The Klein bottle is a non-orientable surface in topology, named after the German mathematician Felix Klein. It is a two-dimensional object that cannot be properly embedded in three-dimensional Euclidean space without self-intersection. The Klein bottle is often described as a closed surface with only one side and no distinct inside or outside. This property makes it non-orientable, meaning it does not have a consistent notion of "up" and "down" across its surface. To visualize a Klein bottle, imagine taking a cylindrical shape and connecting the top and bottom edges but with a twist so that when you trace along its surface, you end up on the opposite side of the starting point. This self-intersecting characteristic is what sets the Klein bottle apart from other surfaces like spheres or tori. Mathematically, the Klein bottle can be represented through various parametric equations or topological constructions. One common representation is by gluing together two Möbius strips along their boundaries. Another method involves identifying points on opposite sides of a square with appropriate twists. The Klein bottle has numerous fascinating properties that make it interesting to mathematicians. For example, it has no true boundary and cannot be smoothly embedded in three-dimensional space without intersecting itself. Additionally, any closed curve drawn on its surface can be continuously shrunk to a point without ever leaving the surface. Although the Klein bottle is primarily studied in mathematics as an abstract concept, its shape has also captured popular imagination. It has appeared in various forms of art, literature, and even as physical objects created through 3D printing technology. # References ```dataview Table title as Title, authors as Authors where contains(subject, "Klein bottle") ```