is the group Spin(3), and represented by the unit quaternions. Thus quaternions are a preferred method for representing spatial rotations – see quaternions and spatial rotation. However, it is often desirable to represent rotations by a set of three numbers, known as Euler angles (in numerous variants), both because this is conceptually simpler for someone familiar with planar rotation, and because one can build a combination of three gimbals to produce rotations in three dimensions. Topologically this corresponds to a map from the 3-torus T3 of three angles to the real projective space RP3 of rotations, and the resulting map has imperfections due to this map being unable to be a covering map. Specifically, the failure of the map to be a local homeomorphism at certain points is referred to as gimbal lock, and is demonstrated in the animation at the right – at some points (when the axes are coplanar) the rank of the map is 2, rather than 3, meaning that only 2 dimensions of rotations can be realized from that point by changing the angles. This causes problems in applications, and is formalized by the notion of a covering space. See also Bethe lattice is the universal cover of a Cayley graph Covering graph, a covering space for an undirected graph, and its special case the bipartite double cover Covering group Galois connection Quotient space (topology) Literature Allen Hatcher: Algebraic Topology. Cambridge Univ. Press, Cambridge, ISBN 0-521-79160-X Otto Forster: Lectures on Riemann surfaces. Springer Berlin, München 1991, ISBN 978-3-540-90617-9 James Munkres: Topology. Upper Saddle River, NJ: Prentice Hall, Inc., ©2000, ISBN 978-0-13-468951-7 Wolfgang Kühnel: Matrizen und Lie-Gruppen. Springer Fachmedien Wiesbaden GmbH, Stuttgart, ISBN 978-3-8348-9905-7 References Algebraic topology Homotopy theory Fiber bundles Topological graph theory