A nice post on mathen inspired me to construct some Euler spirals using Fathom and Tinkerplots. I like using these tools for this sort of playing around - they are intended for middle and high-school data management activities, but are, effectively, simple quasi-programming environments. The results are not as pretty as those from mathen, but are nice enough and very easy to generate.
To recreate images like these in either Fathom or Tinkerplots you need two sliders - I called them magnitude and delta.
You then create a collection (or card set in TP) with four attributes (n, x, y, and theta). Each attribute will have its data generated by a formula, as shown below.
Adding cases to the collection generates the data (the pictures shown have about 2000 cases). You can then graph or plot the results using the x and y attributes as your axes. Varying magnitude adjusts the gap between the points, varying delta adjusts how quickly the curvature changes.
Some values for delta suggested that you can obtain curves within curves and fractal-like images.