Question

Why don’t crystals have 5 fold Symmetry?

Answer 1

Hint: In order to answer this question, to know the reason why crystals don’t have 5 fold symmetry, we should go through the whole concept of crystals formation and then we will discuss some important points related to the crystals.

Complete answer:
A crystal structure is made of atoms. A crystal lattice is made of points. A series of axes makes up a crystal structure. In other words, the structure is a collection of atoms, ions, or molecules that are arranged in a specific order. A unit cell is a building block of a crystal structure that also defines the entire crystal structure and symmetry, as well as the atom positions and principal axes. The length, edges of principal axes, and angle between unit cells are all lattice constants.
We can't pack objects like pentagons or octagons such that they fill up space entirely and that that's one reason there is no 5-fold or 8-fold rotational axis.
Some important points related to the crystals:-
$*$ Crystals do appear to have 5-fold symmetry but these symmetries are not possible.
$*$ Crystals can only exist in the 2, 3, 4 or 6-fold rotational axis. The exterior form of the crystal is determined by the geometric arrangements of atoms, so this is not possible.
$*$ Therefore, crystals cannot have 5, 7, 8, and other higher-fold rotational axes.

Note:
Now, a question arises here: Is 5-fold symmetry possible or not? So, 5-fold symmetry is impossible or almost not possible for a crystal is correct: basically only 3,4 and 6-fold symmetries are possible.