Here is an explanation for the appearance of Golden Angles in Nature.

“The seeds of a sunflower, the spines of a cactus, and the bracts of a pine cone all grow in whirling spiral patterns. Remarkable for their complexity and beauty, they also show consistent mathematical patterns that scientists have been striving to understand.”

The primary pattern is  Fibonacci Numbers. Other patterns like Lucas’ Numbers also appear. Fibonacci is a series with a very simple rule – each number is the sum of the two previous numbers. 1,1,2,3,5,8,13,21,34,55,89,144…
This series forms “Golden Rectangles” that can be subdivided into squares. seen here. Inside the square you can form a spiral with the golden angle of 137.5º.

This would be nothing more than a brainteaser except that it appears in the natural world. The sunflower has 21 clockwise and 34 counterclockwise spirals. So why is this the case? It is unlikely to be a design resulting from evolutionary adaptation of plants.

“Two physicists, Stéphane Douady and Yves Couder from the Laboratory for Statistical Physics in Paris, performed a compelling experiment in 1992 that tied these ideas together. They dropped magnetized drops of ferrofluid into a dish that was magnetized at its edge and filled with silicone oil. The droplets were simultaneously attracted to the edge of the dish and repelled from one another.

When the team dropped the oil in slowly, the droplets moved directly away from each other. But when they increased the speed, two older droplets would repel the new droplet simultaneously. So instead of simply marching to one side or the other, the droplet would move in a third direction—at the golden angle from the line connecting the drop’s landing point with the previous droplet. The resulting pattern formed spirals.”

Here’s the video of the experiment
Forces, like magnetism for metals or biochemistry for plants, push objects outward and downward, forming spirals. Fascinating stuff.

Fibonacci Numbers also play a role in fractal geometry. It can be seen in Mandelbrot’s set for instance.