Cubic Unit Cell Threefold Axes and Tetrahedral Corners

A cubic unit cell is a fundamental building block in crystallography, exhibiting high symmetry. Understanding the symmetry elements within a unit cell, such as axes of rotation, is crucial for describing crystal structures.

An axis of rotation is an imaginary line through the crystal such that rotation about this line by a specific angle leaves the crystal lattice unchanged. A threefold axis of rotation is one where a rotation by 120 degrees (360°/3) results in an identical orientation of the unit cell.

In a cubic unit cell, the threefold axes pass through pairs of opposite corners of the cube. Consider a cube with its origin at (0,0,0) and edges along the x, y, and z axes. The corners are at (0,0,0), (1,0,0), (0,1,0), (0,0,1), (1,1,0), (1,0,1), (0,1,1), and (1,1,1).

A tetrahedron can be inscribed within a cube by connecting four non-adjacent corners. For example, the corners at (0,0,0), (1,1,0), (1,0,1), and (0,1,1) form a tetrahedron. The directions from the center of the cube to the corners of this tetrahedron correspond to the body diagonals of the cube. These body diagonals are precisely the locations of the threefold axes.

Let's identify the pairs of opposite corners that the threefold axes pass through:

• (0,0,0) and (1,1,1)
• (1,0,0) and (0,1,1)
• (0,1,0) and (1,0,1)
• (0,0,1) and (1,1,0)

There are 4 such pairs of opposite corners in a cube, and each pair defines a unique threefold axis. These 4 threefold axes correspond to the four body diagonals of the cube. The directions of these body diagonals also point towards the four corners of a tetrahedron that can be inscribed within the cube by selecting alternate corners.

Therefore, a cubic unit cell has 4 threefold axes, and these axes point towards the corners of a tetrahedron within the cube.