Creating Dynamic Patterns with Complex Functions in R

In this session, we’ll explore how complex functions can be harnessed — first through a mathematical lens, then through the lens of programming in R — to generate visually stunning and intricate patterns. By leveraging R’s capabilities, we demonstrate how concepts like symmetry, frequency, and phase can be translated into dynamic, animated visuals.
Key Mathematical & Theoretical Aspects:

1. Complex Functions as Pattern Generators

• Understanding mappings and how real & imaginary parts interact to create geometry.
• Exploring transformations, rotations, and reflections emerging from complex operations.

2. Symmetry, Frequency, and Phase

• Mathematical representation of symmetry in periodic and rotational systems.
• How frequency and phase modulation shape wave-based or iterative visual forms.
• Linking mathematical periodicity to visual repetition and motion.

3. Iterative & Parametric Visualization

• Iterated transformations and their role in creating self-similar or fractal-like structures.
• Parametric curves and surfaces derived from complex expressions.

4. Mathematics Meets Computation

• Translating analytic expressions into data for visual rendering.
• Numerical evaluation, discretization, and precision control in R.

R Programming Lens — Packages & Libraries We’ll Explore:

| Package / Library | Purpose & Role in the Talk |
| ----------------- | -------------------------- |
| tidyverse | Data manipulation, transformation, and visualization foundation. |
| ggplot2 + ggforce | High-level plotting tools for smooth geometric and parametric rendering. |
| gganimate | Animate mathematical transformations (phase shifts, rotations, iterations). |
| rgl / scatterplot3d | 3D visualization of complex mappings or parametric surfaces. |
| magick | Image manipulation and post-processing of frames. |
| gifski | Exporting smooth, high-quality GIF animations. |
| mathart (Marcus Volz) | A creative toolbox for mathematical art and generative geometry. |
| Rcpp | Accelerating computation for iterative or high-resolution mathematical data generation. |

What You’ll Take Away:

• A theoretical grasp of how complex-valued functions yield structured visual symmetries.
• Practical experience in implementing mathematical ideas using R’s visualization and animation tools.
• An understanding of how to use R as a medium for scientific art, blending computation, mathematics, and creativity.
• Inspiration to extend these concepts toward fractal generation, dynamic systems, and generative aesthetics.