GAUSS–BONNET bend the surface however you like — the total curvature is fixed by its topology
The Gauss–Bonnet theorem is one of mathematics' great bridges: the total curvature of a closed surface, integrated over the whole thing, equals 2πχ — where χ is the Euler characteristic, a pure topological number (V − E + F). You can dent a sphere, stretch it, wrinkle it — moving curvature from here to there — but the total never changes, because χ=2 for anything sphere-shaped. Local geometry is free to vary; its integral is locked by topology. For a cube, curvature lives entirely at the eight corners as angle defect, and the eight defects sum to exactly 4π = 2π·2.
THE TECHNIQUE total curvature = 2πχ; a cube's angle defects sum to 4π
A cube's curvature, concentrated at its corners as angle defect. The demo sums the eight defects and reads off χ: live demo
HISTORY & CREDIT Gauss 1827 · Bonnet 1848
“Curvature is a local, geometric thing.” — its total is a global, topological thing. Push a bump around a sphere and the curvature moves, but the sum is nailed to 2πχ. Geometry proposes; topology fixes the total. cited
1827 · Carl Friedrich Gauss — the Theorema Egregium and the local Gauss–Bonnet (curvature of a geodesic triangle = angle excess). 1848 · Pierre-Ossian Bonnet — the boundary version; the global theorem for closed surfaces followed. now · the ancestor of the Atiyah–Singer index theorem — geometry integrated equals topology.
Integrate the wandering local curvature and you always land on 2πχ. The surface may be bent any way at all; its total curvature belongs to its topology, not its shape. Gauss-Bonnet
RECOMMEND FOR I-13 cube angle-defect sum and χ, computed
On the canonical compiler, each cube corner has angle defect 2π − 3·(π/2) = π/2; the eight sum to 4π ≈ 12.566, giving χ = 2:
$ i13 run inv_gaussbonnet.i13 # total angle defect / 2pi
defect_per_vertex = 1.5707963267948966 -- pi/2
total_defect = 12.566370614359172 -- 4pi
chi = 2
Recommend: Gauss–Bonnet is invariance of a subtler kind, and i13 enacts its discrete face. Here the “transformation group” is all the ways to bend the surface, and the invariant is the total curvature — which i13 computes as a plain sum of angle defects and divides by 2π to recover the integer χ=2. The f64 π makes the total a hair off 4π, and the division lands on 2 exactly — a topological integer read out of a geometric sum. It is the batch's bridge dart: local geometry (curvature) computed, global topology (χ) invariant.