Exploring Fourth Dimension: 4D Art
Exploring the Fourth Dimension
The Art and Philosophy of 4D Geometry
In a world constrained by three dimensions, the concept of 4D art opens doorways to realms beyond ordinary perception. As an artist known as Pardesco (Randall Morgan), my passion lies in visualizing these higher dimensional spaces through innovative techniques that bridge mathematics, technology, and artistic expression.
The Journey into 4D Art
My artistic exploration began with a deep fascination for geometric complexity. While many artists work within the familiar confines of three-dimensional space, I found myself drawn to the mysterious fourth dimension—a realm that exists mathematically but remains beyond direct human perception.
This curiosity led me to spend years developing proprietary techniques for creating detailed geometric visualizations. What started as an examination of models in specialized software like Stella4D 4D evolved into a profound artistic quest to make the invisible visible.
A Technological Breakthrough: Interactive 4D Visualization
For years, I worked with static projections and physical sculptures—frozen moments of four-dimensional forms. But I always envisioned something more: watching these shapes rotate in real-time through four-dimensional space, their edges curving and morphing as they turn through planes we cannot directly perceive.
Until recently, this vision remained frustratingly out of reach. Traditional 3D visualization tools couldn't capture the curved geometry that emerges when 4D objects are projected stereographically into our three-dimensional world. Creating such a system from scratch would require months of mathematical programming—time I didn't have as a solo artist balancing multiple creative pursuits.
The AI-Collaborative Breakthrough
Then came a fundamental shift in how art and technology could merge. Using agentic AI workflows—specifically Claude Code as my technical engineer and Claude App as my creative director—I built something I'd dreamed about for years: an interactive web-based viewer that shows 4D polytopes rotating in real-time with proper curved edges.
This wasn't about AI making the art. It was about AI helping me build the tools to realize my artistic vision. I brought the mathematical understanding, the aesthetic sensibility, and years of studying these forms. Claude Code translated my specifications into working code, implementing the complex stereographic projection mathematics and 4D rotation systems.
The workflow felt like having a brilliant engineer on call 24/7. I'd describe what I needed—"rotate this hypercube around the XW plane" or "the curves should preserve their radial thickness gradient"—and Claude Code would implement it. When something didn't look right, I'd explain the visual problem, and we'd iterate until it matched my vision.
What emerged represents a new horizon in both generative art and AI-collaborative creation. I'm not just showing you static images of 4D geometry anymore. I'm inviting you to interact with these shapes, to rotate them yourself, to see how their structure transforms as they move through dimensions beyond our perception.
Stereographic Projection: Seeing Curves in Higher Dimensions
The key to making 4D shapes come alive lies in understanding how we project them into our 3D world. Most 4D visualization uses perspective projection—the same technique that makes railroad tracks appear to converge in the distance. It's mathematically simple but shows 4D edges as straight lines in 3D space.
I chose a different approach: stereographic projection. This technique, borrowed from cartography and complex analysis, preserves the natural curvature that emerges when higher-dimensional forms intersect our three-dimensional slice of reality. The edges aren't straight—they curve through space like geodesics on a sphere, revealing the true topology of four-dimensional geometry.
The 120-Cell: 600 Vertices of Pure Symmetry
Watch the most complex regular 4D polytope rotate through higher-dimensional space. Each curve you see represents an edge that exists perfectly straight in 4D but appears curved to us—like watching a sphere's surface curve even though the sphere itself is "straight" in its own geometry.
The 120-cell contains 120 dodecahedral cells, 720 pentagonal faces, 1,200 edges, and 600 vertices—the 4D equivalent of a dodecahedron's symmetry elevated to an entirely new dimension.
A Gallery of Higher-Dimensional Forms
Each 4D polytope has its own character, its own way of moving through space. Some are simple and elegant, others impossibly complex. All reveal something profound about how geometry extends beyond our perceptual limits.
The 4D equivalent of a cube—the simplest way to extend a square into the fourth dimension. Watch how its edges curve and intersect as it rotates through planes we cannot perceive.
One of only three regular 4D polytopes with no 3D analogue. Its perfect symmetry exists only in four dimensions—a form that could never be constructed in our 3D world.
The 4D simplex extended—ten tetrahedra joined in perfect 4D symmetry. Its elegant simplicity belies the complex curved relationships visible in stereographic projection.
The simplest regular 4D polytope—the 4D equivalent of a tetrahedron. Even this fundamental form reveals the beautiful curvature that stereographic projection unveils.
The Technical Pipeline: From Vision to Reality
Creating these visualizations required building an entire pipeline—from mathematical calculation to interactive display to final rendered animation. Here's how the process works:
- 4D Geometry Processing: Starting with .off files containing 4D vertex coordinates, I apply stereographic projection formulas to map points from the 4-sphere into 3D space. This preserves the curved relationships that make the geometry come alive.
- Interactive Web Viewer: Built with Three.js, the viewer lets you rotate these shapes through any of the six fundamental 4D rotation planes (XY, XZ, XW, YZ, YW, ZW). You're not just rotating in 3D—you're actually rotating the 4D object before it's projected, revealing aspects impossible to see from any single angle.
- Animation Export: The viewer can export complete rotation sequences as JSON data—every frame capturing the exact 4D state of the polytope.
- Blender Integration: Python scripts convert these animations into Alembic curve files, which import into Blender as animated 3D curves. Geometry Nodes then add tube geometry while preserving the thickness variations that encode dimensional depth.
- Audio-Reactive Rendering: An experimental Blender add-on applies audio-reactive modulation—making the polytopes pulse, glow, and shift color with music, creating a synthesis between mathematical form and emotional experience.
This entire system—from 4D mathematics to final render—was built collaboratively with AI as my technical partner. I provided the mathematical expertise, aesthetic direction, and years of studying these forms. Claude Code provided the engineering implementation, debugging, and optimization. Together, we created something neither of us could have built alone.
From Mathematical Concepts to Physical Sculptures
While the interactive viewer represents the cutting edge of my practice, my journey with 4D art began in more traditional media—though "traditional" hardly describes the challenge of translating four-dimensional mathematics into bronze and silver sculptures.
The complexity of higher-dimensional geometry creates unique obstacles that conventional artistic methods cannot overcome. How do you sculpt something that doesn't fit in three-dimensional space? How do you cast a projection of a form that exists in four dimensions?
Sitphi
3,600 Edges of a 4-Dimensional Object