Open, Flat, Closed
An isotropic and homogeneous space can be described by the metric:
- .
In the limit that the constant of curvature becomes infinitely large, a flat, Euclidean space is returned. It is essentially the same as setting to zero. If is not zero the space is not Euclidean. When the space is said to be closed or elliptic. When the space is said to be open or hyperbolic.
Triangles which lie on the surface of an open space will have a sum of angles which is less than 180°. Triangles which lie on the surface of a closed space will have a sum of angles which is greater than 180°. The volume, however, is not .
Read more about this topic: Curved Space
Famous quotes containing the word closed:
“Don: Why are they closed? They’re all closed, every one of them.
Pawnbroker: Sure they are. It’s Yom Kippur.
Don: It’s what?
Pawnbroker: It’s Yom Kippur, a Jewish holiday.
Don: It is? So what about Kelly’s and Gallagher’s?
Pawnbroker: They’re closed, too. We’ve got an agreement. They keep closed on Yom Kippur and we don’t open on St. Patrick’s.”
—Billy Wilder (b. 1906)