Spirograph+

Formulation of New Pattern Structure through Spirograph in 2D, 3D and 4D with a local rotation

PennDesign Algorithm Seminar

Advisor: Cecil Balmond, Ezio Blasetti

Collaborator: Peng Wang

Fall 2015

Spirograph is a geometric drawing toy that produces mathematical roulette curves of the variety technically known as hypotrochoids and epitrochoids. It was developed by British engineer Denys Fisher and first sold in 1965. Roughly speaking, a roulette is the curve described by a point (called the generator or pole) attached to a given curve as that curve rolls without slipping, along a second given curve that is fixed. We studied the parameters and formal features of four variations of Spirograph: multi-wheel Spirograph, recursive Spirograph, three dimensional Spirograph with spinning track and four dimensional Spirograph by local rotation.

1. MultiWheel

Spirograph has two “wheels”. To generate a three-wheel Spirograph, we add one wheel rolling on the previous rolling wheel. One can keep adding circles this way to make a Multi-wheel Spirograph. Each additional wheel adds three independent parameters to the system: its rolling speed, radius and the drawing arm length.

2. Recursive

When rolling a circle along the Spirograph curve, we can get a new curve. We can roll another circle on the new curve and generate another curve and do this repeatedly. We call this system Recursive Spirograph. Like Spirograph, each additional curve has two parameters. Each new curve is dependent on the previous curve.

3. 3D Spirograph with a Spinning Track

In Three Dimensional Spirograph with Spinning Track, the wheel of Spirograph rotates around the tangent direction of the track at the contacting point.

4. 4D Spirograph by a Local Rotation

In Three Dimensional Spirograph with Spinning Track, the wheel of Spirograph rotates around the virtual local point in w direction of the track at the contacting point.