A checkerboard... that moves?

This recreation of an optical illusion was made using Maple Learn. For an interactive version where you can zoom into the plot and change the colors, check it out here: Optical Illusion: Moving Image. The buttons in the document were made possible by deploying the canvas from Maple using this Maple script.

Changing the colors creates cool visuals - and the motion of the image is apparent using some color schemes more than others.

How Does It Work?

If you zoom really far into the image, you can see that it is made up of a checkerboard in the background, with crosses (plus signs) covering each of the corners:

The checkerboard is made up of unit squares, and to create a cross with hieght=6d, centered at (x,y), polygons were used:

Polygon(
  [x+d,y+3d], [x+d,y+d], [x+3d,y+d],
  [x+3d,y-d], [x+d,y-d], [x+d,y-3d],
  [x-d,y-3d], [x-d,y-d], [x-3d,y-d],
  [x-3d,y+d], [x-d,y+d], [x-d,y+3d]
)

The key to the motion of the image lies in the alternating 1-1-2-1-1-2 pattern of the light/dark crosses along the diagonals.

I was working in my living room.  My computer was upstairs, but I had my phone and tablet.  I'm working on The Book ("Perturbation methods using backward error", with Nic Fillion, which will be published by SIAM next year some time).

I've discovered something quite cool, historically, about the WKB method and George Green's original invention of the idea (that bears other people's names, or, well, initials, anyway).  (As usual.)  Green had written down a PDE modelling waves in a long narrow canal of slowly varying breadth 2*beta(x) and slowly varying depth 2*gamma(x).  Turns out his "approximate" solution is actually an exact solution to an equation of a very similar kind, with an extra term E(x)*phi(x,t).  The extra term depends in a funny way on beta(x) and gamma(x), and only on those.  So a natural kind of question is, "is there a canal shape for which Green's solution is the exact solution with E(x)==0?"  Can we find beta(x) and gamma(x) for which this works?

Yes.  Lots of cases.  In particular, if the breadth beta(x) is constant, you can write down a differential equation for gamma(x).  I wrote it in my notebook using y and not gamma.  I wrote it pretty neatly.  Then I fired up the Maple Calculator on my little tablet, opened the camera, and pow!  Solved.

I wrote the solution down underneath the equation.  It checks out, too.  See the attached image.

Now, after the fact, I figured out how to solve it myself (using Ricatti's trick: put y' = v, then y'' = v*dv/dy, and the resulting first order equation is separable).  But that whole "take a picture of the equation and write down the solution" thing is pretty impressive.

So: kudos to the designers and implementers of the Maple Calculator.  Three cheers!