If I were to draw a straight line from the North Pole to the Equator, then another straight line 45 degrees from that one to the Equator, I’d have a triangle of 225 degrees…agreed? If, then, I started shrinking that triangle, smaller and smaller, it would approach 180 degrees. So, my question is, if I draw a triangle in the dirt in my front yard, is it really 180 degrees or some infinitesimal measurement more? Further, if I drew the same triangle on my kitchen table and held it above the earth, would it be exactly 180 degrees?
A triangle equals 180 degrees in Euclidean geometry, which assumes that it’s drawn on a perfectly flat surface. The flatness is an abstration, though, and the angles of a triangle draw in the dirt may indeed be infinitesmally different from 180 degrees. The same with any actual triangle you draw. But traditional geometry ignores the reality for the ideal.
Spherical geometry doesn’t follow the same rules as plane geometry.
Your construction you summarized was misleading. “If I were to draw a straight line from the North Pole to the Equator” would mean a line that burrows into the sphere of the Earth and re-emerging at the Equator. Two such lines 45[sup]o[/sup] apart with the North Pole at their vertex would form 67.5[sup]o[/sup] angles with a line that went straight between their equatorial endpoints.
So Catenary, your large spherical triangle you described is a birectangular right spherical triangle.
Your triangle on the ground is technically a spherical triangle, but with such short segments it’s hard to tell it from a plane triangle. You’re right that as the sides approach zero, the interior angle sum approaches 180[sup]o[/sup].
Your triangle on the tabletop is a plane triangle, and would have the 180[sup]o[/sup] interior angle sum. But please use paper; your mom will be upset if you start drawing on the furniture. :D:D
Oops, my subject got cut off at the quotes. My definition came from my “Mathematics Dictionary” (4th ed, 1976), James/James.
Given that the sphere this triangle is a portion of doesn’t change size and only the legs’ lengths do.
If we shrink the triangle by shrinking the sphere, though, the angles won’t change, will they?
It seems to me they wouldn’t, but now that I think about it, the curvature is a function of the radius…
Crap. Now I don’t know what I’m talking about.
Right the first time: the angles wouldn’t change if all you did was shrink the sphere. The curvature of the sphere itself would change, as would the area of the triangle and the lengths of its sides, but not the angles.
And who said math wasn’t intuitive? 