  The projections of the lattices, may be used as models of quasicrystals, and
the particular affine extension of the $H_2$ symmetry as a subgroup of $A_4$,
discussed in the work, presents a different perspective to 5-fold symmetric
quasicrystallography. Affine $H_2$ is obtained as the subgroup of the affine
$A_4$. The infinite group with local dihedral symmetry of order 10 operates on
the Coxeter plane of the root and weight lattices of $A_4$ whose Voronoi cells
tessellate the 4D Euclidean space possessing the affine $A_4$ symmetry. It is
shown that the projection of the Voronoi cell of the root lattice tiles the
Coxeter plane with thick and thin rhombuses with the action of the affine $H_2$
symmetry. Projection of the Voronoi cell of the weight lattice onto the Coxeter
plane tessellates the plane with four different tiles: thick and thin rhombuses
with different edge lengths obtained from the projection of the square faces
and two types of hexagons obtained from the projection of the hexagonal faces
of the Voronoi cell. Structure of the local dihedral symmetry $H_2$ fixing a
particular point on the Coxeter plane is determined.
