Triangles On A Sphere: More Complicated Than You Might Think

If any of you have read Steven Hawkings' "The Grand Design", you might have seen the bit about how the straightest distance from A to B on a sphere is actually curved when you draw it on a map. Think about it. The actual straightest distance would have to go through the sphere! But anyways, all of the rules of a triangle that apply to regular triangles don't work because the lines are curved! It's not even a triangle, or even a polygon! It has 3 vertices, but the rest of the lines curve, which makes laws that apply to triangles completely null. The sum of the angles of a triangle on the surface of a sphere must add up to anything greater than 180. Nothing with three vertices can go below 180 degrees then it wouldn't be connected at at least one vertice. Here's an example of a triangle that adds up to something larger than 180. Say Earth is a perfect sphere. Say that China and Mexico are on exactly opposite sides of the earth, and both lie on the equator.. Let those be two of the points and the other point be at the North Pole. The vertice at the North Pole is already 180! Adding the other ones gets roughly 270 degrees. Pretty cool, huh?

On a sphere, draw Vertical lines (Latitudes) at equal intervals and then draw a single horizontal line (Longitude) between any two consecutive latitudes on the surface of a sphere. The figure will represent a triangle on the surface of sphere. Now the 3 angles which are formed, out of which two formed due to intersection with latitudes will form 2 right angles and the third angle is always greater than 0 (based on what is the distance between two latitudes). Hence we have something like this -

Sum of the internal angles = 90 + 90 + (something greater than 0). Hence, the internal angles can be anything but should exceed 180 degrees.

Draw a line going from the top of a sphere to the middle somewhere, and draw a line perpendicular to the other line at the top and bring it to the middle. Then connect the ends of the segments, and you have a triangle with 270 degrees, which is more than 180 degrees.

Spherical surface can be conveniently described by 8 equilateral triangles or faces of 270 degrees each, with 6 crossing lines representing vertices or points, and 12 sides of the triangles. This then makes true the Euler formulae of edges+2= faces+ vertices.... Also it matches the representation of a cube in 4dimensional spacetime. There are 270x8 angles on a cube adding to 2160 deg which the eight triangles on the sphere also yield. And correctly but strangely, the corners of the cube become the faces on the sphere, each of 270 deg, On paper, which is a planar linear surface, we have triangles of 180 degrees, and on a concave surface like a saddle, or the inside gap of three equal sized pennies which touch each other, it is 90 degrees. Think, think, this is the fundamental of science of physics, maths, reason and logic. But it is true only for our world where the numbers 1, _/2, _/3, pi and e are the natural outcome. What's stated here makes evident that our space is closed and finite and will collapse in a cataclysm in moment in future. That's validated and determined and accurately predicted just as the expansion of the cosmos was explained millennium years ago. The expansion will stop abruptly and really that will be the beginning of the end. But that end will give rise to a new universe with different laws and two permanently bifurcated spacetime worlds.