Inscribing a sphere in a parallelepiped I have a parallelepiped with sides, $a$, $b$, and $c$. $\gamma$ is the angle between $a$ and $b$, $\beta$ is the angle between $a$ and $c$, and $\alpha$ is the angle between $b$...--prophetes.ai

Inscribing a sphere in a parallelepiped I have a parallelepiped with sides, $a$, $b$, and $c$. $\gamma$ is the angle between $a$ and $b$, $\beta$ is the angle between $a$ and $c$, and $\alpha$ is the angle between $b$ and $c$, as shown below: ![triclinic unit cell]( My question is, under what circumstances would it be possible to inscribe a sphere into this parallelepiped? And if it is possible, can we express the sides of the parallelepiped in terms of the radius $r$ of the sphere and the angles?

All faces must be equal rhombi, for a sphere to be inscribed. Just think of the three plane sections of the parallelepiped passing through its center and parallel to a couple of opposite faces. The relation between radius and edge is the same as in the case of a rhombus.