Cleavage of minerals
The short answer: cubic cleavage is a specific example of 3D cleavage.

A cleavage plane is a direction along a crystal splits preferentially when exposed to mechanical stress.

3D crystals are objects where the spatial arrangement of atoms is described in  a parallelepiped (a unitcell) with three outer dimensions $\vec{a}$, $\vec{b}$, and $\vec{c}$.  The crystal literally builds by translating this unit cell along $\vec{a}$, $\vec{b}$, and $\vec{c}$.

The relative lengths of $\vec{a}$, $\vec{b}$, and $\vec{c}$ and the angles these vectors enclose define the outer symmetry of this unit cell.  Any of the following seven crystal classes is possible: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic.  In this set, starting from triclinic and moving to cubic, the inherent symmetry increases.

With $\vec{a}$, $\vec{b}$, and $\vec{c}$, equally crystallographic planes may be defined; some of them parallel to / coincide with the cleavage planes observed at macroscopic scale.


Thus, as listed e.g., in Wikipedia you may discern:


Basal or pinacoidal cleavage occurs when there is only one cleavage plane. Graphite has basal cleavage. Mica (like muscovite or biotite) also has basal cleavage; this is why mica can be peeled into thin sheets.

Cubic cleavage occurs when there are three cleavage planes intersecting at 90 degrees. Halite (or salt) has cubic cleavage, and therefore, when halite crystals are broken, they will form more cubes.

Octahedral cleavage occurs when there are four cleavage planes in a crystal. Fluorite exhibits perfect octahedral cleavage. Octahedral cleavage is common for semiconductors. Diamond also has octahedral cleavage.

Rhombohedral cleavage occurs when there are three cleavage planes intersecting at angles that are not 90 degrees. Calcite has rhombohedral cleavage.

Prismatic cleavage occurs when there are two cleavage planes in a crystal. Spodumene exhibits prismatic cleavage.

Dodecahedral cleavage occurs when there are six cleavage planes in a crystal. Sphalerite has dodecahedral cleavage.



Intentionally, the description above was constrained to 3D crystals which eventually are (typically) described in one of 230 3D space groups. Note, there equally are seven planar space groups, and a growing body of work about 2D crystals, too.