  Our understanding of the redshift $z>3$ galaxy population relies largely on
samples selected using the popular "dropout" technique, typically consisting of
UV-bright galaxies with blue colors and prominent Lyman breaks. As it is
currently unknown if these galaxies are representative of the massive galaxy
population, we here use the FourStar Galaxy Evolution (ZFOURGE) Survey to
create a stellar mass-limited sample at $z=3-4$. Uniquely, ZFOURGE uses deep
near-infrared medium-bandwidth filters to derive accurate photometric redshifts
and stellar population properties. The mass-complete sample consists of 57
galaxies with log M $>10.6$, reaching below $M^{\star}$ at $z=3-4$.
  On average, the massive $z=3-4$ galaxies are extremely faint in the observed
optical with median $R_{tot}^{AB}=27.48\pm0.41$ (restframe
$M_{1700}=-18.05\pm0.37$). They lie far below the UV luminosity-stellar mass
relation for Lyman break galaxies and are about $\sim100\times$ fainter at the
same mass. The massive galaxies are red ($R-Ks_{AB}=3.9\pm0.2$; restframe
UV-slope $\beta=-0.2\pm0.3$) likely from dust or old stellar ages. We classify
the galaxy SEDs by their restframe $U-V$ and $V-J$ colors and find a diverse
population: $46^{+6+10}_{-6-17}$% of the massive galaxies are quiescent,
$54^{+8+17}_{-8-10}$% are dusty star-forming galaxies, and only
$14^{+3+10}_{-3-4}$% resemble luminous blue star forming Lyman break galaxies.
This study clearly demonstrates an inherent diversity among massive galaxies at
higher redshift than previously known. Furthermore,we uncover a reservoir of
dusty star-forming galaxies with $4\times$ lower specific star-formation rates
compared to submillimeter-selected starbursts at $z>3$. With $5\times$ higher
numbers, the dusty galaxies may represent a more typical mode of star formation
compared to submillimeter-bright starbursts.

Astrophysics of Galaxies;Cosmology and Nongalactic Astrophysics