I got a book about the philosophy of mathematics, Stephan Körner’s **The Philosophy of Mathematics: An Introductory Essay**. It’s a subject I’m interested in, despite my lack of training. Made it to the second page before I got to something that I had to stop and ponder. 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. He lists some pure-mathematical facts and some applied-mathematical counterparts, among them:

- (2) any (Euclidean) triangle which is equiangular is also equilateral
- (5) if the angles of a triangular piece of paper are equal then its sides are also equal

So where I stopped was: what is the (Euclidean) doing in that first proposition there? Or, its counterpart, about being pieces of paper?

I’m not versed in non-Euclidean geometry. My training brought me to applied-physics applications right away. I never needed a full course in non-Euclidean geometries and have never picked up much on my own. It’s an oversight I’m embarrassed by and I sometimes think to take a proper class. So this bit about equiangular-triangles not necessarily being equilateral was new to me.

Euclidean geometry everyone knows; it’s the way space works on table tops and in normal rooms. Non-Euclidean geometries are harder to understand. It was surprisingly late that mathematicians understood they were legitimate. There are two classes of non-Euclidean geometries. One is “spherical geometries”, the way geometry works … on the surface of a sphere or a squished-out ball. This is familiar enough to people who navigate or measure large journeys on the surface of the Earth. Well. The other non-Euclidean geometry is “hyperbolic geometry”. This is how shapes work on the surface of a saddle shape. It’s familiar enough to … some mathematicians who work in non-Euclidean geometries and people who ride horses. Maybe also the horses.

But! 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. But it’s hard to think about shapes on saddles so I figured to use that only if I absolutely had to. How about on a spherical geometry? And there I got to one of the classic non-Euclidean triangles. Imagine the surface of the Earth. Imagine a point at the North Pole. (Or the South Pole, if you’d rather.) Imagine a point on the equator at longitude 0 degrees. And imagine another point on the equator at longitude 90 degrees, east or west as you like. Draw the lines connecting those three points. 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.

(Which gives me another question that proves how ignorant I am of the history of mathematics. This is an easy-to-understand example. You don’t even need to be an Age of Exploration navigator to understand it. You only need a globe. Or a ball. 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. But I defer to anyone who actually knows something about the history of non-Euclidean geometries to say.)

And that’s fine, but it’s also an equilateral triangle. I can imagine smaller equiangular triangles. Ones with interior angles nearer to 60 degrees each. They have to be smaller, but that’s all right. They all seem to be equilateral, though. The closer to 60 degree angles the smaller the triangle is and the more everything looks like it’s on a flat surface. Like a piece of paper.

So. Hyperbolic geometry, and the surface of a saddle, after all? Oh dear I hope not. Maybe I could look at something else.

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. It can be other nice, smooth shapes. Ellipsoids, for example, spheres that have got stretched out in one direction or other. For example, what if we took that globe and stretched it out some? Leave the equatorial diameter at (say) twelve inches. But expand it so that the distance from North Pole to South Pole is closer to 480 miles. This may seem extreme. But one of the secrets of mathematicians is to consider cartoonishly extreme exaggerations. They’re often useful in getting your intuition to show that something must be so.

Ah, now. Consider that North Pole-Equator-Equator triangle I had before. The North-Pole-to-equator-point distance is right about 240 miles. The equator-point-to-other-equator-point distance is more like nine and a half inches. Definitely not equilateral. But it’s equiangular; all the interior angles are 90 degrees still.

My doubts refuted! And I didn’t have to consider the saddle shape. Very good. “Euclidean” is doing some useful work in that proposition. And the specification that the triangles are pieces of paper does the same work.

And yes, I know that all the real mathematicians out there are giggling at me. This has to be pretty near the first thing one learns in non-Euclidean geometry. It’s so easy to run across, and it so defies ordinary-world intuition. I ought to take a class.

From what I recall, Euclid’s Parallel Postulate was what caused all the trouble with non-Euclidean geometries. Everyone assumed that it could be proven from the other axioms somehow, just that nobody had ever figured out how yet. Considerable effort was spent (fruitlessly) trying to prove it, one famous slacker mathematician offered a proof by contradiction with the assertion of “conduct unbecoming of lines” as the conclusion of his bad proof. I think it was Lobachevsky who bravely took the alternate tack, “What if I assume an alternative parallel postulate and the other axioms and prove that the geometry is still consistent?” and he devised the geometry where the set of lines can be modeled as the set of line segments connecting any two points on the edge of a circle, so that you can have multiple lines through a point that do not intersect a given line not containing the point. Riemann went in the other direction, decreeing that parallel lines do not exist, his set of lines can be modeled as the set of great circles on a hemisphere, clearly all intersecting. And there’s neutral geometry, where no parallel postulate is available, and less can be proven than in any of the other three geometries.

LikeLike

Yes, that’s basically how non-Euclidean geometries developed. I was maybe unclear about what I wondered. What’s got me curious is that non-Euclidean geometries weren’t understood to be a legitimate thing sooner. There were people doing surveys over large enough territories that the curvature of the Earth had to be accounted for by the late 17th century. There were explorers sailing over thousands of miles of ocean and trying to find where things are, and that demands trigonometry done on the surface of a sphere. Before that there were convoys crossing the Arabian and Sahara deserts, working out their paths by astronomical study — itself spherical geometry. And at that, people worked on the geometry of the stars in the sky thousands of years ago.

So why wasn’t it put together sooner? Why wasn’t it noticed that the workings of triangles on the surface of the Earth implied that you could do a legitimate geometry on spheres? Was it misinterpreted as simply the projections of a Euclidean geometry onto a surface, rather than something self-consistent once you made a couple of modifications? (Those being primarily changing talk of ‘straight lines’ to ‘great circles’, giving up the idea that great circles can be infinitely extended, and yes, giving up on the Parallel Postulate.) Why did it take the 19th century to notice what was, literally, under everyone’s feet and shown every night?

There must have been something keeping people from noticing this, or making them dismiss the experience of surveyors and astronomers and navigators as misleading.

LikeLike

If you don’t know this book here it is:

Courant and Robbins “What is Mathematics”

LikeLike

I do know the book! It’s within arm’s reach when I write almost all these pieces. That said I haven’t done more than skim sections in recent years; may be time for a full reread.

LikeLike

In hyperbolic geometry, an equiangular triangle is also equilateral. And saddle shape does not do justice to hyperbolic geometry, it is more like this: http://roguetemple.com/z/hyper/online.php?c=-W+Palace+-rugon+-ruglate (arrows to rotate).

LikeLike

Fascinating temple, thank you. And thanks for the word about hyperbolic geometry. I assume that’s just for regions where the curvature is constant, or does the equilateral property still hold even if the surfaces are … well, I haven’t a better word than “stretched” out more in one direction than another?

LikeLike

Yes, if the curvature is not constant, then the equilateral property no longer holds.

It seems that the terms “hyperbolic/spherical/elliptic/Euclidean geometry” mean slightly different things to various people — does the curvature have to be constant? Is gluing allowed — is a cylinder Euclidean, or a Klein Quartic hyperbolic? According to Wikipedia at least, curvature has to be constant, and gluing is not allowed.

One way to get an equiangular non-equilateral triangle in a geometry with constant curvature, which cheats a bit: pick three vertices on the equator such as (0 E, 120 E, 110 W) — the angles are all 180 degrees, and the edges correspond to 120, 110, and 130 degrees. However, are 180 degree angles really allowed?

Another way: stand on a sphere of circumference 40000 km; walk 10000 km, rotate 90 degrees CW, walk 10000 km, rotate 90 degrees CW, walk 50000 km, rotate 90 degrees CW — and you are in the starting point and orientation! A similar construction could be used on a cylinder or another glued surface. You would have to be very liberal with your definitions to call it a triangle, though.

LikeLike

There’s probably some slightly different schools of thought about what elliptical and hyperbolic geometries do mean. It’s a good reminder for people who want to talk up the objectivity and absolutely true nature of mathematics. A mathematical fact might be independent of the humans who noticed it and the culture in which they grew. But that they found it interesting is a result of social constructs.

My instinct, untarnished by experience with actually working in non-Euclidean geometries, would say that non-constant curvatures are fine, but that gluing one part of a surface to another would not be. But that is an uninformed opinion about what should be. And I do see where it’d be useful to study uniform and non-uniform curvatures for these geometries, and distinguish between the kinds.

Your equator ‘cheat’ with a triangle of three 180 degree angles is a nice one. It seems like a construction that isn’t explicitly ruled out by earlier work, but again, I say that without knowing what spherical geometers figure is normally allowed. For an analogy, it’s obvious to anyone not doing number theory that 1 ought to be a prime number. It’s obvious to a number theorist that having 1 be a prime number is more trouble than it’s worth. There may be something equivalent going on with such a triangle-of-the-equator. But I do like noticing and thinking about cheats, or possible cheats, like these.

LikeLike