  We examine the geometry of the Cayley graph of the lamplighter group with
respect to the generating set rising from its interpretation as an automata
group due to Grigorchuk and Zuk. We find some metric behavior with respect to
this generating set analogous to the metric behavior in the standard wreath
product generating set. The similar metric behavior includes expressions for
geodesic paths and families of `dead-end' elements, which are endpoints of
terminating geodesic rays. We also exhibit some different metric behavior
between these two generating sets related to the existence of `seesaw'
elements.
