Spirograph
(2 · cos(u) +(4 + cos(t)) · cos(2u/3), 0, 2 · sin(u) −(4 + cos(t)) · sin(2u/3))
Open in the app The dials and keys named below are the app's.
What it draws
The second expression is 0, the height, so the whole figure lies flat on the ground plane and Top shows it as a drawing on paper. Sideways and in depth the point is a sum of two turning arms. One has length 2 and goes round once as u advances by 2π. The other has length 4 + cos(t) and goes round two thirds as fast in the opposite sense, which is what the minus sign in the third expression marks.
That is a hypotrochoid: the track of a pen fixed in a circle of radius 3 rolling inside a fixed circle of radius 5, with the pen 4 + cos(t) from the rolling circle's centre. The two rates only agree again after u has covered 6π, which is why Turns of u opens at 3. Advancing u by 6π/5 turns the figure onto itself, so it has five lobes.
What the clock does
Only the pen distance carries t. It breathes between 3 and 5 and back every 2π ≈ 6.28 seconds at Speed 1, and the figure swells with it. The outer tips reach 7 from the centre at the widest and 5 at the narrowest. The pen sits outside the rolling circle for most of that swing, so each of the five tips carries a small loop. At the narrowest moment the pen distance equals the rolling radius, 3, the two speeds cancel exactly, and the five loops close into five cusps on the circle of radius 5.
History
Curves like this were studied long before the toy, but the name is the toy's. The British engineer Denys Fisher built a drawing set of toothed rings and wheels and showed it in 1965. Kenner sold it in the United States from 1966, and the gear ratios of its wheels are the ratios in this formula.
Try
- Top: straight down on the flat figure, the view the toy gives.
- Replace 4 + cos(t) with 3 in both expressions. The breathing stops and the five cusps stay, the case where the pen sits on the rim of the rolling circle.
- Change both 2u/3 to 3u/4, with Turns of u at 4 so the path still closes: seven lobes instead of five.
- Turn on trace to follow the pen around the five lobes while the figure breathes.