Visualizations
Every interactive built for this project, in lesson order. Each one lives inside its lesson — commissioned right after I learned the theorem it illustrates, to make that specific idea tangible. This page holds all of them, in order.
Grid Deformation Under f(z) = z²
Watch a square grid deform under z². Vertical and horizontal lines map to two families of parabolas that stay perpendicular everywhere — except at the origin, where f′(0) = 0 and conformality visibly breaks down.
Open interactive → Lesson 0The Big Picture: Domain Coloring & Conformal Maps
A complex function maps a plane to a plane — a literal graph would need 4 dimensions. This opener introduces the two substitute pictures used all through the subject: domain coloring (color each input by the angle of its output) and side-by-side input/output planes.
Open interactive → Lesson 1Open Sets Explorer
The topology vocabulary of ℂ, made draggable: test points with ε-disks to see what interior, boundary, open, and closed actually mean — the definitions every later theorem quietly relies on.
Open interactive → Lesson 1Roots of Unity
The n solutions of zⁿ = 1 sit at the vertices of a regular n-gon on the unit circle. Explore how polar form turns multiplication into rotate-and-scale, and why raising each root to the nth power walks it around the circle back to 1.
Open interactive → Lesson 2Conformality & the Jacobian
Why do holomorphic maps preserve angles? Zoom in at a point and see the Jacobian act as a pure rotation-plus-scaling when the Cauchy-Riemann equations hold — the geometric content of complex differentiability.
Open interactive → Lesson 3The Complex Logarithm & Its Branch Cut
log z is multivalued: every loop around the origin adds 2πi. Explore the spiral of values, choose a branch, and see exactly what a branch cut removes to force a single-valued function.
Open interactive → Lesson 3Geometric Series: Partial Sums in the Plane
Drag z and watch the partial sums of Σzⁿ trace a path: a spiral forced inward to 1/(1−z) when |z| < 1, an unbounded outward spiral when |z| > 1, and endless circling right on the boundary. An ε-disk around the limit makes the ε–N definition of convergence literal — find the N past which the path never leaves.
Open interactive → Lesson 4Contour Deformation & the Cauchy Integral Formula
Two panels: deform integration contours freely without changing the integral (as long as no singularity is crossed), then step through the actual derivation of the Cauchy Integral Formula, including the shrinking-circle limit that makes it work.
Open interactive → Lesson 5From CIF to the Fundamental Theorem of Algebra
A single proof chain, animated end to end: the Cauchy Integral Formula gives derivative formulas, which give Cauchy's estimates, which give Liouville's theorem, which proves that every nonconstant polynomial has a root. Each link pairs a visualization with a step-through proof. Lesson 5 embeds a shorter scrolly recap of this same chain (9 steps) — this standalone page is the long-form version.
Open interactive → Lesson 6Isolated Zeros — the Floor Argument
Why continuity forces g(z) away from 0 near z₀, so a zero of a holomorphic function can never have another zero arbitrarily close by. Drag z and watch the reverse-triangle-inequality floor hold live, synced to the general proof it's demonstrating.
Open interactive → Lesson 6Singularity Zoo — Domain Coloring
Three isolated singularities, same domain-coloring convention, one shared zoom slider: removable stays smooth, a pole brightens to a clean double-wound white center, and an essential singularity cycles through every color at every zoom level — Casorati-Weierstrass made visible.
Open interactive → Lesson 6Vary the Circle — Why R = 3, Exactly
Deriving, not asserting, that a power series' radius of convergence equals the distance to the nearest singularity. Grow a certifying circle toward the singularity for the R≥3 half; watch |f(z)| blow up approaching it for the R≤3 half.
Open interactive → Lesson 7The Crosscut Construction
Turning a region with holes into one connected boundary, so Cauchy's theorem applies. Shrink the crosscut's width to zero and watch the two opposite-direction traversals cancel exactly — an orientation-reversal identity, not a limiting argument.
Open interactive → Lesson 8Circles and Lines Under Inversion: the A↔D Swap
Every Möbius map sends circles and lines to circles and lines. Drag a line or circle in the z-plane and watch its image in the w-plane under w=1/z — a line missing the origin becomes a circle through it, and the algebra's A↔D coefficient swap plays out live below the panels.
Open interactive → Lesson 8Conformal Indistinguishability & the Riemann Mapping Theorem
The Riemann Mapping Theorem says every simply connected proper subset of ℂ is conformally the unit disk. Morph a lobed region — or a Koebe slit domain stretched past 25× the disk's radius — back to 𝔻 along a genuine closed-form conformal map, with a probe showing angles held at exactly 90° throughout. A third preset shows why ℂ itself is the one exception.
Open interactive → Lesson 8Möbius Maps on the Riemann Sphere
Every Möbius map is a rigid rotation of the Riemann sphere. Grab the sphere and spin it — the induced transformation plays out on the plane below, 1/z is a 180° flip, and lines and circles trade places because on the sphere they're all just circles. Scalings and translations are the maps that need a non-rigid flow instead — the reason the sphere's rotations are only a subgroup of all Möbius maps.
Open interactive → Lesson 9The Gamma Recursion Crawl
The integral defining Γ(s) converges only on Re(s) > 0. Step the recursion Γ(s) = Γ(s+1)/s and watch the domain extend one strip left at a time, each step putting a new pole at the next non-positive integer. Analytic continuation as something built rather than asserted.
Open interactive → Lesson 11Stirling's Formula, Two Routes
Laplace's method on the real line beside steepest descent in the complex plane, sharing one n slider. The real integrand has a peak; the complex integrand has a saddle. Swing the contour off the descent direction and the decaying bump turns into growth while the integral's value stays put.
Open interactive →