The structural and optical properties of controlled-heterostructure-potential, low-dimensional GaAs/AlGaAs nanostructures self-formed during organometallic chemical vapor deposition in tetrahedral pyramids etched in (111) B-GaAs substrates, are investigated using electron microscopy, cathodoluminescence, photoluminescence (PL), photon correlation spectroscopy, and theoretical modeling. Quantum wires/dots with AlGaAs cores with growth-controlled dimensions are formed, with a system of well-defined, low-dimensional nanostructure barriers around them. Transitions between carrier states confined in the AlGaAs quantum wires and dots are identified in the PL spectra, with features in good agreement with model calculations. Emission of single-photons and bunched-photon pairs is observed using temporal photon correlation spectroscopy. This self-formed nanostructure system provides new ways for shaping low-dimensional quantum structures and their heterostructure environment.