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Science Fiction World , v. 1, issue 4, August 1946

Page 2

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SCIENCE FICTION WORLD (2)
A ready picture of this may be had by visualizing the great circles of longitude on the earth. You might wonder if the parallels of latitude on the earth are not parallel lines despite the Elliptic promise of intersection. The answer to that one is that the parallels of latitude are not straight lines; The shortest distance between any two points on a parallel does not lie along the parallel but on a great circle. In elliptic, as in ordinary Euclidean geometry, a straight line is the shortest distance between two points. Dont, at this point, ask me how things work out on an ellipsoidal figure. It just happens that an Euclidean sphere is the handiest figure on which restricted examples of [Elliptic-goo?] can be constructed. Let's take up [Hyper-geo?] first and see what some of the results are when you have any number of non-intersecting lines. There are Euclidean figures upon which, with restrictions, representations of H-G can be constructed. One of these is the pseudosphere.....
[image]
roughly ----
As I said before, in this [geo?] parallel lines and non-intersecting lines are different entities. Parallel lines converge and diverge in opposite directions so that the distance between them, in one instance, is smaller than any assigned distance, and, in the second instance, is larger than any assigned distance. They do not meet.
[image]
Parllels in H-G
Two non-intersecting lines have one and only one common perpendicular. They diverge continuously in both directions away from it
[image]
non-intersecting lines
One of the most amazing things about HG is the fact that similar polygons of different sizes do not exist! If three triangles have all three of their angles equal to one another's the triangles are congruent and exactly alike. The extension of the theorem covers all polygons. Also we find; the sun of the angles of a triangle is less than two right angles; the construction of rectangals or cubical figures is impossible because two intersecting sets of parallel lines cannot enclose a figure with four right angles and lastly we find HG approaching Euclidean as the value of constants involved in problems increases to infinity. It might be well to point out here that in HG as in Euclidean, the infinitude of the line is accepted. In Elliptic-geo the straightline is 'finite yet boundless'.

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