Hyperbolic geometry is a non-Euclidean geometry that describes the properties of objects in a space with a constant negative curvature. In this geometry, the sum of the angles of a triangle is always less than 180 degrees and parallel lines never intersect.
A hyperbolic rectangle is a four-sided figure with four right angles that exists in a hyperbolic space. Unlike in Euclidean geometry, the opposite sides of a hyperbolic rectangle are not equal in length and the angles formed between the sides are not congruent.
A Euclidean rectangle exists in a flat, two-dimensional space and has two pairs of equal sides and four right angles. A hyperbolic rectangle exists in a space with a constant negative curvature and has two pairs of unequal sides and four angles that are less than 90 degrees.
No, a hyperbolic rectangle cannot be constructed in the real world as it exists in a space with a constant negative curvature, which is not possible in our three-dimensional world. However, it can be represented and studied mathematically.
Hyperbolic geometry has many applications in fields such as architecture, art, and physics. It is used to create unique designs in buildings, sculptures, and art pieces. It also has applications in the study of black holes and the universe, as well as in computer graphics and game development.