  We propose a new approach to the value distribution theory of entire
holomorphic curves. We define a ``packing density'' of an entire holomorphic
curve, and show that it has various non-trivial properties. We prove a ``gap
theorem'' for holomorphic maps from elliptic curves to the complex projective
space, and study the relation between theta functions and our packing problem.
Applying the Nevanlinna theory, we investigate the packing densities of entire
holomorphic curves in the complement of hyperplanes.
