Hypercube

Hypercubes were a communication technology used by the Time Lords. They resembled cubes and were used by thinking or speaking to them. They were 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 it 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 on his own quick enough. Unlike the Hypercube he would find in the future, the Doctor constructed this one from six separate white cards and merged them with his mind. Calling upon his people for help proved to be a poor choice for the Doctor. It led to his Exile on Earth as the Third Doctor and the separation from his companions Jamie McCrimmon and Zoe Heriot. (DW: The War Games) This would mark the only time it was seen 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 remained alive by the time other Time Lords arrived; House killed them for parts to repair Auntie and Uncle. This produced TARDISes to feed House for five thousand years. The last Time Lord 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 into the main universe and found the Doctor's TARDIS because there were no other Time Lords left in 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. All the Doctor found was a cabinet filled with hypercubes House had stolen from the dead Time Lords. (DW: The Doctor's Wife)

The Doctor kept a hypercube in the TARDIS Drawing Room. (VG: The Gunpowder Plot)

Behind the scenes

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