There is always a great deal of symbolism in the appearance ascribed to a deity, but what can we learn from. the fact that Ganesha, the elephant headed god in the Hindu religion, has only one tusk? One might think it represents some imperfection, but in reality it is exactly the opposite.
It happened once upon a time that the Sage Vyasa felt inspired to dictate the great epic the Mahabharata, and asked Ganesha to write down the verses. Ganesha agreed to the task, but requested that once he began his divine flow of words, Vyasa would not stop; Ganesha wished to receive the flow of inspiration uninterrupted.
Vyasa agreed to the request of Ganesha, but in turn he requested that Ganesha would not only write each verse but understand it as well. To this, Ganesha also agreed, and so the dictation began, Vyasa transmitting the divine stream, and Ganesha writing with a quill pen. And whenever Vyasa needed to catch his breath, he would deliver a particularly challenging verse, so that Ganesha has to reflect upon it before he was ready to continue.
So they continued, through the scores of thousands of verses of the Mahabharata, but the quill pen soon wore out, and Ganesha, filled with delight at the wisdom he was hearing, and obedient to the promise he had made, snapped off one of his tusks and used it to complete the task.
Therefore the broken tusk of Ganesha represents the sacrifice by which knowledge is gained.