Each Ca atom has 6 closest oxygen atoms. The CO3 groups are arranged in a flat triangle with the C in the middle. These groups are all aligned perpendicular to the c-axis.
Calcite has a perfect rhombohedral cleavage that is the result of the crystal structure.
The unit cell is twice as long in the c direction than shown. The initial view is down the c-axis. Each Ca atoms has 9 closest oxygen atoms. In aragonite, the flat CO3 groups are aligned in the same plane but point in two directions. Ca is relatively large for the structure at room pressure and temperature, so aragonite is metastable at room conditions.