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. A. circle B. regular octagon C. regular pentagon D. rhombus

GATE 2023 · General Aptitude · Patterns In Three Dimensions · medium

Answer: D. rhombus

1. Check circle: Circles have no straight edges and leave curved (crescent-shaped) gaps when placed side by side. A circle cannot tessellate the plane.
2. Check regular octagon: 360/135 = 2.67, which is not an integer. Regular octagons alone cannot tile the plane without gaps (square holes remain). Option B is eliminated.
3. Check regular pentagon: 360/108 = 3.33, not an integer. Regular pentagons cannot tessellate the plane. Option C is eliminated.
4. Check rhombus: A rhombus is a parallelogram. Any parallelogram tessellates the plane by translation (sliding). Place rows of rhombuses where each row is offset by half a rhombus. The four angles meeting at each interior vertex sum to exactly 360 degrees, leaving no gaps. Option D is correct.