Moebius
Group together anything on the Moebius strip concept, including Moebius ladder, which is an interesting animated graphic defined in points


Moebius strip
Top A Möbius strip, Möbius band, or Möbius loop, also spelled Mobius or Moebius, is a surface with only one side (when embedded in threedimensional Euclidean space) and only one boundary. The Möbius strip has the mathematical property of being unorientable. It can be realized as a ruled surface. Its discovery is attributed to the German mathematicians August Ferdinand Möbius and Johann Benedict Listing in 1858, though a structure similar to the Möbius strip can be seen in Roman mosaics dated circa 200–250 AD. An example of a Möbius strip can be created by taking a paper strip and giving it a halftwist, and then joining the ends of the strip to form a loop. However, the Möbius strip is not a surface of only one exact size and shape, such as the halftwisted paper strip depicted in the illustration. Rather, mathematicians refer to the closed Möbius band as any surface that is homeomorphic to this strip. Its boundary is a simple closed curve, i.e., homeomorphic to a circle. This allows for a very wide variety of geometric versions of the Möbius band as surfaces each having a definite size and shape. For example, any rectangle can be glued to itself (by identifying one edge with the opposite edge after a reversal of orientation) to make a Möbius band. Some of these can be smoothly modeled in Euclidean space, and others cannot. A halftwist clockwise gives an embedding of the Möbius strip different from that of a halftwist counterclockwise – that is, as an embedded object in Euclidean space, the Möbius strip is a chiral object with right or lefthandedness. However, the underlying topological spaces within the Möbius strip are homeomorphic in each case. An infinite number of topologically different embeddings of the same topological space into threedimensional space exist, as the Möbius strip can also be formed by twisting the strip an odd number of times greater than one, or by knotting and twisting the strip, before joining its ends. The complete open Möbius band is an example of a topological surface that is closely related to the standard Möbius strip, but that is not homeomorphic to it. Finding algebraic equations, the solutions of which have the topology of a Möbius strip, is straightforward, but, in general, these equations do not describe the same geometric shape that one gets from the twisted paper model described above. In particular, the twisted paper model is a developable surface, having zero Gaussian curvature. A system of differentialalgebraic equations that describes models of this type was published in 2007 together with its numerical solution.[6] The Euler characteristic of the Möbius strip is zero.


Moebius ladder
Top In graph theory, the Möbius ladder Mn is a cubic circulant graph with an even number n of vertices, formed from an ncycle by adding edges (called "rungs") connecting opposite pairs of vertices in the cycle. It is sonamed because (with the exception of M6 = K3,3) Mn has exactly n/2 4cycles[1] which link together by their shared edges to form a topological Möbius strip. Möbius ladders were named and first studied by Guy and Harary (1967). Contents 1 Properties 2 Graph minors 3 Chemistry and physics 4 Combinatorial optimization 5 See also 6 Notes 7 References 8 External links Properties Every Möbius ladder is a nonplanar apex graph, meaning that it cannot be drawn without crossings in the plane but removing one vertex allows the remaining graph to be drawn without crossings. Möbius ladders have crossing number one, and can be embedded without crossings on a torus or projective plane. Thus, they are examples of toroidal graphs. Li (2005) explores embeddings of these graphs onto higher genus surfaces. Möbius ladders are vertextransitive – they have symmetries taking any vertex to any other vertex – but (again with the exception of M6) they are not edgetransitive. The edges from the cycle from which the ladder is formed can be distinguished from the rungs of the ladder, because each cycle edge belongs to a single 4cycle, while each rung belongs to two such cycles. Therefore, there is no symmetry taking a cycle edge to a rung edge or vice versa. When n = 2 (mod 4), Mn is bipartite. When n = 0 (mod 4), it is not bipartite. The endpoints of each rung are an even distance apart in the initial cycle, so adding each rung creates an odd cycle. In this case, because the graph is 3regular but not bipartite, by Brooks' theorem it has chromatic number 3. De Mier & Noy (2004) show that the Möbius ladders are uniquely determined by their Tutte polynomials. The Möbius ladder M8 has 392 spanning trees; it and M6 have the most spanning trees among all cubic graphs with the same number of vertices.[2] However, the 10vertex cubic graph with the most spanning trees is the Petersen graph, which is not a Möbius ladder. The Tutte polynomials of the Möbius ladders may be computed by a simple recurrence relation.[3]
