  Algebraic integers in totally imaginary quartic number fields are not
discrete in the complex plane under a fixed embedding, which makes it
impossible to visualize all integers in the plane, unlike the quadratic
imaginary algebraic integers. In this note we consider a naturally occurring
discrete subset of the algebraic integers with similar properties as lattices.
For the fifth cyclotomic field, we investigate those integers with absolute
values under a fixed embedding in a given bound. We show that such integers
form a discrete set in the complex plane. It is observed that this subset has
quasi-periodic appearance. In particular, we also show that the distance
between a fixed point to the most adjacent point in this subset takes only two
possible values.
