Consider a solid rectangular block (cuboid) with distinct dimensions $a \neq b \neq c$ (none of the three edge lengths are equal). Which of the following statements regarding its spatial symmetry elements is/are TRUE?
Options:
(A) It possesses exactly $3$ mutually perpendicular planes of reflection symmetry.
(B) It possesses exactly $3$ two-fold ($180^\circ$) axes of rotational symmetry passing through the centers of opposite faces.
(C) It possesses a center of inversion symmetry at its geometric center.
(D) It possesses $4$ three-fold ($120^\circ$) axes of rotational symmetry along its main space diagonals.
Answer Key:
A, B, C
Step-by-Step Solution & Explanation
**Method 1: Point Group Symmetry Analysis ($D_{2h}$)**
For an orthorhombic cuboid where $a \neq b \neq c$:
- (A) True: There are 3 mutually orthogonal mirror planes parallel to each pair of opposite faces bisecting the cuboid.
- (B) True: There are 3 orthogonal $C_2$ ($180^\circ$) rotational axes passing through the centers of opposite faces.
- (C) True: The inversion operation $(x, y, z) \to (-x, -y, -z)$ about the geometric centroid maps the body onto itself.
- (D) False: 3-fold rotational axes ($C_3$) along body diagonals exist only in a regular cube ($a = b = c$, cubic symmetry $O_h$). When $a \neq b \neq c$, rotating $120^\circ$ about a body diagonal does not preserve the rectangular shape.
Therefore, options **A**, **B**, and **C** are true.