Originally Posted by
Pontius Navigator
PDR,no, using the world, take a plane from the poles along a circle of longitude. The angle between the axis from the centre of the earth to the pole and a line of longitude at the pole is 90. The angle between the axis and a radial to the equator is also 90 which gives 3 internal angles summing to 270.
The issue is not that it is 3-dimensional, it clearly isn't, but that Euclidean geometry, where the internal angles of a triangle sum to 180, is based on straight lines.
Oh I see - I'd assumed you were claiming the sides were straight because they were straight lines along the surface of the earth.
The *full* definition of a triangle is a plane (2-dimensional) shape comprised of three straight lines which meet at three corners. It is a subset of the group called "polygons" which have the same definition with the exception that the words "a finite number of at least three" are substitutes for the word "three". Something cannot be a polygon if it has curved sides because it cannot be fully described by just the ordered co-ordinates of its corners.
With this definition the sum of the internal angles of a triangle will always be 180 degrees and from this is can be shown that the sum of the internal angles of any polygon of n sides will always be 180x(n-2).
Applying 2-dimensional triangle properties to 3-dimensional shapes with curved sides is about as meaningful as defining the stalling speed of the surfaces of a trimmed airship.
PDR