![]() The triangle's angles summed to three right angles and the circle's circumference was only four times the radius. Their properties were radically different from Euclidean triangles and circles. We can now return to the triangles and circles visited earlier. The great circles are the routes taken by ships and airlines over the surface of the earth, whenever possible, since they are the paths of least distance. Since it is the great circle, it is the curve of least distance in the surface of the sphere between A and B. It connects A and B by a path that deviates to the North. The great circle passing through points A and B is shown in the second figure. For geodesics are produced by the intersection of the sphere with planes that pass through the center of the sphere. However it is not the analog of straight line in this geometry, a geodesic. The parallel of latitude is a parallel to the equator. In the figure below, points A and B of the same latitude are connected by a parallel of latitude. People sometimes mistake a parallel of latitude for a great circle. All pairs of great circles intersect somewhere. In such a geometry, there are no parallel lines. In short, the new geometry of 5 NONE is just the geometry of of great circles on spheres. They are the " great circles." That is, they are the circles produced by the intersection of the sphere with a plane that passes through the center of the sphere. There is a simple way of creating geodesics on the surface of a sphere. The curve that implements the shortest distance in the surface is known as a "geodesic". There is no burrowing into the earth to get a shorter distance between two points. But now we are forced to remain on the surface of the sphere in finding the shortest distance. A straight line between two points A and B is still the shortest distance between two points. We do need to adjust our notion of what a straight line is. The perpendiculars we erected to it in the last chapter then just become lines of longitude all of which intersect at the North Pole, that is, at O. To see how the connection to the geometry of 5 NONE works, we need only identify the line AGG'G'' with the equator. That just means that the curvature is everywhere the same. The surface of a sphere has constant curvature. Of a sphere, such as is our own earth, to very good approximation. The geometry of 5 NONE proves to be very familiar it is just the geometry that is natural to the surface The New Geometry of 5 NONE is Spherical Geometry We now risk a new danger: we may just overlook the fact that we are really dealing with new and different geometries. It is a trick that makes visualizing the geometries of 5 NONE and 5 MORE easy. We will see in this chapter how this arises. Recognizing that fact makes it easy to visualize these new geometries and one rapidly develops a sense of the sorts of results that will be demonstrable in them. Naturally in surfaces of constant curvature. The geometry of 5 NONE and the geometry of the other postulate 5 MORE turn out to be the geometries that arise The surprising thing is that this is not so. You would be forgiven for thinking that the new geometry of 5 NONE is a very peculiar and unfamiliar geometry and that there is no easy way to comprehend it as a whole. ![]() So the sum of its angles is three right angles (and not the two right angles dictated by Euclid's geometry). The triangle OGG', for example, has three angles, each of one right angle. Each of its quadrants are triangles with odd properties. ![]() Its cirumference is both a circle and a straight line at the same time. Its circumference is only 4 times is radius (and not the 2π times its radius dictated by Euclid's geometry). We constructed a circle with center O and circumference G, G', G'', G'''. The outcome was a laborious construction of circles and triangles with some quite peculiar properties. We drew lines and found points only as allowed by the various postulates. ![]() In the last chapter, we explored the geometry induced by the postulate 5 NONE by means of the traditional construction techniques of geometry familiar to Euclid. The New Geometry of 5 NONE is Spherical GeometryĮuclid's Postulates and Some Non-Euclidean Alternatives.
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