![]() ![]() Consider that North Pole-Equator-Equator triangle I had before. They’re often useful in getting your intuition to show that something must be so.Īh, now. But one of the secrets of mathematicians is to consider cartoonishly extreme exaggerations. But expand it so that the distance from North Pole to South Pole is closer to 480 miles. For example, what if we took that globe and stretched it out some? Leave the equatorial diameter at (say) twelve inches. Ellipsoids, for example, spheres that have got stretched out in one direction or other. So while I, and many people, talk about spherical geometry, it doesn’t have to be literally the geometry of the surface of a sphere. Hyperbolic geometry, and the surface of a saddle, after all? Oh dear I hope not. The closer to 60 degree angles the smaller the triangle is and the more everything looks like it’s on a flat surface. They have to be smaller, but that’s all right. Ones with interior angles nearer to 60 degrees each. ![]() I can imagine smaller equiangular triangles. But I defer to anyone who actually knows something about the history of non-Euclidean geometries to say.)Īnd that’s fine, but it’s also an equilateral triangle. So why did it take so long for mathematicians to accept the existence of non-Euclidean geometries? My guess is that maybe they understood this surface stuff as a weird feature of solid geometries, rather than an internally consistent geometry. You don’t even need to be an Age of Exploration navigator to understand it. (Which gives me another question that proves how ignorant I am of the history of mathematics. That’s a triangle with three right angles on its interior, which is exactly the sort of thing you can’t have in Euclidean geometry. Draw the lines connecting those three points. And imagine another point on the equator at longitude 90 degrees, east or west as you like. (Or the South Pole, if you’d rather.) Imagine a point on the equator at longitude 0 degrees. How about on a spherical geometry? And there I got to one of the classic non-Euclidean triangles. But it’s hard to think about shapes on saddles so I figured to use that only if I absolutely had to. Maybe also the horses.īut! Could someone as amateur as I am in this field think of an equiangular but not equilateral triangle? Hyperbolic geometries seemed sure to produce one. It’s familiar enough to … some mathematicians who work in non-Euclidean geometries and people who ride horses. This is how shapes work on the surface of a saddle shape. The other non-Euclidean geometry is “hyperbolic geometry”. This is familiar enough to people who navigate or measure large journeys on the surface of the Earth. One is “spherical geometries”, the way geometry works … on the surface of a sphere or a squished-out ball. There are two classes of non-Euclidean geometries. It was surprisingly late that mathematicians understood they were legitimate. Non-Euclidean geometries are harder to understand. So this bit about equiangular-triangles not necessarily being equilateral was new to me.Įuclidean geometry everyone knows it’s the way space works on table tops and in normal rooms. It’s an oversight I’m embarrassed by and I sometimes think to take a proper class. ![]() I never needed a full course in non-Euclidean geometries and have never picked up much on my own. My training brought me to applied-physics applications right away. I’m not versed in non-Euclidean geometry. So where I stopped was: what is the (Euclidean) doing in that first proposition there? Or, its counterpart, about being pieces of paper? (5) if the angles of a triangular piece of paper are equal then its sides are also equal.(2) any (Euclidean) triangle which is equiangular is also equilateral.He lists some pure-mathematical facts and some applied-mathematical counterparts, among them: I thought to share that point and my reflections with you, because if I had to think I may as well get an essay out of it. Made it to the second page before I got to something that I had to stop and ponder. It’s a subject I’m interested in, despite my lack of training. I got a book about the philosophy of mathematics, Stephan Körner’s The Philosophy of Mathematics: An Introductory Essay. ![]()
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