About the Animated GCD Arcs Project
The Animated GCD Arc plots are digital drawings plotting the greatest common divisor [1] of integer pairs as geometric shapes within a grid. See examples here: Animated GCD Arc. This project, one in a series of Grid projects, is based on my GCD and Coprimes projects which were based on an image from Wolfram's MathWorld of relatively prime numbers [1].
The column and row numbers are reduced to a single small integer by their greatest common divisor, GCD, or the GCD of formulas in which the column and row number figure. The result is assigned one of four orientations for an image. In the animated versions a program cycles through the grid moving images from adjacent cells into cells with no image.
The polar grid was arbitrarily chosen. It has no particular significance, though it slightly complicates the math. It's a pleasing shape. The cells are arranged like discs in sectors and tracks around a center point, with axes dividing sectors and concentric circles dividing tracks. The animations of images are done by one of two transformations. When an image moves to an adjacent sector it amounts to a mathematical reflection about an axis. When a image moves to an adjacent track it represents an oblique or scaling reflection.
The images are designed to be rotated to one of four orientations. An animation may contain one or more related images which are rotated in quarter-turns to three other orientations. When the animations contain multiple images the image selection is random, but the initial orientation is calculated using the greatest common divisor method described above. The resulting animations represent dynamic, arc shaped plane symmetry groups.
The diagram below was a necessary part of the programming process. There are four possible orientations for images in a single grid cell. There are sixteen possible transformations from an image cell to an empty cell. See the animation here.
