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What are some applications of elliptic geometry (positive curvature)?

This is a discussion post, please follow principles answered in MyPost doc file to fulfill the following student work: “you are expected to initiate topics and provide substantive response to the student. A substantive response will move our understanding forward through comments, questions or new resources. Remember that your claims must be supported by properly cited sources.”

Respond to this students work:

“In what physical context(s) does it make sense to think of curved space?

A curved space may make sense when pictured as a “saddle or a Pringle (Mastin, 2010, pp. 4).” Some real physical examples of this type of geometry are the curvatures or ruffles observed in lettuce or kelp (The Institute for Figuring, 2014). One way that helped me picture it is to think of negative space from a cut-out three dimensional object. Hyperbolic geometry is like dealing with the surface of a donut and elliptic geometry is like dealing with the surface of a donut hole.

What are some applications of elliptic geometry (positive curvature)?

This type of geometry is used by pilots and ship captains to navigate the globe (Castellanos, 2007, pp. 5). Elliptic geometry or spherical geometry is just like applying lines of latitude and longitude to the earth making it useful for navigation.

What are some applications of hyperbolic geometry (negative curvature)?

Imagine that you are riding in a taxi. You realize you’re running late so you ask the driver to speed up. You know you are moving faster because you can feel the change in speed and see the measurement on the speedometer or on a radar sign. Now, imagine that your speed is being measured by the radar sign but instead of measuring the speed of the car, the radar detects the light from the headlights. If we knew the given speed of light then it’s speed would change the same amount as the moving car, right? Actually, this is not true. Light always has a fixed speed, no matter how we observe it (671 million mph). In order for this measurement to be true, the surface of space must not be absolute and the existence of time must not be absolute; the speed of light is constant so other things must adjust to accommodate it. These adjustments are what make up the shape of space-time. Instead of space being measured as a flat surface or a box, it’s a box that bends, ripples and twists. This causes the measurements taken on it’s ‘surface’ to bend, ripple and twist. This relativity of space to time is Einstein’s Theory of General Relativity (Greene, 2011). Hyperbolic geometry is very useful for describing and measuring such a surface because it explains a case where flat surfaces change thus changing some of the original rules set forth by Euclid.

Where can elliptic or hyperbolic geometry be found in art? Compare at least two different examples of art that employs non-Euclidean geometry.

Introduced to the concept by Donal Coxeter in a booklet entitled ‘A Symposium on Symmetry (Schattschneider, 1990, p. 251)’, Dutch artist M.C. Escher explores hyperbolic symmetries in his work “Circle Limit (The Institute for Figuring, 2014, pp. 4-5).” In these tessellations, symmetrical objects increase endlessly from a center point while decreasing in size “to approach an infinite number of points on the boundary of an enclosed region (Schattschneider, 1990, p. 250-252).” Escher essentially captured the essence of infinity in his tessellations.

At the culmination of cubist, surrealist and expressionist art at the turn of the 20th Century, there were many artists who used non-Euclidean geometry in their artwork in a less obvious fashion. Dali, a well known surrealist used non-Euclidean geometry to reject the usual rules of artwork in such works as The Persistence of Memory is his display of melting clocks in a scalding desert (Henderson, 2013). Though Dali’s work was less straightforward in it’s use of non-Euclidean geometry than Escher’s, the viewer still has a sense of something being other-worldly when viewing their work. It’s a world that appears distorted as if some elements are reflected in water while other elements appear flattened. Both artists certainly create works that change one’s view of the world.

Are there examples of non-Euclidean geometry in non-European cultures?

Though early non-Eucledian geometry was studied by Bolyai in the Transylvanian mountains of eastern Europe, Lobachevski of Russia simultaneously, and independently of Bolyai, explored geometry omitting Euclid’s fifth postulate and arriving at similar conclusions about hyperbolic geometry. This early non-Euclidian geometry may be referred to as Bolyai-Lobachevskian geometry, as the two share the credit in it’s creation despite Gauss’ claims (Mastin, 2010 & Google Maps, n.d.).

Mastin (2010) mentions that there are earilier claims of work in similar mathematic fields credited to Omar Khayyam in the 11th Century, et al., but this work was “speculative and inconclusive in nature (pp. 8).”

Works Cited:

Castellanos, Joel. (2007). What is Non-Euclidean Geometry? Retrieved from http://www.cs.unm.edu/~joel/NonEuclid/noneuclidean.html

Google Maps. (n.d.) Retrieved from https://www.google.com/maps/@43.2417996,-77.6192335,15z

Greene, B.; MacLowry, R.; McMaster, J. (Writers), & MacLowry, R. (Director). (November 2, 2011). Fabric of the Cosmos: What is Space? [Season 39, Episode 6]. In P. Apsell (Producer), NOVA. Boston, MA: PBS.

Henderson, Linda Dalrymple. (2013). The Fourth Dimension and Non-Euclidean Geometry in Modern Art, Revised Edition. Retrieved from https://mitpress.mit.edu/books/fourth-dimension-and-non-euclidean-geometry-modern-art

The Institute for Figuring (IFF). (2014). Hyperbolic Space: Physical Models of Hyperbolic Space. Retrieved from http://www.theiff.org/oexhibits/oe1d.html

Mastin, Luke. (2010). 19th Century Mathematics – Bolyai and Lobachevsky. Retrieved from http://www.storyofmathematics.com/19th_bolyai.html.

Schattschneider, Doris. (1990). M.C. Escher: Visions of Symmetry. New York, NY: Harry N. Abrams, Inc.”


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