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

Abstract

Non-Euclidean geometry can be taught to students 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 lower-level college students. We teach students how to calculate lengths of mountain ranges or areas of craters on the Moon's surface and introduce ideas of geodesics and spherical triangles. Students can see that accurate measurements cannot be obtained using flat geometry. Instead, by using three-dimensional curved geometry, estimates of lengths and areas can be computed with less than 4% error.

Non-Euclidean geometries in and for the mathematics teacher education: A literature review

“…En cuanto a las estrategias didácticas utilizadas y recomendadas desde la investigación, dado Relativo al conocimiento didáctico, se propone una enseñanza por medio de la interdisciplinariedad y contextualidad, en otras palabras, a través de diferentes contextos de aplicación que favorezcan el acercamiento a estas geometrías (Aparecida;Romualdo, 2015;Caerols;Carrasco;Asenjo, 2021;Conceição, 2018;Pataki, 2003;Pinto;Portella;Machado, 2017;Song;Schwenz, 2013);…”

“…Las geometrías no euclidianas en y para la formación del profesorado de matemáticas: una revisión de literatura Olhar de professor, Ponta Grossa, v. 27, p. 1-25, e-22500.021, 2024. Disponível em <https://revistas.uepg.br/index.php/olhardeprofessor> 16 por ejemplo, para la geometría esférica, algunos de estos contextos son: la geografía (Conceição, 2018;Pataki, 2003;Pinto, 2013), arte-fotografía (Caerols;Carrasco;Asenjo, 2021), navegación aérea y marítima (APARECIDA; ROMUALDO, 2015;PINTO, 2013), la geología -principalmente en el paleomagnetismo que estudia el campo magnético de la Tierra- (Aparecida;Romualdo, 2015), la astronomía (Aparecida;Romualdo, 2015;Caerols;Carrasco;Asenjo, 2021), la propia matemática (Pinto;Carneiro, 2011), y el Sistema de Posicionamiento Global (GPS) (Song;Schwenz, 2013).…”

Measuring the elliptical form of the Moon’s terminator curve and its illuminated area by direct and indirect methods

“…In [3] we have already discussed how to use the Moon for teaching non-Euclidean geometries. Since the Moon is a spherical body, what we are actually observing in a photo is the projection of part of its spherical surface illuminated in a plane that passes through the centre of the Moon.…”

“…For lunar photography there are several recommendations, we suggest the reader review the indications in [3] where we have put a special section on the subject. When taking pictures with a smartphone, the greater the number of pixels in the camera, the error associated with the measurements will decrease.…”

“…The Moon can also be an excellent object to make different astronomical calculations and measurements in order to prove the powerfulness of physics and mathematics [2]. For example, selenography can be used to introduce non-Euclidean geometries to undergraduate students [3]. The measurements can be done using simply a photography taken with a smartphone and a geometrical software.…”