Atomic nuclei are usually deformed except for a few magic nuclei which are spherical in shape. Modern nuclear experiments have discovered that some nuclei may have a pear-like shape, with reflection asymmetry. According to theory, other nuclei may also have a pyramid-like shape with tetrahedral symmetry, but this new nuclear symmetry has not yet been discovered experimentally. This paper aims to explain the formation mechanism of the tetrahedral symmetry as well as give a theoretical description based on the symmetry and symmetry-breaking of a quantum strong interaction many-body system. The basic ways to search for tetrahedral nuclei and recent experimental developments are also discussed.