The tortoise is believed to carry the world on its back and its long life no doubt accounts for its universal appeal as the symbol of longevity. Like Matsya, it is also replete with iconographic symbolism throughout Indic culture and philosophy. We'll try to list a few!
Kurma is able to withdraw its limbs within its shell. This imagery gives rise to the concept of the tortoise as a model of the self-restrained human, who has command over their indriyas (senses) and is able to withdraw from sense objects.
In Indian cosmology, Kurma is one of the avatars of Vishnu, a mount of river goddess Yamuna and also a lanchana (cognizance) of certain Jain Yakshas and Tirthankaras. The tortoise also symbolises the Sun, the waters and is the Lord of creation in Vedic cosmology.
Because of its shape, the tortoise is regarded as a symbol of the three worlds; its lower shell is this terrestrial world, upper shell the sky and in between is the atmosphere. It is a familiar sight to see a tortoise at the central place in a Hindu temple, either at the entrance, or in the centre of the main hall. The symbol has traveled far and wide, and the Balanese Hindu temples, called Meru, show a large tortoise at the basement, with its feet, mouth and tail protruding out.
Fascinating fact! In Rangoli, Kurma is commonly depicted in hexagonal shape, which is another symbol for the Kurma yantra used in Vedic astrology as a remedy for the malefic effects of Saturn. Incidentally, the north pole of Saturn is hexagonal in shape as per images released by NASA, whattt!