THE COMPLETE JOURNEY
e
iπ
+ 1 = 0
FROM INTEGER LATTICES TO THE MOST BEAUTIFUL EQUATION
ALL COMPUTED USING ONLY INTEGER OPERATIONS
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MILESTONE 1
π (Area)
π = lim 4·N_k/2^(2k)
3.14159265...
MILESTONE 2
e (Growth)
e = lim (1+2^(-k))^(2^k)
2.71828182...
MILESTONE 3
√2 (Metric)
√M = lim A_p/2^p
1.41421356...
MILESTONE 4
exp(x)
exp(x) = lim (1+x/2^k)^(2^k)
Euler's method
MILESTONE 5
log(x)
log(x) = y : exp(y) = x
Binary search
MILESTONE 6
sin/cos
CORDIC: shifts + adds
No multiplication!
MILESTONE 7
Complex i
z = (a, b), i² = -1
(0, 1)
MILESTONE 8
e^(iπ) = -1
exp(iθ) = (cos θ, sin θ)
THE GOAL!
VERIFICATION
exp(iπ) =
-1.00000000
+
0.00000000
i
Three refinement invariants meet at one point:
e
(growth) ·
π
(half-turn) ·
i
(rotation)
All derived from integer operations on dyadic lattices.