Question

Which of the following symmetry does not exist:

• A. one fold symmetry
• B. two fold symmetry
• C. four fold symmetry
• D. five fold symmetry

Hint:

Remember the allowed rotational symmetries in crystals are 1, 2, 3, 4, and 6. Any other number, most commonly 5 or anything greater than 6, is forbidden by the crystallographic restriction theorem. Note that five-fold symmetry is observed in quasicrystals, which are ordered but not periodic.

Answer 1

The Correct Option is D

Step 1: Concept Overview:
This question addresses rotational symmetry within crystal structures. Due to the periodic nature of crystal lattices, only specific rotational symmetries are permissible, as defined by the crystallographic restriction theorem.
Step 2: Detailed Explanation:
The crystallographic restriction theorem limits the order (n-fold) of rotational symmetry in crystals to 1, 2, 3, 4, or 6.


One-fold symmetry (360\(^\circ\) rotation): Trivial; present in all objects.
Two-fold symmetry (180\(^\circ\) rotation): Allowed.
Three-fold symmetry (120\(^\circ\) rotation): Allowed.
Four-fold symmetry (90\(^\circ\) rotation): Allowed.
Six-fold symmetry (60\(^\circ\) rotation): Allowed.
Five-fold rotational symmetry (72\(^\circ\) rotation) is forbidden. Pentagons cannot tile space periodically in two or three dimensions without gaps or overlaps, conflicting with the repeating structure of crystal lattices.
Step 3: Conclusion:
Therefore, five-fold symmetry is not observed in periodic crystals.