#kuavataide — Public Fediverse posts

Back in the day, I made a couple of demos where a Hamiltonian path is carved out on a polyhedron. Looking back, I started to wonder about the shape left around the path, and what it means in terms of graph theory. I call this shape the "dual complement" of the path.

The dual of a polyhedron is essentially the result of turning faces into vertices and vice versa. This is shown in the first clip with a snub dodecahedron and its dual, the pentagonal hexecontahedron; to keep the view cleaner, I'm only showing the edges of one at a time.

The duality transformation also affects the edges, but their number remains the same, and there's a 1:1 mapping between the original and dual edges. Each dual edge "cuts through" the original. To make the dual complement of a path, I remove the dual counterpart of each edge in the path, leaving only the stuff on the sides. It's like driving a snow plough along the path, leaving walls of snow on the sides.

For the final view, I combine original Hamiltonian paths with their dual complements.

#graphtheory #hamiltonianpath #hamiltoniancycle #dualpolyhedron #dualcomplement #snubdodecahedron #pentagonalhexecontahedron #3dgraphics #digitalsculpture #pythoncode #numpy #opengl #creativecodeart #algorithmicart #algorist #mathart #laskutaide #computerart #ittaide #kuavataide #iterati

Rule/Yule 30 cellular automaton with a "band-pass" filter every few iterations. Done in the Bash shell using bitwise math, so each row/state is a single number of 63 bits.

#cellularautomata #cellularautomaton #rule30 #bitwiseoperators #textmode #textmodeart #oldskool #retrocomputing #unixshell #shellprogramming #xterm #joulu6 #xmastree #algorithmicart #algorist #mathart #laskutaide #ittaide #kuavataide #iterati