Structure-property relationships have always been guiding principles in discovering new materials.Here we explore the relationships to discover novel two-dimensional (2D) materials with the goal of identifying 2D magnetic semiconductors for spintronics applications. In particular, we report a density functional theory + U study of single-layer antiferromagnetic (AFM) semiconductor CoS2 with the pentagonal structure forming the so-called Cairo Tessellation. We find that this singlelayer magnet exhibits an indirect bandgap of 1.06 eV with light electron and hole effective masses of 0.03 and 0.10 m0, respectively, which may lead to high carrier mobilities. The hybrid density functional theory calculations correct the bandgap to 2.24 eV. We also compute the magnetocrystalline anisotropy energy (MAE), showing that the easy axis of the AFM ordering is out of plane with a sizable MAE of 153 µeV per Co ion. We further calculate the magnon frequencies at different spin-spiral vectors, based on which we estimate the Néel temperatures to be 20.4 and 13.3 K using the mean field and random phase approximations, respectively. We then apply biaxial strains to tune the bandgap of single-layer pentagonal CoS2. We find that the energy difference between the ferromagnetic and AFM structures strongly depends on the biaxial strain, but the ground state remains the AFM ordering. Although the low critical temperature prohibits the magnetic applications of single-layer pentagonal CoS2 at room temperature, the excellent electrical properties may find this novel single-layer semiconductor applications in optoelectronic nanodevices.

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Pentagonal geometries are attractive to professional and amateur mathematicians, for whom one long-standing question is how many types of irregular, convex pentagons can tessellate a plane leaving no gaps or overlaps. It is well known that a regular pentagonal tiling on the Euclidean plane is impossible because its internal angle is not a divisor of 360°. But the number of families of irregular, convex pentagons that tile a plane have grown to 15 since the discovery of the first type by Reinhardt in 1918 [1]. Figure 1 illustrates 15 types of pentagons that have been discovered so far. This list is recently claimed to be complete according to the proof by Rao [2].The 15 types of pentagons are distinguished from each other by the constraints of the five angles and the five side lengths. For example, the sum of angles B and E in type 10 pentagon is 180°, whereas there is no such a geometrical relation in type 3 pentagon. Each of these 15 monohedral tiling types possesses more than one degree of freedom, except types 14 and 15 whose interior angles are all fixed. Apart from different constraints, one can distinct these 15 types by their primitive units and symmetry groups of tiling topologies. For instance, the primitive cell of type 13 tiling plane contains eight pentagons while six pentagons make up the primitive cell of type 14 tiling plane.Tessellating a plane with irregular, convex pentagons is like playing a jigsaw puzzle. However, encoding the puzzle with laws of quantum mechanics by placing atoms at the vertices of each piece of the puzzle will likely alter the shape of the piece owing to the electron-electron, electron-ion, and ion-ion interactions. We will show that ab initio density functional theory (DFT) calculations can be used to play this imaginary puzzle.Ab initio DFT simulations are an ideal tool to optimize the atomic positions of a system until it reaches a configuration of local energy minimum of the corresponding high-dimensional energy surface. Essentially, DFT is an eigenvalue solver giving information of total energy that can be used to derive a number of properties of the

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