Hypercube

Hypercubes were a communication technology used by the Time Lords which resembled cubes; they were used by thinking or speaking to them and then sent through time and space to Gallifrey or to someone they trusted enough to rescue them. They could be used by humans as well as Time Lords. (EDA: Vampire Science, The Ancestor Cell) They were theoretically impossible to intercept, but a Little Brother of Faction Paradox intercepted a cube the Eighth Doctor sent to the Time Lords. They were mainly owned by members of official Gallifreyan organisations. (EDA: Unnatural History)

History of usages
Irving Braxiatel once used a hypercube to contact his people. (BNA: Tears of the Oracle)

The First Doctor collected his when he worked for the Bureau of Possible Events, but left them behind on Gallifrey when he escaped with Susan. It was returned to him by his mentor K'anpo Rimpoche shortly before his first regeneration along with his Five Hundred Year Diary. (ST: The Three Paths)

The Second Doctor used a hypercube to ask the Time Lords to return the soldiers kidnapped by the War Lords to their own time periods as he could not do so own his own quick enough. Unlike the Hypercube he would find in the future, the Doctor constructed this one from six seperate white cards and merged them with his mental energy. However, calling upon his people for help proved to be a poor choice for the Doctor as it lead to his Exile on Earth as the Third Doctor and the seperation from his companions Jamie McCrimmon and Zoe Heriot. (DW: The War Games) This would mark the only time it was used until Series 6 of the revived series.

Hundreds of Time Lords, stranded in the bubble universe by House, used hypercubes to send distress calls. However, none of them were able to remain alive by the time other Time Lords arrived as House killed them for parts needed to repair Auntie and Uncle. This set of the TARDIS food supply for House in the duration of 5000 years. The last unfortunate soul to arrive on House was the Doctor's good friend Corsair. He recorded his hypercube, but was killed before he could send it. The Corsair's hypercube was sent out into the main universe and found the Doctor's TARDIS as there were no other Time Lords left in throughout time and space. After the Doctor arrived in the bubble universe, he heard the distress calls on Nephew's translation sphere and thought this meant there were other living Time Lords besides Corsair. To his dismay and fury, all the Doctor found was a cabinet filled with hypercubes House stole from the dead Time Lords. (DW: The Doctor's Wife)

The Doctor kept a hypercube in the TARDIS Drawing Room. It is most certainly the very same one from The Corsair, as he had no time to reach the others in storage. It was most likely kept by the Doctor as a memento of his old friend. (VG: The Gunpowder Plot)

Behind the scenes

 * In real-world mathematics, a hypercube is a higher-dimensional analogue to a cube, the former having four dimensions, the latter three.