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In the last post I introduced the "dual complement" idea for polyhedral graphs. I'm not sure if it has any mathematical significance, but I've made a fun discovery: the dual complement of a spanning tree is another spanning tree.
This result is rather intuitive and I don't have a rigorous proof for it yet, but here are the main supporting ideas. First, a spanning tree over v1 vertices has v1 - 1 edges. We can then show, using basic duality relations and Euler's polyhedral formula, that the dual complement has v2 - 1 edges that connect all of its v2 vertices. The complement doesn't have any cycles, since those would "capture" parts of the original graph, which we know is a single component.
The original polyhedron here is a {3,5+}_2,1 geodesic, so the dual is a Goldberg polyhedron.
No AI, no apps, just my original Python + OpenGL code.
#graphtheory #dualpolyhedron #dualcomplement #spanningtree #geodesicpolyhedron #goldbergpolyhedron #3dgraphics #digitalsculpture #pythoncode #numpy #opengl #creativecodeart #algorithmicart #algorist #mathart #laskutaide #computerart #ittaide #kuavataide #iterati
Investigating the S3 Virge’s Reputation as a 3D Decelerator Card
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
Cerchi un visualizzatore 3D veloce e leggero per Linux? MegaMesh punta su Vulkan, prestazioni elevate e compatibilità con i formati più diffusi. #Linux #OpenSource #Vulkan #3DGraphics #CAD #Software
Antes de que las GPUs modernas dominaran el escritorio, 𝐒𝐢𝐥𝐢𝐜𝐨𝐧 𝐆𝐫𝐚𝐩𝐡𝐢𝐜𝐬 ya estaba mostrando hacia dónde iba la computación visual.
No eran solo computadoras. Eran estaciones para construir el futuro.
#retrocomputingmx #SiliconGraphics #SGI #onyx2 #octane #sgio2 #mips #OpenGL #retrocomputing #vintagecomputing #computerhistory #3dgraphics #workstations
En sus máquinas se crearon mundos virtuales, escenas imposibles y herramientas que marcaron a toda una generación de artistas, ingenieros y desarrolladores.
Su hardware era tan distintivo: gabinetes azules, morados y turquesa; arquitectura MIPS; gráficos OpenGL; multiprocesamiento; y una estética industrial que todavía hoy parece salida de una película de ciencia ficción.
#onyx2 #octane #sgio2 #mips #OpenGL #retrocomputing #vintagecomputing #computerhistory #3dgraphics #workstations
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