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  1. 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

  2. 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

  3. Years ago, 2021, I was playing around with @drawBot These are vectors that can not be divided into smaller fractions, like primes. It made an interesting image, fine. Then recently I look for something else in The Computer in Art, Jasia Reichardt from 1971. It shows work of M. R. Schroeder. It is *almost* identical! I had no idea. Some small differences, curious why, but generally: math is universal and will be discovered over and over. #math #computerart #primes #mathematics

  4. Years ago, 2021, I was playing around with @drawBot These are vectors that can not be divided into smaller fractions, like primes. It made an interesting image, fine. Then recently I look for something else in The Computer in Art, Jasia Reichardt from 1971. It shows work of M. R. Schroeder. It is *almost* identical! I had no idea. Some small differences, curious why, but generally: math is universal and will be discovered over and over. #math #computerart #primes #mathematics

  5. Thinking about traditional moorish art, and the distinction between "geometric pattern" versus "algorithmic pattern". These visual designs are geometric. But definitely not algorithmic: humans make these by heuristically matching/filling the pattern, not by rule-following. In computer art we often conflate the two, since most programs to make these would proceed algorithmically. #computerart #MoorishArchitecture

  6. Thinking about traditional moorish art, and the distinction between "geometric pattern" versus "algorithmic pattern". These visual designs are geometric. But definitely not algorithmic: humans make these by heuristically matching/filling the pattern, not by rule-following. In computer art we often conflate the two, since most programs to make these would proceed algorithmically. #computerart #MoorishArchitecture

  7. Austin art friends - I am showing one my earliest fractal art pieces in the Bee Cave Arts Foundation's Members Show, beginning next Saturday, May 9 in the Bee Cave Gallaria (same weekend as the Pecan Street Fair). The show is located in the BCAF space, The Hive, across from H&M and Dillards. Hope to see you there!

  8. Austin art friends - I am showing one my earliest fractal art pieces in the Bee Cave Arts Foundation's Members Show, beginning next Saturday, May 9 in the Bee Cave Gallaria (same weekend as the Pecan Street Fair). The show is located in the BCAF space, The Hive, across from H&M and Dillards. Hope to see you there! #abstractart #digitialart #computerart #art #beecavearts