Exploration of non-Euclidean Geometries
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Definition
Spherical Geometry arose from a rewording of Playfair's postulate, where it states;
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Given a line and a point not on that line, it is possible to draw no lines through that point that is parallel to the first line.
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Methods
The goal of my exploration was to see how Euclid's first 31 propositions change when on a sphere. To accomplish this a physical sphere was used and two examples were tested on the sphere, purposely trying to find a counter example. If one was not found it was deemed a possible proposition.
Results
From my exploration of spherical geometry using the sphere, these are the results found.
Propositions
In exploration I found counter examples to three propositions (16,17,21), proving by counter example that these do not hold in spherical geometry.
Lines
Lines on a sphere are extensions of geodesics. Since there is only limited space around the sphere, when the geodesic is extended it runs into itself on the other side. In this way, lines are not infinite, but instead are defined as great circles, and in this way the term great circle and line are interchangeable.
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Circles
Angles
In spherical geometry lines are curved as they are circles. This creates an issue in measuring angles as angles can only be measured from the distance in between two straight lines. There is a method that is used to create straight lines so that angles can be measured. A tangent plane to the surface of the sphere is created and lines are added that relate to the lines on the sphere. Then the angle between these two lines is measured and is related the the measurement on the sphere.
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Triangle Angle Sum
The triangle angle sum proposition in spherical geometry does not hold. Instead of all triangles in spherical geometry having an interior angle sum equal to 180°, the proposition is reworded to:
“In spherical geometry, the sum of all interior angles in any triangle is greater than 180°, and less than 540°.” |
Saccheri Quadrilaterals
Saccheri quadrilaterals are still created the same way in spherical, but does not have the sum measurement for summit angles. As we showed in Euclidean geometry, a Saccheri quadrilateral's summit angles can be found by cutting the quadrilateral into two equal triangles. When this is done on a spherical quadrilateral the angles turn out the be obtuse angles instead of right angles.
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Gallery
Below is a gallery of the pictures taken in proposition exploration.