Mandelbrot

An interactive Mandelbrot set explorer for iOS, built with Metal compute shaders.

Features

• Real-time rendering using Metal compute shaders
• Pinch-to-zoom with focal point preservation (zoom centers on your pinch location)
• Pan gestures to explore the fractal
• Deep zoom support via perturbation theory, enabling zooms beyond 10^14
• Adaptive quality - lower resolution during gestures for smooth interaction, full quality on release
• Dynamic iteration scaling - automatically increases iterations at deeper zoom levels
• Two color modes: HSV gradient and palette-based coloring

Rendering Modes

Standard Mode

Used at shallow zoom levels (scale > 10^-6). Computes the Mandelbrot iteration directly using single-precision floats on the GPU. Includes optimizations:

• Cardioid and period-2 bulb early-out checks
• Periodicity detection to identify interior points quickly
• Smooth iteration coloring with continuous potential

Perturbation Mode

Automatically activates at deep zoom levels (scale < 10^-6) where single-precision floats lose accuracy. Uses perturbation theory:

1. Computes a reference orbit at the screen center using Double precision on the CPU
2. For each pixel, computes only the perturbation delta from the reference using the recurrence:

   δₙ₊₁ = 2·zₙ·δₙ + δₙ² + δc
   

3. Since deltas remain small, Float precision suffices for GPU computation

This extends the zoom range from ~10^7 (float limit) to ~10^14 (double limit).

UI Indicators

The top bar displays:

• Zoom depth: Shows magnification as "10^X" at deep zooms or "Nx" at shallow zooms
• Rendering mode: Blue dot for Standard, green dot for Perturbation
• Color mode picker: Toggle between HSV and Palette coloring

Interesting Locations to Explore

Location	Coordinates	Suggested Zoom
Seahorse Valley	(-0.743643887037151, 0.131825904205330)	10^-10
Elephant Valley	(0.250006, 0.0)	10^-8
Main cardioid cusp	(-0.75, 0.0)	10^-3
Period-3 bulb	(-0.125, 0.75)	10^-2

Technical Details

Dependencies

• swift-numerics - Provides Complex<Double> for reference orbit computation

Architecture

• MandelbrotView.swift - SwiftUI/UIKit bridge with gesture handling
• Renderer.swift - Metal rendering pipeline with dual-mode support
• ReferenceOrbit.swift - CPU-side reference orbit computation
• MandelbrotKernel.metal - Standard GPU shader
• MandelbrotPerturbation.metal - Perturbation theory GPU shader

Requirements

• iOS 18.0+
• Device with Metal support

Building

Open Mandelbrot.xcodeproj in Xcode and build for an iOS device or simulator.

Background

This project was created to learn shader programming in Metal, and also made for my kids to explore the beauty of fractals.