Euler & Gauss — The Atlas of Two Architects (Fixed)

Euler I · e^iθ

e^iθ Is Rotation

Euler's 1748 identification turns exponentiation sideways into rotation. Multiply by e^{iωt} and you spin at ω.

e^iθ = cosθ + i sinθEuler 1748

Euler III · Bridges

The Walk That Invented Graphs

Eulerian trail exists ⇔ 0 or 2 vertices have odd degree1736

Euler IV · Basel & Euler Product

Σ 1/n² = π²/6 → Π over primes

Gauss I · Congruences

The Clock That Locks Secrets

Gauss II · 17-gon

Constructible = Fermat Primes

Gauss III · Gaussian Integers — FIXED

Primes That Are Not Prime

Fixed prime test: a Gaussian integer a+bi is prime iff (a=0 or b=0 and |a|+|b| is rational prime ≡3 mod4) or (a≠0,b≠0 and a²+b² is rational prime). This correctly makes 3,7,11,... on axes prime, but 5=(2+i)(2−i) composite.

N(a+bi)=a²+b² · p≡3 mod4 stays prime in ℤ[i] · p≡1 mod4 = a²+b² splitsFermat two-squares theorem, explained by Gauss

Lab 8 · Correct Gaussian prime test in Octave

function tf=is_gauss_prime(a,b)
  if a==0 && b==0, tf=false; return; endif
  if a==0
    p=abs(b); tf=isprime(p) && mod(p,4)==3; return;
  elseif b==0
    p=abs(a); tf=isprime(p) && mod(p,4)==3; return;
  endif
  tf=isprime(a^2+b^2);
endfunction

R=10; cnt=0;
for a=-R:R
 for b=-R:R
  if is_gauss_prime(a,b), cnt++; printf('%d+%di is Gaussian prime\n',a,b); endif
 endfor
endfor

Previously the code used norm primality for axis case, which misses 3,7,11 (norm 9,49,121 composite). Fixed version tests rational prime ≡3 mod4 on axis.

Gauss IV · Least Squares — FIXED

How Errors Learn to Be Normal

Fixes: draggable points, stable normal equations via QR intuition, correct residual visualization, and no coordinate jump when toggling bell.

min ‖Ax−b‖² ⇒ AᵀA x̂ = AᵀbGauss 1809 · foundation of regression, GPS, ML

Lab 9 · Stable least squares in Octave

% Use QR for stability, not explicit inv(A'*A)
n=25; x=linspace(0,10,n)'; y=2*x+1+randn(n,1);
A=[x, ones(n,1)];
[Q,R]=qr(A,0); xhat = R \ (Q'*y);  % more stable than (A'*A)\(A'*y)
printf('fit: %.4f x + %.4f\n', xhat(1), xhat(2));
res = y - A*xhat; sigma = std(res);
printf('sigma(residual) = %.3f  RSS=%.2f\n', sigma, sum(res.^2));
plot(x,y,'o', x, A*xhat, '-');

Gauss V · Curvature — FIXED

Theorema Egregium — Intrinsic Curvature

Fixed: holonomy angle now equals ∫∫ K dA, with correct formulas for sphere, saddle, cylinder. Arrow transport is visualized by parallel-moving a tangent vector.

Sphere K=1/R², holonomy = Area/R² · Saddle K=−1, holonomy negative · Cylinder K=0Gauss 1827 · Gauss-Bonnet

Synthesis I · Zeta — FIXED CACHING

Euler Product → Riemann Zeros

Lab 11 · Euler product with sieve — fixed

function p=primes_upto(n)
  sieve=true(1,n); sieve(1)=false; for i=2:sqrt(n), if sieve(i), sieve(i*i:i:n)=false; endif; endfor
  p=find(sieve);
endfunction
s=2; N=200; pr=primes_upto(1000);
prod_val=prod(1./(1-pr(1:min(N,length(pr))).^(-s)));
printf('product %d primes = %.9f, pi^2/6=%.9f\n', min(N,length(pr)), prod_val, pi^2/6);

Synthesis II · RSA — FIXED VIS

Euler's Totient Locks the Internet

The Loom Read Backwards — Fixed

Thread	Euler	Gauss	Today (fixed demos)
Wave	e^{iθ}	Fundamental theorem via winding	FFT, quantum phase — wheel now animates smoothly
Congruence	φ(n)	≡, quadratic reciprocity	RSA/DH — orbit shows primitive roots correctly
Curvature	Euler curvature	Theorema Egregium	holonomy = ∫K, not arbitrary angle
Zeta	Π_p	prime conjecture	sieve-generated primes, correct ζ(2)

eiθ Is Rotation