Dimensional transcendentalism

Dimensional transcendentalism was the state wherein an object's interior was bigger than its exterior, an effect made possible by transdimensional engineering. (TV: The Robots of Death) It is in opposition to being dimensionally immanent, where something's interior is smaller than its exterior. (PROSE: Cold Fusion)

The First Doctor was the first to attempt to explain at how his TARDIS was bigger on the inside to Ian Chesterton and Barbara Wright by saying the TARDIS box was just an entrance chamber to a fourth dimension and that a reversal of time acceleration occurs at the inner door. Therefore, all but the box was invisible to the outside world. (COMIC: The Secrets of the Tardis) Another account stated that the interior of the TARDIS existed "outside [humans'] world of size" altogether. (PROSE: Who is Dr Who?)

The Fourth Doctor once explained the principle to Leela by using the analogy of how a larger cube might appear to be able to fit inside a smaller one were the larger cube further away, yet immediately accessible at the same time. He described this as "a key Time Lord discovery". (TV: The Robots of Death) Rory Williams later surprised the Eleventh Doctor with his understanding of the principle, that the inside was "another dimension" unto itself; (TV: The Vampires of Venice) the Thirteenth Doctor told Yaz that her brain would have to itself be "bigger on the inside" for her to understand how it worked. (PROSE: TARDIS Tour) Though some humans found it hard to understand, even if they were brilliant minds. Nikola Tesla immediately understood "the interior dimensions transcend the exterior", unlike Thomas Edison, who stated that "I don't understand it." (TV:  Nikola Telsa's Night of Terror) Nardole once explained that once one understands how the process worked, it was possible to create dimensional transcendentalism with little effort. (TV: The Pilot)

The relationship between the interior and exterior sizes of a TARDIS could be controlled via the dimensional control, (TV: "Checkmate") as well as a series of dimension dams. (TV: The Name of the Doctor) Dimensionally transcendental objects other than TARDISes included Dalek time machines, (TV: The Chase) the SIDRATs constructed by the War Chief, (TV: The War Games) the Genesis Ark, (TV: Doomsday) by some accounts, the Celestial Omnibus, (PROSE: Just the Ticket, et al) the Doctor's pockets, (TV: Genesis of the Daleks, The Runaway Bride, The Vampires of Venice), the Towers of Canonicity and Likelihood on Gallifrey, (PROSE: The Gallifrey Chronicles) and clown cars from the planet Semtis. (PROSE: Shroud of Sorrow)

Sometimes when a TARDIS was dying, its dimension dams would start to break down in what was called a "size leak". The Eleventh Doctor described it as "All the bigger on the inside starts leaking to the outside. It grows." This happened to the Doctor's TARDIS, as it was dying on Trenzalore. Here it served as the Doctor's tomb in an alternative timeline in which the Time Lords never granted the Doctor a new cycle of regenerations at the conclusion of the Siege of Trenzalore. Prior to his battle, the Eleventh Doctor and Clara Oswald discovered the ruined TARDIS relative to his future when they went to Trenzalore. It still appeared as a police box but was much larger. Clara described it as "one hell of a monument." (TV: The Name of the Doctor)

The Daleks were capable of creating dimensionally transcendent ships, (AUDIO: The Apocalypse Element) including their factory ships. (PROSE: War of the Daleks)

The Seventh Doctor temporarily expanded the interior dimensions of a boarding house in Cheldon Bonniface to allow all the guests at Bernice Summerfield's wedding to stay there. (PROSE: Happy Endings)

Jack Harkness once suggested that a dimensionally transcendental chameleon circuit was placed on a particular spot in Roald Dahl Plass, where it welded its perception properties to the Cardiff Space-Time Rift, allowing for the invisible lift used to enter Torchwood Cardiff. (TV: Everything Changes) Indeed, Jack had been present in one instance where the Doctor's TARDIS landed on Roald Dahl Plass, where it used the rift to refuel. (TV: Boom Town)

Upon meeting the Teselecta, "a robot worked by tiny people", the Eleventh Doctor, wondering how they could all fit inside, briefly suggested that it was bigger on the inside until he discovered it was via basic miniaturisation sustained by a compression field. (TV: Let's Kill Hitler)

used St Paul's Cathedral as a base for 3W, making it bigger on the inside in order to create an army of Cybermen out of the dead which she stored in tombs filled with dark water. (TV: Dark Water / Death in Heaven)

It's bigger on the inside
Many of the Doctor's companions were in awe of the police box that was larger within, often commenting that "it's bigger on the inside." (TV: The Three Doctors, Robot, Rose, The Runaway Bride, Smith and Jones, The Eleventh Hour, The Bells of Saint John, The Pilot)

The Doctor looked forward to this, (TV: The Vampires of Venice) and was excited at an opportunity to exclaim it himself. (TV: The Husbands of River Song) Rose Tyler expected Adam Mitchell to say the phrase, and said it for him; the Ninth Doctor mused she could learn some things from Adam when he instead gave the proper name for it. (AUDIO: The Other Side) Kevin, due to being a robotic t-rex, found the inside rather claustrophobic. (COMIC: Space Squid) When Bill Potts, who had initially believed the TARDIS to be a secret room and then a lift, finally shouted the phrase, the Doctor and Nardole expressed glee at finally hearing her say it. (TV: The Pilot) However, Clara Oswin Oswald surprised the Doctor by saying the reversed phrase, "It's smaller on the outside." (TV: The Snowmen) Journey Blue later said this phrase. (TV: Into the Dalek)

After Jack Harkness, Gwen Cooper and Rex Matheson entered a PhiCorp warehouse, Jack stated that it was bigger on the inside. (TV: Dead of Night)

Behind the scenes

 * The Discontinuity Guide made the claim that, having only survived as a result of the Fourth Doctor's interference with the creation of the Daleks, Davros' presence reduced the Dalek Empire to "a mess of squabbling factions" which were "incapable of the unity needed to develop dimensionally transcendental time travel.