Part 6: Recursive Lo Shu Expansion – Building Rank-n Magic Squares

How a 3×3 square grows into a harmonic grid engine

🧮 The Proto-Lo Shu Trick

We begin with the classical 3×3 Lo Shu magic square:

[4 9 2]
[3 5 7]
[8 1 6]

It uses digits 1 through 9, with a magic constant of 15.

But here’s the trick:

Subtract 1 from each value.

This gives us the Proto-Lo Shu:

[3 8 1]
[2 4 6]
[7 0 5]

This 0-based matrix becomes the key to recursive expansion.

🔁 Recursive Expansion via Substitution

To build the next-rank square:

Multiply every Proto-Lo Shu value by 9
Add each product to the original Lo Shu cell values (1–9)
Position these 3×3 grids according to the shape of the original Lo Shu

Each subgrid becomes part of a 9×9 square with a magic constant of 369.

Repeat this recursively:

At rank 3, you get a 27×27 square
At rank 8, you have a 6561×6561 square with values from 1 to 6561²
All cells are unique and the square maintains its magic sum across all rows, columns, and both diagonals

✨ Properties of the Expansion

Each rank-n Lo Shu square:

Has side length = 3ⁿ
Has a magic constant = (3ⁿ × ((3ⁿ)² + 1)) / 2
Preserves the sum in all major directions
Embeds fractal-like sub-structure

This means: order, harmony, balance, and recursion from a single 3×3 grid.

Info Box
Okay, okay!! I hear you say, “but what are they good for?”
Well, you can ask A.I., Google, or Semantic Scholar. One of my favorite recent finds is Magic Squares: A children’s puzzle meets quantum physics

🔄 Closed-Form Generation

For squares built like this, there is also a deterministic way to compute any cell value at position (r, c) in a rank-n square without needing to recursively building the lower ranks.

It involves:

A base-3 representation of the row and column numbers
Mapping that onto the Proto-Lo Shu index
Combining weighted subcomponents

While the full function is complex, it’s reversible, fast, and perfect for rendering sections of large squares on the fly.

🔍 Why It Matters

This shows that magic squares:

Aren’t just hand-crafted novelties
Can emerge from structured, repeatable, and scalable processes
Might hint at deeper symmetry between arithmetic, geometry, and harmony

You don’t need mysticism to be mesmerized.

🧰 Coming Soon: Code + Visualization

In a future post, we’ll:

Publish runnable JS and Python demos
Let readers pan/zoom through giant Lo Shu fractals
Explore whether this recursive grid technique applies to non-3-based primes

And then, we bring the whole system full circle—combining PTS, magic structure, and musical mapping.

Until then:

The bigger the grid, the deeper the hum.