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

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

edited by Pita Arreola, Corinna Gardner, Melanie Lenz – Digital Art (Victoria and Albert Museum): 1960s–Now

https://neural.it/2026/02/edited-by-pita-arreola-corinna-gardner-melanie-lenz-digital-art-victoria-and-albert-museum-1960s-now/

#mediaarthistory #artandtechnology #arttech #audiovisualart #audiovisuals #interactiveinstallation #interactiveart #mediainstallation #newmediaart #newmediaartist #mediaart #mediaarts #mediaartist #digitalart #digitalartist #timebasedart #digitalsculpture #digitalaesthetics #cyberart #computerart #electronicart #onlineart #netart #netartist #webart #webartist #internetart#artreview #neuralmag #neuralmagazine

https://archive.neural.it/init/default/show/3330

Following my recent 2D image morphing demos, I've been thinking of visualizing the process as a single 3D shape. Since it starts with pixelated images, a voxel approach seemed the most straightforward.

While I wasn't quite happy with the blocky look, the parallel columns of voxels caught my attention, and they became a key stylistic element; they reminded me of the "fibre optic" mineral ulexite. I imagined carving a sculpture of such a fibrous material. So I replaced the piecewise columns by these long tubes.

Fox drawing courtesy of @noira_musti

#morphing #3dgraphics #digitalsculpture #pythoncode #numpy #opengl #algorithmicart #algorist

My client #3DPrinted Ford #Monster inspired by the #art by #EdRoth, made with #maya #zbrush

#creature #3d #3dart #3dartist #gameart #american #usa #retro #creature #vehicles #arttoys #toys #sculpture #sculpture #hotrodart #digitalsculpture #kustomkulture #weirdohs #commission #freelancer

A #commission work Ford #Monster inspired by the #art by #EdRoth, made with #maya #zbrush and #render with #keyshot split by parts for easy #3Dprint

#creature #3d #3dart #3dartist #gameart #american #usa #retro #creature #vehicles #arttoys #toys #sculpture #sculpture #hotrodart #digitalsculpture #kustomkulture #weirdohs

I'm currently working on a 3D sculpture dedicated to Ayrton Senna — a tribute to a legendary figure whose intensity and presence continue to inspire. This project is a personal exploration of emotion through form, aiming to capture the essence of Senna with subtlety and respect.

More to come soon.

#3DArt #DigitalSculpture #AyrtonSenna #WorkInProgress #ArtAndEmotion #DesignInMotion