Polyhedra, Moduli, and Quantum Geometry · Section 9

Volume Hessians and boundary response

A small worked example connects a hyperbolic volume function to a finite-element matrix, an electrical boundary response, and a sum over rooted forests.

Computational companion

A rectangle with two interior vertices

Divide a 3 by 2 rectangle into six unit squares, then divide each square along its northeast diagonal. There are twelve right-isosceles triangles and two interior vertices, numbered 5 and 6. Every face has counterclockwise orientation.

For an oriented triangle (a,b,c), the angle at a changes by ub − uc, with the same rule cyclically. Set u = 0 on the boundary. These leading–trailing angle coordinates preserve each triangle sum and each prescribed opposite-edge angle sum.

The square diagonals have zero cotangent weight. This is a non-strict Delaunay example with flat diagonals, not a certificate of a strictly simplicial convex polyhedron.

Triangulated 3 by 2 rectangleTwelve triangles form six squares. Interior vertices 5 and 6 are highlighted. Dashed square diagonals have zero network conductance.

Solid grid edges carry the network. Dashed diagonals carry zero weight. Interior vertices are teal.

The negative volume Hessian is twice the stiffness matrix

Let V be the sum of Lobachevsky functions over all corner angles. At the geometric configuration, adjoining the ideal point at infinity to each planar triangle gives twelve ideal tetrahedra. Their total volume is 12G, where G is Catalan’s constant.

−Du2V = 2LII = K.
LII =
4−1
−14
K =
8−2
−28
,   det K = 60.

These matrix entries and determinant are checked with rational arithmetic directly from the integer-coordinate geometry.

The computation assembles L in two independent ways: from cotangents, and by integrating products of barycentric basis gradients. It also differentiates an independent high-precision Clausen representation of V.

Numerical volume: .

The extra check at u = (0.05, −0.03) tests the Hessian identity on an abstract positive angle structure. It does not assert that the deformed triangles close to form a planar mesh.

Eliminate the interior to obtain boundary currents

For this separate electrical problem, prescribe boundary voltages vB. The interior harmonic values and the Dirichlet-to-Neumann matrix are

vI = −LII−1LIBvB,   N = LBB − LBILII−1LIB.

The program uses a linear solve, not an explicitly formed inverse. N is symmetric and annihilates constant boundary voltages. With v = x/3, the interior values are exactly 1/3 and 2/3, and the Dirichlet energy is 2/3.

This control scales the checked linear solution. It is not a new optimization or an angle deformation.

Boundary currents
VertexVoltageCurrent

The same elimination in the volume quadratic form

Eliminating the first interior coordinate from K leaves Keff = 8 − (−2)²/8 = 15/2. The determinant factorization is exact: det K = 8 × 15/2 = 60. These angle variables and the electrical boundary voltages are different variables; it is the elimination operation that they share.

The determinant counts rooted forests

A boundary-rooted spanning forest has exactly one boundary vertex in each component. A contributing forest here has two edges. Nine choices attach each interior vertex to a boundary vertex separately. Six choices join the two interior vertices and attach the pair to one of their six boundary neighbors.

∑F rooted at B ∏e ∈ F we = 9 + 6 = 15;   det K = 2² × 15 = 60.

All fifteen nonzero contributions have weight one. Exhaustive enumeration in rational arithmetic agrees with the grounded determinant. This exact calculation concerns this finite fixture; the general identity is the theorem explained in the monograph.

Reproduce and inspect the evidence

Numerical checks and declared tolerances
CheckMaximum errorToleranceResult

From the consolidated repository, with its toolkit installed in a supported Python environment, or from the extracted reproduction bundle after installing its requirements:

python experiments/section9/run.py

For a repository run without an editable installation, add PYTHONPATH=packages/volume-toolkit to that command. The ZIP contains the complete experiment and a labeled export of the maintained numerical modules; detached execution needs only NumPy, SciPy and mpmath. The result records actual source and input hashes, dependency versions, and methods. Detached runs explicitly lack a Git working-tree status and may omit the book assets. The volume uses the binary64 Bloch–Wigner kernel. The independent derivative comparison uses 50 decimal digits; neither is an interval enclosure.

Individual Python links are inspection copies; use the complete ZIP or repository to run the experiment. Canonical source labels: eq:ipolyAngleCoordinates, eq:ipolyHessian, eq:ipolyDtN, eq:ipolySchur, eq:ipolyForestDeterminant. When included, the full PDF is the imported reference edition, not a newly reviewed release. This is an authored worked-example companion, not a conversion of the complete chapter. It makes no global-optimality, strict-realization, or nearly-linear-solver claim.