  We study properties of an array of numbers, called "the triangle," in which
each row is formed by rotating all the numbers in the previous row to the left
by $m$ positions in cyclical fashion, then appending a number to the end of the
row. We show that a number's position in the triangle is uniquely determined by
the infinite sequence of column positions--called "rotation remainders"--which
we track as the number repeatedly rotates back to the first $m$ columns. The
rotation remainders can be viewed as the digits in a "base $m/(m+1)$" expansion
in an "$m$-adic" topological ring of a number encoding a given position in the
triangle. Properties of these expansions are used to prove interesting claims
about the triangle, such as the aperiodicity of any sequence of rotation
remainders.
