Equilateral right triangles

I am definitely not the type of person who enjoys math, and although I can usually figure out the problems my math book throws at me, I'm not one who takes great pleasure in proving random theorems. However, once in a blue moon, (that is to say, quite infrequently), there is something which hits my mathematical funny bone and for some reason strikes me as being funny, interesting, and even intriguing. The last time this happened was when I was doing algebra, and I learned about the imaginary numbers (What's the square root of negative one? i. What is i? The square root of negative one.) Now that I'm doing geometry, something else has emerged from the pages of my math book which is quite fascinating, at least to me.

I am nearing the end of my geometry book (indeed, I have but to take the final test and I shall be finished), and the last chapter was labeled with the caliginous title "Non-Euclidean Geometries." "Non-Euclidean" was scary enough, but then to learn that there is more than one of these exotic geometries was almost too much for me to handle. But I pressed on through this foreboding chapter, and learnt about Spherical Geometry, Lobachevskian Geometry (did you know the sum of the angles of a triangle is less than 180°?), and Reimannian Geometry. The latter two of these were quite odd, for I could not visualise them, but at least I could understand Spherical Geometry. It was in the first lesson that I encountered something that actually made me laugh (quite extraordinary for a math book). There, right there on the page was a picture of an equilateral right triangle!

Now please allow me to explain. Unless you've done geometry before, this may not quite make sense, indeed, it may not even if you have taken geometry. Please bear with me, however. First of all, equilateral triangles are triangles whose sides are all the same length. Also you may know that all equilateral triangles happen to be equiangular, which simply means all its angles have the same measure. Now for the second part. Right triangles are triangles which have one right angle, right meaning that it measures 90°. Now think about it for a moment. This equilateral right triangle has one angle of 90°, and since it's also equiangular, the other two angles must be 90° as well, which is the difference between vertical and horizontal. Don't worry if you can't picture it, as in Euclidean Geometry it's impossible.

That's why it was so funny to see one in Spherical Geometry! You see, this type of geometry is done on the curved surface of a sphere (a ball) instead of a flat surface. Thus this type of triangle is entirely possible. Just so you can see what I've been talking about, I found this picture on Wikipedia which demonstrates the concept. As you can see, every one of the lines of triangle ABC is equal, and each one is 90°! Now just wait till you see the triangles in Lobachevskian geometry...