Pi and phi (both irrational numbers) visualized as turn ratios. Also compare e (another irrational number). Select a turn angle ratio. Select pi, or phi (golden ratio). Spacebar to pause. Phi fills space most efficiently and has inherent Fibonacci spirals. Pi spirals around, but is less space filling, because it isn't as irrational a number as Phi. Using seven colors shows the closest denominator approximating pi is seven (22/7 is an approximation to Pi). With Phi, the spirals number as Fibonacci numbers. The ratio of clockwise to clockwise spirals roughly approximate Phi. There are many ways to show off the spirals with colors. If your turn ratio has a whole number N in the denominator, the clones will go out in N straight arms. But pi, golden angle and e, don't have a whole number in the denominator, so what will happen? We also compare 22/7, commonly used to approximate π.
A turn of "1" is just a full 360 degree turn, 0.25 is 90 degrees, etc. Buttons select irrational numbers π, φ or e (pi, golden ratio, or e). With the colors, notice the interesting spiral patterns you can find with Phi. Count of clockwise and counterclockwise spirals both are Fibonacci numbers and their ratio approximates phi. But what patterns can you find with pi (none). Yet even pi spirals around filling space a bit (easier to see in turbowarp, allowing infinite clones, needed with smaller step). With pi, seven colors work with the seven spiral arms, because the closest denominator is 22/7. Clones move out from center, each new clone turns the amount you specified, relative to the previous clone. If you set it to 0.2 (1/4 a turn) or 0.75 (3/4) and four arms result. Pick 1/8 or 5/8 of a turn, eight arms result. 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. This comparison of π vs φ shows that that φ (golden ratio) is more irrational. Clones making golden ratio turns always find a new place in between others clones, 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. 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. For better resolution, use turbowarp with step set to 0.1 https://turbowarp.org/544454370/fullscreen?clones=Infinity
