  A beta expansion is the analogue of the base 10 representation of a real
number, where the base may be a non-integer. Although the greedy beta expansion
of 1 using a non-integer base is in general infinitely long and non-repeating,
it is known that if the base is a Pisot number, then this expansion will always
be finite or periodic. Some work has been done to learn more about these
expansions, but in general these expansions were not explicitly known. In this
paper, we present a complete list of the greedy beta expansions of 1 where the
base is any regular Pisot number less than 2, revealing a variety of remarkable
patterns. We also answer a conjecture of Boyd's regarding cyclotomic co-factors
for greedy expansions.

Number Theory;Numerical Analysis