Hyperbolic Tessellation and Geometric Art is a mosaic of changing symmetrical patterns
of the Poincare's model of hyperbolic geometry.
Hyperbolic geometry (also called saddle
geometry or Lobachevskian geometry) is a nonEuclidean geometry in which it
is assumed that through any point there are two or more parallel lines that
do not intersect a given line in the plane.
A tessellation or tiling of the plane is
a collection of plane figures that fills the plane with no overlaps and no
gaps.
The Poincare disk is a model for
hyperbolic geometry in which a line is represented as an arc of a circle
whose ends are perpendicular to the disk's boundary. Two arcs which do not
meet correspond to parallel rays, arcs which meet orthogonally correspond to
perpendicular lines, and arcs which meet on the boundary are a pair of
limits rays.
A pattern, whether in nature or art,
relies upon three characteristics: a unit, repetition, and a system of
organization.
Symmetry is a fundamental organizing
principle in nature and in culture. The analysis of symmetry allows for
understanding the organization of a pattern, and provides a means for
determining both invariance and change. A mathematical operation, or
transformation, that results in the same figure as the original figure (or
its mirror image) is called a symmetry operation. Such operations include
reflection, rotation, double reflection, and translation.
