Slantedors

Slantedors are a fictional class of planar polygonal tiles introduced in speculative geometry to illustrate how tile orientation influences tiling behavior. They are defined for instructional purposes and do not correspond to a standard object in established mathematical literature.

Definition: A slantedor is a convex polygon whose edges lie along two fixed directions that form an angle θ (0 < θ < 180 degrees). The two directions are represented by two families of parallel lines, and a slantedor boundary alternates between edges from each family.

Construction and parameters: Slantedors can be generated from a lattice spanned by two non-collinear vectors meeting at angle θ. By selecting polygons whose sides lie along the lattice directions, one obtains a family of tiles parameterized by θ and by the polygon's side-lengths. The slant of the tile increases as θ departs from 90 degrees.

Tilings and properties: When placed in a plane with translations that respect edge directions, slantedors can form periodic tilings. The symmetry and tile counts depend on θ and on how many sides align with each direction. As θ approaches 0 or 180 degrees, tiles become elongated; near 90 degrees, they resemble rectangular or rhombic forms.

Applications and relevance: In teaching and visualization, slantedors provide a simple model to study how edge-direction constraints affect tiling possibilities, symmetry classes, and density. They appear in computer graphics for anisotropic texture patterns and in architectural prototypes exploring slanted-grid facades.

History: The term slantedor was coined by Jane Doe in 2024 as a pedagogical construct to illustrate concepts in tiling theory.

See also: tiling theory, parallelogon, rhomb tiling, lattice, isohedral tiling.