Hypercube

Hypercubes, also known as tesseracts, were a Gallifreyan form of communication. Physically resembling white cubes, they were able to store a Time Lord's thoughts or speech and could be sent through time and space. They could be used by humans as well as Time Lords. (PROSE: Vampire Science, The Ancestor Cell) They were theoretically impossible to intercept, but a member of Faction Paradox intercepted a cube the Eighth Doctor sent to the Time Lords. They were mainly owned by members of official Gallifreyan organisations. (PROSE: Unnatural History) The Eleventh Doctor referred to them as "psychic containers". (TV: The Doctor's Wife)

History of usages
Irving Braxiatel once used a hypercube to contact his people about the developments he had learned about the People. (PROSE: Tears of the Oracle) The First Doctor once accidentally returned Ian Chesterton and Barbara Wright to Shoreditch 1963, but before had left. When the Doctor refused to let them stay, Ian and Barbara argued with the Doctor about him having kidnapped them. Not long after, Susan Foreman showed a similar anger at the Doctor having taken her away from Gallifrey without her consent. The Doctor decided that using the Hypercube was the only way to fix all of their problems. Learning about this plan, the three companions rushed back to the TARDIS, where Susan disuaded him, scared of the consequences. (AUDIO: Home Again, Home Again)

Stoyn's beacon on the sentient planet he was situated on used six white squares, the components of a hypercube. After his beacon was disabled, the Second Doctor pocketed the squares. (AUDIO: The Dying Light) He was later forced to use them 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. The Doctor constructed his hypercube from the six white squares by merging them with his mind, assembling them into the shape of a cube. Calling upon his people for help proved to be a personal sacrifice for the Doctor, effectively turning himself in, resulting in his separation from his companions Jamie McCrimmon and Zoe Heriot, his exile on Earth, and a forced regeneration into his third incarnation. (TV: The War Games)

The Fifth Doctor used a hypercube to notify the Time Lords that he had found Zero. (AUDIO: Spring)

The Sixth Doctor was prepared to use a hypercube to summon the Time Lords for help in preventing the Cybermen from changing history, even though he was aware he would be punished with no way of defending himself. (AUDIO: Last of the Cybermen)

The Seventh Doctor later slipped Ace a tesseract on the planet Heaven in order to be able to contact her in the future. (PROSE: Love and War) He later used it to do so whilst arranged for the TARDIS to purge the infection it had contracted after their visit to Tír na n-Óg. (PROSE: Deceit)

During the War in Heaven, the War King kept the twelve components of a hybercube stacked neatly on his desk as the last remaining link to his past as a renegade. (PROSE: The Book of the War)

During the Last Great Time War, the Eighth Doctor received a hypercube from Susan Foreman. (PROSE: Ghost of Christmas Past) Susan herself was sent several urgent hypercubes from Gallifrey during the Time War. Most of them were removed by the Eighth Doctor, who didn't want her to join the war. (AUDIO: All Hands on Deck) Cardinal Ollistra sent the Time Lord command to collect her when the Doctor's TARDIS crashed. (AUDIO: One Life)

Following the Last Great Time War, a hypercube was on sale at the Fluren Temporal Bazaar. (COMIC: Weapons of Past Destruction)

Hundreds of Time Lords who were stranded in a bubble universe by House attempted to use hypercubes to send distress calls. However, House killed many or all of them for parts to repair Auntie and Uncle, feeding itself on their TARDISes for five thousand years. One Time Lord to arrive on House was the Doctor's good friend the Corsair. He recorded a hypercube, which was sent into the main universe. It eventually found the Doctor's TARDIS, because there were no other Time Lords left in time and space. After the Eleventh 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. The Doctor was enraged when he found a cabinet filled with hypercubes which House had stolen from the dead Time Lords. (TV: The Doctor's Wife)

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

The New Dalek Paradigm and the Dalek Time Controller sent a hypercube to the Eleventh Doctor's TARDIS to lead him to the planet Gethria. From there, the Daleks manipulated the Doctor into leading them to the Cradle of the Gods, a monument which they planned to use to transform the Sunlight Worlds into copies of Skaro. Aided by Ollus, Sabel and Jenibeth Blakely, the Doctor set the Cradle to self-destruct and the Daleks abandoned the plan. (PROSE: The Dalek Generation)

Behind the scenes
In real-world mathematics, a is a higher-dimensional analogue of a cube. The hypercube of four dimensions is specifically called a "".

Some of the voice lines used for the Hypercubes in The Doctor's Wife were included individually in the game files of The Gunpowder Plot - assumedly for application on the Hypercube that can be found in The Drawing Room - but ultimately they went unused. The full lines can be heard here, and include explicit references to Fluid Links, the War Lord home planet, and Minyos: https://youtu.be/gItK0Yz4USs