Non-Euclidean geometry experiment: Please do try this at home!

SoniaC, wielding her protractor, measuring the angles inside the triangle on her table tennis ball.

Looking for an intellectually meaty graphic novel that blends historical, technological, and mathematical elements into a quirky steampunk masterpiece? Have a hankering for intensively footnoted fiction? You’re in luck! We recently wrapped up our reading of Sydney Padua’s The Thrilling Adventures of Lovelace and Babbage: The (Mostly) True Story of the First Computer and can confidently and enthusiastically recommend it to, well, nearly everybody. Yes, it truly is that good.

After wringing readers’ hearts with the true tale of the unfulfilled promise of Ada Lovelace and Charles Babbage‘s Victorian-era mechanical computing collaboration, Padua spins a series of delightful counterfactual yarns set in a “pocket universe” in which Babbage’s mechanical computer is constructed and he and Lovelace become something like joint-CTOs of the United Kingdom. Padua has written many prominent nineteenth-century authors, engineers, and mathematicians into these stories as guest stars and she works aspects of their real-world accomplishments into the narrative in ways that ensure curious readers will pore over her magnificently crafted footnotes and endnotes and then, to learn more, head off to Wikipedia and Google.

Answers to the elliptical-space triangle quiz question in one section of our Upper Intermediate I classes.

The Thrilling Adventures of Lovelace and Babbage‘s final two stories are a short vignette featuring George Boole and introducing Boolean logic and a longer riff on the Alice novels of Lewis Carroll (aka Oxford mathematician Charles Dodgson). In the latter, Padua introduces non-Euclidean geometry. That’s why the reading comprehension portion of the quiz administered to our Upper Intermediate I students last weekend challenged them to correctly use Boolean logic’s three operators (AND, OR, and NOT) and also to measure the magnitudes of the three angles inside a triangle drawn in a non-Euclidean space. That’s what Sonia is doing in the image atop this post. Her and her classmates’ solutions are shown above. All of their answers were correct, within a small margin of error to allow for the difficulty inherent in using a semirigid plastic protractor to measure angles on a fiddly little ping-pong ball.

RyanY carefully aligning his protractor with one of the vertices of his non-Euclidean triangle.

A sphere’s outer surface is a commonly-used model in explanations of elliptical geometry and, as you’ve likely guessed if you’ve scrutinized the images included in this post (like this closeup of Ryan aligning his protractor with one of the vertices of his elliptical-space triangle), the non-Euclidean space which we utilized for the quiz happened to be the oh-so-prosaic exterior of a table tennis ball. We prepared enough balls, each bearing a differently-proportioned triangle, to ensure that every student in each class would be measuring a unique set of angles.

HarrisY measuring his elliptical-space triangle's interior angles.

Drawing a triangle and summing its interior angles is one, but not the only, way to discover whether one is dealing with Euclidean or a non-Euclidean geometry. In familiar flat-plane (i.e. Euclidean) geometry, the magnitudes of the interior angles of any triangle always add up to exactly 180°. The angles in a triangle drawn in a non-Euclidean space add up to either less than 180° (in hyperbolic space) or more than 180° (in elliptical space). This was borne out by the results of our in-class experiment-disguised-as-a-quiz-question. In the answers to this question shown above, the sums of the triangles’ angles ranged from 245° to 365°.

We had read and discussed the relevant The Thrilling Adventures of Lovelace and Babbage endnote in class but, as indicated by the hand-crafted graphite shocked-face emoji that graced one quiz paper, nothing beats hands-on experience!