From keypad wanderings to harmonic structure
🧭 The Setup: A Simple 3×3 Grid
Imagine a standard phone keypad:
[7 8 9]
[4 5 6]
[1 2 3]
Each of the 9 positions becomes a starting point. We define directional steps using coordinate pairs:
- N = (0, -1)
- NE = (1, -1)
- E = (1, 0)
- SE = (1, 1)
- S = (0, 1)
- SW = (-1, 1)
- W = (-1, 0)
- NW = (-1, -1)
- O = (0, 0) (stand still)
We take 3 steps in any of these directions from each cell, wrapping around the edges.
For example, starting at 7 and moving East:
- Step 1: 7 → 8 → 9 → wrap to 7 → …
The digits visited form a 3-digit number. This is repeated across all positions and directions.
🔢 Result: 81 Unique Three-Digit Numbers
Every combination generated follows a toroidal path and uses only digits 1–9.
Surprising outcomes:
- Every result is divisible by 3
- All values reduce to 3, 6, or 9
- Includes classic angel numbers (e.g., 111, 222, …)
- Captures the Solfeggio group all together in one round (396, 417, etc.)
Sorted, the PTS looks like:
[111, 123, 132, 147, 159, 168, 174, 186, 195,
213, 222, 231, 249, ... up to 999]
🧠 Code Demo Tabs (Placeholder)
Python Tab:
# Define the 3x3 grid
# Define all directional vectors
# Walk from each position in each direction
# Collect and sort unique 3-digit results
JavaScript Tab:
// Create grid array
// Define directions as [dx, dy]
// Use modular arithmetic to wrap
// Build three-digit numbers
Live Demo (Placeholder): [ GridWalker.js demo with interactive starting points + path lines ]
🎼 Why It Matters
The structure that emerged wasn’t random. It hinted at something underlying—not just a pattern, but a harmonic container.
From this, further patterns emerged:
- 9×9 magic square substitutions with preserved sums
- Lower and Higher-order PTS extensions (2PTS, 4PTS)
- A numeric analog to tone systems like 12TET
The numbers didn’t just align—they sang, subtly but with seeming intention.
Next up: We map the PTS into the framework of the 9×9 Lo Shu-inspired magic square—and discover why 4995 might just be the most harmonious number you’ve never heard of.