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A triangle whose interior angles sum to zero

135 points by tzury

by abhashanand1501

8 subcomments

>In spherical geometry, the interior angles of a triangle add up to more than π. And in fact you can determine the area of a spherical triangle by how much the angle sum exceeds π. On a sphere of radius 1, the area equals the triangle excess
To all the flat earthers out there, this property can be used to find out earth is not flat, just by drawing a giant triangle on the surface, without leaving the earth. Historically, to prove the earth is round, people have relied on the sun shining directly overhead on wells in different cities. But this approach proves it without the need to refer the sun.

by ethmarks

2 subcomments

> Note also that the triangle has infinite perimeter but finite area.
How common is this property in geometry? I know that fractals like the Koch Snowflake also have infinite perimeter over finite area, but I don't know what else does.

by anigbrowl

1 subcomments

by gloftus

1 subcomments

Worth noting that the hyperbolic triangle in the article contains "points at infinity" which are not actually a part of the hyperbolic plane, so this is really an "improper triangle" as the article notes. One could construct a similar improper triangle in the Euclidean plane that consisted of two parallel lines meeting at infinity. Such a triangle would still have 180 degrees of internal angle but it's area and perimeter would be infinite.

by gorfian_robot

0 subcomment

by vismit2000

0 subcomment

Girard's Theorem - Spherical Geometry - Deriving The Formula For The Area Of A Spherical Triangle: https://youtu.be/Y8VgvoEx7HY T = r^2 (alpha + beta + gamma - pi)

0 subcomment

by dkdcio

5 subcomments

by MathMonkeyMan

0 subcomment

by ericol

2 subcomments

That's not a fucking triangle.
(It's Friday night people it's a joke and I have no idea what the article is talking about just looked at the picture)