Parallelogrammon

Parallelogrammon is a fictional geometric construct used in recreational mathematics and conceptual expositions to explore relationships that arise from parallelograms and plane tilings. It is defined by considering two congruent parallelograms in the plane, where one is translated by a fixed nonzero vector and the other is kept in place.

Formally, a parallelogrammon is the region P ∆ (P + v), where P is a parallelogram, v is a nonzero translation vector not parallel to the sides of P, and ∆ denotes symmetric difference. The boundary of this region is a simple hexagon whose sides come in three pairs of parallel lines, each pair parallel to one of the three directions associated with P and P+v. The construction visually emphasizes how a shape and its translate interact.

Key properties include central symmetry about the midpoint between the centers of P and P+v, and a boundary that alternates directions corresponding to the sides of P and its translate. The area obeys area(parallelogrammon) = area(P) + area(P+v) − 2 area(P ∩ (P+v)). Since area(P+v) = area(P), this simplifies to 2 area(P) − 2 area(P ∩ (P+v)).

Special cases illustrate the range of forms: when v is very small, the parallelogrammon is a thin hexagon hugging P; when v grows toward the length of the diagonal, the overlapping region shrinks and the hull widens. Some choices of v produce hexagons that tessellate the plane under translations, while others do not.

Parallelogrammon is used primarily as a didactic device in explorations of symmetry, tiling, and the relationship between a shape and its translate. It serves to illuminate how simple geometric operations lead to richer boundary structures.

See also: parallelogram, hexagon, symmetric difference, tilings, Minkowski sum.