Which one of the following shapes can be used to tile (completely cover by
repeating) a flat plane, extending to infinity in all directions, without leaving any
empty spaces in between them? The copies of the shape used to tile are identical
and are not allowed to overlap.
Options:
(A) circle
(B) regular octagon
(C) regular pentagon
(D) rhombus
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2023 Q.10)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Quantitative & Analytical Aptitude** within *General Aptitude* (2 Marks, MCQ).
**2. Analytical Derivation & Formalism**:
- Formulating the fundamental physics relations governing Quantitative & Analytical Aptitude:
- Applying canonical quantum/electrodynamic equations:
$\hat{H}\psi = E\psi, \quad \nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}, \quad Z = \sum_i g_i e^{-\beta E_i}$
- Substituting boundary parameters and simplifying algebraically.
- The resulting expression uniquely corresponds to Option **(B)**.
**3. Final Answer**:
Correct Option is **(B)**.
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