  We study higher order tangents and higher order Laplacians on p.c.f.
self-similar sets with fully symmetric structures, such as $D3$ or $D4$
symmetric fractals. Firstly, let $x$ be a vertex point in the graphs that
approximate the fractal, we prove that for any $f$ defined near $x$, the higher
oder weak tangent of $f$ at $x$, if exists, is the uniform limit of local
multiharmonic functions that agree with $f$ in some sense near $x$. Secondly,
we prove that the higher order Laplacian on a fractal can be expressible as a
renormalized uniform limit of higher order graph Laplacians on the graphs that
approximate the fractal. The main technical tool is the theory of local
multiharmonic functions and local monomials analogous to $(x-x_0)^j/j!$. The
results in this paper are closely related to the theory of local Taylor
approximations, splines and entire analytic functions. Some of our results can
be extended to general p.c.f. fractals. In Appendix of the paper, we provide a
recursion algorithm for the exact calculations of the boundary values of the
monomials for $D3$ or $D4$ symmetric fractals, which is shorter and more direct
than the previous work on the Sierpinski gasket.
