ZenoRogue | Non-Euclidean Sunflowers @ZenoRogue | Uploaded April 2020 | Updated October 2024, 1 hour ago.
A simulation of sunflower growth. Each new seed is pushed out from the center at a fixed angle α from the previous seed. Spiral patterns emerge; for the "best" values of α, if we count the spirals, we get Fibonacci numbers.

Since Fibonacci numbers grow exponentially, the number of rows to get each new Fibonacci number increases exponentially. However, this is because sunflowers are roughly flat. In the second part, we see what happens when we change the curvature of our sunflower.

The hyperbolic plane grows exponentially. In a hyperbolic sunflower, we only need to add a fixed number of rows to get the next Fibonacci number!
Many biological processes produce such negatively curved surfaces, e.g., kale or lettuce leaves, jellyfish tentacles.

We have been using surfaces of (approximately) constant negative curvature. For completeness, the last part shows the result of this simulation on a surface of constant positive curvature. This is a good way to distribute points on the sphere regularly.

Made with the HyperRogue engine: roguetemple.com/z/hyper -- don't watch videos, play HyperRogue or join the HyperRogue discord to understand non-Euclidean geometry!