The Box was a seamless, violet contianer with a DNA response system, so that it could only open when touched by a certain individual.
Mira sent Leah to inflitrate Terra Nova and retreive the box. She found it hidden under the floor of Karem and Selena McMillen's home. However she was caught before she brought the box back to Mira, and the box was stored in the Clinic, so that it could be examined by Malcolm Wallace. ("The Runaway") After the Sixers used an Empirosaur as a diversion, they were able to escape with the box. Mira delivered it to Lucas, who succesfully opened the container. This revealed projections of equations, similar to those he had painted on the rocks near the waterfall. ("Nightfall") The equations were later revealed by Commander Nathaniel Taylor to be a part of an attempted recreation of formulas created by Lucas, after the orignals were destroyed. The formula some how allowed one to open a portal that led from the past to the future. ("Vs.")
The use for the box is finally discovered when Lucas Taylor uses it as a remote control to manipulate the portal. ("Within")