Step 1

Start with a triangle wave, or at least the first six components of its Fourier series: enough to create a fairly pointy curve. I wanted five points, so we pick the range to generate two-and-a-half waves. (The fundamental is sin x, so we have to plot over x = 0 to 5p.)

Step 2

Square the function to generate an all-positive curve. This also has the effect of introducing greater curvature at the zero-touching points, which is what we want.

Step 3

To create points of varying size, we can multiply the curve by another, which acts as a shaping envelope, shown dotted. To get a smooth rise and fall over the five peaks, we choose a sine function with a wavelength five times the fundamental of the triangle series (the 2 is just to heighten the overall curve shape)...

Step 4

...and this is the result.

Step 5

Having created our five-point curve, we want to raise the central points away from the axis. This we do by adding the previous shaping envelope).

Step 6

We have half the desired result. Plot the mirror image (the negative of the function) to get our final curve: a mathematical holly leaf.