When a toroidal number scale lands squarely in ancient math territory
🧩 Magic Squares 101 (and a Quick Lo Shu Recap)
Magic squares are grids where the sum of every row, column, and diagonal equals the same “magic constant.”
The classic 3×3 Lo Shu magic square:
[4 9 2]
[3 5 7]
[8 1 6]Has a magic constant of 15, using the digits 1 through 9 without repetition.
The ancient Chinese Lo Shu square is revered for its simplicity and balance—so much so, it’s been echoed in architecture, alchemy, and mysticism.
But what happens when you scale up?
Enter the 9×9 square.
🧮 The Presh Talwalkar Construction
Educator Presh Talwalkar once demonstrated a beautiful 9×9 grid assembled using Lo Shu-inspired principles. By layering, rotating, and compacting 3×3 subsquares, he built a larger structure with similar traits—though its underlying sum was not preserved from the 3×3.
The constant sum in his original version was 369—an interesting number in its own right.
This inspired a natural question:
What if we substitute the values from the Preston Toroidal Scale (PTS) into that same structure?
🎯 Substitution: The PTS Takes the Square
Using the sorted 3-digit PTS set:
[111, 123, 132, 147, ..., 999]We performed a 1-to-1 replacement of numbers 1 through 81 in the Talwalkar 9×9 grid. Each cell originally labeled 1 became 111, 2 became 123, 3 became 132, and so on.
After substitution, we recalculated the row, column, and diagonal sums.
And what we found was… unexpected:
Every column and diagonal still summed to the same value.
That value? 4995
🔁 Why 4995?
Let’s do a quick calculation:
- The sum of integers 1 to 81 is: (81 × 82) / 2 = 3321
- Multiply 3321 by the average growth from unit steps in the PTS (approx. 1.5×): boom — we land near 4995
But more interestingly:
- 4995 is divisible by 3, 5, 15, 111
- 4995 is the total sum of each column in the PTS-substituted square
- It mirrors the harmonic coherence of the PTS itself
The structure carried through. Even though the PTS values have no natural sequential ordering like [1, 2, 3, …], they maintained a deep internal balance when substituted into a historically sacred grid.
✨ More Than Coincidence?
Was it luck? Math magic? Selective perception?
We can check that by running the same substitution into every isomorphic variant of the 9×9 square—rotated, flipped, transposed—and we find the same result: 4995 everywhere.
Even more, the Lo Shu and the Pythagorean keypad grid (with [7 8 9] on top) are the only two known 3×3 arrangements to produce this consistent 3PTS walk outcome.
👀 What’s Next
The PTS doesn’t stop at three digits. In the next article, we’ll:
- Extend the walk to 4-digit cycles (4PTS)
- Examine digit roots and symmetry
- Analyze the logarithmic shape of PTS across 1–4 digit groupings
- Explore connections to tone frequencies and 12TET
Until then, know this:
4995 isn’t just a sum. It’s a whisper from the grid.
And we’ve only just begun to listen.