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.
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