPSHUNYAYA/SSUM-Finite-Structural-Area-Experiment
This blog is the narrative entry point.
The Observatory remains the single source of truth for execution.
📘 License & Attribution (SSUM)
Open Standard — provided as-is.
You may read, study, reference, and build upon the concepts.
Optional attribution (not mandatory):
“Implements concepts from Shunyaya Structural Universal Mathematics (SSUM).”
⚠️ Disclaimer
Research and observation only.
Not intended for optimisation claims, safety decisions, or engineering execution.
OMP
SHUNYAYA 