SGDL AI
Boustrophedon — a robot that creates art through reasoning, not imitation
Boustrophedon is a physical robot and software stack whose "creative brain" is SGDL AI, an AI of Form Intelligence designed to think like a geometer. Where mainstream generative‑AI systems rely on vast image datasets and statistical pattern‑matching, SGDL AI is built on Synthetic Reasoning: it proves that a form can exist, then constructs it from first principles. The method is documented in the XXVI Generative Art Conference paper "How to teach Concrete Art to a robot?" written by Alex Kummerman and Jean-François Rotgé and summarised below.
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Key idea: Arithmetic of Forms
Arithmetic of Forms turns every topological notion (vertex, edge, face, region) into pure integers. By working in projective rather than Euclidean space, the robot:
  1. Assigns an integer "density" to each point in space.
  1. Uses lambda‑calculus expressions (implemented today in Julia and Scheme) to manipulate those integers logically.
  1. Re‑projects the result as a concrete geometric object ready for CAD/CAM export.
This arithmetisation lets SGDL AI avoid the floating‑point errors and data‑structure bloat that usually plague computational topology.
The creative pipeline SGDL AI follows
1
Knowledge search
Scans mathematical and art‑historical sources to confirm that, say, "a convex polyhedron with identical but non‑regular pentagonal faces" is possible.
Grounds the work in proven theory, not guesswork.
2
Topological mapping
Establishes vertex–edge–face counts and adjacency graphs.
Defines a skeleton immune to later geometric distortions.
3
Projective proof
Uses projective geometry to guarantee coplanarity, convexity and symmetry.
Supplies the formal proof Max Bill demanded of Concrete Art.
4
Metric realisation
Converts the projective skeleton to exact Euclidean coordinates; exports STL/STEP.
Links pure theory to CNC milling, 3‑D printing or resin casting.
(All four steps are visible in the flow‑chart on page 3, Fig. 3 of Rotgé and Kummerman paper.)
Tools in the robot's "studio"
1
Skew‑truncation algorithm
Adapted from the work of geometer J.‑F. Rotgé was later Instilled to AI by Alex Kummerman; generates snub‑type polyhedra and their duals.
2
Space‑filling metacurves (Gray/Péano/Hilbert)
Drive plane partitions, chaotic colour fields and "metapixels" without resorting to random noise.
3
Jordan‑curve–based topological stencils
Partition any lattice or freeform grid into interior/boundary/exterior regions for colouring.
4
SFC‑attractor colour engine
Produces controlled "chaos" reminiscent of Vera Molnar's late work but computed deterministically.
These modules form a toolbox the artist can pilot by adjusting only a handful of high‑level parameters (e.g., golden‑ratio divisions, Tribonacci constants, recursion depth).
From Renaissance polyhedra to Boustrophedon 5–7
1
Historical dialogue
Leonardo's and Dürer's polyhedra were proof‑of‑concepts for perspective; SGDL AI extends that quest with modern rigour.
2
Material expression
Boustrophedon 5 (110 cm, polished resin) and the iridescent Boustrophedon 6 & 7* translate those digital proofs into tangible, light‑catching sculptures whose facets echo their formal derivation.
3
Architectural potential
Like the Structural Topology Group's experiments in the 1980s, each piece is a scale model of a load‑bearing shell, not just a gallery object.
Why it matters
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3
1
Ethical originality
No scraping of copyrighted images; every vertex is computed, not copied.
2
Explainable process
Each decision is a line in a proof tree, exportable as code or LaTeX.
3
Interdisciplinary bridge
Merges symbolic AI, advanced geometry, and Concrete‑Art aesthetics into a reproducible method.
Putting it in one sentence
"Boustrophedon is a robotic artist whose SGDL AI 'thinks in geometry': it proves a form can exist through arithmetic logic, then incarnates that proof as a physical polyhedron or plane composition—continuing the lineage from Renaissance draftsmen to Concrete Art, but with 21st‑century mathematical rigour."