The bug on the sphere

Thursday, 28 December 2023 The bug on the sphere Amelia Carolina Sparavigna This paper examines the classic problem of a hypothetical two-dimensional inhabitant, a "bug on a sphere," attempting to determine the intrinsic curvature of its world. Following a method inspired by Feynman, the bug measures the circumference (C) and the radial distance (s) of a circle traced on the sphere's surface. The discovery that the measured circumference is less than the Euclidean prediction (2πs) is used to quantify the spatial distortion. By analyzing the limit of the difference between the actual and predicted radius as the measured radius approaches zero, the paper demonstrates the calculation of the Gaussian Curvature. The curvature of the sphere is analytically determined to be 1/R2, where R is the sphere's radius. The curvature as determined by a bug on a sphere. Problem: a bug on a sphere wants to evaluate the curvature of his world. "The bug might make a circle like the one shown in" the following figure "and measure circumferences". https://www.feynmanlectures.caltech.edu/II_42.html “He would discover that the circumference is less than 2π times the radius [here in this figure, the radius of the circumference is "r"]. (You can see that because from the wisdom of our threedimensional view it is obvious that what he calls the “radius” [here in the figure, "radius" is "s"] is a curve which is longer than the true radius of the circle.) Suppose that the bug on the sphere had read Euclid, and decided to predict a radius by dividing the circumference C by 2π. Then he would find that the measured radius was larger than the predicted radius. Pursuing the subject, he might define the difference to be the “excess radius,” … and study how the excess radius effect depended on the size of the circle." (Feynman). The curvature is: determined by: This is also the Gauss curvature. More at: Sparavigna, A. C. (2021). Metrica e Curvature di Gauss e Riemann. Zenodo. https://doi.org/10.5281/zenodo.4724602 Conclusions The analysis confirms that the intrinsic curvature of a sphere, 1/R2, can be determined entirely by local measurements taken within its two-dimensional surface, without requiring reference to the embedding three-dimensional space. The discrepancy found between the measured circumference (C) and the Euclidean expectation (2πs) effectively serves as a local, geometrical signature of the surface's deviation from flatness. This foundational problem beautifully illustrates how the physical reality of a curved manifold dictates the principles of measurement and motion, where the path of the "bug" seeking the shortest distance is, in essence, a geodesic. This result serves as a cornerstone for more generalized studies in Riemannian Geometry.