  The fractal properties of four-dimensional Euclidean simplicial manifold
generated by the dynamical triangulation are analyzed on the geodesic distance
D between two vertices instead of the usual scale between two simplices. In
order to make more unambiguous measurement of the fractal dimension, we employ
a different approach from usual, by measuring the box-counting dimension which
is computed by counting the number of spheres with the radius D within the
manifold. The numerical result is consistent to the result of the random walk
model in the branched polymer region. We also measure the box-counting
dimension of the manifold with additional matter fields. Numerical results
suggest that the fractal dimension takes value of slightly more than 4 near the
critical point. Furthermore, we analyze the correlation functions as functions
of the geodesic distance. Numerically, it is suggested that the fractal
structure of four-dimensional simplicial manifold can be properly analyzed in
terms of the distance between two vertices. Moreover, we show that the behavior
of the correlation length regards the phase structure of 4D simplicial
manifold.
