This paper presents the discovery of a three-dimensional (3D) Dirac semimetal in the material β-cristobalite BiO₂. The authors show that the pseudo-relativistic physics of graphene near the Fermi level can be extended to 3D materials. They demonstrate that certain space groups allow for 3D Dirac points as symmetry-protected degeneracies. Using ab initio calculations, they show that β-cristobalite BiO₂ exhibits three Dirac points at the Fermi level and is metastable, making it a 3D analog to graphene. In a Dirac semimetal, the conduction and valence bands touch at discrete points in the Brillouin zone (BZ) and disperse linearly around these points. In 2D, graphene exhibits such point-like degeneracies. In 3D, the analogous Hamiltonian is given by $\hat{H}(\mathbf{k}) = v_{ij}k_{i}\sigma_{j}$. This Hamiltonian is robust against perturbations because it uses all three Pauli matrices. The robustness of a Weyl point is quantified by the Chern number of the valence band on a sphere surrounding the point. However, the total Chern number associated with the entire Fermi surface must vanish, leading to the existence of two more Weyl points of opposite Chern number. The Fermi surface of a Dirac semimetal consists entirely of such point-like degeneracies. 3D Dirac semimetals are predicted to exist at the phase transition between a topological and a normal insulator when inversion symmetry is preserved. If either inversion or time reversal symmetry is broken, a Dirac point separates into Weyl points, leading to a Weyl semimetal. The authors show that certain double space-groups allow Dirac points at high symmetry points on the boundary of the BZ. As an example, they present ab initio calculations of β-cristobalite BiO₂, which exhibits Dirac points at three symmetry-related X points on the boundary of the FCC BZ. This system realizes a Dirac degeneracy first encountered in a tight-binding model of s-states in diamond. The authors also discuss the criteria necessary for a 3D Dirac point to exist. These include the presence of four-dimensional irreducible representations (FDIRs) at some point k in the BZ such that the four bands degenerate at k disperse linearly in all directions around k and the two valence bands carry zero total Chern number. They apply these criteria to two important space-groups, showing that the X point in space-group 227 is a candidate to host a Dirac semimetal if its FDIR can be elevated to the Fermi level. Indeed, they show that β-cristobalite BiO₂ exhibits such a Dirac point atThis paper presents the discovery of a three-dimensional (3D) Dirac semimetal in the material β-cristobalite BiO₂. The authors show that the pseudo-relativistic physics of graphene near the Fermi level can be extended to 3D materials. They demonstrate that certain space groups allow for 3D Dirac points as symmetry-protected degeneracies. Using ab initio calculations, they show that β-cristobalite BiO₂ exhibits three Dirac points at the Fermi level and is metastable, making it a 3D analog to graphene. In a Dirac semimetal, the conduction and valence bands touch at discrete points in the Brillouin zone (BZ) and disperse linearly around these points. In 2D, graphene exhibits such point-like degeneracies. In 3D, the analogous Hamiltonian is given by $\hat{H}(\mathbf{k}) = v_{ij}k_{i}\sigma_{j}$. This Hamiltonian is robust against perturbations because it uses all three Pauli matrices. The robustness of a Weyl point is quantified by the Chern number of the valence band on a sphere surrounding the point. However, the total Chern number associated with the entire Fermi surface must vanish, leading to the existence of two more Weyl points of opposite Chern number. The Fermi surface of a Dirac semimetal consists entirely of such point-like degeneracies. 3D Dirac semimetals are predicted to exist at the phase transition between a topological and a normal insulator when inversion symmetry is preserved. If either inversion or time reversal symmetry is broken, a Dirac point separates into Weyl points, leading to a Weyl semimetal. The authors show that certain double space-groups allow Dirac points at high symmetry points on the boundary of the BZ. As an example, they present ab initio calculations of β-cristobalite BiO₂, which exhibits Dirac points at three symmetry-related X points on the boundary of the FCC BZ. This system realizes a Dirac degeneracy first encountered in a tight-binding model of s-states in diamond. The authors also discuss the criteria necessary for a 3D Dirac point to exist. These include the presence of four-dimensional irreducible representations (FDIRs) at some point k in the BZ such that the four bands degenerate at k disperse linearly in all directions around k and the two valence bands carry zero total Chern number. They apply these criteria to two important space-groups, showing that the X point in space-group 227 is a candidate to host a Dirac semimetal if its FDIR can be elevated to the Fermi level. Indeed, they show that β-cristobalite BiO₂ exhibits such a Dirac point at