The spiral of the nautilus reveals that it conforms to the shape of the Golden Ratio spiral, what is known as a "sacred geometry". Based on the Fibonacci series: that all living things have this ratio in common and are therefore inherently linked or all parts of a greater whole. 
As a symbol, the chambered nautilus is a powerful metaphor for one's spiritual evolution.  We may find that once we grow, we have mentally moved into a new box, we can never go back to the smaller one - it will no longer fit. Thus the spiral suggests that we can grow forever - since there is never a design for a final chamber. A truly meaningful and significant shape.