+1 862 207 3288 History of Mathematics: Non-Euclidean Geometries and Curved Space (M5AR)
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.”
The 19th century saw great ferment in geometry. Mathematicians had long been troubled by Euclid’s Parallel Postulate.
Gauss, Lobachevsky and Bolyai showed that there was nothing, in a formal sense, to prevent them from postulating the negation of Euclid’s parallel posulate. In other words, we may develop alternative geometries in which, given any line L and any particular point P not on L, either
1. there is more than one line that passes through P and is parallel L (Hyperbolic geometry), or
2. there is no line that passes through P and is parallel L (Elliptic geometry).
This site called The Geometry of Spacetime will help you to visualize curved space (elliptic and hyperbolic non-Euclidean space).
At first, mathematicians thought that the non-Euclidean geometries were purely theoretical, but 20th century thinkers (including mathematicians, physicists and artists) found numerous applications.
• In what physical context(s) does it make sense to think of curved space?
• What are some applications of elliptic geometry (positive curvature)?
• What are some applications of hyperbolic geometry (negative curvature)?
• Where can elliptic or hyperbolic geometry be found in art? Compare at least two different examples of art that employs non-Euclidean geometry.
• Are there examples of non-Euclidean geometry in non-European cultures?
Answered Discussion:
In what physical context(s) does it make sense to think of curved space?
It makes sense to think of curved space in the context of large masses and their ability to bend light. Large masses are known to bend light beams such as in the displacement of star images around an eclipsed sun (Kelly et al. 1127). Kelly et al. give an example of the gravitational lens (group of galaxies), that bends light as it travels from a distant source towards an observer (1127).The effect is that curved arcs and spots can be observed from the bended light. Einstein predicted the effect in his relativity theory
What are some applications of elliptic geometry (positive curvature)?
Elliptic geometry finds use in the determination of minimal submanifolds (Dajczer and Vlachos 1). For example, a family of an elliptic surface is used to clarify the geometry around its associated family of minimal submanifolds (Dajczer and Vlachos 1). Elliptic geometry also finds application in Riemannian Manifolds and minimal maps. For example, Huang and Ruan indicate that elliptic equations find use in the determination of theorems applicable to Riemannian Manifolds (189). Savas-Halilaj and Smoczyk add to this application by using the maximum elliptic principle to obtain Bernstein type theories that find application in computations involving minimal maps found in between Riemannian manifolds (550).
What are some applications of hyperbolic geometry (negative curvature)?
Hyperbolic geometry finds use in quantum chaos and cosmology computations. (Bölte and Steiner 249). For example, hyperbolic geometry is used to compute temperature fluctuations in cosmic microwave background (Bölte and Steiner 249) and is also used in Maass waveforms computations. Another source, Bao and Chen, also indicate that hyperbolic geometry is used in the characterization of Lambert Quadrilaterals and Saccheri Quadrilaterals (6).
Where can elliptic or hyperbolic geometry be found in art? Compare at least two different examples of art that employs non-Euclidean geometry.
Non-Euclidean geometry is found in art such as in Alexander Calder’s mobiles and Ben Nicholson’s reliefs (Malloy 1). Calder developed abstract structures using thin wires to explore abstracted and at times intersecting planes, and sometimes having spheres overlapping each other (Malloy 4). Calder incorporated kinetic movement in his sculptures resulting in a series of kinetic mobiles that moved according to the air currents in the surrounding environment (Malloy 4). Nicholson’s white reliefs were a sculptural series produced between 1934 and 1937. The reliefs emphasized the relationship of abstracted spatial planes through the curved out layers of a wooden board (Malloy 5). Nicholson uses a variety of effects that range from recession to projection, and interaction and separateness among elements in his works. (Malloy 5).
Are there examples of non-Euclidean geometry in non-European cultures?
All evidence in the available literature suggests that the origin of Non-Euclidean geometry is in European cultures. There are two types of non-Euclidean geometries, hyperbolic and elliptic geometries. Hyperbolic geometry is thought to have been developed by Bolyai (Hungarian), and Lobachevski, a Russian (Storyofmathematics.com). Elliptic geometry is believed to have been developed by Bernhard Reinmann a German (Storyofmathematics.com). However, non-Euclidean geometry has since spread to other non-European countries and is used for various applications such as art, architecture and mathematics.
Works Cited
Bao, Hui, and Xingdi Chen. ‘A New Hyperbolic Area Formula of a Hyperbolic Triangle and Its Applications’. Journal of Mathematics 2014 (2014): 1-8. Web. 13 July 2015.
Bölte, Jens, and F Steiner. Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology. Cambridge, UK: Cambridge University Press, 2012. Print.
Dajczer, Marcos, and Theodoros Vlachos. ‘The Associated Family of an Elliptic Surface and an Application to Minimal Sub manifolds’. Geometriae Dedicata (2015): n.1-20. Web.
Huang, Qin, and Qihua Ruan. ‘Applications of Some Elliptic Equations in Riemannian Manifolds’. Journal of Mathematical Analysis and Applications 409.1 (2014): 189-196. Web.
Kelly, P. L. et al. ‘Multiple Images of A Highly Magnified Supernova Formed By An Early-Type Cluster Galaxy Lens’. Science 347.6226 (2015): 1123-1126. Web. 14 July 2015.
Malloy, Vanja. ‘Non-Euclidean Space, Movement and Astronomy in Modern Art: Alexander Calder’s Mobiles and Ben Nicholson’s Reliefs’. EPJ Web of Conferences 58 (2013): 1-5. Web. 14 July 2015.
Savas-Halilaj, Andreas, and Knut Smoczyk. ‘Bernstein Theorems for Length and Area Decreasing Minimal Maps’. Calc. Var. 50.3-4 (2013): 549-577. Web.
Storyofmathematics.com. ‘Riemann – 19Th Century Mathematics – The Story Of Mathematics’. N.p., 2015. Web. 14 July 2015.
Storyofmathematics.com. ‘Bolyai And Lobachevsky – 19Th Century Mathematics – The Story Of Mathematics’. N.p., 2015. Web. 14 July 2015.
