Each firefly follows a circular path, but also listens to a shared rhythm. When you tap the screen, you move the center they are dancing around.
This behavior comes from angles, circles, and repeating motion. Fireflies, waves, music, and even planets use similar patterns.
Each moving point leaves behind a pink trail showing where it has actually traveled. This is not a prediction — it is history.
In mathematics, this is called an integral of motion. Position is built by adding up many tiny changes over time.
When several paths begin to align, collective structure appears without any central controller.
These are polygons: shapes made from straight edges and corners. Tapping the screen changes how many sides the polygon has.
The shapes gently pulse and twist because their size and angle are changing over time. This comes from angles, rotation, and waves.
By stacking many similar polygons, small changes turn into rich motion. Simple rules create complex patterns.