•Octagonal-C (-Si) and dodecagonal-C (-Si) are four novel two-dimensional carbon or silicon monolayer materials with sp2-like hybrid chemical bonds.•Octagonal-C (-Si) and dodecagonal-C (-Si) have good mechanical and dynamic stability.•Dodecagonal-C has very small band gap of 0.09 eV, and it can be a candidate for commercial optical filters and infrared detectors. Two-dimensional materials play increasingly important roles in the fields of electronic and photoelectric applications. Four two-dimensional carbon and silicon monolayers with sp2-like hybrid chemical bonds have been predicted by the first-principles calculations based on density functional theory in the current work. One type of two-dimensional monolayer materials is composed of four- and eight-membered rings called octagonal-C and octagonal-Si, and another type of two-dimensional structures consists of regular triangles and twelve-membered rings named dodecagonal-C and dodecagonal-Si. The theoretical phonon dispersions show that all predicted structures are dynamically stable. The molecular dynamics simulation confirms that octagonal-C (-Si) and dodecagonal-C have good thermal stability to maintain their crystal structures lower than 1000 K, while dodecagonal-Si has slightly worse thermal stability. The analysis of electron density differences manifests that the covalent C-C and Si-Si bonds are dominant in these two-dimensional monolayer structures. All these four structures are mechanically stable, and the mechanical properties of octagonal-C are superior to the other three. The electronic band structures indicate that octagonal-C (-Si) and dodecagonal-Si are metallic. Interestingly, dodecagonal-C has a very small band gap of 0.09 eV, and it maybe have potential applications in infrared optical devices such as commercial optical filters and infrared detectors.