CORE INSIGHT
π emerges from counting squares!
No circles needed. No floats needed.
Just: (2i+1)² + (2j+1)² ≤ 2^(2k+2)
Pure integer arithmetic → π as limit.
No circles needed. No floats needed.
Just: (2i+1)² + (2j+1)² ≤ 2^(2k+2)
Pure integer arithmetic → π as limit.
CURRENT LEVEL k = 4
GRID SIZE
16×16
TOTAL CELLS
256
INSIDE (Nk)
201
BOUNDARY
24
INTEGER BOUNDS TRAPPING π
3.125
≤
π
≤
3.1875
Gap: 0.0625 = O(2-k)
CONVERGENCE TO π
3.0
π ≈ 3.14159
3.3
THE INTEGER TEST
// Cell (i,j) center in integer coords:
X = 2i + 1
Y = 2j + 1
// Inside quarter-disk test:
X² + Y² ≤ 2^(2k+2)
// Area estimate:
A_k = 4 × N_k / 2^(2k)
// Definition of π:
π = lim(k→∞) A_k
X = 2i + 1
Y = 2j + 1
// Inside quarter-disk test:
X² + Y² ≤ 2^(2k+2)
// Area estimate:
A_k = 4 × N_k / 2^(2k)
// Definition of π:
π = lim(k→∞) A_k
ERROR ANALYSIS
Error at level k: O(2-k)
Each refinement adds ~1 bit of precision
k=4: ~1 decimal digit
k=7: ~2 decimal digits
k=10: ~3 decimal digits
k=7: ~2 decimal digits
k=10: ~3 decimal digits
VISUAL LEGEND
Inside cells (counted)
Boundary cells (uncertain)
Outside cells
True quarter circle