**Beiträge zur Algebra und Geometrie
**

Contributions to Algebra and Geometry

Vol. 39, No. 2, pp. 379-393 (1998)

#
The Angle Defect for Arbitrary Polyhedra

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Ethan D. Bloch

Bard College, Annandale-on-Hudson, NY 12504, e-mail: bloch@bard.edu

**Abstract:** The classical polyhedral Gauss-Bonnet Theorem for surfaces uses the angle defect to measure curvature. Using a natural stratification for all polyhedra (not necessarily manifolds), the angle defect is generalized to arbitrary polyhedra in all dimensions. A Gauss-Bonnet type theorem is then proved for arbitrary polyhedra, using a modified Euler characteristic based on this stratification rather than the standard Euler characteristic.

**Keywords:** polyhedra, Gauss-Bonnet Theorem, curvature, stratification, Euler characteristic

**Classification (MSC91):** 52A25

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