Torus (3, 2) · living lattice
R = 1.00 r = 0.38 V = 128 χ = 0
A torus is the surface swept by a circle of radius r orbiting an axis at distance R. Node colour encodes Gaussian curvature K: positive on the outer ring, zero on the top and bottom circles, negative in the inner throat, integrating to zero over the whole surface. The winding curve is a (p,q) torus knot. Tokamak plasma confinement uses exactly this geometry.
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χ = V − E + F = 0
Every quad mesh on a torus obeys the Euler characteristic zero: each node carries exactly two edges (one u, one v) and one face. Genus 1 — one handle, one hole.
Two independent cycles
The first homology group is ℤ ⊕ ℤ: the meridian (around the tube) and the longitude (around the axis). Neither can be contracted to a point, which is why toroidal fields are self-sustaining.
Curvature sums to zero
K = cos v / (r(R + r cos v)) is positive on the outer equator, zero on the top and bottom circles and negative on the inner throat. Gauss–Bonnet: ∫K dA = 2πχ = 0.
Ring vs spindle
R > r gives a ring torus, R = r a horn torus (the hole closes to a point), R < r a self-intersecting spindle torus.
(p, q) torus knots
A closed curve winding p times around the axis and q times through the hole. When gcd(p, q) = 1 it is a single knot — (2,3) is the trefoil; otherwise it splits into gcd(p, q) linked strands.
Flat topology
Cutting along both cycles unrolls the torus to a rectangle with opposite edges identified — the same wrap-around space in which the node lattice above is periodic.
