Interactive visualizations of 4D polytopes using
stereographic projection and true 4D rotation
Rotate in 6 fundamental planes (XY, XZ, XW, YZ, YW, ZW)
Beautiful curved edges from true 4D geometry
Mouse, keyboard, and touch controls for exploration
Understanding stereographic projection and higher-dimensional geometry
Unlike standard perspective projection which flattens 4D shapes into straight lines, stereographic projection preserves the angles of the geometry, resulting in curved edges. This allows you to see the internal structure of a hypercube (tesseract) without visual distortion.
When you rotate a 4D polytope in four-dimensional space, the curves morph and transform--revealing the hidden symmetries that make these objects beautiful.
In 3D space, objects rotate around axes (X, Y, Z). In 4D space, objects rotate around planes. This viewer implements rotation across all 6 fundamental planes: XY, XZ, XW, YZ, YW, and ZW.
The "pure 4D" rotations (those involving the W-axis) have no 3D equivalent--they produce motion that is literally impossible to achieve with any 3D transformation.
The six regular convex 4-polytopes are the 4D analogs of Plato's solids. The most famous is the tesseract (8-cell), but the 120-cell and 600-cell are far more complex, with hundreds of faces and thousands of edges.
Our library includes over 1,700 uniform 4-polytopes--computed from mathematical research--making this the most comprehensive 4D visualization tool available online.
Perspective projection (like in Stella4D or 3D modeling software) renders 4D edges as straight lines by projecting from a fixed viewpoint. This is fast but loses geometric information.
Stereographic projection maps the 4D sphere onto 3D space using a conformal transformation, preserving angles and creating curved edges. This reveals the true structure of 4D polytopes--especially their internal symmetries.
Yes! The Creator tier ($49/year) unlocks:
These exports capture the exact 4D rotation state you're viewing, making it easy to create renders in Blender or 3D print physical sculptures.
The 24-cell is a regular convex 4-polytope with no 3D equivalent. It's composed of 24 octahedral cells, 96 triangular faces, 96 edges, and 24 vertices.
What makes it unique: It is self-dual (its dual polytope is itself) and has the symmetry group F4. In 3D, the analog would be... nothing--there is no 3D shape with these properties.
The 120-cell (also called the hyperdodecahedron) has:
It's the 4D analog of the dodecahedron and one of the most complex regular polytopes.
The edges appear curved because of stereographic projection. In true 4D space, the edges are straight lines. But when we project them down to 3D (so we can see them), the projection introduces curvature--similar to how a straight line on a globe becomes an arc when mapped to a flat surface.
This curvature is not an artifact or rendering error--it's the mathematically correct way to visualize 4D geometry in 3D space while preserving angles.
Absolutely! Many VJs and live visual artists use our viewer to:
The 4D rotation creates impossible-looking animations perfect for immersive visual experiences.
Yes! Students and educators get 50% off with a verified .edu email. Email us for your discount code.
This 4D polytope viewer is completely free and open for everyone to explore. Want to remove watermarks and export high-quality images? Upgrade to Creator Tier!
Upgrade to Creator TierWatermark-free exports, high-resolution images, and priority support
Special thanks to the Polytope Discord Server community for their invaluable support and feedback during the development of this viewer.
