Q.10 An object is said to have an n-fold rotational symmetry if the object, rotated by an angle of 2π/n, is identical to the original.
Which one of the following objects exhibits 4-fold rotational symmetry about an axis perpendicular to the plane of the screen?
Note: The figures shown are representative.

Introduction

Rotational symmetry is an important concept in physics and geometry, especially in questions related to symmetry operations. An object is said to possess n-fold rotational symmetry if rotating it by an angle of 360°/n results in an appearance identical to the original.

In this article, we analyze a multiple-choice question asking which object exhibits 4-fold rotational symmetry about an axis perpendicular to the plane of the screen, and we explain why only one option is correct.

Understanding 4-Fold Rotational Symmetry

An object has 4-fold rotational symmetry if it looks exactly the same after rotations of:

• 90°

• 180°

• 270°

• 360°

This means the object repeats itself four times during one full rotation.

Correct Answer

✅ Correct Option: (C)

Explanation of Each Option

Option (A): ❌ Incorrect

• Although the shape resembles a cross, the positions of the black dots are not symmetric under a 90° rotation.

• After rotation, the dots shift to different relative positions.

• The object may show 2-fold symmetry, but not 4-fold symmetry.

👉 Fails the 90° rotation test

Option (B): ❌ Incorrect

• The arms of the shape are unequal in dot placement.

• Rotating the figure by 90° does not reproduce the original configuration.

• Symmetry breaks due to asymmetric dot distribution.

👉 Not invariant under quarter-turn rotation

Option (C): ✅ Correct

• The object is perfectly symmetric in shape and dot placement.

• Rotating it by 90°, 180°, or 270° gives the same appearance.

• All four arms are identical and equally spaced.

👉 Satisfies the definition of 4-fold rotational symmetry

Option (D): ❌ Incorrect

• The vertical and horizontal arms are not equivalent.

• Dot placement is asymmetric across different directions.

• Only 180° rotation preserves similarity.

👉 Exhibits 2-fold symmetry, not 4-fold

Why Option (C) Is the Only Correct Choice

Only Option (C) remains unchanged under all required rotations.

Key Takeaway

• 4-fold rotational symmetry requires perfect repetition every 90°.

• Both shape and internal features (like dots) must align after rotation.

• Option (C) uniquely satisfies this condition.

Conclusion

Understanding rotational symmetry helps solve a wide range of physics and geometry problems efficiently. By carefully checking symmetry under rotation angles, we can confidently conclude that Option (C) is the correct answer for this question.

📌 Final Answer: Option (C)