  Quasicrystals can be described as projections of sections of higher
dimensional periodic lattices into real space. The image of the lattice points
in the projected out dimensions, called the perpendicular space, carries
valuable information about the local structure of the real space lattice. In
this paper, we show that perpendicular space projection can be used to analyze
the elementary excitations of a quasicrystal. In particular, we consider the
vertex tight binding model on the two dimensional Penrose lattice and
investigate the properties of strictly localized states using perpendicular
space images. Our method reproduces the previously reported frequencies for the
six types of localized states in this model. We also calculate the overlaps
between different localized states and show that the number of type-5 and
type-6 localized states which are independent from the four other types is a
factor of golden ratio $\tau=(1+\sqrt{5})/2$ higher than previously reported
values. Two orientations of the same type-5 or type-6 which are supported
around the same site are shown to be linearly dependent with the addition of
other types. We also show through exhaustion of all lattice sites in
perpendicular space that any point in the Penrose lattice is either in the
support of at least one localized state or is forbidden by local geometry to
host a strictly localized state.
