  The fiducial cross section for $\Upsilon$(1S) pair production in
proton-proton collisions at a center-of-mass energy of 13 TeV in the region
where both $\Upsilon$(1S) mesons have an absolute rapidity below 2.0 is
measured to be 79 $\pm$ 11 (stat) $\pm$ 6 (syst) $\pm$ 3 ($\mathcal{B}$) pb
assuming the mesons are produced unpolarized. The last uncertainty corresponds
to the uncertainty in the $\Upsilon$(1S) meson dimuon branching fraction. The
measurement is performed in the final state with four muons using proton-proton
collision data collected in 2016 by the CMS experiment at the LHC,
corresponding to an integrated luminosity of 35.9 fb$^{-1}$. This process
serves as a standard model reference in a search for narrow resonances decaying
to $\Upsilon$(1S)$\mu^+\mu^-$ in the same final state. Such a resonance could
indicate the existence of a tetraquark that is a bound state of two b quarks
and two $\bar{\mathrm{b}}$ antiquarks. The tetraquark search is performed for
masses in the vicinity of four times the bottom quark mass, between 17.5 and
19.5 GeV, while a generic search for other resonances is performed for masses
between 16.5 and 27 GeV. No significant excess of events compatible with a
narrow resonance is observed in the data. Limits on the production cross
section times branching fraction to four muons via an intermediate
$\Upsilon$(1S) resonance are set as a function of the resonance mass.
