Circular network, fully connected

Here's my version of a design taken from the Jared Tarbell talk that I referenced earlier.

31 points, radius = [1., 1.8, 2.2, 3.1, 3.5, 3.7, 4.]

The recipe is simple:
* Fully connect points that are equispaced around the circumference of a circle
* Repeat for circles of various radii

This exercise posed an interesting challenge. Full details in my repository here, but the short version is that I thought I could take a clever approach using Processing's scale transform and—well—it didn't pan out.

So it was pretty obvious from the symmetry of the pattern that I would want to do my math in polar coordinates and then convert to cartesian at the last moment. And rather than recalculating the points around the edge of the circle whenever I change the radius of the circle, I can leave the points' r and theta (and their equivalent x and y) untouched and use Processing's scale() transform to change the underlying basis vectors and hence the points' position on the screen. Neat! No recalculations!

Unfortunately, scale() scales everything. It's not just that all the basis vectors are scaled, the actual screen representation of a pixel is scaled too. So if I call scale(8) all of a sudden any line (say) that used to be one pixel wide is now 8 pixels wide. And although I can downscale the resulting line widths, I cannot do so below the new representation of 1 pixel. For example, the minimum value that I can pass to strokeWeight() is 1. This is 1 transformed pixel, not 1 actual pixel. So when I use a large value in scale(), my lines suddenly start to get thick and they stay thick. Here's a detail from an early iteration of the design:

The solution is not to use scale(): instead simply recalculate the points

More curves; or "how do I google something whose name I don't know?"

I have a vague childhood memory of visiting a museum and seeing some Victorian contraption of pendulums and pulleys that would produce lovely swirling, swooshing designs from the pen clamped at its center. The designs weren't Lissajous figures, they were far more complex than that. What were they? And what were the devices that produced them?

Sand pendulums? No, I think that was from "Vision On" with Tony Hart. But searching for "sand pendulums" did take me to the archive of the Bridges Organization, which "oversees the annual Bridges conference on mathematical connections in art, music, architecture, education, and culture". OK, I feel I should have known about that and am ashamed that I did not. Moving on…

My point of entry into the archive was a paper by Douglas McKenna, "From Lissajous to Pas de Deux to Tattoo: The Graphic Life of a Beautiful Loop". Victory! It mentions the device I was trying to recall. The harmonograph. Yes, those are the patterns I remember, those are the devices.

Googling for examples is much easier now I know what the things are called. It turns out that several artists are working in this space and producing delightful work. Here's an example from Rick Hall (it's also the source of the image at the top of this post):

Source: http://rickhall.co.uk/harmonograph-2/

Now I just need to work out how to build one of these creatures.

Spirograph math

Yesterday's post about Mary Wagner's giant spirographs prompted two lines of thought:

  1. Wouldn't it be cool to actually make some of those cogs?
  2. Wouldn't it be cool to prototype them in code?

It didn't take me long to put the first of these on ice. My local maker space has all the necessary kit to convert a sketch on my laptop into something solid but oh my, gears are hard. There seem to be all sorts of ways of getting a pair of gears to grind immovably into one another. I suspect I'd get to explore that space thoroughly before actually getting something to work. I don't have the patience for that. (If you are more patient than me, check out this guide to gear constrution from Make.)

Let's turn to the code then.

There are plenty of spirograph examples out in the wild (the one by Nathan Friend is probably most well known). But if I want to build one of my own, that's going to need some math.

Wikipedia didn't let me down. Apparently the pattern produced by a spirograph is an example of a roulette, the curve that is traced out by a point on a given curve as that curve rolls without slipping on a second fixed curve. In the case of a spirograph we have two roulettes:

  1. A circle rolling without slipping inside a circle (this is a hypotrochoid); and
  2. A circle rolling without slipping outside a circle (this is an epicycloid)

Here's a hypotrochoid:

Source: https://commons.wikimedia.org/wiki/File:HypotrochoidOutThreeFifths.gif

And here's an epicycloid:

Source: https://commons.wikimedia.org/wiki/File:EpitrochoidOn3-generation.gif

The math is straightforward (here and here) so no real barrier to coding this up, right?

Hmm. As usual one line of enquiry opens up several others. Sure, spirographs look nice but they are rather simple, no? What are our options if we want something more complex but still aesthetically satisfying? Wouldn't it be more fun to play with that? I mean the math couldn't be much harder, right?

More on that tomorrow…