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Paper Surface Geometry: Surveying a Locally Euclidean Universe

Andrew D. Hwang
The American Mathematical Monthly
Vol. 120, No. 6 (June–July 2013), pp. 487-499
DOI: 10.4169/amer.math.monthly.120.06.487
Stable URL: http://www.jstor.org/stable/10.4169/amer.math.monthly.120.06.487
Page Count: 13
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Paper Surface Geometry: Surveying a Locally Euclidean Universe
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Abstract

Abstract The concepts of parallel transport and intrinsic (Gaussian) curvature arising in the differential geometry of surfaces may be pleasantly and concretely investigated using paper models and familiar notions of length and angle. This paper introduces parallel transport and curvature in the context of “locally Euclidean” surfaces: polyhedra, for which curvature is concentrated at isolated points, and “polycones”, for which curvature is concentrated along circular arcs. The geometry of polycones is used to recover a strikingly simple intrinsic formula for the Gaussian curvature of a surface of rotation. We give instructions for building paper models of the catenoid and surfaces of constant Gaussian curvature.

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