More than any other book that I know of, Theodore Andrea Cook's The Curves of Life shows the extent of our fascination with spirals. First published in 1914, it is an odd blend of 19th-century natural history, amateur mathematics, and art history. On the mathematics of spirals, it is not the best source, Conway and Guy's The Book of Numbers has a better overview on spirals in plants, but it is unmatched as a compendium of all things spiral.
I was thinking about the allure of spirals while I finally got around to attempting some better renderings of spirals from earlier posts. The older pictures in this blog were made with Fathom, which worked well, but these drawn using Processing look a bit nicer I think, and the code is easier to play with.
The spiral below is a quadratic spiral displaying the triangular and hexagonal numbers, originally from this post.
This other spiral is a phyllotaxis spiral like the ones described here. The picture at the top of the post is based on the one below - with edges between points shown instead of the points themselves.
For what it's worth, the Processing code for these and other similar spirals is here. If Processing isn't your thing, you can find Mathematica and Python versions of polygonal-numbers-on-quadratic-spirals at Walking Randomly.