Cardioids for Valentines Day


A cardioid is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. The typical technique for constructing them is by complex equations, but here is another technique for producing the pattern that only use simple straight lines.


Choose some small angle, say θ = 3° and a natural number n > 1. Draw the line segments joining the points on the unit circle with coordinates {Cos[r θ], Sin[r θ]} and {Cos[n r θ], Sin[n r θ]} for all 0 < r < k (where k depends on n). A cardioid appears (each of the straight lines is tangent to a cardioid) and is called the envelope of the family of straight lines. Here is the code.