  The Cairo tessellation refers to a pattern of type 2 pentagons that can pave
an infinite plane without creating a gap or overlap. We reveal the hidden,
layered Cairo tessellations in the pyrite structure with a general chemical
formula of $AB_2$ and space group $pa\bar3$. We use this hidden tessellation
along with density functional theory calculations to examine the possibility of
obtaining a two-dimensional (2D) material with the Cairo tessellation from the
bulk, using PtP$_2$ as an example. Unlike previously reported single-layer
materials such as PdSe$_2$ with a buckled, pentagonal structure-strictly
speaking, not belonging to the Cairo tessellation, we find that single-layer
PtP$_2$ is completely planar exhibiting dynamically stable phonon modes. We
also observe a reduction in the bandgaps PtP$_2$ from bulk to single layer
using the Heyd-Scuseria-Ernzerhof (HSE) hybrid density functional, and the
bandgap type switches from indirect to direct. By contrast, using the standard
Perdew-Burke-Ernzerhof functional leads to the conclusion of single-layer
PtP$_2$ being metallic. We further study the bonding characteristics of this
novel single-layer material by computing the Bader charge transfer, the
electron localization function, and the crystal orbit overlap population, which
show mixed P-P covalent bonding and Pt-P ionic bonding, with the former being
stronger. Finally, we study the surface states of single-layer PtP$_2$ and
consider the spin-orbit coupling. We observe no spin-helical Dirac cone states,
therefore ruling out single-layer PtP$_2$ as a topological insulator. We expect
the example demonstrated in this work will stimulate interest in
computationally identifying novel 2D materials from a variety of bulk materials
with the pyrite structure.
