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.
REFERENCES
1.
William M.
Smart
and
Robin M.
Green
, Textbook on Spherical Astronomy
(
Cambridge U. P.
,
Cambridge
, 1977
).2.
Christian R.
Hayes
,
Allison M.
Matthews
,
Yiqing
Song
,
T. Sean
Linden
,
Robert F.
Wilson
,
Molly
Finn
,
Xiaoshan
Huang
,
Kelsey E.
Johnson
,
Anne M.
McAlister
,
Brian
Prager
,
Richard
Seifert
,
Sandra E.
Liss
,
Andrew M.
Burkhardt
, and
Nicholas
Troup.
“
First results from the dark skies, bright kids astronomy club draw-a-scientist test
,” Phys. Rev. Phys. Educ. Res.
16
(1
), 010131-1
–010131-12
(2020
).3.
Nafis I.
Karim
,
Alexandru
Maries
, and
Chandralekha
Singh
, “
Do evidence-based active-engagement courses reduce the gender gap in introductory physics?
,” Eur. J. Phys.
39
(2
), 025701
(2018
).4.
Wei
Lee
,
Heidi L.
Gilley
, and
Joshua B.
Caris
, “
Finding the surface temperature of the Sun using a parked car
,” Am. J. Phys.
65
(11
), 1105
–1109
(1997
).5.
Luisa M.
Rebull
,
Debbie A.
French
,
Wendi
Laurence
,
Tracyanne
Roberts
,
Michael T.
Fitzgerald
,
Varoujan
Gorjian
, and
Gordon K.
Squires
, “
Major outcomes of an authentic astronomy research experience professional development program: An analysis of 8 years of data from a teacher research program
,” Phys. Rev. Phys. Educ. Res.
14
(2
), 020102-1
–020102-21
(2018
).6.
Malcolm
Wells
,
David
Hestenes
, and
Gregg
Swackhamer
, “
A modeling method for high school physics instruction
,” Am. J. Phys.
63
(7
), 606
–619
(1995
).7.
Afrodita L.
Boldea
and
Ovidiu
Vaduvescu
, “
Teaching students about informatics and astronomy using real data for detection of asteroids
,” Eur. J. Phys.
38
(5
), 055706-1
–005706-13
(2017
).8.
Benjamin
Farr
,
GionMatthias
Schelbert
, and
Laura
Trouille
, “
Gravitational wave science in the high school classroom
,” Am. J. Phys.
80
(10
), 898
–904
(2012
).9.
Snapseed. Snapseed (
2020
).10.
GeoGebra. GeoGebra Classic 5.0 User Manual, <https://www.geogebra.org/> (
2019
).11.
Hugo
Caerols
and
Felipe A.
Asenjo
, “
Estimating the moon-to-earth radius ratio with a smartphone, a telescope, and an eclipse
,” Phys. Teach.
58
(7
), 497
–501
(2020
).12.
Gazetteer of Planetary Nomenclature, International Astronomical Union (IAU) Working Group for Planetary System Nomenclature (WGPSN)
(2020
).13.
14.
Rodrigo A.
Carrasco
,
Hugo
Caerols
, and
Felipe A.
Asenjo
, Moon Geometry Repository v.1.3. <https://github.com/raxlab/moonGeometry> (2021
).15.
16.
John
Wilkinson
, The Moon in Close-up: A Next Generation Astronomer's Guide
(
Springer Science & Business Media
,
Berlin
, 2010
).17.
Yang
Cai
, “
How many pixels do we need to see things?
,” in Lecture Notes in Computer Science, ICCS 2003
, edited by
P. M. A.
Sloot
,
D.
Abramson
,
A. V.
Bogdanov
,
Y. E.
Gorbachev
,
J. J.
Dongarra
, and
A. Y.
Zomaya
(
Springer
,
Berlin
, 2003
), pp. 1064
–1073
.18.
Antonio
Torralba
, “
How many pixels make an image?
,” Visual Neurosci.
26
(1
), 123
–131
(2009
).© 2021 Author(s). Published under an exclusive license by American Association of Physics Teachers.
2021
Author(s)
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