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Wolfram Alpha » Explore anything with the first computational knowledge engine. Wolfram Demonstrations Project » Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. The flatworm's answer, however, might be totally different from ours. If we start with three points on the flatworm's universe, we can take a shortcut through space and connect them by straight line segments, and the angles of the triangle we obtain will sum to degrees.
The flatworm, on the other hand, might claim that the sum of the angles is not constant and that it depends on the size of the triangle. A spherical triangle can have three right angles beetween its pairs of great circle sides, even though the triangle with straight sides determined by the same vertices has angles of 60 degrees.
Consider the particular case where the flatworm is confined to the surface of a sphere. He could construct a large triangle on the sphere running from the north pole to a point on the equator, then one-quarter of the way around the equator, and back to the north pole.
Each of the angles of this spherical triangle is 90 degrees, so the angle sum is much greater than degrees. For us who can leave the surface of the sphere to connect the points in space, the north pole and the two points on the equator form an equilateral triangle with each angle 60 degrees.
The geometric study of the sphere has a very long history, but by and large it was considered a subtopic in solid geometry. People spoke about great circle arcs, and they even knew in some sense that these represented the shortest paths on the surface of the globe.
But they did not think of them as the same sorts of objects as the segments that provided the shortest distances in plane geometry. In ancient times, Ptolemy certainly knew that three great circle arcs forming a spherical triangle would determine angles adding up to more than degrees, and in fact he was able to prove that the bigger the area of the triangle, the larger the angle sum.
With proper choice of units this relationship could be made explicit: the area of a triangular region on the sphere is precisely the amount by which its angle sum exceeds degrees. Why didn't Ptolemy realize that this was an example of a non-Euclidean geometry, where the important Euclidean theorem that the angle sum equals degrees simply does not hold?
The answer is that he did not think of the relationships among points of a sphere and great circle arcs as a geometry. To qualify as a geometry, a system would have to have elements corresponding to points and lines, and the first four axioms would have to be satisfied.
The system consisting of points on a sphere and lines given by great circle arcs did satisfy the third and fourth axiom, and even the second if we interpret it correctly, but it did not satisfy the first axiom. Although two nearby points on the sphere determine a unique great circle arc, there are point pairs for which this is not true. More than one great circle arc joins the north and south poles, and in fact there are infinitely many half-circles of longitude joining these two points, all of the same length.
Thus spherical geometry did not qualify as a non-Euclidean geometry, although later on in this chapter we will see that it was closely related to one. Each of them realized that it was possible to construct a two-dimensional geometry with points and shortest distance lines satisfying the first four axioms of Euclidean geometry, but not the fifth.
The other possibility, viz. Saccheri, as all the others, assumed probably, correctly that Euclid had in mind exactly this interpretation: straight lines may be extended so as to have infinite length.
Riemann was the first to notice that, although meant by Euclid, this interpretation does not necessarily follow from the postulate : A piece of straight line may be extended indefinitely.
Riemann wrote:. The unboundedness of space possesses But its infinite extent by no means follows from this.
Circles can be extended indefinitely since they have no ends. However, circles are of finite extent. Implicit in the first postulate - A straight line may be drawn between any two points - was the assumption that such a line is unique.
So Riemann modified Euclid's Postulates 1, 2, and 5 to. It is very easy to envisage the objects points, lines, and the plane of a geometry that satisfies the five modified postulates. Plane is a sphere, lines are the great circles, i.
Furthermore, if this is true of a single triangle, this is also true of all possible triangles. With this definition, any two points determine a unique line so that the traditional form of Euclid's first postulate is restored. Thus modified, spherical geometry became what Klein called elliptical geometry. Spherical trigonometry This is the branch of spherical geometry dealing with the ratios between the sides and angles of spherical triangles. Spherical astronomy , of importance in positional astronomy and space exploration, is the application of spherical trigonometry to determinations of stellar positions on the celestial sphere.
Generalization to elliptical geometry It was Felix Klein who first saw clearly how to rid spherical geometry of its one blemish: the fact that two lines have not one but two common points.
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