Clifford Torus — 4D Projection
T² ⊂ S³ ⊂ ℝ⁴ — stereographic projection to ℝ³
Rotation plane (4D)
XW
XW rotation
YW rotation
ZW rotation
XY rotation
XZ rotation
4D rotation speed
0.5
Grid lines u
16
Grid lines v
16
Color mode
UV coordinates
W coordinate
3D depth
Pause / Play
Reset View
Clifford torus
: parametrized as
(cos u, sin u, cos v, sin v) / √2 ⊂ S³ ⊂ ℝ⁴
It is a
flat
torus — intrinsic curvature = 0! Unlike a donut embedded in ℝ³ which is curved.
Stereographic projection
from S³ to ℝ³:
(x,y,z,w) → (x,y,z)/(1−w)
As the 4D rotation sweeps w, the projected shape morphs continuously between a torus and two interlocked circles (Villarceau circles).
Drag to rotate the 3D view. The 4D plane control picks which 4D rotation to animate.