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Cambridge University Science Magazine

The Inhabitance of SPACE IN GENERAL


This Work is Dedicated

By a Humble Native of Flatland

In the Hope that

Even as he was Initiated into the Mysteries


Having been previously conversant


So the Citizens of that Celestial Region

May aspire yet higher and higher

To the Secrets of FOUR FIVE or EVEN SIX Dimensions

Thereby contributing

To the Enlargment of THE IMAGINATION

And the possible Development

Of that most and excellent Gift of MODESTY

Among the Superior Races


-- Edwin A. Abbott, in his dedication of Flatland

While we are most of the time comfortable with a world of three

dimensions, it may be intriguing to ask why exactly that is the

number. But perhaps it isn't. For roughly a century, scientists have

been introducing higher dimensions into their theories of nature. Often

it has merely been a mathematical convenience, for example the notion

of time as the fourth dimension among the three of space. However, the

practical results of Einstein's General Relativity suggest that space

behaves as if it were curved in a further dimension, the fourth

spatial dimension.

An analogue can be drawn with the most familiar of examples, the

surface of Earth. Only for a few thousands of years the Earth has been

known to be a sphere. Did we not know of the sun and the planets, most

everyday observations would show the world as flat. In the larger

scale, effects of curvature become apparent: two vertical poles of

equal height will cast shadows of slightly different length, if they

are separated by a sufficient distance. This is what made the ancient

Greeks conjecture the idea of a the world as a sphere.

The geometry of the surface of a sphere leads to the practical

conclusion that, although the surface has a definite area, it has no

borders. Traveling in any direction will eventually bring you to where

you started. If this is the case with the universe, being curved in a

higher spatial dimension, it might naively look like it would solve

the problem of the border of the universe, and what lies beyond, for

there would be no beyond.

On the contrary, there indeed is a border. On the Earth, we are aware

of the third dimension and hence the curvature of ''space'' is not

something mystical. Once we lift off the ground, we are outside the

two-dimensional space. The situation would be more interesting if we

actually perceived only the two dimensions of the surface. The kind of

world is described by Edwin A. Abbot in his fascinating classic of

science fiction, Flatland. If we were two-dimensional beings on the

surface of the Earth, we would have no direct way of observing the

curvature, like that used by the ancient Greeks. In fact, our brains

would be so accustomed to the two dimensions that it would take a

giant leap of faith even to consider the idea of the third dimension,

let alone perform any measurements. The important lesson the book

attempts to convey is that of a three-dimensional being who contacts

the main character, doing his best to explain what his world of higher

dimensionality is like. In fact, Abbott was a mathematics teacher and

wrote the novel in order to give a hint of a four-dimensional

existence to us in three-space.

If some of us (in the two-dimensional world) had the arrogance to

suggest a third dimension, there would, in fact, be a rather simple

method for determining the curvature. If you draw a circle on a

sphere, the circumference is generally less than the familiar pi times

the diameter - note that the diameter is measured along the surface,

if you think you only know of two dimensions. Alternatively, if the

surface resembles a saddle, the circumference will be greater than


So far, such an experiment has not shown any signs of the curvature of

our three-space in the fourth dimension. However, as predicted by

General Relativity, the curvature will be larger in the vicinity of

heavy objects such as the Sun. Indeed, the first direct evidence for

the theory was obtained during the solar eclipse of 1945 when the

position of a distant star was observed to differ from what it usually

was. The ray of light from the star had been slightly bent, because of

the curvature of space caused by the Sun. From the two-dimensional

analogy of a membrane curved in the third dimension, the universe

which we perceive is thought to exist on a ''brane'' contained in a

space of higher dimensions.

Given the experimental evidence, why is it that we only observe three

dimensions? More precisely, why does so much of physics work as if

there were only three of them? For example, the inverse square law of

gravitation is a direct consequence of three dimensions: the flux of

gravity from a massive object is evenly spread over an area

proportional to the square of the distance. Similarly, in an imaginary

four-dimensional world, the force of gravity would follow an inverse

cube law.

Surprisingly, recent theories (originally by Arkani-Hamed, Dimopoulos

and Dvali in 1998) suggest that, of all the fundamental forces, only

gravity might actually be aware of higher dimensions. This would

require, however, that space be ''curled up'' in the higher dimensions

on a very small, subatomic scale, leaving only three dimensions

accessible to anything practically observable. According to the

theory, gravitational flux from a mass would expand into all

dimensions (by an inverse nth power law, if there are n+1 dimensions)

in the small scale, but revert to the usual inverse square behaviour

in all measurements in practical scales. This would explain why

gravity is so much weaker than any other fundamental force, for

instance the electromagnetic. Moreover, the gravitational law has only

been tested down to the millimetre scale, while the scale of the

curled up dimensions is supposed to be significantly smaller than that

of fundamental particles.

Fundamental as electrons and quarks may seem, there are currently

known to be twenty-four different ''elementary'' constituents of matter,

so it is no surprise that scientists are working on a more elegant and

unified theory. A bold idea about as old as the higher dimensions, the

string theories are based on a single kind of entity which would

appear as different particles when in different working modes. These

would be open or closed loops of ''string'', and the kinds of vibration

of the string, for instance the frequency, would give the different

particle modes. There are several slightly different string theories,

all of which require the existence of ten or more dimensions to

work. As is the case with the above theory of gravitation, the extra

dimensions would be curled up in scales of the so-called Planck length

- about 1019 times smaller than the atomic nucleus. In

order to explore such details, we would need particle accelerators

with about 1016 times more energy than the present

facilitites, so there is currently no hope of directly testing string


Zooming up to the cosmological scale, there are yet more mysteries to

be solved. One of the currently most important is the notion of the

dark matter. The observed amount of matter in the universe appears to

be notably less, than what is required by General Relativity, to

account for recent experiments of the expansion of the universe. These

have shown that the universe has a critical mass density - i.e. if it

were any higher, gravity would eventually draw everything into a Big

Crunch. This result, by the way, also proves that in the large scale,

the space is flat instead of being curved. To compensate fof the mass

deficiency ''dark matter'' has been proposed. It is thought to be very

different from ordinary matter, for instance in that it does not

gravitate on itself to form clusters. This may be because the dark

matter resides on a different brane - which, from our point of view,

is truly a parallel universe. As expected by the new theory of

gravitation, the flux of gravity may spread between two branes,

although nothing else (e.g. light) can pass through. It may just be

ordinary matter, but on this brane the only effect we see is that of

gravity, which is why it appears so distorted in relation to the rest

of matter.

Probably the most famous known anomalies created by gravity are black

holes. In the context of branes, these are places of extreme curvature

created by ultra-dense masses. In the two-dimensional analogy, they

are points very sharply stretched away from the plane of the

membrane. It may happen that this stretch hits an adjacent brane and

we get a ''wormhole'' into the parallel world. Alternatively, the

wormhole may form a shortcut into another place on our brane. The term

was originally coined from the two-dimensional example of the surface

of an apple. A worm, living in this two-space, find a shortcut to the

opposite side by digging through in the third dimension. Some day,

wormholes may be viable method of transport, but so far the problem

with the extreme gravitational fields exerted on the passengers has

not been solved.

A further consideration of black holes, suggested by Lee Smolin and

others, is that in the ''inside'' of a black hole, a sub-universe might

start expanding into something not unlike our known brane of

universe. In the two-space, the end of the stretch would expand into a

sphere, while the connection to the ''mother brane'' would stay

infinitely narrow. The latter is what makes the holes black to us, for

no information can flow out into this world. Smolin believes this is a

widely occurring mechamism for the evolution of universes, and the

origin of our world. The Big Bang would correspond to the creation

of the black hole in the first place, but the origin of the first

universe remains a question, probably in the shadowy realms of quantum


With the brave new theories of branes in higher levels of reality, it

can be said that physics has once again peeked at a whole new world of

phenomena that could hardly have been predicted, until some

inquisitive minds had the arrogance to question the obvious. While the

work on higher dimensions appears like an intellectual game, it is

likely that we see real-world applications some day. Like quantum

mechanics, which was certainly perceived as a mathematical amusement

by some people at the time it was invented, has brought us electronics

and computing. Maybe some day when this planet gets too crowded, we

can just take the tube to the next brane...


Risto A. Paju is an Undergraduate in Physics at Queens'