Sidesboth

Sidesboth is a geometric term describing a property of polygonal shapes. A polygon is said to have the sidesboth property if there exists a line of symmetry that maps the polygon onto itself, and among its boundary edges there is a pair of sides located on opposite sides of the symmetry axis that are parallel and have equal length. The name emphasizes the presence of two sides on both sides of the axis.

Origin and usage: The term sidesboth emerged in online geometry discussions to distinguish bilateral symmetry with a specific parallelism constraint from more general symmetric figures. It is used mainly in descriptive classifications of polygons and in computational geometry where explicit boundary correspondences matter.

Formal definition: Let P be a polygon in the plane and ℓ a line. If P is invariant under reflection across ℓ, and there exist two distinct boundary sides s1 and s2 such that s1 is the image of s2 under the reflection, and s1 and s2 are parallel with equal length, then P has the sidesboth property. In practice, many regular polygons and isosceles trapezoids satisfy this condition.

Examples: A rectangle, an isosceles trapezoid, and a regular hexagon admit a sidesboth axis of symmetry. More generally, any polygon with bilateral symmetry in which a pair of opposite sides are parallel and congruent qualifies.

Relation and contrasts: Sidesboth is related to, but distinct from, general line symmetry; it emphasizes a concrete pair of opposite, parallel, equal-length sides.

See also: symmetry, polygon, bilateral symmetry.