Polynomial & Fractal Atlas

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Fractal

Mandelbrotz²+c

Julia Setz²+c

Burning Ship|z|²+c

Newton z³z−f/f'

Sierpinskid≈1.585

Koch Snowd≈1.262

Dragond=2

IFS CustomAx+b

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Iter120

Cycle10

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abcdef

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IFS dH: —

Properties

Type: Mandelbrot Set

Formula: z → z² + c

d_H: 2.0 (boundary)

Domain: c ∈ ℂ

Mandelbrot Set

iter: 120

box d: —

zoom: 1.0×

— + —i

Dimension

Poly Connection

Box Count Plot

Hausdorff / Box Dimension

d = lim[ε→0] log N(ε) / log(1/ε)

Click "Box Count" to estimate d from current render

Similarity Dimension

d = log N / log(1/r)

d = 1.5850

Moran Equation (IFS)

∑ r_i^d = 1

r_i

e.g. Sierpinski: 0.5,0.5,0.5

Lyapunov / KY Dimension

d_KY = j + ∑λ_i/|λ_{j+1}|

λs

—

Chebyshev Nodes & Julia Sets

The Chebyshev nodes x_k = cos((2k-1)π/2n) on [−1,1] are exactly the optimal interpolation points that minimise the Runge oscillation — the same oscillatory instability that appears at the boundary of fractal basins. Use the overlay to see nodes placed on the Mandelbrot boundary.

T_n(x_k) = 0 ⇔ x_k are equidistributed w.r.t. arcsine measure
= equilibrium measure of Julia set for z²+c near c=0

Hermite Polynomials & Brownian Fractals

The Hermite functions ψ_n(x) = H_n(x)e^(-x²/2) are eigenfunctions of the quantum harmonic oscillator. Brownian motion paths have Hausdorff dimension 2, and their quadratic variation connects to the Gaussian weight w(x) = e^(-x²) that defines Hermite orthogonality.

d_H(Brownian path) = 2 = 2 × 1 (parameter dim)
E[|B(t)|²] = t ⇐⇒ Gaussian measure

Legendre & Mandelbrot Boundary

The conformal map from the exterior of the Mandelbrot set to the exterior of the unit disk has Laurent expansion whose coefficients relate to generalised Legendre / Faber polynomials. Shishikura proved d_H = 2 for the Mandelbrot boundary using estimates on these polynomial expansions.

d_H(∂M) = 2 (Shishikura 1998)
Proof via Beurling-Ahlfors estimates on Faber polys

Potential Theory Bridge

The Green's function of a filled Julia set K is the limit of (1/d^n) log|p_n(z)| where p_n are the Mandelbrot iterates. This is a polynomial-theoretic object — the logarithmic capacity of K equals the transfinite diameter, and orthogonal polynomial asymptotics (Szego's theorem) describe the boundary measure.

G_K(z) = lim (1/2^n) log|f^n(z)|
cap(K) = exp(-min energy) = Chebyshev const

Run box count to see log-log plot and regression data.

Family

LegendreP_n(x)

HermiteH_n(x)

LaguerreL_n(x)

ChebyshevT_n(x)

GegenbauerC_n^λ

JacobiP_n^(α,β)

Controls

Degree n3

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Properties

Family: Legendre

w(x): 1 on [−1,1]

Rec: (n+1)P_{n+1}=(2n+1)xPn−nP_{n-1}

Orth: ∫PmPn dx=2/(2n+1)δ

PDF: Uniform[−1,1]

Legendre P_n(x,y)

mode: product

drag to rotate · scroll to resize

2D Curves

Zeros & Weights

Recurrences

Zeros of P_n(x)

All n zeros ∈ (−1,1) for Legendre, Chebyshev; interlacing property

Select a family and degree

Gauss Quadrature Weights

w_k = 2/((1-x_k²)[P'_n(x_k)]²) — Gauss-Legendre

—

—

Family ↔ Distribution

Degree n3

P_n On

w·P² On

Info

Dist: Uniform

PDF: f(x)=½

Weight: w(x)=1

Polynomials uncorrelated under this distribution

Legendre / Uniform

Interpretation

Norm Values

Legendre / Uniform[−1,1]
E[P_m · P_n] under Uniform = (1/2)∫PmPn dx = 0 for m≠n
Gauss-Legendre quadrature exactly integrates degree 2n-1 polynomials using n Uniform-spaced-optimal nodes
Hermite / Normal N(0,½)
H_n are uncorrelated under Gaussian: E[H_m H_n] = 0
Foundation of Wiener chaos expansion in stochastic analysis

—

Bridges Between Polynomials & Fractals

The deep mathematical connections linking orthogonal polynomial families with fractal geometry, dimension theory, and complex dynamics

Chebyshev Polynomials & Julia Sets

T_n(cosθ) = cos(nθ) — the Chebyshev polynomial is the unique monic polynomial of degree n with smallest sup-norm on [−1,1]. This is the minimax problem whose solution is controlled by the same logarithmic potential theory that governs Julia set geometry. The equilibrium measure of the Julia set of z^n is exactly the arcsine measure dθ/π — the orthogonality measure for Chebyshev polynomials.

cap([−1,1]) = ½ = Chebyshev constant
Equil. measure = arcsine = Chebyshev T weight

Legendre & Conformal Maps on M

The Riemann map Φ: ext(M) → ext(D) from the exterior of the Mandelbrot set to the exterior of the unit disk has a Laurent expansion involving Faber polynomials generalising Legendre-type constructions. Shishikura's proof that d_H(∂M) = 2 uses deep estimates on how these polynomial approximations fail to converge — the boundary is too complex to be captured by any polynomial system.

d_H(∂M) = 2 (Shishikura 1998)
via Beurling-Ahlfors & polynomial capacity estimates

Hermite & Fractal Stochastic Analysis

The Hermite polynomials H_n(x) with Gaussian weight e^(-x²) are the natural basis for expanding functionals of Brownian motion (Wiener chaos). Since Brownian paths have d_H = 2 almost surely (Lévy's theorem), Hermite expansions encode the fractal geometry of Gaussian processes. The Ornstein-Uhlenbeck process — the prototype for Brownian motion on fractals — is diagonal in the Hermite basis.

F(B) = ∑ c_n H_n dB — Wiener chaos
d_H(B[0,1]) = 2 a.s. ⇐⇒ Gaussian weight

Laguerre & Random Matrix Eigenvalues

The eigenvalues of a Wishart matrix (W = X^T X, X random Gaussian) are distributed according to the Laguerre ensemble, with joint density proportional to ∏|λ_i|^α e^(-λ_i). As n→∞, the empirical spectral distribution follows the Marchenko-Pastur law — a distribution whose orthogonal polynomial system is exactly the generalised Laguerre family L_n^(α). The support of this measure has fractal-like boundary structure.

LUE: weight x^α e^(-x) = Laguerre measure
Marchenko-Pastur = limiting spectral density

Potential Theory: Capacity = Chebyshev Constant

The logarithmic capacity cap(K) of a compact set K equals its transfinite diameter, which in turn equals 1/||T_n||^(1/n) where T_n is the monic Chebyshev polynomial for K (the polynomial minimising the sup-norm on K). This is the deepest bridge: the fractal geometry of a set (capacity, Hausdorff dimension) and the polynomial approximation theory of that set (Chebyshev polynomials) are two faces of the same logarithmic potential.

cap(K) = lim ||p_n||_K^(1/n)^(-1)
= transfinite diameter = Chebyshev constant

Gegenbauer & Spherical Harmonics on Fractals

Gegenbauer polynomials C_n^λ(cosθ) appear as the angular part of solutions to Laplace's equation in d = 2λ+1 dimensions. Analysis on fractal sets embedded in S^d — like the Sierpinski sphere or Cantor dust on a sphere — uses these polynomials as a spectral basis. The parameter λ acts like a continuous "effective dimension", interpolating between integer-dimensional geometries.

C_n^λ: effective dim = 2λ+1
λ=½ → Legendre (3D) · λ→0 → Chebyshev T

Hausdorff Dimension & Orthogonality Measures

For a self-similar measure μ on a fractal K with IFS ratios r_i and probabilities p_i, the Hausdorff dimension of μ is d_H = ∑p_i log p_i / ∑p_i log r_i (information dimension). The orthogonality measure of a polynomial sequence supported on K is absolutely continuous with respect to μ, and the asymptotic zero distribution of the orthogonal polynomials converges to the equilibrium measure of K — connecting dimension theory directly to spectral asymptotics.

d_H(μ) = ∑ p_i log p_i / ∑ p_i log r_i
zero distrib(P_n) → equil. measure of K

Bernstein & Fractal Approximation

Bernstein polynomials B_n f converge to f at rate O(1/n) — slower than Chebyshev at O(e^(-cn)) for analytic f. This slower rate reflects the fractal-like equidistribution of Bernstein nodes (k/n) under the uniform measure, versus the Chebyshev nodes under arcsine. The de Casteljau algorithm for computing Bézier curves is a discrete subdivision — the same structure as IFS fractal generation, and cubic Bézier curves can approximate (but never exactly reproduce) fractal curves.

B_n rate ~ 1/n vs Cheb rate ~ e^(-cn)
de Casteljau ≡ IFS subdivision scheme

Weierstrass ℘ & Elliptic Curves

The Weierstrass ℘-function parametrises y²=4x³−g₂x−g₃ via x=℘(z), y=℘′(z). Since ℘ is even (℘(−z)=℘(z)) and ℘′ is odd, the group-law inverse (x,y)→(x,−y) corresponds exactly to z→−z on the torus ℂ/Λ. The j-invariant j(Λ) classifies lattices up to homothety and extends to a modular function on the upper half-plane.

℘(−z)=℘(z) (even) ℘′(−z)=−℘′(z) (odd)
Taniyama-Shimura: every E/ℚ is modular (→ Wiles, FLT)

Dimension & Polynomial Calculators

IFS Hausdorff (Moran)

∑ r_i^d = 1

r_i

e.g. Sierpinski: 0.5,0.5,0.5 → d≈1.585

Similarity Dimension

d = log N / log(1/r)

d = 1.5850

Box Count from Data

d = −Δlog N / Δlog ε

(ε,N) pairs

Enter pairs sep by semicolons

Lyapunov Exponent

λ = (1/N) ∑ log |f'(x_n)|

Map

—

Kaplan-Yorke Dimension

d_KY = j + ∑λ_i / |λ_{j+1}|

λ list

—

Polynomial Zeros

P_n(x_k) = 0 — n zeros in support

Family

Degree n

—

Polynomial Norm ||P_n||

∫ P_n² w(x) dx = h_n

Family

—

Correlation Dimension

d_c = lim log C(r) / log r

Points

Enter comma-sep 1D points

Weierstrass Elliptic Curve

y² = x³ + ax + b

a = b =

▶ Plot

−x −3x+1 −1 (j=0) secp256k1

    Click Plot to render.
  

Key Properties

Δ = −16(4a³+27b²)
j  = −1728(4a)³/Δ
Non-singular: Δ ≠ 0
Even in y: (x,y)↔(x,−y)

Group Law

Identity: 𝒪 (point at ∞)
Inverse: −P = (x, −y)
P+Q: chord ∩ curve, reflect
2P: tangent ∩ curve, reflect

    General Weierstrass: y²+a₁xy+a₃y = x³+a₂x²+a₄x+a₆

    Short form (char≠2,3): y² = x³+ax+b  —  complete the square & cube

Fractal Dimensions

Fractal	d_T	d_H
Cantor Set	0	log2/log3 ≈ 0.631
Sierpinski Δ	1	log3/log2 ≈ 1.585
Sierpinski Carpet	1	log8/log3 ≈ 1.893
Menger Sponge	1	log20/log3 ≈ 2.727
Koch Curve	1	log4/log3 ≈ 1.262
Dragon Curve	1	2.0
Mandelbrot boundary	1	2.0 (Shishikura)
Brownian path	1	2.0 a.s.
Lorenz attractor	2	≈ 2.062
UK Coastline	1	≈ 1.25

Polynomial Families

Family	Weight w(x)	PDF
Legendre P_n	1	Uniform[−1,1]
Hermite H_n	e^(-x²)	Normal N(0,½)
Laguerre L_n^(α)	x^αe^(-x)	Gamma(α+1)
Chebyshev T_n	(1-x²)^(-½)	Arcsine
Chebyshev U_n	(1-x²)^½	Wigner semicircle
Gegenbauer C_n^λ	(1-x²)^(λ-½)	Beta(λ+½,λ+½)
Jacobi P_n^(α,β)	(1-x)^α(1+x)^β	Beta(α+1,β+1)
Charlier C_n	Poisson(a)	Poisson(a)
Krawtchouk K_n	Binomial(N,p)	Binomial

Weierstrass Equation & Elliptic Curves

Form	Equation	Notes
General Weierstrass	y²+a₁xy+a₃y=x³+a₂x²+a₄x+a₆	any characteristic
Short Weierstrass	y² = x³ + ax + b	char ≠ 2,3
Discriminant Δ	−16(4a³+27b²)	≠ 0 for smooth curve
j-invariant	−1728(4a)³/Δ	isomorphism class
j = 0	a = 0: y²=x³+b	CM by ℤ[ω], ω=e⁶²𝜋ⁱ∕³
j = 1728	b = 0: y²=x³+ax	CM by ℤ[i]
Real components	Δ>0: two Δ<0: one	topology of E(ℝ)
Inverse of P=(x,y)	−P = (x, −y)	even in y

Named Curve	a	b	j
secp256k1 (Bitcoin)	0	7	0
P-256 (NIST)	−3 (mod p)	b₃(mod p)	—
Curve25519	Montgomery form By²=x³+Ax²+x
y²=x³−x	−1	0	1728
y²=x³−1	0	−1	0
Congruent number	−n²	0	1728
Wiles (FLT proof)	Frey curve: y²=x(x−aⁿ)(x+bⁿ)

          Parametrisation over ℂ: x=℘(z), y=℘′(z) where ℘ is the Weierstrass ℘-function, a doubly-periodic meromorphic function on ℂ/Λ (Λ=ℤω₁+ℤω₂ a lattice).

          ℘ is even: ℘(−z)=℘(z).   ℘′ is odd: ℘′(−z)=−℘′(z).
        
          Laurent expansion near z=0: ℘(z) = 1/z² + ∑(2k+1)G₂ₚ₊₂ z²⁽ where G₂ₚ are Eisenstein series. The curve equation y²=4x³−g₂x−g₃ uses g₂=60G₄, g₃=140G₆, connecting ℘ to the lattice invariants.

Annals of Mathematics · Vol. 141 · May 1995

Fermat’s Last Theorem

xⁿ + yⁿ ≠ zⁿ for any n ≥ 3

No solution in positive integers · 358 years from conjecture to proof

The Complete Logical Chain

Visual overview

✗ Assume: a^p + b^p = c^p for prime p ≥ 5, integers a,b,c > 0

Frey (1984) — §2

1 Construct Frey curve E: y² = x(x − a^p)(x + b^p)

conductor, semistability — §2

2 E is semistable N_E = 2·rad(abc) Δ = 2⁻⁸(abc)^2p

Wiles + Taylor (1995) — §5

3 ρ̄_E,p is modular of level N_E (E is modular)

Ribet (1990) × ω(abc) steps — §4

4 ρ̄_E,p is modular of level 2

genus of X₀(2) = 0 — §4

⊥ ⊥ S₂(Γ₀(2)) = 0 — no such modular form exists — CONTRADICTION

✓ ∴ FLT holds for all prime p ≥ 5 (combined with Fermat’s n=4 proof) ■

History

Timeline

1637

Fermat writes: “I have a truly marvellous proof, which this margin is too narrow to contain.” Almost certainly incorrect for n≥5.

1753–1839

Euler (n=3), Dirichlet & Legendre (n=5), Lamé (n=7). Each case required a separate argument.

1847

Kummer discovers Lamé’s proof fails (ℤ[ζ_p] lacks unique factorisation). Invents ideal numbers; proves FLT for all regular primes (p<100 except 37, 59, 67).

1984

Frey observes that a counterexample a^p+b^p=c^p produces an elliptic curve with anomalously small conductor.

1985–90

Serre formulates the ε-conjecture. Ribet proves it (1990): if the Frey curve is modular, we can level-lower to level 2. But S₂(Γ₀(2))=0.

1993

Wiles announces a proof at Cambridge. A gap is found in the Euler system / Iwasawa theory component during peer review.

1995

Wiles & Taylor close the gap with the patching argument. Complete proof in Annals of Mathematics, May 1995. 358 years after the margin note.

§1 Reduction to Prime Exponents

It suffices to prove FLT for n=4 and for all odd primes p≥5.

Reduction tree

Any n ≥ 3
┬───────────────┼───────────────┤
4 | n
↓
FLT for n=4
Fermat: x⁴+y⁴=w² impossible
(infinite descent)
odd prime p | n
↓
FLT for p ≥ 5
Wiles 1995
(the deep case)
If n=ab, any solution (x,y,z) for n gives (xb,yb,zb) for a  ⇒  FLT for a ⇒ FLT for n.

Fermat’s infinite descent for n=4: Suppose x⁴+y⁴=z⁴. Fermat showed the stronger statement that x⁴+y⁴=w² has no solution in positive integers. If a right triangle with integer sides has square area, construct a strictly smaller one with the same property — contradiction. This handles n=4 and any n divisible by 4.

§2 The Frey Curve

Suppose a^p+b^p=c^p (prime p≥5, gcd(a,b,c)=1, b even, a≡−1 mod 4). Frey’s construction:

  Ea,b,c:   y² = x(x − ap)(x + bp)     roots: 0, ap, −bp

Illustrative Frey-type curve (a=1, b=2, p=3: y²=x(x−1)(x+8), not a real counterexample)

Discriminant:
Δ = 2−8(abc)2p
Each odd prime ℓ|abc appears in Δ to the power 2p ≥ p — key for Ribet.
Conductor:
NE = 2 · rad(abc) = 2·∏ℓ|abc oddℓ
Semistable: multiplicative reduction at all bad primes.

The even symmetry y→−y (visible in the plot) is the group-law inverse: −P=(x,−y). This is the same property we discussed for the general Weierstrass equation.

§3 Galois Representations

The bridge between elliptic curves and modular forms is built via Galois representations. To each prime ℓ we attach:

  ρE,ℓ : Gal(ℚ̄/ℚ) → GL2(ℤℓ)    via action on Tℓ(E) = lim← E[ℓn]

How the Galois group acts on E[p] ≅ (ℤ/p)²

INPUT
σ ∈ Gal(ℚ̄/ℚ)
(absolute Galois group)
→
OUTPUT
ρ̄E,p(σ) = a   b
c   d
matrix in GL2(𝔽p)

det(ρ̄E,p) = χp   (cyclotomic character)    — always
For Frey curve: at each odd prime ℓ|abc, ρ̄E,p|Gℚℓ ≅
      χp   *
0    1
        (ℓp | Δ forces deep unramifiedness)
    
⇒ Serre conductor of ρ̄E,p = 2

A representation ρ is modular if ρ ≅ ρ_f,p for some weight-2 newform f. If E is modular (which Wiles proves), then ρ̄_E,p is modular of level N_E.

§4 Modular Forms & Ribet’s Theorem

A modular form of weight 2 and level N is a holomorphic f on ℍ satisfying f((az+b)/(cz+d))=(cz+d)²f(z) for all [[a,b;c,d]] ∈ Γ₀(N). Cusp forms vanish at cusps; their space is S₂(Γ₀(N)).

Key Fact
S2(Γ0(2)) = 0
The modular curve X0(2) has genus 0 (it is isomorphic to ℙ1). Genus-0 curves have no holomorphic differentials, hence no non-zero weight-2 cusp forms.

Ribet’s theorem: Let E/ℚ be semistable with conductor N, p≥5 prime, p∪2N. If ρ̄_E,p is modular of level N, and q||N with q^p|Δ_E, then ρ̄_E,p is also modular of level N/q.

Level-lowering pipeline: stripping primes from N_E

Each prime ℓ|abc satisfies ℓ||N_E (semistable) and ℓ^2p|Δ ⇒ ℓ^p|Δ ⇒ Ribet applies. Strip ω(abc) primes one by one until only level 2 remains.

§5 Wiles’s Modularity Theorem

Theorem (Wiles 1995; completed with Taylor)

Every semistable elliptic curve over ℚ is modular.

The proof establishes an isomorphism between a deformation ring R and a Hecke algebra T, showing every Galois deformation is modular.

5.1 The R = T Strategy

Deformation ring vs Hecke algebra

Fix mod-p rep ρ̄.   R = universal deformation ring.   T = Hecke algebra on S2(Γ0(N)).
Natural map φ: R → T exists (every modular lift is a deformation).   Wiles proves φ is an isomorphism.
R ≅ T   ⇔   every deformation of ρ̄ is modular   ⇔   E is modular.

5.2 The 3–5 Switch

ρ̄E,3 irreducible
Use 3-adic deformation theory. Base case: GL2(𝔽3) is solvable ⇒ Langlands–Tunnell applies ⇒ ρ̄E,3 is modular.
ρ̄E,3 reducible
For semistable E, ρ̄E,5 is irreducible (Mazur). Find auxiliary E′ with ρ̄E′,5≅ρ̄E,5 and ρ̄E′,3 irreducible. Prove E′ modular; transfer via shared mod-5 rep.

5.3 Taylor–Wiles Patching

To prove R=T, Wiles’s original Euler system approach had a gap. Taylor and Wiles replaced it with a patching argument.

Patching: inverse system of augmented problems

Q‪1
g primes
Q‪2
g primes
Q‪3
g primes
⋯
Q‪∅
empty
↓
↓
↓
↓
RQ‪1≅TQ‪1
RQ‪2≅TQ‪2
RQ‪3≅TQ‪3
⋯
R≅T ■
Each Qn = set of g auxiliary primes q≡1 (mod pn). Augmented problem RQn≅TQn holds (Selmer group trivialised). Inverse limit R∞≅T∞≅ℤp[[x1,…,xg]] ⇒ R≅T.

§6 Closing the Argument

Proof checklist

Assume aᵖ+bᵖ=cᵖ (prime p≥5, gcd=1, b even, a≡−1 mod 4).
Construct E: y²=x(x−aᵖ)(x+bᵖ). Check E is semistable with N_E=2·rad(abc).
Wiles–Taylor: E is semistable ⇒ E is modular ⇒ ρ̄_E,p is modular of level N_E.
Δ=2⁻⁸(abc)^2p contains each odd prime ℓ|abc to power 2p≥p, and ℓ||N_E.
Ribet ×ω(abc): strip each odd prime factor. ρ̄_E,p becomes modular of level 2.
Contradiction: S₂(Γ₀(2))=0 — no weight-2 newform of level 2 exists.
The assumption fails. FLT holds for all primes p≥5.
Combined with Fermat’s proof for n=4: FLT holds for all n≥3. ■

For every integer n≥3, the equation xⁿ+yⁿ=zⁿ has no solution in positive integers x, y, z. ■

∞

§7 Key Objects

Object	Definition	Key Property
Frey curve E	y²=x(x−aᵖ)(x+bᵖ)	Semistable; Δ=2⁻⁸(abc)²ᵖ
T_ℓ(E)	lim← E[ℓⁿ] ≅ ℤ_ℓ²	Free ℤ_ℓ-module, rank 2
ρ_E,ℓ	G_ℚ→GL₂(ℤ_ℓ)	ℓ-adic Galois representation
ρ̄_E,p	G_ℚ→GL₂(𝔽_p)	Serre conductor = 2
S₂(Γ₀(N))	Wt-2 cusp forms level N	S₂(Γ₀(2)) = 0
R (deformation ring)	Universal lift of ρ̄	Represents deformation functor
T (Hecke algebra)	End(S₂(Γ₀(N)))	R≅T ⇒ all lifts modular
Q_n (auxiliary primes)	Primes q≡1 (mod pⁿ)	Trivialise Selmer; enable patching