What happens if you scribble back and forth on a white board that's spinning underneath your marker? What if you scribble in a figure-8 instead? Or a square pattern? This computer simulation by Jason Schattman shows the amazing and beautiful curves that arise from repeatedly drawing a simple shape on a spinning surface. It's similar to a Spirograph but with a wider range of generator shapes. Even the tiniest variation in the speed of the wheel, the size of the pattern you're drawing, or the placement of the pattern within the wheel produces a completely different curve. This video shows just a handful of the infinitude of possibilities.

I coded these animations in the Processing programming language. In each frame of the animation, a new point is added to the curve at the current location of the marker. To make the curve look smoother, adjacent points are connected using cubic splines. The path of the marker is computed using simple trigonometry and the formulas for Lissajous curves. The continuous rotation of the curve that has been drawn so far is computed by multiplying each point in the curve by a common rotation matrix, the angle of which is growing at a constant rate as the wheel spins.

To play with the program and create your own fun shapes, download the application from here:

Windows:
https://drive.google.com/file/d/1uPzM...

Mac OS:
https://drive.google.com/file/d/10TzY...

More math animations from my channel
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On the beautiful geometry of imaginary numbers & complex functions (can be enjoyed even without knowing what that means)
   • Twisting the Plane with Complex Numbers  

All 6 trig functions on the unit circle
   • All 6 Trig Functions on the Unit Circle  

Mathematical art using the idea of epicycles
   • Amazing Epicycles  

Optical illusions made using trigonometric functions
   • Optical Illusions  

Fancy "card tricks" animated using mathematical pretzels (called Lissajous curves)
   • Video  

Sound waves in an oval room:
   • The Physics of Sound Waves in an Ellipse  

Fly through the 3D Sierpinski pyramid:
   • Fly Through the Sierpinski Pyramid Fr...