Rhomboedrisk

Rhomboedrisk is a proposed quantitative descriptor used in geometry and crystallography to characterize the distortion of a rhombus (rhomboid) from a perfect square. It is defined for a rhombus formed by two equal-length vectors a and b in the plane, with the interior angle theta between a and b. The rhomboedrisk R is defined as R = |cos theta| = |a · b| / (|a||b|).

Properties of rhomboedrisk include scale invariance and a range between 0 and 1. A value of R = 0 corresponds to a square rhombus (theta = 90°), while R approaching 1 indicates greater deviation toward a degenerate shape (theta approaching 0° or 180°).

The measure relates to area through the side length s: the area of a rhombus is A = s^2 sin theta. Since sin^2 theta = 1 − cos^2 theta = 1 − R^2, the area can be expressed as A = s^2 sqrt(1 − R^2). Thus, rhomboedrisk couples angular distortion directly to area for a given side length.

Computation is straightforward: given two equal-length vectors a and b, compute the dot product to obtain cos theta, take the absolute value, and obtain R.

Applications of rhomboedrisk include quantifying lattice distortion in rhombohedral crystals, assessing skew in computer graphics and tiling with rhombic tiles, and providing a simple metric for comparing rhombic shapes in design and architecture. Examples: theta = 60° yields R = 0.5; theta = 100° yields R ≈ 0.1736.

See also: rhombus, rhombohedron, lattice distortion.