parallelogrammil

Parallelogrammil is a hypothetical geometric object described here as a family of parallelograms obtained by applying a shear transformation to a unit square. The construction is parameterized by a real number m, referred to here as the mil. For a given m, the parallelogram has vertices at (0,0), (1,0), (m,1), and (1+m,1). The top and bottom edges are horizontal and parallel, and the left and right edges are parallel, making the figure a parallelogram for every m.

Properties of the parallelogrammil include that its area is preserved by the shear and equals 1, the same as the unit square. The bottom edge length is 1 and the top edge length is also 1; the left and right edges have length sqrt(m^2+1). The bottom-left interior angle is arctan(1/m) for m ≠ 0, with the special case m=0 yielding a rectangle. By varying m from negative to positive values, the parallelogrammil ranges from skewed to skewed in the opposite direction; large absolute values of m yield a more pronounced skew while maintaining area.

Variants and use: The parallelogrammil is sometimes used in introductory geometry to illustrate affine transformations and invariants under shear. It also serves as a simple example in computer graphics for testing rendering of skewed parallelograms and in tiling problems where unit-area shapes are required.

Relation to other concepts: It is a particular affine image of the unit square; every parallelogram is an affine image of a square, so parallelogrammils form a one-parameter subfamily corresponding to a fixed shear direction.

Notes: Parallelogrammil is a fictional name used here to illustrate a standard mathematical idea; it is not a widely recognized term in formal geometry.