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:
D
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2023 Q.10)**
**1. Governing Concept**:
- **Topic**: Geometry & Monohedral Plane Tiling (Tessellation)
- For identical copies of a polygon to tile a 2D Euclidean plane without overlaps or gaps, the interior angles meeting at each vertex must sum to $360^\circ$.
- Any arbitrary parallelogram (including a **rhombus**) can tile the plane, because adjacent angles sum to $180^\circ$, and 4 rhombi meeting at a vertex form $2(\theta) + 2(180^\circ - \theta) = 360^\circ$.
- Regular pentagons (interior angle $108^\circ$, not dividing $360^\circ$) and regular octagons ($135^\circ$) cannot tile the plane alone. Circles cannot tile due to interstitial gaps.
**2. Option Analysis**:
- **Option (A) [circle] is INCORRECT**: Leaves gaps between kissing circles.
- **Option (B) [regular octagon] is INCORRECT**: $135^\circ$ requires squares to fill gaps (semi-regular tiling).
- **Option (C) [regular pentagon] is INCORRECT**: Regular pentagons cannot tile the plane.
- **Option (D) [rhombus] is CORRECT**: All quadrilaterals, including rhombi, tile the Euclidean plane seamlessly.
**3. Final Answer**:
- **Correct Choice**: **D**
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