Problem 12 - A Complete Resource Book in Chemistry for JEE Main

Question

The total elements of symmetry in a cube are (a) 23 (b) 14 (c) 25 (d) 16

Step-by-Step Solution

Answer

The total elements of symmetry in a cube are 16 (option d).

Step 1: Understanding Symmetry Elements

A cube has several elements of symmetry, which include planes of symmetry, rotational axes, and combinations thereof. We will calculate each of these components step-by-step.

Step 2: Count Planes of Symmetry

A cube contains three different perpendicular planes that pass through the centers of opposite faces. Thus, there are three planes of symmetry in a cube.

Step 3: Count Rotational Axes

The cube has axes of rotational symmetry about which the cube can be rotated and map onto itself. There are 3 fourfold axes (about the faces), 4 threefold axes (about the diagonal corners), and 6 twofold axes (through the midpoints of the edges). This gives a total of 3 (fourfold) + 4 (threefold) + 6 (twofold) = 13 rotational symmetries.

Step 4: Total Elements of Symmetry

To find the total elements of symmetry, we add the number of planes of symmetry to the number of rotational symmetries: 3 planes of symmetry + 13 rotational symmetries = 16. Therefore, a cube has a total of 16 elements of symmetry.

Key Concepts

Planes of Symmetry

A plane of symmetry divides an object into two mirror-image halves. For a cube, these planes of symmetry are particularly easy to visualize. There are three principal planes of symmetry in a cube:

• The first plane cuts the cube horizontally across the center, slicing it into two equal top and bottom halves.
• The second plane cuts vertically down the middle from front to back, creating left and right halves.
• The third plane runs vertically from side to side, separating the cube into front and back halves.

Each of these planes is perpendicular to one of the cube's coordinate axes, showcasing how symmetry provides a beautiful balance within geometric shapes.

Rotational Symmetry

Rotational symmetry occurs when an object can be rotated around an axis and still look the same. In the case of a cube, this symmetry is quite elaborate due to its regular shape.

• The cube possesses three fourfold rotational symmetries. This means you can rotate the cube 90°, 180°, 270°, or 360° around an axis through the centers of any face pair, and the cube will look identical at each step.
• Four threefold rotational symmetries are observed when rotating about the cube’s corners. Each rotational motion of 120° around a trio of diagonally connecting corners maintains the cube's appearance.
• Lastly, the cube’s six twofold symmetries let you rotate 180° around any axis running through the midpoints of opposite edges.

Rotational symmetry helps to understand how objects can maintain their appearance despite being spun around, a handy feature not just in geometry, but in design and engineering too.

Axes of Rotation

Axes of rotation are imaginary lines around which an object can rotate. In a cube, these axes are crucial to its rotational symmetry.

• The three fourfold axes pass through the center of opposite faces. When rotating a cube 90° around these axes, each face swaps places with the adjacent face, perfectly preserving its orientation.
• The four threefold axes run through diagonal corners. Rotating around one of these axes causes a seamless swap between sets of three faces meeting at the corners.
• Finally, the six twofold axes travel through the midpoints of opposite edges. Each 180° turn around these allows the cube to look the same since opposite faces merely exchange positions.

Understanding these axes of rotation enables one to grasp how multidirectional symmetry operates in regular solids, offering insights into both mathematics and natural patterns.