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The eight regions of space defined by the eight possible combinations of signs for $(+/- , +/-, +/-)$ for $x$, $y$, $z$ are called octants.

Given a ball of radius 1 centered in the origin $(0, 0, 0)$. How are the eight sections called obtained by the intersection of the ball with the octants?

More generally, given a (regular) triangulation of the sphere (I hope such a thing exists). Now connecting such a spherical triangle with the origin how are the pieces cut out of the (unit) ball this way called?

Pointer to literature on these topics are appreciated.

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    @Rahul: Great idea! Makes it seem more exotic. :-) But perhaps a Greek scholar should be consulted...2011-08-25

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I've heard the terminology spherical pyramid used to refer to the solid formed by a spherical polygon and the center of the sphere. So perhaps you can call what you are looking at a "triangular spherical pyramid" or a "spherical pyramid with triangular base."