THE HOLONOMY carry an arrow around a loop and it comes back rotated — by the area you enclosed
Parallel transport moves a vector along a surface while keeping it “as straight as possible.” On a flat plane, carry an arrow around a closed loop and it returns pointing the same way. On a curved surface, it comes back rotated — and the rotation, the holonomy, measures the curvature enclosed by the loop. On a sphere, transport an arrow around a triangle of three right angles (one-eighth of the sphere) and it returns turned by exactly π/2 — the triangle's spherical excess, which is its area. The holonomy is what a Foucault pendulum measures, and the geometric phase (Berry phase) a quantum system picks up going around a loop. The path can wander any way inside the loop; the holonomy depends only on the area enclosed.
THE TECHNIQUE transport around a loop returns rotated by the enclosed area (excess)
Parallel transport around a spherical triangle with three right angles. The demo computes the holonomy as the angle excess — the enclosed area: live demo
HISTORY & CREDIT Levi-Civita 1917 (excess: Harriot 1603)
“An arrow carried around and back points the same way.” — on a curved surface it comes back rotated, by the curvature you enclosed. The rotation is the surface telling you it is not flat. cited
1603 · Thomas Harriot (and Girard, 1629) — a spherical triangle's area equals its angle excess. 1917 · Tullio Levi-Civita — parallel transport; holonomy as the signature of curvature. now · the Foucault pendulum, gauge theory's Wilson loop, Berry's geometric phase.
Go around the loop and the arrow turns by the area you enclosed — wander the path however you like inside it, the holonomy only counts the region. Curvature, read off a round trip. Levi-Civita 1917
RECOMMEND FOR I-13 holonomy = spherical excess, computed
On the canonical compiler, a spherical triangle with three right angles has angle sum 3π/2, so the holonomy (excess over π) is π/2 ≈ 1.5708 — its area:
$ i13 run inv_holonomy.i13 # angle sum - pi
angle_sum = 4.71238898038469 -- 3pi/2
holonomy = 1.5707963267948966 -- pi/2, the enclosed area
Recommend: the holonomy is invariance's inverse image — the exact failure of an arrow to return unchanged, and that failure is itself an invariant of the loop (its enclosed area). i13 computes it as an angle sum minus π, landing on π/2. It pairs with Gauss–Bonnet (dart 281): holonomy is curvature integrated over a region, the same way total curvature is integrated over the whole surface. Where most of the batch shows a quantity that a transformation cannot change, this one shows a transformation (the round trip) whose net effect is fixed by the area — the geometric phase, computed in f64.