  We introduce an extension of cone splines and box splines to fractional and
complex orders. These new families of multivariate splines are defined in the
Fourier domain along certain $s$-dimensional meshes and include as special
cases the three-directional box splines \cite{article:condat} and hex splines
\cite{article:vandeville} previously considered by Condat, Van De Ville et al.
These cone and hex splines of fractional and complex order generalize the
univariate fractional and complex B-splines defined in
\cite{article:ub,article:fbu} and investigated in, e.g.,
\cite{article:fm,article:mf}. Explicit time domain representations are derived
for these splines on $3$-directional meshes. We present some properties of
these two multivariate spline families such as recurrence, decay and
refinement. Finally it is shown that a bivariate hex spline and its integer
lattice translates form a Riesz basis of its linear span.
