The Orthographic Orator: The TARDIS from First Principles

Originally published Mistfall issue 17, November 1995

by Anthony B. Martin

"Well, aren't you going to say 'It's bigger on the inside than the outside'?  Everybody else does."

"That's pretty obvious, isn't it?  Anyway, nothing to do with you surprises me now, Doc."

-- The Doctor and Sgt. Benton, The Three Doctors part one

It is not beyond the bounds of theoretical possibility that an object like a TARDIS could exist. This article attempts to explain why. Here's some basic geometry to start.

A one-dimensional object has just one dimension.

Imagine a straight line with no width and no height, lust length. It has no side, no corners and no angles. Such a thing is a theoretical idea and cannot "exist" except in the mind. It is an abstract idea. The concept is valid however as a necessary component of two-dimensional objects.

A two-dimensional object has length and width.

Imagine a drawing of a square. It has length equal to width, four sides, four corners or vertices and four angles of 90 degrees each. Two-dimensional objects can be many other shapes too, but we'll stick to cubes and squares for simplicity. It is important to remember that two-dimensional images are an abstract idea also. Even a picture drawn on paper has some physical height.

A three-dimensional object has length, width and depth.

Imagine a cube of wood. It has length, width and depth all equal, and eight corners or vertices all consisting of 90-degree angles.

A four-dimensional object... is impossible to build since the abstract of a fourth dimension has no physical meaning. Or does it? What can we guess about it...?

A three-dimensional cube has six sides, a two-dimensional square has four sides and a 1D line has just one side. Similarly, a 3D cube has eight corners (or vertices). A four-dimensional cube-based object would have 12 sides and 16 corners. Four sides would meet at each corner. Each of the four edges would be at right angles in space to the other three.

It's called a tesserect.

Images "on paper" are two-dimensional. Normal objects are three-dimensional. How can an object be larger on the inside than on the out? The TARDIS is at least a four-dimensional object. Huh?

The classic explanation sequence from Doctor Who that nearly everyone relies on when trying to explain the phenomenon of the TARDIS' dimensional transcendentalism is this one --

LEELA: I know. There is no such thing as magic.

THE DOCTOR: Exactly. To the rational mind nothing is inexplicable, only unexplained.

LEELA: Then explain to me how this... TARDIS of yours is larger on the inside than on the outside.

THE DOCTOR: Well, it's because Inside and Outside aren't in the same dimension.

(LEELA looks puzzled.)

THE DOCTOR: All right, I'll show you.

(THE DOCTOR produces two small wooden boxes, one solid and one hollow. He holds them up for LEELA to see.)

THE DOCTOR: Now, which box is larger?

LEELA: (Indicating the solid box) That one.

(THE DOCTOR puts the solid box down atop the console, walks over to LEELA, and holds the hollow box up in front of her eyes. The solid box now appears to be inside the hollow one.)

THE DOCTOR: Now which is larger?

LEELA: (Again indicating the solid box on the console) Still that one.

THE DOCTOR: But it looks smaller, doesn't it?

LEELA: That's only because it's farther away.

THE DOCTOR: Exactly. If you could keep that box exactly the same distance away and have it here, then the large box would fit inside the small one.

LEELA: That's silly.

THE DOCTOR: That's Transdimensional Engineering, a key Time Lord discovery.

-- The Robots of Death part one

What does this say? It gives a name to the area to Time Lord knowledge that has worked it out. Quite how "engineering" (the application of already discovered knowledge) can be a "discovery" is a bit mysterious but it is a pedantic point that I won't pursue. Perhaps this is just sloppy writing by Mr Boucher and Mr Holmes?

This discussion implies that the outside appearance of the TARDIS is a 3D object and the inside is part of another dimension. It implies that the inside dimensions are an illusion of sorts perhaps due to the multi-dimensional nature of the actual TARDIS object. More likely the inside dimensions are a feature of a multi-dimensional object that cannot be explained in a 3D perspective.

Clearly the key to understanding the problem is having a complex perspective. That is, not to approach matters only from how they first appear.

The above passage contains a very clever concept but it doesn't explain how transcendental dimensions might work, either theoretically or practically.

How is this helpful in understanding the transcendental dimensions of the TARDIS? Well... that's a hard question. Let's try an easy one... well, easier anyway...

In general, how does a higher-dimensional object appear in a lower dimension? This question can be answered by analogy.

A famous attempt to explain four-dimensional geometry is the essay Flatland by Aldous Huxley, published in the 1920s. Flatland is a model universe where there are just two dimensions. All objects are two-dimensional and the population has a limited 2D perspective. That is, the people of Flatland can only see in two dimensions. Many strange phenomena can be generated in the 2D universe by understanding a 3D perspective.

For instance, if the distance between two objects in Flatland is increasing, it might be due to a 2D force accelerating one object away from the other. The 2D perspective only allows this explanation. A 3D perspective allows the objects to be placed on the expanding surface of a curved object, such as a balloon. As the balloon expands the distance between the two objects increases but the reason for it cannot be explained from a 2D perspective. To the people of Flatland, the two objects are just moving away from each other for no apparent reason. They cannot influence the mechanism by which the two objects are being moved because they are unable to understand it.

An observer with a four-dimensional perspective probably would be able to explain many three-dimensional physical conundrums quite simply (eg the wave and particle behaviour of light, Cheryncov radiation, the photo-electric effect, the Heisenberg Uncertainty Principle, etc., etc.). I am unable to speculate on what these 4D explanations might be. Let imagination run wild...

A three-dimensional cube can be represented in a distorted fashion by a two-dimensional diagram.

This representation can be thought of as a "projection" of the cube onto a two-dimensional surface. It is a shaped shadow of the cube. It is like a cube but it is not a cube because of certain geometric anomalies. For instance, the corners are not all at 90 degrees.

By rearranging the three-dimensional cube the two-dimensional projection can change shape. By reorienting the cube the shadow can appear differently.

It is possible to view a 3D shadow of a tesserect. How?

It is possible to make a three-dimensional "shadow" of a tesserect. It has certain geometric anomalies. For example it has four edges meeting at each of eight corners but not at 90 degrees.

There are several possible ways to meet the geometric criteria required for a tesserect by bending just one or two of them.

Here's one. Just try to imagine.

Imagine a cube within a cube.

Each exterior vertex (corner) of the inner cube is joined by a "side" to the interior vertex (corner) of the outer cube. The object has 12 faces (six on each cube), 16 corners and 4 sides joining at each vertex (corner). It is possible to have each side the same length. The areas of each face are then unequal and the angles between each side are 90 degrees in only three ways, not four.

Alternately, the sides could be unequal and the areas of each face equal.

Assume that the dimensional anomalies of a 4D object viewed in a 3D perspective are analogous to the dimensional anomalies of a 3D object viewed in 2D perspective. Obviously they are more complex. You then have a generalised model for the appearance of higher-dimensional objects on lower dimensions. Perhaps every object can be viewed in higher dimensions?

We now know some of the geometric properties of the tesserect. We are no closer to describing quite how a tesserect could be built. It is beyond the bounds of current technology, of course.

This analysis suggests a philosophical path that may lead to an understanding of multi-dimensional principles.

We can further extrapolate that the dimensional anomalies of a 5D object in three dimensions are likely to be quite astounding. For a 5D object like a TARDIS, one feature might be that it looks bigger on the inside. Using a method from earlier in this article, a 5D TARDIS shell would have 32 surfaces... Another might be that it can change its shape by reorienting its projection. In Doctor Who that's usually called "the Chameleon Circuit" but that name simply means nothing.

The TARDIS is a remarkable creation of Science-Fiction fantasy. It is truly the only non-"classical physics"-based spaceship in popular SF.