Streamlit Applications¶

This work is licensed under a Creative Commons Attribution 4.0 International License.

Streamlit applications provide Python-based interactive tools for exploring fractals and computing fractal dimensions. These applications are ideal for scientific exploration and analysis.

Running Streamlit Apps¶

Quick Start¶

cd apps
pip install -r requirements.txt
streamlit run mandelbrot.py

Available Applications¶

Application	File	Description
Mandelbrot Explorer	`mandelbrot.py`	Interactive Mandelbrot set with zoom
Julia Set Generator	`julia.py`	Julia set visualization with parameter control
Branching Tree	`branching_tree.py`	Self-affine branching structure
Pythagoras Tree	`pythagoras_tree.py`	Classic Pythagoras tree fractal

Mandelbrot Explorer¶

Interactive exploration of the Mandelbrot set with:

Zoom controls: Click to zoom into regions of interest
Iteration depth: Adjust maximum iterations for detail
Color schemes: Multiple coloring algorithms
Export: Save high-resolution images

Usage¶

streamlit run apps/mandelbrot.py

Mathematical Background¶

The Mandelbrot set consists of complex numbers \( c \) where the sequence:

\[ z_0 = 0, \quad z_{n+1} = z_n^2 + c \]

remains bounded (i.e., \( |z_n| < 2 \) for all \( n \)).

Julia Set Generator¶

Generate Julia sets for any complex parameter \( c \).

Features¶

Parameter selection: Slider controls for real and imaginary parts of \( c \)
Connected vs disconnected: Visualize how the Julia set changes
Escape time coloring: Multiple algorithms
Animation: Animate through parameter space

Mathematical Background¶

For a fixed \( c \), the Julia set is the boundary of points \( z_0 \) for which:

\[ z_{n+1} = z_n^2 + c \]

remains bounded.

Branching Tree¶

Generate self-affine branching structures that model plant vascular networks.

Parameters¶

Parameter	Description	Range
Branching angle	Angle between child branches	0°-90°
Length ratio	Ratio of child to parent length	0.5-0.9
Width ratio	Ratio of child to parent width	0.5-0.9
Iterations	Number of branching levels	1-15

Self-Affinity¶

When length ratio ≠ width ratio, the tree exhibits self-affine scaling:

\[ \gamma = \frac{l_{k+1}}{l_k} \neq \xi = \frac{r_{k+1}}{r_k} \]

This matches the predictions of Metabolic Scaling Theory.

Pythagoras Tree¶

The classic Pythagoras tree fractal using self-similar squares.

Features¶

Angle control: Adjust the branching angle (default 45°)
Depth control: Number of iterations
Symmetric vs asymmetric: Toggle symmetric branching
Color by depth: Visual hierarchy

Construction¶

Each iteration: 1. Draw a square on each leaf 2. Construct two smaller squares forming a right triangle 3. Repeat

With angle \( \theta \), the scaling factor is \( r = \cos\theta \).

Development Guide¶

Creating New Streamlit Apps¶

Create a new .py file in apps/:

import streamlit as st
import numpy as np
import matplotlib.pyplot as plt

st.title("My Fractal App")

# Parameters
iterations = st.slider("Iterations", 1, 100, 50)

# Computation
# ... your fractal algorithm ...

# Display
fig, ax = plt.subplots()
ax.imshow(result)
st.pyplot(fig)

Run with streamlit run apps/your_app.py

Dependencies¶

Key packages used:

streamlit: Web interface
numpy: Numerical computation
numba: JIT compilation for speed
matplotlib: Static visualization
plotly: Interactive visualization
scipy: Scientific algorithms
pillow: Image processing

Performance Tips¶

Use @st.cache_data for expensive computations
Use numba JIT for numerical loops
Consider WebGL alternatives for real-time interaction

Docker Deployment¶

For production deployment:

cd docker
docker build -t streamlit-fractals .
docker run -p 8501:8501 streamlit-fractals

The Dockerfile includes all dependencies and exposes port 8501.