Diagonally

Diagonally is an adverb describing a direction that follows a diagonal, meaning a line or path that is neither purely horizontal nor vertical. It typically refers to movement or orientation from one corner to the opposite corner of a shape, such as a square or rectangle, or to a line that crosses from one side to the opposite side at an angle.

Etymology wise, diagonally derives from the noun diagonal, which comes from Latin diagonalis and Greek diagon, formed from dia- meaning through and gonia meaning angle. The suffix -ly yields the adverb.

In geometry and mathematics, a diagonal is a line segment connecting non-adjacent vertices of a polygon, most commonly a quadrilateral. The term diagonally describes things aligned with or moving along such lines. In linear algebra, the main diagonal of a matrix consists of elements with equal row and column indices, while the anti-diagonal runs across the opposite corners. The concept of diagonals also appears in geometry, tiling, and various algorithms that reference diagonal directions or distances.

In everyday language and practical use, diagonally appears in phrases such as moving diagonally across a street, cutting diagonally through a park, or placing objects diagonally to create a dynamic composition. In games and grid-based systems, a diagonal move typically refers to stepping to a diagonally adjacent cell, as seen in board games like chess or checkers, and in pathfinding algorithms that permit diagonal traversals.

See also: diagonal, main diagonal, anti-diagonal, diagonal matrix, diagonalization.