The most unsymmetrical crystal system is

Triclinic

The correct answer is option 3. Triclinic.

Crystal Systems and Their Symmetries

1. Tetragonal

Axes: Two axes are equal in length (let's call them \(a\)) and perpendicular to each other, and the third axis (\(c\)) is of a different length but also perpendicular to the \(a\) axes.

Angles: All angles are 90°.

Symmetry: The tetragonal system has more symmetry compared to the triclinic system because of its equal \(a\) axes and right angles between all axes.

2. Hexagonal

Axes: Three axes (\(a\)) are of equal length and lie in the same plane, forming 120° angles with each other. The fourth axis (\(c\)) is perpendicular to this plane and can be of a different length.

Angles: The three in-plane angles are 120°, and the angle between the \(c\) axis and the \(a\) axes is 90°.

Symmetry: The hexagonal system has a higher degree of symmetry due to the 120° angles between the three equal \(a\) axes and the perpendicular relationship with the \(c\) axis.

3. Triclinic

Axes: All three axes are of different lengths. None of the angles between the axes are 90°.

Angles: The angles between the axes are all different and not right angles.

Symmetry: The triclinic system is the least symmetrical because there are no constraints on the lengths of the axes or the angles between them. This lack of constraint means it has the lowest symmetry of all the crystal systems.

4. Rhombohedral

Axes: All three axes are of equal length, but the angles between them are not 90°.

Angles: The angles between the axes are equal but not 90°, making them different from those in the tetragonal or orthorhombic systems.

Symmetry: The rhombohedral system has a notable symmetry due to the equal lengths of all three axes.

Why the Triclinic System is the Most Unsymmetrical

Axis Length: In the triclinic system, all three axes are of different lengths (\(a \neq b \neq c\)).

Angles Between Axes: None of the angles between the axes are right angles (\(\alpha \neq \beta \neq \gamma \neq 90^\circ\)).

Symmetry Operations: Due to the lack of equal axis lengths and non-right angles, there are fewer symmetry operations that can be applied, leading to the lowest symmetry of all crystal systems.

In summary, the triclinic crystal system has the most unsymmetrical arrangement of axes and angles, making it the least symmetric of all crystal systems. This lack of symmetry results in a more complex and less regular crystal structure.