The voter model is one of the standard interacting particle systems.
Two related problems for this process are to analyze its behavior, after large
times $t$, for the sets of sites (1) sharing the same opinion as the site 0,
and (2) having the opinion that was originally at 0. Results on the sizes of
these sets were given by Sawyer (1979)and Bramson and Griffeath (1980). Here,
we investigate the spatial structure of these sets in $d \geq 2$, which we show
converge to quantities associated with super-Brownian motion, after suitable
normalization. The main theorem from Cox, Durrett and Perkins (2000) serves as
an important tool for these results.