Transdimensional engineering
Transdimensional engineering, (TV: The Robots of Death) or simply dimensional engineering, (TV: Death in Heaven) was the technology of creating dimensionally transcendental objects, which the Fourth Doctor proclaimed a "key Time Lord discovery". It was used to make TARDISes bigger on the inside. (TV: The Robots of Death)
Principles
While on the planet Desolation, the Thirteenth Doctor claimed that the ability to engineer dimensions was beyond human comprehension. (TV: The Ghost Monument)
However, the Fourth Doctor had once attempted to demonstrate the principles of transdimensional engineering to Leela, by using two boxes of different size. He placed the larger box on the TARDIS console, and then walked to stand next to Leela and asked which one was bigger, which was pointed out by Leela, to which the Doctor stated it was smaller which was because of the distance. The Doctor said that if the large box was kept away at that distance, and had it nearby the larger box would fit inside the smaller box. Leela called that "silly," much to the Doctor's annoyance.(TV: The Robots of Death)
One account suggested that the First Doctor had been able to make his TARDIS bigger on the inside than on the outside was because the Ship had allowed him to personally observe the dimensions and multi-dimensions which all "existed together in one place with all the infinite others". From this enlightened point of view, physical size in one particular dimension became simply irrelevant. (PROSE: The Equations of Dr Who)
History
Dimensional engineering was perfected by the people of Gallifrey in Rassilon's days. A temporally-displaced Eleventh Doctor, under the alias of the Other, bore witness to the "fitting" of huge interior dimensions into a Type 1 TARDIS, against which the sentient timeship was trying to rebel. (COMIC: The Lost Dimension)
The War Chief, a High Councilmember-turned-Renegade Time Lord, (PROSE: The Legacy of Gallifrey) taught the War Lords how to construct dimensionally-transcendetal timeships known as SIDRATs. The scale and configuration of their interior dimensions could be altered from the outside by simply sliding cursors on the War Lords' central control panels. (TV: The War Games)
Shortly before the Dalek Civil War, the Dalek Emperor's Daleks constructed Dalek time machines which displayed dimensional transcendentalism similar to the Doctor's TARDIS's. (TV: The Chase, The Evil of the Daleks) However, after the Last Great Time War, the Cult of Skaro still described the technology which had permitted them to conceal an entire Dalek army in their Void Ship, namely the stolen Gallifreyan prison ship dubbed the Genesis Ark, "Time Lord science". (TV: Doomsday) At any rate, Mawdryn's people, who sought to emulate the Time Lords, also employed such techniques for their transmat capsules. (TV: Mawdryn Undead)
The Ux, who were capable of altering the physical world with a thought, were known to the Thirteenth Doctor through hearsay as "faith-driven dimensional engineers". She deduced that they had used this ability to build T'zim Sha his Shrine on the planet Ranskoor Av Kolos. The Ux later proved familiar with dimensional transcendentalism. (TV: The Battle of Ranskoor Av Kolos)
When Missy settled down the 3W Institute inside St Paul's Cathedral, with a group of 91 Cybermen, the Twelfth Doctor explained that nobody noticed thanks to "dimensional engineering — one space folded inside another". (TV: Death in Heaven)
