Tessellation: the tiling of a surface with shapes such that no gaps or overlaps occur. Famous examples: (1) BATHROOM TILES — squares, rectangles, hexagons tessellate the floor; (2) HONEYCOMB — bees build hexagonal cells that tessellate perfectly; (3) BRICK WALLS — bricks tessellate (with offset for strength); (4) ESCHER ARTWORKS — complex tessellations of birds, fish, lizards.
Which shapes tessellate: (a) ALL TRIANGLES and ALL QUADRILATERALS tessellate (you can prove this mathematically); (b) REGULAR POLYGONS tessellate only if 360°/interior angle is a whole number: equilateral triangle (60° × 6 = 360°), square (90° × 4 = 360°), regular hexagon (120° × 3 = 360°); (c) IRREGULAR POLYGONS may or may not tessellate; (d) CIRCLES do NOT tessellate (they leave gaps). RELEVANCE to ASVAB AO: in FITTING-PARTS questions, the disassembled pieces must tessellate (fit together without gaps or overlaps) to form the assembled figure. If the pieces couldn't tessellate, the answer choice is wrong.
Strategy: (1) Look at the disassembled pieces and ask 'could these fit together without gaps?' If a piece has a curve that doesn't match a curve on another piece, they probably can't fit without gaps; (2) Look at the assembled answer and check for gaps (small holes between pieces); the pieces must have shared edges that line up exactly. RELATED concept: PERIMETER vs AREA. Tessellated pieces have boundaries (edges) that match each other (shared edges); the perimeter of the assembled shape is composed of the NON-SHARED edges.
Example: two equilateral triangles sharing one full edge form a rhombus; the rhombus's perimeter has 4 edges (two from each triangle), not 6. Counting the perimeter edges of the assembled shape and comparing to the number of free (non-shared) edges of all pieces verifies the tessellation logic.