The current work investigates a novel three-dimensional boron nitride called bulk B4N4 and its corresponding two-dimensional monolayer B4N4 based on the first-principles of density functional theory. The phonon spectra prove that bulk B4N4 and monolayer B4N4 are dynamically stable. The molecular dynamics simulations verify that bulk B4N4 and monolayer B4N4 have excellent thermal stability of withstanding temperature up to 1000 K. The calculated elastic constants state that bulk B4N4 and monolayer B4N4 are mechanically stable, and bulk B4N4 has strong anisotropy. The theoretically obtained electronic structures reveal that bulk B4N4 is an indirect band-gap semiconductor with a band gap of 5.4 eV, while monolayer B4N4 has a direct band gap of 6.1 eV. The valence band maximum is mainly contributed from B-2p and N-2p orbits, and the conduction band minimum mainly derives from B-2p orbits. The electron transitions from occupied N-2p states to empty B-2p states play important roles in the dielectric functions of bulk B4N4 and monolayer B4N4. The newly proposed monolayer B4N4 is a potential candidate for designing optoelectronic devices such as transparent electrodes due to its high transmissivity.