by lengthwise slices of Möbius strips with varying widths, are sometimes called paradromic The Möbius strip can be cut into six mutually-adjacent regions, showing that maps on the surface of the Möbius strip can sometimes require six colors, in contrast to the four color theorem for the Six colors are always enough. This result is part of the Ringel–Youngs theorem, which states how many colors each topological surface The edges and vertices of these six regions form Tietze's graph, which is a dual graph on this surface for the six-vertex complete graph but cannot be drawn without crossings on a plane. Another family of graphs that can be embedded on the Möbius strip, but not n the plane, are the Möbius ladders, the boundaries of subdivisions of the Möbius strip into rectangles meeting These include the utility graph, a six-vertex complete bipartite graph whose embedding into the Möbius strip shows that, unlike in the plane, the three utilities problem can be solved on a transparent Möbius The Euler characteristic of the Möbius strip is zero, meaning that for any subdivision of the strip by vertices and edges into regions, the numbers , , and of vertices, edges, and regions satisfy . For instance, Tietze's graph has vertices, edges, and regions; Constructions There are many different ways of defining geometric surfaces with the topology of the Möbius strip, yielding realizations with additional geometric properties. Sweeping a line segment One way to embed the Möbius strip in three-dimensional Euclidean space is to sweep it out by a line segment rotating in a plane, which in turn rotates around one of its For the swept surface to meet up with itself after a half-twist, the line segment should rotate around its center at half the angular velocity of the plane's rotation. This can be described as a parametric surface defined by equations for the Cartesian coordinates of its points, for and where one parameter describes the rotation angle of the plane around its central axis and the other parameter describes the position of a point along the rotating line segment. This produces a Möbius strip of width 1, whose center circle has radius 1, lies in the -plane and is centered at The same method can produce Möbius strips with any odd number of half-twists, by rotating the segment more quickly in its plane. The rotating segment sweeps out a circular disk in