  This paper explores properties and applications of an ordered subset of the
quadratic integer ring $\mathbb{Z}\left[\frac{1+\sqrt{5}}{2}\right]$. The
numbers are shown to exhibit a parity triplet, as opposed to the familiar
even/odd doublet of the regular integers. Operations on these numbers are
defined and used to generate a succinct recurrence relation for the
well-studied Fibonacci diatomic sequence, providing the means for generating
analogues to the famed Calkin-Wilf and Stern-Brocot trees. Two related fractal
geometries are presented and explored, one of which exhibits several identities
between the Fibonacci numbers and golden ratio, providing a unique geometric
expression of the Fibonacci words and serving as a powerful tool for
quantifying the cardinality of ordinal sets. The properties of the presented
set of numbers illuminate the symmetries behind ordinals in general, as well as
provide perspective on the natural numbers and raise questions about the
dynamics of transfinite values. In particular, the first transfinite ordinal
$\omega$ is shown to be logically consistent with a value whose cardinality is
dual: both zero and one. Considerations of these points and opportunities for
further study are discussed.
