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: Understanding the Concept:
This question relates to rotational symmetry in crystallography. In a periodic crystal lattice, not all rotational symmetries are possible. The crystallographic restriction theorem dictates which rotational symmetries can exist.
Step 2: Detailed Explanation:
The crystallographic restriction theorem states that if a crystal has rotational symmetry, the order of rotation (n-fold) can only be 1, 2, 3, 4, or 6.


One-fold symmetry (360\(^\circ\) rotation): This is trivial and exists in all objects.
Two-fold symmetry (180\(^\circ\) rotation): Exists.
Three-fold symmetry (120\(^\circ\) rotation): Exists.
Four-fold symmetry (90\(^\circ\) rotation): Exists.
Six-fold symmetry (60\(^\circ\) rotation): Exists.
A five-fold rotational symmetry (72\(^\circ\) rotation) is not allowed because pentagons cannot tile a two-dimensional or three-dimensional space without leaving gaps or overlapping. This would violate the periodic, repeating nature of a crystal lattice.
Step 3: Final Answer:
Therefore, five-fold symmetry does not exist in periodic crystals.