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Torus (3, 2) · living lattice

x = (R + r·cos v)·cos u, K = cos v / [ r(R + r·cos v) ]

R = 1.00 r = 0.38 V = 128 χ = 0

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node 0-0 · K = 1.907node 0-1 · K = 1.467node 0-2 · K = 0.000node 0-3 · K = -2.545node 0-4 · K = -4.244node 0-5 · K = -2.545node 0-6 · K = -0.000node 0-7 · K = 1.467node 1-0 · K = 1.907node 1-1 · K = 1.467node 1-2 · K = 0.000node 1-3 · K = -2.545node 1-4 · K = -4.244node 1-5 · K = -2.545node 1-6 · K = -0.000node 1-7 · K = 1.467node 2-0 · K = 1.907node 2-1 · K = 1.467node 2-2 · K = 0.000node 2-3 · K = -2.545node 2-4 · K = -4.244node 2-5 · K = -2.545node 2-6 · K = -0.000node 2-7 · K = 1.467node 3-0 · K = 1.907node 3-1 · K = 1.467node 3-2 · K = 0.000node 3-3 · K = -2.545node 3-4 · K = -4.244node 3-5 · K = -2.545node 3-6 · K = -0.000node 3-7 · K = 1.467node 4-0 · K = 1.907node 4-1 · K = 1.467node 4-2 · K = 0.000node 4-3 · K = -2.545node 4-4 · K = -4.244node 4-5 · K = -2.545node 4-6 · K = -0.000node 4-7 · K = 1.467node 5-0 · K = 1.907node 5-1 · K = 1.467node 5-2 · K = 0.000node 5-3 · K = -2.545node 5-4 · K = -4.244node 5-5 · K = -2.545node 5-6 · K = -0.000node 5-7 · K = 1.467node 6-0 · K = 1.907node 6-1 · K = 1.467node 6-2 · K = 0.000node 6-3 · K = -2.545node 6-4 · K = -4.244node 6-5 · K = -2.545node 6-6 · K = -0.000node 6-7 · K = 1.467node 7-0 · K = 1.907node 7-1 · K = 1.467node 7-2 · K = 0.000node 7-3 · K = -2.545node 7-4 · K = -4.244node 7-5 · K = -2.545node 7-6 · K = -0.000node 7-7 · K = 1.467node 8-0 · K = 1.907node 8-1 · K = 1.467node 8-2 · K = 0.000node 8-3 · K = -2.545node 8-4 · K = -4.244node 8-5 · K = -2.545node 8-6 · K = -0.000node 8-7 · K = 1.467node 9-0 · K = 1.907node 9-1 · K = 1.467node 9-2 · K = 0.000node 9-3 · K = -2.545node 9-4 · K = -4.244node 9-5 · K = -2.545node 9-6 · K = -0.000node 9-7 · K = 1.467node 10-0 · K = 1.907node 10-1 · K = 1.467node 10-2 · K = 0.000node 10-3 · K = -2.545node 10-4 · K = -4.244node 10-5 · K = -2.545node 10-6 · K = -0.000node 10-7 · K = 1.467node 11-0 · K = 1.907node 11-1 · K = 1.467node 11-2 · K = 0.000node 11-3 · K = -2.545node 11-4 · K = -4.244node 11-5 · K = -2.545node 11-6 · K = -0.000node 11-7 · K = 1.467node 12-0 · K = 1.907node 12-1 · K = 1.467node 12-2 · K = 0.000node 12-3 · K = -2.545node 12-4 · K = -4.244node 12-5 · K = -2.545node 12-6 · K = -0.000node 12-7 · K = 1.467node 13-0 · K = 1.907node 13-1 · K = 1.467node 13-2 · K = 0.000node 13-3 · K = -2.545node 13-4 · K = -4.244node 13-5 · K = -2.545node 13-6 · K = -0.000node 13-7 · K = 1.467node 14-0 · K = 1.907node 14-1 · K = 1.467node 14-2 · K = 0.000node 14-3 · K = -2.545node 14-4 · K = -4.244node 14-5 · K = -2.545node 14-6 · K = -0.000node 14-7 · K = 1.467node 15-0 · K = 1.907node 15-1 · K = 1.467node 15-2 · K = 0.000node 15-3 · K = -2.545node 15-4 · K = -4.244node 15-5 · K = -2.545node 15-6 · K = -0.000node 15-7 · K = 1.467

T(R=1.00, r=0.38) · knot (3, 2)

Nodes V
128
Edges E
256
Faces F
128
Euler χ
0
Genus g
1
Aspect R/r
2.63
Area 4π²Rr
15.002
Volume 2π²Rr²
2.850
Knot strands
1

Ring torus: the hole is open, every node lies on a smooth embedded surface. Nodes are coloured by Gaussian curvature — outer ring positive, top/bottom circles zero, inner throat negative; their integral over the whole surface is exactly 0.

1.00
0.38
16
8
3
2
62
24

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.