# Hyperbolic Tessellation and Geometric Art

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 non-Euclidean 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.