Beautiful mathematics, ten feet tall. A native Apple TV app that turns the living-room screen into an interactive mathematical exhibition — built in Swift and SwiftUI, driven entirely by the Siri Remote.
The opening exhibit is a deep exploration of the logistic map, xₙ₊₁ = r·xₙ(1−xₙ): the one-knob population model whose bifurcation diagram contains period-doubling cascades, universal constants, chaos, and infinitely many worlds within worlds.
Ten curated stops fly the camera through the diagram with cinematic zooms: the whole story, the period-doubling cascade, Feigenbaum universality, the edge of chaos at r∞, band mergers and veils, the period-5 and period-3 windows, the self-similar microcosm, the interior crisis, and solvable chaos at r = 4. Each stop pairs the picture with the mathematics that explains it. Press ⏯ for an ambient auto-tour.
Free-flight over the (r, x) plane: pan with the remote, zoom to ×10⁹ magnification (Double-precision limited), with a live readout of the crosshair's parameter, state, local Lyapunov exponent, and magnification — plus a minimap and toggleable veil curves, the forward images of the critical point x = ½ that trace every band edge and caustic in the chaos.
A live cobweb plot with an animated staircase, fixed-point stability markers, a time-series strip, and a field guide of hand-picked parameters — watch intermittency breathe just below the period-3 tangency, or full chaos at r = 4.
Every bifurcation view is backed by a progressive CPU density renderer (700 → 14 000+ samples per column, antialiased in both axes, with an attracting-cycle shortcut that makes periodic regions essentially free) tone-mapped through a log/percentile transfer curve into the "Ember" palette. An aligned Lyapunov strip under the plot shows λ(r) — order below zero, chaos above — with cusps at every bifurcation. Landmark annotations appear progressively as you zoom.
Every constant displayed on screen is re-derived from its defining equation by the test suite (Packages/TVMathKit/Tests/LogisticCoreTests). If a number on screen is wrong, CI fails.
| Value | r | Method |
|---|---|---|
| First period doubling | 3 (exact) | 2-var Newton on cycle + multiplier = −1 |
| Second doubling | 1+√6 ≈ 3.449490 | exact form, Newton-checked |
| 4→8, 8→16, 16→32 | 3.5440904, 3.5644073, 3.5687594 | Newton |
| Feigenbaum point r∞ | 3.5699456719 | literature (OEIS A098587), cross-checked by δ-extrapolation of the cascade |
| Feigenbaum δ, α | 4.6692016, 2.5029079 | OEIS A006890/A006891; δ re-measured from the computed cascade |
| Band merges 4→2, 2→1 | 3.5925722, 3.6785735 | Newton on Misiurewicz conditions (f⁵(½) hits the 2-cycle; f³(½) hits the fixed point) |
| Period-6 / period-5 windows | 3.6265532, 3.7381724 | bisection on attracting-cycle onset |
| Period-3 window | 1+√8 ≈ 3.828427 (exact) | tangency verified numerically on both sides |
| Superstable 2- and 3-cycles | 1+√5, 3.8318741 | Newton on fᵖ(½) = ½ |
| Interior crisis | ≈ 3.8568007 | bisection on attractor width |
| λ at r = 4 | ln 2 | measured against the sin²(2ⁿπθ) conjugacy |
| Mode | ◀ ▶ ▲ ▼ | Select | ⏯ | Menu |
|---|---|---|---|---|
| Tour | ◀▶ stops · ▲ toolbar · ▼ orbit lab (linked stops) | take the controls (explore) | auto-tour | toolbar / back |
| Explore | pan | zoom in ×2 | zoom out ×2 | toolbar / back |
| Orbit lab | ◀▶ tune r · ▲ toolbar · ▼ field guide | cycle speed | pause | toolbar / back |
Requirements: Xcode 16+ with the tvOS SDK (developed against Xcode 26 / tvOS 26.5).
open TVMath.xcodeproj # select an Apple TV simulator and Runor from the command line:
xcodebuild -project TVMath.xcodeproj -scheme TVMath -configuration Release \
-destination 'platform=tvOS Simulator,name=Apple TV 4K (3rd generation) (at 1080p)' buildUse Release when judging visuals — the renderer is ~10× faster than Debug.
Deep links for development and screenshots:
xcrun simctl launch booted dev.jlanej.tvmath -mode tour -poi 6
xcrun simctl launch booted dev.jlanej.tvmath -mode explore -viewport 3.7,3.9,0.1,0.98
xcrun simctl launch booted dev.jlanej.tvmath -mode cobweb -cobweb-r 3.8275On a real Apple TV: open the project in Xcode, pair the device (Xcode ▸ Window ▸ Devices and Simulators, with the Apple TV in Settings ▸ Remotes and Devices ▸ Remote App and Devices), set your development team on the TVMath target, and Run.
# Math core: dynamics, renderer, and constant-verification (runs natively on macOS)
swift test -c release --package-path Packages/TVMathKit
# Interaction grammar: simulated Siri Remote driving the real app
xcodebuild test -project TVMath.xcodeproj -scheme TVMath \
-destination 'platform=tvOS Simulator,name=Apple TV 4K (3rd generation) (at 1080p)'TVMath.xcodeproj thin shell (app target + UI-test target)
App/ @main entry, asset catalog (icon is generated — see below)
UITests/ XCUIRemote walkthrough tests
Packages/TVMathKit/
Sources/LogisticCore/ pure math: map dynamics, Newton/bisection solvers,
verified constants, progressive density renderer,
colormaps, critical curves, viewport algebra
Sources/TVMathUI/ tvOS SwiftUI: home gallery, experience shell,
bifurcation stage + overlays, tour content,
explore HUD, orbit lab (all `#if os(tvOS)`)
Sources/assetforge/ macOS tool that renders the layered tvOS app icon
and top-shelf art from the real renderer:
swift run -c release --package-path Packages/TVMathKit assetforge .
Tests/LogisticCoreTests/
Almost everything lives in the Swift package, so the hand-written project.pbxproj never needs to change as the app grows. The rendering pipeline is progressive and cancellable: the camera flies by re-projecting the last image on the GPU while background passes refine the density field, so navigation stays at 60 fps.
The lobby already reserves spots. To add one: create Sources/TVMathUI/<Exhibit>/ with an experience view, add a Route case in RootView, and swap its "in the workshop" card for a featured card. LogisticCore's viewport algebra, colormaps, and progressive-render scaffolding are exhibit-agnostic.
On the bench, in order of affection:
- The Mandelbrot Set — the natural sequel: the logistic map is conjugate to z² + c on the real line, so this diagram is the Mandelbrot set's spine unfolded — its periodic windows are the mini-Mandelbrots on the real axis.
- The Lorenz Attractor — chaos in continuous time.
- Chladni Resonances — standing waves and the geography of sound.
MIT — see LICENSE.