# Mathographics 2

further exercises from
Dixon, R. (1991). Mathographics (Dover ed.). New York: Dover Publications

2. False daisies with polygons. Use a hundred minimum polygons. Adjust K (spiral tightness) until the polygons are nearly touching.
2a. C= SQR (2)
2b. C = e = 2.718
2c. c= pi
2d. C=cube root 2
2e. C=5
2f. C=5/8

Fermat’s spiral,

$R=\sqrt{\theta}$

in polar coordinates where, R is the radial distance, theta is the angle from the starting horizontal line.
Polygons are placed on the spiral (not actually drawn) at degree intervals that Dixon calls Divergence and defines as

$D=360/C$

Dixon’s method (1980s) was to use a fixed polygon and experiment with a constant K to enlarge the spiral. We can use a polygon sub routine that allows variable size polygons in addition to varying the size of the spiral with a constant.

dixon says the “true daisy” is accomplished by a radial spacing of 360/1.618 or 222.49 degrees. 1.618 is the famous number,

phi $\phi$ or tau $\tau$

the beautiful and scholarly “The Algorithmic Beauty of Plants” by Prusinkiewicz and Lindenmayer uses the number 137.5 degrees which is 360/1.618^2

http://algorithmicbotany.org/papers/abop/abop.pdf

here are both numbers for comparison.