Phi: A whole H of a lot cooler than Pi

Warning: This post contains no chemistry. Now with that off my chest I wanted to share something that I just discovered all on my own (That's not to say it wasn't known before I discovered it it just means that no one showed it to me.)

I want to make an original tessellation like M.C. Escher's. I then started sketching different regular polygons that can tessellate a plane. When I got to the pentagon I knew that it couldn't tessellate alone but I didn't have a good idea of what else I needed to make it work. So I started drawing pentagons. The first thing I drew looked like this:

I thought "Oh, it makes another pentagon. I could turn this into a fractal" and so I did and I made this:

I then wanted to know what fraction of the area of the large pentagon is taken up by the smaller pentagons. My train of thought went something like this: To do this I needed to find the length of the side of the new large pentagon. That's 2 times the first pentagon plus the space in between. The space in between is the leg of a triangle. The angle of the triangle is 36o so half of it is 18o. So sin 18 is 0.309, double it so that's 0.618. WAIT! If I add that to the one of the pentagon's sides that is 1.618 that is the Golden Ratio!
So, after looking on line all of these things were known before I started but I had fun discovering them.


10binary said...

Did you draw those pictures in inkscape or did you use another program?

David said...

I drew the images in CorelDRAW. Though Inkscape would have worked.