Starts at angle 180 degrees then, as angle steadily decreases, approximates a parabolic spiral. Then, rational number angles yield straight arms, with a count equal to the fraction's denominator, e.g. 160°/360° = 4/9, so yields 9 straight arms. There are far more irrational number angles (no whole number in denominator). Those yield curved arms. The golden angle is an irrational ratio which most evenly fills the space with dots. Spirals continue from irrational to rational angles, and from clockwise to counterclockwise.
Places dots starting from the center and turning at "angle" as distance from center increases. The angle gradually decreases. Angle starts at 180° which yields two arms, but then when angle decreases to just under 180° the arms curve, then spiral around. at first approximating a parabolic spiral (Fermat's spiral). Then we have a dance between various rational and irrational dance. When angle is 90° (1/4 a turn) , four arms result. 1/8 (or 3/8, 5/8, or 7/8) of a turn, eight arms result. A 10° turn results in 36 arms (360° divided by 10°). Those are rational numbers. Irrational numbers, like pi or the golden ratio, have no whole number denominator, so you don't get straight arms. However, pi is closely approximated by 22/7, so 7 arms result, albeit curved. But the golden ratio isn't so closely approximated by any one or two small whole number fractions. φ (golden ratio angle) is more irrational than π (pi). When dots are placed at golden ratio turns, they always find a new place in between others, just as the golden ratio falls in between ratios of successive Fibonacci numbers. More irrational than π but less irrational than φ is the number e, also called Euler's number, which distributes dots fairly evenly, but not as well as the golden angle. We compare also 22/7 because it is often used as a rough approximation to π. Notice it results in 7 perfectly straight arms rather than the 7 curved arms resulting from π. Those are just the famous irrational numbers. There are actually more irrational numbers than there are rational. That's why we see so many spirals rather than straight armed stars. In sum, the golden ratio translated into degrees (. 137.50776405.... degrees), most evenly spaces the maximal dots and without overlap. This "golden angle" is the best way to fill the space, and causes the number of clockwise and counterclockwise spirals to be Fibonacci numbers (such as 5, 8, 13, 21, 34, 55, 89, 144....). Sunflowers, daisies and other plants with numerous little florets on the flowerhead have golden angle spiral patterns. The pen dots represent the florets. But here the pen size is very small, much smaller than the seeds relative to the total circular head, accommodating more spirals. Modifies my earlier projects by changing angle automatically, as well as changing number of colors. Music, copyright free: Bengo Latino - Jimmy Fontanez
