Baltor Get started

Gallery / 4D geometry

Clifford torus in the 3-sphere

In your harness4D geometry toolMIT

Parametrize the flat Clifford torus in S3, check its flat metric, area 2 pi^2, the two solid tori of volume pi^2 it bounds and its linked core circles, and write its stereographic image turning in 4D as a looping glTF morph animation.

Render of Clifford torus in the 3-sphere
The render its check recorded in Godot 4.7.2, 256 by 256 pixels.

What it takes and gives

Entry points
point, metric, area, solid_torus_volumes, is_clifford, stereographic_from, core_linking, image_radii and 6 more
Outputs
example.gltf
Coordinates
(cos u, sin u, cos v, sin v)/sqrt(2) in x, y, z, w; stereographic image from (0, 0, 0, 1), Godot/glTF +Y up

Limits. The animation stores 16 keyframes per loop as morph targets (positions and normals), so between keyframes vertices move along chords and the surface shrinks slightly. Rotation planes containing w are refused because they carry points through the projection pole. The solid torus volumes come from Simpson's rule (error near 1e-10).

Details

Kind
4D geometry tool
Engine
Python 3.10 or later
Files
10 files, 295.2 KB
Licence
MIT
Declared effects
reads files, runs commands, writes files
Checked with
Godot 4.7.2, gl_compatibility (opengl3, Mesa llvmpipe under Xvfb)
Checks
cli rebuilds example, gltf structure example, cli example runs, cli example gltf structure, godot loads example, godot loads cli clifford yz, preview render
Techniques
Clifford torus, induced metric of a parametrized surface, Hopf coordinates and the volume element of S3, stereographic projection to a torus of revolution, Dupin cyclides, Gauss linking integral
Tags
4d, clifford torus, flat torus, 3-sphere, stereographic projection, dupin cyclide, morph targets, gltf
Identity
library.supply.creative_originals.e2ce13a478f966d4d7a2a0da.773080b31ff49d14
Version
773080b31ff49d1415c2acda9a00d4f21c47f469ae4299fd6741b573ef45bf5a

The version is the SHA-256 of the item your harness downloads; every file in it names its own checksum.

Something wrong with this item, its rights or its safety? Report it on the support page, or from your harness after a download.