In the genus lithops the head or single plant body consists of two opposite leaves that have thickened through evolution.  These leaves are fused along the outer edges and are foreshortened to form a obconic, turbiniform body (an upside-down cone shape).  The fissure across the top of the plant divides it into two nearly identical halves.  Most plants are fused (1 - 4 mm deep) along the outer edges.  The line of adhesion is very thin and closes the fissure.  Below the fissure is the cleft and it extends downwards between the adpressed inner lobes to the growing point (meristem) of the plant which is situated just above the junction between the main plant body and roots.  When the plant flowers it breaks through the line of adhesion in the middle of the fissure whih opens the cleft.
Figure 1:  Illustration of the location of the cleft, line of adhesion, fissure and new body.

Every year a single head or two (new body) develops, this is usually accompanied by a flower that develops from the meristem.. The bud finds its way up the cleft on a pedicel where it breaks through the line of adhesion until the pedicel has reached its full extent and the petals open.  The plant usually goes dormant after flowering (winter and early summer).  During this period of dormancy the old body usually shrivels and dries up.  This period of dormancy is actually quite a busy time for the new body, it draws nutrients from the old body  to aid in its own rapid development.  The old leaves are usually just dried husks at the end of the dormancy period, the new body will burst through these "husks" after the first rains or watering.  The fissure on the body/bodies is approximately at right-angles to the old growth, if two heads are formed the division of the plants is usually in line with the fissure of the old plant.