Figure 1617a (a) shows a four-fold rotation axis in crystallography, and Figure 1617a (b) shows the two-fold, three-fold and four-fold axes in a cubic structure.

Figure 1617a. (a) A four-fold axis, and (b) The two-fold, three-fold and four-fold axes in a cubic structure.

Figure 1617b shows that two parallel 4-fold axes of rotation generate translational symmetry at the same distance of separation as in the original pair.

Figure 1617b. Two parallel 4-fold axes of rotation generating translational symmetry.

Figure 1617c shows the CBED pattern of β-pyrochlore oxide superconductor KOs₂O₆ along the [001] zone axis. [1] The square array with small dark disks near the center is zero-order Laue zone (ZOLZ) and the surrounding circle formed by the highly contrasted disks is first-order Laue zone (FOLZ). The magnified image of the inset presents a four-fold rotational symmetry along the c* axis and two mirror symmetries m_a and m_b, indicating that the whole pattern (WP) has 4mm symmetry.

Figure 1617c. CBED pattern taken from a KOs₂O₆ crystal along [001] zone axis. [1]

[1] Jun-Ichi Yamaura, Zenji Hiroi, Kenji Tsuda, Koichi Izawa, Yasuo Ohishi, Satoshi Tsutsui, Re-examination of the crystal structure of the β-pyrochlore oxide superconductor KOs2O6 by X-ray and convergent-beam electron diffraction analyses, Solid State Communications 149 (2009) 31-34.