Rolling Polygons Mandalas

Exploring Geometric Mandalas

My latest software art project is an interactive Observable notebook exploring beautiful geometric patterns. All of these mandalas are created by rolling smaller circles around the inside of a larger circle and connecting points within each of the smaller circles, as well as related points across the smaller circles.

Try the interactive Observable Notebook here!

How It Works

The mathematical structure relies on three main concepts:

• Vertices: The total number of points in the overall structure.
• Star Density: A value determining the shape, which must be less than the number of vertices.
• Polygons: The number of smaller circles is (Vertices - Star Density). Inside each smaller circle, there is a polygon with Star Density sides. The intra-circle polygons themselves have (Vertices - Star Density) sides.

The notebook provides interactive sliders to adjust these properties and checkboxes to toggle various elements on or off. Experiment. Have fun. Relax. Enjoy.

Rainbow Color Animations

The notebook includes detailed controls for intra-circle and inter-circle polygon color animations in "Rainbow" mode.

The red, green, and blue components of each color are controlled independently. For each component, you can pick a minimum percentage value and a maximum percentage value. The component then follows a sine wave oscillating between the minimum and maximum values.

You can control the period of the sine wave separately for each color component. Making the periods relatively prime (like 9, 11, and 13 seconds) will significantly increase the variety of colors generated over time.

Credits

• The original, and brilliant, animation code was written by Kieran Clancy and can be found here. It was adapted for Observable and enhanced with the addition of color animations by me.
• This project was inspired by a fascinating video by Mathologer (Burkard Polster). Follow him on YouTube—he is brilliant!