Dork Scratchings

It occurred to me the other day while looking at a set of LED traffic lights that they could be improved quite easily.  The layout of the LEDs in all the ones I've seen seems to favour some radial directions over others and/or result in a higher concentration of bulbs at some radii than at others. My solution involves placing LEDs a increasing distances from the centre and as each LED is placed rotating by the golden angle .  This ensures that no radial direction is preferred over any other.  In addition to this, setting $r_n = \sqrt{n}$ ensures that each bulb covers a roughly similar area. By "area covered" I am referring to the portion of space which is closer to that bulb than to any other.  This is the Voronoi cell generated by a point belonging to a set of points.  The picture above shows the Voronoi cells generated by a set of LEDs placed in the manner described, and the code below shows how I obtained it.  I think it would make a pretty stain glass

This post will attempt to explain a strange phenomenon in nature.  If you look into a sunflower, daisy, cactus or fir cone you always see a spiral pattern like the one shown above.  If you look carefully you in fact see two: one spiralling clockwise and the other spiralling anti-clockwise.  And, bizarrely, if you count the number of spiral arms of each you always find they form a consecutive pair from the Fibonacci sequence. To recap, the Fibonacci sequence is the sequence you get if you start off with $F_1 =1$ and $F_2 = 1$ and then iterate using $F_{n+2} = F_{n+1}+F_n$.  So: $$ (F_n) = 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,... $$ This all seems highly contrived, but somehow turns up in seed and leaf patterns.  How? The answer has to do with Phyllotaxis.  This is the manner in which new leaves or seeds are formed.  Imagine a long stem plant growing upwards and generating new leaves as it grows.  The tip of the plant is known as the bud and embryonic leaves known as primordia

The number $\phi = \frac{1+\sqrt{5}}{2}$, known as the Golden Ratio , turns up in a lot of places.  This one that I read about in National Geographic surprised me.  The following coordinates describe an icosahedron: $$ (\phi,1,0), (-\phi,1,0), (-\phi,-1,0), (\phi,-1,0) \\ (0,\phi,1), (0,-\phi,1), (0,-\phi,-1), (0,\phi,-1) \\ (1,0,\phi), (1,0,-\phi), (-1,0,-\phi), (-1,0,\phi) \\ $$I thought I'd have a go at figuring out why, and the above picture is the result.  All you have to do is observe that the vertices of an icosahedron are the same as those of a set of three identical intersecting rectangles.  The proof then reduces to showing that the ratio of the long side to the short side is $\phi$ - i.e. that these are golden rectangles.  The proof above takes the short side to be of length 1 and the long side to be of length $x$. So, why does $\phi$ turn up so often in maths and geometry?  I think the answer is the same as the answer to the question "why does $\frac{1}{2}$