Fractal Scene

Overview

The Fractal Scene is one of the most powerful features in MathArt, allowing you to explore and create stunning mathematical art. A fractal is a geometric shape with self-similarity, meaning no matter how much you zoom in, you'll see similar pattern structures. With this scene, you can easily create classic Mandelbrot sets, Julia sets, and many other fascinating fractal patterns.

[Screenshot placeholder: Displaying a beautiful Mandelbrot fractal example]

Quick Start

Creating Your First Fractal

1. Select Fractal Type: In the "Geometry" section of the left inspector panel, click the "Formula" dropdown menu and select the fractal type you want (such as Mandelbrot, Julia, etc.).

2. Adjust View: Use the mouse scroll wheel to zoom, and drag with the left mouse button to pan the view and explore fractal details.

3. Customize Colors: In the "Appearance" section, adjust the gradient colors to change the fractal's color presentation.

[Screenshot placeholder: Showing inspector panel geometry and appearance settings]

That's it! You've created your first fractal artwork. Now, let's dive deeper into each setting option.

Geometry Settings

Geometry settings are the core of the Fractal Scene, determining the basic form and mathematical characteristics of the fractal.

Formula Selection

MathArt provides a rich variety of fractal formulas for you to choose from:

Classic Fractals

• Mandelbrot: The most classic fractal pattern, displaying infinite details on the complex plane
• Julia: Closely related to Mandelbrot, each Julia set corresponds to a point in the Mandelbrot set

Advanced Fractals

• Newton: Based on Newton's root-finding algorithm, displaying the beautiful patterns of mathematical equation solving
• Nova: A variant of Newton fractal with unique symmetry and color expression
• Phoenix: Has a flame-like appearance with rich color layers
• Magnet: Includes Magnet I and Magnet II, shaped like magnetic attraction
• Lambda: Displays the complex structure of parameter space
• Complex Sin: Fractal based on complex sine functions
• Spiral Septagon: Unique fractal with spiral and septagon features

[Screenshot placeholder: Showing comparison of different formula fractal effects]

Variant Selection

Some fractal formulas support multiple variants:

• Mandelbrot Variant: Explore parameter space on the complex plane, each point corresponds to a Julia set
• Julia Variant: Uses fixed Julia seed values, displaying the Julia set pattern for that seed

When a formula supports both variants, you can switch between them using the "Variant" dropdown menu.

Parameter Adjustment

Each fractal formula has its specific parameters that directly affect the fractal's form:

Common Parameters

Iterations

• Controls the depth of fractal calculation
• Larger values produce more details but slower rendering speed
• Recommended range: 100-2000, default is usually 100-500
• 💡 Tip: Use lower iteration counts during exploration, then increase for more detail after finding a composition you like

Zoom

• Controls the magnification of the view
• Larger values reveal more details
• Can quickly adjust with mouse scroll wheel

Center X/Y

• Defines the center coordinates of the view
• Can be adjusted by mouse dragging

Formula-Specific Parameters

Mandelbrot/Julia Parameters

• Power: Controls the basic form of the fractal, default is 2 (classic form)
• Bailout: Threshold for determining if a point escapes, default is 4.0
• Start Point: Starting point for iteration, default is (0, 0)

Julia-Specific Parameters

• Julia Seed: Defines the form of the Julia set, a complex value
• 💡 Technique: Click on a point in the Mandelbrot set to quickly set that point as the Julia seed

Newton/Nova Parameters

• Exponent: Controls the power of the equation
• Relaxation: Affects convergence speed and pattern form

Phoenix Parameters

• Phoenix Param: Controls the intensity of the flame effect

Spiral Septagon Parameters

• Singularity Guard: Prevents numerical calculation anomalies

[Screenshot placeholder: Showing parameter panel and effects of different parameter values]

Appearance Settings

Appearance settings give you complete control over the fractal's visual presentation, from colors to coloring algorithms, creating unique artistic works.

Formula Display

You can choose to display the mathematical formula on the screen during rendering:

• Show Main Equation: Check this option to display the fractal's mathematical expression during rendering
• Position Adjustment: Set the X, Y coordinates for the formula
• Scale: Adjust the size of the formula text
• Color: Set the color of the formula text

💡 Tip: Formula display is perfect for creating educational videos or showcasing the mathematical beauty of fractals.

Gradient Control

Gradient (color mapping) is the core of fractal art, determining the fractal's color presentation. MathArt provides three gradient modes:

1. Palette Mode

The palette mode lets you precisely control each color stop:

[Screenshot placeholder: Showing palette editor interface]

• Color Stops: Support up to 16 color stops
• Add Color: Click the "+" button below the palette to add a new color
• Delete Color: Double-click a color stop to delete it (minimum 2 colors required)
• Adjust Color: Click a color block to open the color picker
• Adjust Position: Drag color stops to change their position in the gradient

💡 Tips:

• Use high-contrast colors to highlight fractal details
• Adjacent similar tones create smooth transition effects
• Try warm-cool color contrasts for stunning effects

2. Cosine Palette Mode

The cosine palette generates smooth gradient effects using mathematical formulas:

[Screenshot placeholder: Showing cosine palette settings]

• Color Cycle: Controls the frequency of color cycling
• RGB Channel Parameters: Control the three color channels separately
  • A (Offset): Base color value
  • B (Amplitude): Color variation amplitude
  • C (Frequency): Color variation frequency
  • D (Phase): Color variation starting phase

💡 Tip: Cosine palette is especially suitable for creating rainbow-like smooth color transitions.

3. Curve Palette Mode

The curve palette provides the most flexible color control:

[Screenshot placeholder: Showing curve palette editor]

The curve editor allows precise adjustment of color channels through control points:

• Add Control Point: Click on an empty area of the curve
• Delete Control Point: Double-click a control point (minimum 2 points required)
• Move Control Point: Drag control points to adjust curve shape
• Channel Switching: Edit red, green, and blue channels separately

💡 Tips:

• Use S-curves for smooth color transitions
• Try different curve shapes for different channels to create unique color effects

Coloring Algorithm

The coloring algorithm determines how iteration results are mapped to colors. MathArt provides multiple professional coloring algorithms:

[Screenshot placeholder: Showing comparison of different coloring algorithm effects]

Inside Coloring

Inside coloring applies to points inside the fractal set (points that don't escape):

• None: Use solid fill color
• Basic: Simple coloring based on iteration count
• Smooth Mandelbrot: Smooth color transitions, eliminates banding
• Decomposition: Angle-based coloring, shows complex phase information
• Binary Decomposition: Sign-based binary coloring
• Distance Estimator: Coloring based on distance to set boundary
• Direct Domain: Use complex values directly as colors
• Direct Orbit Trap: Coloring based on orbit traps
• Direct Normalized Z: Normalized complex value coloring

Outside Coloring

Outside coloring applies to points outside the fractal set (escaping points):

Provides the same algorithm options as inside coloring, but typically with different parameter settings.

💡 Tips:

• For Mandelbrot/Julia sets, outside coloring is usually more important than inside coloring
• Smooth Mandelbrot is the most commonly used outside coloring algorithm for smooth color transitions
• Distance Estimator is suitable for highlighting boundary details

Coloring Parameters

Each coloring algorithm has its specific parameters:

Common Parameters

Color Density

• Controls the frequency of color variation
• Larger values create more color cycles
• Recommended range: 0.1-10.0

Transfer Function

• Controls the mapping from iteration values to color indices
• Options:
  • None: Use raw values directly
  • Linear: Linear mapping
  • SQR: Square mapping, enhances high-value areas
  • SQRT: Square root mapping, enhances low-value areas
  • CUBE: Cube mapping, strongly enhances high-value areas
  • CUBEROOT: Cube root mapping, strongly enhances low-value areas
  • LOG: Logarithmic mapping, compresses high-value areas
  • EXP: Exponential mapping, expands high-value areas
  • SIN: Sine mapping, creates periodic variations
  • ARCTAN: Arctangent mapping, smooth transitions

💡 Tips:

• Using LOG transfer function reveals more details
• SQR and CUBE are good for highlighting main fractal structures
• SQRT and CUBEROOT are good for revealing fine details

Gradient Offset

• Adjusts the starting position of colors in the gradient
• Range: Any real number
• Used for fine-tuning color distribution

Repeat Gradient

• Checked: Colors cycle repeatedly
• Unchecked: Use solid color for out-of-range areas

Solid Color

• The color used for out-of-range areas when not repeating gradient
• Click the color block to choose a color

Algorithm-Specific Parameters

Basic Coloring

• Metric: Select the metric to use for coloring
  • ITERATION: Iteration count
  • REAL: Real part value
  • IMAGINARY: Imaginary part value
  • SUM: Sum of real and imaginary parts

Smooth Mandelbrot Coloring

• Exponent: Usually same as fractal's power
• Power: Affects smoothness
• Bailout: Should match the fractal's bailout setting

Binary Decomposition Coloring

• Decomposition Type:
  • TYPE_1: Based on real part sign
  • TYPE_2: Based on imaginary part sign

Distance Estimator Coloring

• Power: Should match the fractal's power

[Screenshot placeholder: Showing coloring parameter settings panel]

Timeline Animation

The Fractal Scene supports powerful animation features, allowing you to create dynamic fractal art.

Parameter Animation

You can create animations for various fractal parameters:

1. Open Timeline Panel: Click the "Timeline" tab at the bottom
2. Add Animation Clip: Drag a parameter animation from the animation library to the timeline
3. Set Keyframes: Set parameter values at different time points on the timeline
4. Preview Animation: Click the play button to preview the effect

[Screenshot placeholder: Showing timeline panel and animation settings]

Gradient Animation

Gradient animation makes colors change over time:

1. Add a "Gradient Animation" clip to the timeline
2. Set color configurations at different time points
3. MathArt will automatically smoothly interpolate between keyframes

Zoom Animation

Create stunning zoom animations:

1. Add a "Zoom Animation" clip
2. Set start and end zoom levels
3. Adjust animation curve to control zoom speed

💡 Tips:

• Using exponential zoom curves creates more natural zoom effects
• Combining parameter animations and zoom animations creates complex dynamic effects

Advanced Features

Rendering Methods

MathArt provides three rendering methods that balance quality and performance:

One Pass

• Renders full resolution in one go
• Suitable for final output or quick preview after parameter adjustment
• Longest render time but highest quality

Multi Pass

• Renders lower resolution first, then progressively increases
• Suitable for interactive exploration, quick preview
• Balances responsiveness and render quality

Guessing

• Uses smart algorithm to predict details
• Fastest rendering method
• Suitable for quick preview but may lose some details

💡 Tip: Use Guessing or Multi Pass during exploration, switch to One Pass for final output to get the best quality.

Perturbation Algorithm

Perturbation algorithm is an advanced rendering technique allowing ultra-high precision fractal rendering:

• Precision Level:
  • Low: Fast rendering, lower precision
  • Medium: Balanced precision and speed
  • High: High precision rendering
  • Ultra: Highest precision, suitable for deep zooming

💡 Note: Perturbation algorithm requires more computational resources, only enable when ultra-high precision is needed.

Palette Space

Choose the color space for color interpolation:

• RGB: Interpolate in RGB color space, direct color transitions
• HSV: Interpolate in HSV color space, more natural color transitions

💡 Tip: HSV space typically produces more harmonious color transitions.

Operation Tips

Mouse Operations

• Scroll Wheel: Zoom view
• Left Click + Drag: Pan view
• Double Click: Reset view to default position

Keyboard Shortcuts

• Space: Play/Pause animation
• R: Reset view
• S: Save current settings

Performance Optimization Suggestions

1. Iterations: Use lower values during exploration, increase for final rendering
2. Rendering Method: Use Multi Pass for interaction, One Pass for output
3. Resolution: Lower preview resolution, increase for final output
4. Coloring Algorithm: Simpler algorithms render faster

FAQ

Q: Why does my fractal look blurry?

A: Possibly too few iterations or too high zoom level. Try increasing iterations or using perturbation algorithm for higher precision.

Q: How to create smooth color transitions?

A: Use Smooth Mandelbrot coloring algorithm and choose LOG or SQRT transfer function. Also, using HSV palette space produces more natural color transitions.

Q: What if fractal rendering is too slow?

A:

1. Reduce iteration count
2. Use Guessing or Multi Pass rendering method
3. Lower preview resolution
4. Simplify coloring algorithm

Q: How to find interesting fractal details?

A:

1. Start exploring from the Mandelbrot set
2. Zoom into boundary areas
3. Look for spirals, branches and other structures
4. Use different coloring algorithms to highlight details

Creative Inspiration

Classic Works

1. Mandelbrot Deep Zoom: Explore infinite details of the Mandelbrot set
2. Julia Set Gallery: Try different Julia seed values
3. Newton Fractal: Explore the beauty of mathematical equation solving

Artistic Creation

1. Color Experiments: Use different gradients and coloring algorithms
2. Animation Creation: Combine parameter animations and zoom animations
3. Formula Combinations: Try different fractal formulas and parameter combinations

[Screenshot placeholder: Showing examples of excellent fractal art]

Next Steps

• Explore other scene types: Parametric Surface, Implicit Surface, Strange Attractor, Perlin Noise
• Learn Core Features for more advanced techniques
• Check Interface Guide to familiarize yourself with all panels and tools

---

💡 Tip: Fractal art is a field that requires patience and exploration spirit. Don't be afraid to try different parameter combinations—often the most unexpected settings produce the most stunning results!