  We report on a search for mutually unbiased bases (MUBs) in 6 dimensions. We
find only triplets of MUBs, and thus do not come close to the theoretical upper
bound 7. However, we point out that the natural habitat for sets of MUBs is the
set of all complex Hadamard matrices of the given order, and we introduce a
natural notion of distance between bases in Hilbert space. This allows us to
draw a detailed map of where in the landscape the MUB triplets are situated. We
use available tools, such as the theory of the discrete Fourier transform, to
organise our results. Finally we present some evidence for the conjecture that
there exists a four dimensional family of complex Hadamard matrices of order 6.
If this conjecture is true the landscape in which one may search for MUBs is
much larger than previously thought.
