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Spirograph

The Spirograph is a geometric drawing toy that produces mathematical curves (hypotrochoid). It was invented by the British engineer Denys Fisher, who exhibited it in 1965 at the Nuremberg International Toy Fair (source: Wikipedia).

 

A basic version of the Spirograph consists of :

  • a plastic frame (stator) which has two rings, of different radius, one next to the other
  • plastic disks (gears, rotors) of different radius, each one having holes.

 

kabylografos

 

What is special about the rounded constuctions, is that the rings' and disks' perimeter is not smooth, but all edges have teeth to engage any other piece, so that the disk (gear) rotates around the inside edge of the ring, without slipping.

The spiral order of the holes on the disks has no direct effect on the drawn curve. The different spots that we are interested in could be brought together on a radius of the disk. However, since each spot corresponds to a hole of countable diameter, in terms of construction, the spiral order gives us few more spots than those we may construct on a radius or diameter of the disk and arranged in a bit more impressive way in comparison with any other order on the disk (e.g. a triangle).

 

On the following applet, parameters' values can be changed by respective sliders:

.R radius of ring, .r radius of disk, .d distance between hole and disk's centre, and finally disk's rotations (counted in ring's rads).

You may also hide all colored curves (redgreenblue) by setting off the respective tick box, in order to study the basic construction and its operation (in white colour).

Note that for R = 2 r, ellipse is drawn, while for d = r , the ypocycloid.

Let's also remind that with Spirograph we have R > r > d (constraint that derives from construction).

On the applet we can draw curves for R, r και d values that do not satisfy this condition.

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