Exploration of non-Euclidean Geometries
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Description
Hyperbolic Geometry arrises from a rewording of Playfair's postulate to state;
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Given a line and a point not on that line it is possible to draw infinitely many lines through the point that is parallel to the first line.
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Model
For this geometry a Poincaré Disk was used. This model is a disk that has a distortion on the edge so that all points on the edge are infinitely far apart. This can be seen best in the picture on the right. All the fish in the picture are the same size however as the images grow closer to the edge distortion becomes greater making them look visually smaller.
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Methods
The goal of exploration was to see how Euclid's first 31 propositions change when on a Poincaré Disk. To accomplish this, a computer program called Non-Euclid was used, and I tested two examples on the disk, purposely trying to find a counter example. If one was not found it was deemed a possible proposition.
Results
From exploration of hyperbolic geometry using Non-Euclid, these are the results found.
Propositions
In exploration counter examples were found to two propositions (29,30), proving by counter example that these do not hold in hyperbolic geometry.
Lines
Lines in hyperbolic geometry are still extended geodesics, but now there are two different kinds of lines. Diameters and circle intersections. Diameters, modeled in blue in the related image, are lines that cut directly through the disk. Circle intersections, modeled in red, are lines that were created by circles intersecting with the disk at 90°.
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Circles
Circles themselves take on a different shape. Because of the properties of the Poincar disk, and therefore the way distance distorts, the circles on the center of the disk, modeled in green, will look like Euclidean ones, while circles on the edge, modeled in blue, will have a center closer to the edge of the disk. When the circle is located on the center of the disk distance is being distorted equally on all points. When it is located on the edge, distance is being distorted more on one side and therefore draws the center closer to that side.
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Angles
Angles in hyperbolic geometry are also going to be measured in degrees or radians, however the process of measuring the angles is again going to be different. Instead of drawing a tangent plane, we can draw tangent lines that go through the point, tangent to the curves. Then measure to angle via Euclidean, and call it the hyperbolic angle.
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Triangle Angle Sum
Saccheri Quadrilaterals
Saccheri quadrilaterals are still created the same way in hyperbolic geometry, but do not have the same 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 hyperbolic quadrilateral the angles turn out to be acute angles instead of right angles.
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Gallery
In exploration of the propositions pictures were taken that showed a counter example if one was found, or an example of a proposition that still holds. In examples where measurements were important, on the side of the picture a menu is located that shows the values of different angles and sides.