Projections from a Higher Space

Each form is a real four-dimensional solid — a uniform polychoron whose vertices are exact coordinates in four-space. You cannot sculpt such a thing. The only honest way to see it is to build the machine that projects it — so I did.

Working with an agentic-AI engineering partner, I coded a four-dimensional viewer from the mathematics up: it turns the polytope through a plane of four-space, projects it into three dimensions, and carries each point's depth in the fourth dimension forward as the thickness of the light that draws it. The rotation is exact linear algebra — validated to one part in a quadrillion — and every loop closes on itself perfectly, repeating without a seam.

The hard part is what no tutorial covers: projection singularities where points race to infinity, a stable point-count as edges sweep the pole, a turn that must return exactly to its origin. That geometry flows through Universal Scene Description into Blender, where geometry nodes grow crystal along the curves and a GPU path-tracer renders it as refractive glass, lit from a single point at the form's own center.

To my knowledge, no one has rendered these four-dimensional solids at this fidelity — not the wireframes the fourth dimension is usually drawn with, but physically path-traced crystal: hundreds of self-intersecting edges bending light like a cut gem.

Here the three things I have always chased stop being separate: the mathematics is the truth, the technology the instrument, the agentic AI the leverage that lets one person hold an entire studio. The AI is not the artist — it is what turns a lifetime's fascination with higher dimensions into something you can hold in your hand, and behold.