epäplanaarisuuden

Epäplanaarisuus, a Finnish term, translates to non-planarity or non-planar geometry. In mathematics and geometry, it refers to the property of a figure or a set of points that cannot be contained within a single plane. A planar figure can be drawn on a flat surface without any part lifting off it, while a non-planar figure requires three-dimensional space to be fully represented.

The most fundamental example of a non-planar figure is a tetrahedron, a pyramid with a triangular base. While each face of a tetrahedron is a triangle and thus planar, the four vertices and edges collectively cannot lie on the same plane. Similarly, any three-dimensional object, such as a cube, sphere, or a human body, is inherently non-planar.

In graph theory, a graph is considered non-planar if it cannot be drawn on a plane without any edges crossing each other. The Kuratowski's theorem provides a definitive criterion for determining if a graph is planar or non-planar. It states that a graph is non-planar if and only if it contains a subgraph that is a subdivision of either the complete graph K5 (five vertices, each connected to every other vertex) or the complete bipartite graph K3,3 (two sets of three vertices, with every vertex in one set connected to every vertex in the other). The concept of non-planarity has implications in various fields, including circuit design, topology, and computational geometry.