Real-time visualization of explicit and implicit complex functions in four dimensions, running in the browser on WebGPU compute shaders. Explicit functions are embedded directly in R⁴, with Riemann sphere and S²×S² product compactifications available; implicit relations are extracted as isosurfaces of a 4D field via a marching-tesseract scheme, in point-cloud and connected-mesh form.
Accompanying paper: Rendering Explicit and Implicit Complex Functions in 4D: A WebGPU-Based Approach, Web3D '26.
npm install
npm run devWebGPU is required. Supported on all major browsers except Firefox; a Chromium-based browser is recommended. For integrated GPU's, for some reason Mozilla Firefox is more stable.
The following flags should be enabled:
- Unsafe WebGPU Support —
enabled - WebGPU Developer Features —
enabled - enable-vulkan —
enabled(Linux only)
If you're interested in the mathematical core of the project, the key files are:
| File | Description |
|---|---|
src/engine/render/compute/complex4d.wgsl |
Core 4D logic |
src/engine/render/compute/complex-functions.wgsl |
Library of complex functions used in the visualizations |
src/implicit/evaluator.wgsl |
Marching-tesseract extraction |
src/implicit/hypercube-march.wgsl |
Freudenthal Triangulation Code |
src/implicit/marching-pong.wgsl |
Implicit Point-Cloud Generation |
Most other files are generic rendering pipeline elements and are less relevant to the mathematics.
The next step is parsing input and converting the project into a full complex graphing calculator; A good start would be the Algebrite library. Additional improvements are bounding-boxes with annotated axes. In the future, visualizing more advanced topics from Complex Analysis like contour integration is planned. Long term we hope this can be an open-source Desmos alternative, as the pipeline is modular enough to accommodate normal functions as-well.
Admir Huseini and Art Saiti. 2026. Rendering Explicit and Implicit Complex Functions in 4D: A WebGPU-Based Approach. In The 31th International Conference on 3D Web Technology (Web3D ’26), October 13–15, 2026, Doha, Qatar. ACM, New York, NY, USA, 10 pages. https://doi.org/10.1145/3822516. 3834886
Art Saiti — WebGPU architecture, marching-tesseract extraction, simplicial connectivity scheme; repository maintainer
Admir Huseini — numerical methods, S²×S² compactification, mathematical formulation
MIT