Laboratory
Two Circulations
A living torus with two flows on one surface: the long way around the ring, the short way around the tube, and a guided walk from geometry into practice.
Drag to turn the figure. Green travels the long way; copper wraps the tube. The chapters beside the torus (or the numbered steps on a phone) walk you from pure geometry into the practice of outward care and inward return. Prefer less motion? The figure still teaches with static arrows. Without WebGL, a clear diagram stands in.
A torus of revolution is a tube bent into a ring. Toroidal circulation travels the long way around the ring. Poloidal circulation wraps the short way around the tube, passing through the hole. A hero particle can follow both motions at once along a helical path that does not cross itself. In practice, the long way figures outward care and work; the short way figures inward stillness and reflection. Both belong to one life.
Chapter 1 · One form
On a toroidal surface, two independent directions of motion can run at the same time without cancelling each other. Watch the green stream take the long way around the ring while the copper stream wraps the tube. One brighter particle rides both at once.
Chapter 1 · One form
On a toroidal surface, two independent directions of motion can run at the same time without cancelling each other. Watch the green stream take the long way around the ring while the copper stream wraps the tube. One brighter particle rides both at once.
Chapter 2 · The long way
Toroidal motion travels the long way around the main ring, the way a bead slides along a bracelet. Scrub the marker below to feel that single circuit. The copper stream stays faintly present so the whole form remains legible.
Chapter 3 · The short way
Poloidal motion wraps the short way around the tube: through the center, over the top, and back. Drag the marker around the tube while the inset shows that path as a simple circle. Loops bloom all around the ring.
Chapter 4 · Both at once
When both circulations run together, a single particle traces a helix that never crosses itself and never stops. Independent directions share one form. Change the speed ratio and watch the helix tighten or open.
Chapter 5 · Surface and solid
What you have been watching is a two-dimensional surface curved through space. A solid torus is the three-dimensional volume enclosed inside the tube. Sweep the cut-plane to see the hollow passage, then fill the interior.
Chapter 6 · In practice
Green becomes outward reach: work, care, the kept promise. Copper becomes inward return: stillness, gratitude, honest revision. Both belong to one life. Sit with the figure as long as you need.