Using smartphone photographs of the Moon to acquaint students with non-Euclidean geometry

TL;DR

This paper demonstrates how smartphone photographs of the Moon can be used to teach non-Euclidean geometry concepts to students by enabling practical calculations of lunar features with minimal error, thus making abstract ideas more tangible.

Contribution

It introduces a novel educational approach using astronomical images to concretely teach non-Euclidean geometry to high school and college students.

Findings

Students can accurately estimate lunar feature sizes with less than 4% error.

The method effectively illustrates the difference between Euclidean and spherical geometry.

Students gain a tangible understanding of non-Euclidean concepts through practical calculations.

Abstract

Although they are sometimes considered problematic to grasp by students, the concepts behind non-Euclidean geometry can be taught using astronomical images. By using photographs of the Moon taken with a smartphone through a simple telescope, we were able to introduce these concepts to high-school students and college newcomers. By recognizing different Moon geological structures within the photograph, we teach students how to calculate distances of mountain ranges or areas of craters on the Moon's surface, introducing the notions of geodesics and spherical triangles. Furthermore, students can empirically see that the correct estimations for the actual values cannot be obtained using flat geometry. Instead, by using three--dimensional curved geometry, precise estimates of lengths and areas of geological elements in the Moon can be computed with less than 4\% error. These procedures help students understand, concretely, non-Euclidean geometry concepts.