Hyperbolic fBM Volumes & Probabilistic Space Modeling
"Hyperbolic Cathedral"
Major milestone achieved, rendering the phase transition of hydrogen 1s and 2p_z eigenstates. The resulting oscillation is driven by phi_time in the node graph for continuous realtime looping of eigenstate transitions. Can be viewed in cutaway as well!
As part of my larger goal for this to act as a substrate for visualizing QFT equations in Hilbert space, I put together a Python script for a single mode quantum harmonic oscillator fed into the W input of the fBM noise--W(t)=W 0+Acos(ωt+ϕ 0)
"Coterminous Recursion"
Scaling FOV from 200mm to 18mm while moving from one end of the volume to another in reverse. Transformations are happening procedurally from camera movement (vectors mapped to screen space)
Extreme FOV scaling (200mm to 2mm) while moving backward through volume. Recursion happens at multiple vanishing points.
A quick and simple compilation of various field behaviors.
Wavelength input is animated from 380–750nm while sweeping the W axis of a 4D fBM volume. The result is a spectrally-driven visualization of field evolution, where emission corresponds to localized threshold crossings in a pseudo-random scalar field.
Matcap preview to show the camera flight path of the next two volumes and give spatial context. Note spherical anchor at the center which assists in the field's hyperbolic projection.
Probably my most favorite one. Here, the tex coord of an invisible sphere is in the center as a tether point. I fly through at a steady speed while pitching downward.
Same camera movement as previous, but different node setup. Forms only shift when camera moves and rotates, and it's happening entirely on its own.
Extreme FOV fly through from one end of a volume to the other, using variations of camera and object vectors as field transforms.
EEVEE fly through of volume. I turn around at the end. Transforms are mapped to camera space.
Same EEVEE fly through as previous. Transforms are mapped to camera space, but parameters are changed and wavelength is shifted progressively.
Removed observer dependency, animated “W” to influence movement within the volume while flying through.
Scaling wavelength 380 NM to 780 as I slowly fly through. Also experimenting with dot product fresnel emissives on the sphere. Transforms mapped to sphere in center which rotates 360 degrees as I fly through.
Attaching camera vectors to various transforms to achieve observer dependent perspective shifts of form. Was still using stepped planes, which wasn't great framerate wise.
The first iteration of this involved a masked out stepped plane volume, much like what you'd see with POM shaders. This wasn't an efficient rendering method but it was good enough for initial experimentation.
This is the point where I began playing with mesh anchors that influence transforms hyperbolically.
Switched to volume rendering here, evolved from the stepped plane system. Animated lacunarity, roughness as well as wavelength to visualize how a phase shift in a quantum system could be represented.
Scaling through "W" as well as wavelength. The W parameter transforms the field simultaneously across multiple perspectives, causing some forms to vanish entirely while others spontaneously appear. In effect, it functions as a four-dimensional delta-time.
Here I've reintroduced the mesh anchor to control volume transforms such as scaling and rotation. Scaling the anchor down shrinks the field uniformly across its dimensional vectors.
WASD fly-through with mesh anchor at the center. Multiple perspective shifts are happening at once as I fly through. The anchor at the center causes the transforms to briefly spike, before completing the fly through.
A strange cocktail of camera, object and texcoord generated vectors produced this incredible higher dimensional fluiditiy.
Interesting node configuration where spherical volume had an influence point that I scaled down, which produced a gorgeous recursion of transforms the smaller I scaled it.
Early volume renders, exploring recursion that happens when screen space is plugged into field transforms.
Spherical influence
Older stepped plane volume, note the lack of visibility directly on the horizon due to critical view angle threshold. Planes at eye level become too flat relative to viewer.
Setting epsilon to a very low number created nice colorful vector lines. Using stepped plane system here--line volume depended on how many planes I used for volume projection.
This project is an ongoing visual research initiative exploring how complex, high-dimensional physical concepts can be intuitively communicated through procedural volumetric systems. Developed through independent research and cross-disciplinary practice spanning game development, studio art, and physics, the work focuses on perceptual translation—using spatial intuition and mathematical structure to make abstract field behaviors visually legible.
At its core, the system is built around hyperbolic fractal Brownian motion (fBM) evaluated across four dimensions and rendered as a dynamic scalar field. Rather than relying on keyframed animation, all transformations emerge in real time through procedural parameter control, allowing the field to continuously evolve without predefined states.
Key parameters—including lacunarity, roughness, wavelength, and emissive thresholding—are animated to explore behaviors analogous to field interference, phase transitions, emergence, and collapse.
A fourth input parameter ("W") functions as a continuous global transform, simultaneously affecting the scalar field across all spatial coordinates. In practice, this produces non-local changes in structure: some forms vanish entirely while others spontaneously emerge, depending on both parameter state and observer perspective. Conceptually, this axis functions as a four-dimensional delta-time—a mechanism for visualizing transformation across an abstract dimension rather than conventional temporal animation.
The W parameter is currently driven by a harmonic oscillator encoding coherent-state dynamics—mathematically equivalent to the expectation value evolution of a quantum harmonic oscillator. This was initially developed through artistic intuition and later identified as structurally analogous to a known physical formalism — consistent with the project's broader thesis on tool-first abduction: that building an expressive tool first, and recognizing its formal resonances afterward, can be a legitimate pathway to cross-disciplinary insight. The architecture supports extension to multi-mode superposition, enabling visualization of beating, interference, and energy exchange between coupled oscillator modes.
Several iterations of the system incorporate observer-dependent coupling, where screen-space vectors and camera orientation are fed back into the volumetric coordinate space. This creates a perceptual dependency between viewpoint and structure, reinforcing the idea that observation itself participates in how the field is resolved—a principle central to many modern interpretations of quantum and complex systems.
While originally developed as a perceptual tool rather than a numerical simulation, the system's architecture has proven structurally analogous with physically accurate field dynamics, demonstrating strong potential as a visual and educational instrument for communicating core concepts in quantum field theory—including emergent structure, probabilistic behavior, entropy-like diffusion, and recursive spatial transformation. The emphasis is on intuition, pattern recognition, and perceptual insight as potential pathways to formal understanding.
In 2025, this work was reviewed by Prof. Dr. Fabrizio Carbone (EPFL), a leading researcher in ultrafast electron scattering and quantum field imaging. He noted the system's effectiveness in visualizing complex behaviors and its potential value in scientific communication—particularly where artistic intuition intersects with hard scientific concepts to stimulate direct sensory understanding.
This project continues to evolve as a bridge between visual art, procedural animation, and scientific inquiry, investigating how far perceptual systems can go in making the invisible structures of nature thinkable, seeable, and felt. Full correspondence available upon request.