Shifted to the right (or left)
Last month I thoroughly explored the applications of geometric tessellations using the polar coordinate filter. After doing a bit more research I discovered how similar these “amazing circles” appear to be very similar to the sinusoidal curves of a Polar Rose. They use the equation:
In the original process I used to construct last month’s Polar Roses each interation added one tile to the entire panorama. I believe this represents the a in the equation. And after each iteration, there would be a new pedal added.
Today’s discovery is the realization of how these panoramas are wrapped around a cylinder to produce the pedals when the polar coordinate system is used. I’ve been working in whole numbers only, yet if the tiles are wrapped in a cylindrically fashion using a half shift, there will be a change in the pedal’s layout.
So to create this shift, half of one tile is removed on each end (notice above), and when they connect after being wrapped around a cylinder they still form a perfect tessellation and thus new pedal configuration when the filter is applied.
The result of this polar shift is quite interesting- a 90 degree rotation for each iteration.
View the 90 degree difference:
Shifted 90 degrees:
I can now apply this periodic shift to each of the geometries to produce a completely new set of Polar Roses. This should be interesting.