Swayambhunath is among the oldest religious sites in Nepal. The dome at the base represents the entire world. When a person awakes (represented by eyes of wisdom and compassion) from the bonds of the world, the person reaches the state a bit higher. The thirteen pinacleson the top of it symbolises that sensient beings have to go through the thirteen stages of enlightenment to reach Buddhahood.
Above each pair of eyes is another eye, the third eye. Saying goes that when Buddha preaches, cosmic rays emanate from the third eye which actsas message to heavenly beings, so that those interested can come down to earth to listen to the Buddha. The hellish beings and beings below the human realm cannot come to earth to listen to the Buddha's teaching, however, the cosmic ray relieves their suffering when Buddha preaches.