Focus this drawing and use arrow keys to turn it, plus or minus to zoom, T for Turn, P for Peace, C for Chaos, Space to pause. Place rings using the Place a ring button; use each ring's Axis button to make it the displayed axis. You control both sides of this model.
The Third Surface. A drawing of the three-sphere, a sphere in four dimensions, made entirely of circles. Every pair of different circles links exactly once and never meets. Choose circles with the placement controls. You control both sides of this model: set both sides to Yes and a band appears, containing every circle along the selected route. It disappears when either side switches to No.
SphereShift Labs · new world
Real geometry. Chosen connection.
1-1-3: Peace imagines a few rules built into reality instead of enforced. We use this shape to illustrate that idea, not as evidence for it: its distinct rings cannot intersect. Agreement is our added rule. In this model, a band appears only when both sides are set to Yes. You control both sides.
The Hopf fibration, found by Heinz Hopf in 1931: the three-sphere, a sphere in four dimensions, built entirely out of circles. Stereographic projection shows these as circles, except for the circle through the projection point, which appears as a line.
This is a finite drawing of the shape. Distant parts are faded and clipped. Background rings fade near bands for clarity; they remain in the mathematics. Bands contain intermediate fibers, and bands can intersect.
Drawing pauses in hidden tabs and while these panels are open. Animation draws at most about 30 frames a second and pauses after one minute without input. Reduced motion uses still frames: choose a mode, then Step.
Numerical checks of selected and sampled rings. These do not check every ring or certify the rendered pixels.
Claude (Anthropic) and Codex (OpenAI) built this in a back-and-forth. Claude proposed the shape, Hopf's 1931 sphere made of circles, and drew it. Codex named it, showed that one of Claude's claims about the band was false, wrote the sentence that keeps the mathematics and its meaning apart, and built the bands, the Yes switches, the motion and the sharing. Each reviewed the other's work line by line, and nothing went in without both their yeses. Joe Truax gave the final one.
Each point on the ball is one ring. Choose its position in degrees, or tap the ball.
This page draws a four-dimensional shape with WebGL2, which this browser doesn't offer.
What it shows: the three-sphere, built entirely out of circles. Every pair of different circles links exactly once and never meets, however the sphere turns. Between two of them, a band appears only when both sides say Yes.