The ant is back on the third hyperbolic paraboloid, but this time under it and standing upside-down . ( Compare with Klein bottle .)
42.
Franklin proved that the chromatic number of a graph embedded in the Klein bottle can be as large as 6, but never exceeds 6.
43.
The Franklin graph can be drawn on the Klein bottle in a way that forms six mutually-adjacent regions, showing that this bound is tight.
44.
The homology is worked as an example in Mayer Vietoris sequence # Klein bottle . talk ) 22 : 04, 1 May 2011 ( UTC)
45.
Some of the various are well-known groups . is the free abelian group on two generators, and is the fundamental group of the Klein bottle.
46.
For parity issues, I think it is possible to have such paths if space can be imagined to twist in Mobius / Klein bottle configurations.
47.
More formally, the Klein bottle is the square [ 0, 1 ] ?[ 0, 1 ] with sides identified by the relations for and for.
48.
The behaviour of the system on the Klein bottle is much more similar to the one on the torus than the one on the real projective plane.
49.
In 1934, Franklin disproved the Heawood conjecture for the Klein bottle by showing that any map drawn on the Klein bottle can be coloured with at most six colours.
50.
In 1934, Franklin disproved the Heawood conjecture for the Klein bottle by showing that any map drawn on the Klein bottle can be coloured with at most six colours.
