Beyond space and time: Fractals, hyperspace and more

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Beyond space and time: Fractals,
hyperspace and more

(Image: Ryan Wills)


We don't have any trouble coping with three dimensions

or four a
t a pinch. The 3D world of solid objects and
limitless space is something we accept with scarcely a second thought. Time, the fourth dimension, gets a little
trickier. But it's when we start to explore worlds that embody more

or indeed fewer


that things
get really tough.

These exotic worlds might be daunting, but they matter. String theory, our best guess yet at a theory of
everything, doesn't seem to work with fewer than 10 dimensions. Some strange and useful properties of solids,
such a
s superconductivity, are best explained using theories in two, one or even no dimensions at all.

Prepare your mind for boggling as we explore the how, why and where of dimensions.


On the dot

Surely, with no dimensions there's no room for anything, so a 0D space must amount to nothing at all



Walk the line

Add one dimension, and phy
sics starts to look a little familiar


Fractal landscapes

Welcome to the irregular landscapes between the familiar worlds of one, two and three dimensions


Vistas of flatland

Physics in one dimension is too simple to be satisfying, and three dimensions are complicated and messy.
dimensional "flatland" is jus
t right


We're here because we're here?

Flatland and multi
dimensional hyperspace make fine playgrounds for the mind, but our bodies seem stuck in a
space of three dimensions


Time, the great deceiver

Space consists of three dimensions. Time, we are told, is also a dimens
ion. So how come it is so different?


Into the unseen

By adding a fifth dimension to space
time, it is possible to show that gravity and electromagnetism are two
aspects of one and the same force



Whenever physicists invoke extra dimensions, they always seem to mean the space kind. Why can't we have
more time?


Surfer's paradise

Eight dimensions is a rarefied space that i
s home to the octonions

"the crazy old uncle nobody lets out of the


String country

Ten dimensions, and we finally reach the fabled land of string theory

Beyond space and time: 0D

On the dot

26 August 2009 by
Richard Webb

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(Image: laverrue / Ludovic Bertron)

The idea of some
thing having zero dimensions carries a whiff of the emperor's new clothes. Surely, with no
dimensions there's no room for anything, so a 0D space must amount to nothing at all

mustn't it?

Not necessarily. Some of the hottest properties in physics are 0D
semiconductor structures known as
. Anything from nanometres to micrometres across, they do admittedly have a size, but electrons can be
into them so tightly that they have no dimensions to move in at all.

"It's a zero
dimensional trap for charges," says
Leo Kouwenhoven

of Delft University of Technology in the
Netherlands. Electrons confined in this way start to behave very strangely indeed, a
nd quite usefully.

For a start, any energy you pump into a quantum dot cannot be used to shuffle the electrons around, but can
only be released as light. This makes quantum dots promising as a highly efficient low
power light source.
Because they are so sm
all, the dots might also serve as fluorescent markers to label biological molecules such
as antibodies in order to track their progress through a living organism.

Kouwenhoven admits that this is still some way off; first we'll have to fabricate quantum dot
s from proven non
toxic materials, he says. His own research focuses on another potentially hot application. Each excited electron
trapped in a quantum dot produces exactly one photon, and so information can be transferred reliably back and
forth between t
he electron and photon. That could make quantum dots just the right medium in which to
manipulate and store data in a first generation of
quantum computers


the aw
esomely powerful devices which,
if one big enough can ever be built, promise to transform how we process information.

"We'll have a proof of principle probably a few years down the line," says Kouwenhoven, "and maybe
commercial applications in a decade or
so." Encouragement enough, perhaps, that it is not always nothing that
comes of nothing.

Read more:

Beyond space and time

Beyond space and time: 1D

Walk the line

26 August 2009 b
Michael Brooks

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(Image: Andrew Dunsmore / Rex Features)

One is the dimension in which physics starts to look a little famili
ar. A single dimension is just a straight line, the
perfect environment for the operation of such classical concepts as Newton's laws of motion.

But it is in quantum theory that 1D physics really comes to life. "You get completely novel effects

things th
you don't get in any other dimensionality," says
Thierry Giamarchi
, a specialist in 1D materials at the University
of Geneva in Switzerland.

Take the behaviour of electrons. They will nor
mally do anything to keep out of each other's way, but trapped in
a 1D channel where they can only move backwards and forwards they begin to interact and move as one. Get
the conditions right, though, and things go the other way: a confined electron can be

made to act as if it were
two particles, one with the electron's charge, and one with its spin. "There are a host of phenomena like this,"
says Giamarchi.

Such tricks are the delight of physicists, but their significance goes way beyond that. As electroni
c devices get
ever smaller, 1D will increasingly become the place to be. 1D carbon nanotubes can be made fully conducting
or semiconducting at will, and are a hot prospect for wiring up
future generations of computer chips

Beyond space and time: 1½D

Fractal landscapes

26 August 2009 by
Valerie Jamieson

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We live in a world of three
onal objects bounded by two
dimensional surfaces and outlined by one
dimensional lines. All in all, a comforting, intelligible, whole
number sort of world.

Or do we? As the mathematician
noit Mandelbrot

pointed out in his 1982 book
The Fractal Geometry of
, clouds are not spheres, mountains are not cones and coastlines are not circles. The dimensions of the
raw, rough real world do not, it turns out, come in tidy integers.

for example, tracing the delicate outline of a snowflake. As you zoom in, you find yourself following an
ever more intricate pattern, and the closer you get, the longer the line you trace becomes. Your drawing is still
a line, but its
crinkles embrace far more of the space on the page

than a straight line. And yet a line, however
hard it squirms, can never be more than a 1D object. Or can it?

Welcome to fractal dimensions, the irregu
lar landscapes between the familiar worlds of one, two and three
dimensions. Fractal dimensions are not the same as the left
right, back
front and up
down directions we're
used to, but they are intimately related: they describe how much more space a comple
x object fills as you look
it at finer scales and measure more of its detail (see diagram).

It's not just about snowflakes. Many natural objects have fractal geometry: river networks, branching lightning,
clouds, broccoli. You might even claim to live in a

fractal landscape, more or less so depending on where you
are in the world. The rugged coastline of Great Britain, for example,
differs in length wildly

according to whether
you measure it with a yardstick or calipers. It has been calculated to have a fractal dimension of about 1.25.
Smooth South Africa, on the other hand, is only slightly rougher than a strai
ght line, with a fractal dimension of

Beyond space and time: 2D

Vistas of flatland

26 August 2009 by
Michael Brooks

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(Image: Fiona Bradley)

"Two dimensions are golden," says
Andre Geim

of the University of Manchester, UK. Physics in one dimension
is too simple to be satisfying, and three dimensions are compli
cated and messy. Two
dimensional "flatland" is
just right, with just enough room for interesting and useful things to arise. "As a physicist, this is the dimension
you would like to live in," says Geim.

He would say that. Geim was one of the team that in 2
004 produced the first 2D material, sheets of carbon one
atom thick known as
. Graphene could indeed be incredibly useful, with e
lectrons shooting across its
sheets almost unhindered. If 1D nanotubes are the wires of future computers, graphene could be their circuit

That's not all. Take
temperature superconductors
. We already know of materials that conduct with
lutely no resistance at temperatures up to around 130 kelvin, just under halfway from absolute zero to
room temperature. We'd love to know how they do it, but after 20 years of head
scratching all we know is that
the effect seems to arise from the formatio
n of 2D

of electrical charges. If we could fully fathom the
physics behind that, it could set us on the path to superconductors that work even at room temperatur

So flatland is practical, but it is also profound. When electrons are confined by powerful magnetic fields to a 2D
layer of semiconducting material cooled to less than one
third of a degree above absolute zero, electrons

which were long considered fun
damental, indivisible particles

appear to break down into particles each with
just a fraction of the electron's charge.This phenomenon is known as the
l quantum Hall effect
, and the
resulting particles are ambiguous characters dubbed anyons.

Anyons not only force us to rethink the nature of the electron, but might also, like zero
dimensional quantum
dots, represent a
great hope for building an ultrapowerful quantum computer
. If we could get such a machine to
work on a significant scale, it would perform astonishing feats of information
processing, and could also
faithfully model the behaviour of quantum systems. In short, flatland may open grand vistas on everything from
new drugs to
parallel uni

Beyond space and time: 3D

We're here because we're here?

26 August 2009 by
Stephen Battersby

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(Image: Mila Zink

2D flatland and multi
dimensional hyperspace make fine playgrounds for the mind, but our bodies seem stuck
in a space of three dimensions. Why not two or four, or five or more? Recently, physicists trying to meld gravity
and quantum theory, and so exp
lain the nature of space and time, have begun to revisit this old question.

String theory, one route to quantum gravity, gives an unsatisfactorily vague answer: space can have anything
from zero to 10 dimensions. That drives theorists to anthropic argument
s: universes of all possible
dimensionalities exist, but we see what we see because beings like us require a 3D habitat.

In 2005, Andreas Karch of the University of Washington, Seattle, and
Lisa Randall

of Harvard University came
up with a more mechanistic explanation of the mystery of threeness. They created a model in which many
universes of different dimensions float around inside an expanding 10
dimensional hypers
pace of the kind
popular in string theory. When these universes collide, they annihilate one another. The
calculations showed

that three and seven
dimensional universes
are the ones most likely to survive such catastrophes.

If you accept the premise, that almost answers the question

but why shouldn't we live in a spacious realm of
seven dimensions instead of our cramped 3D universe?

That might be explained by looking at

space not as a uniform whole, but as a construction built up from tiny
pieces. A European team
has done just that

with higher
dimensional analogues of the triangle, the

simplest unit
that can be stuck together in different ways to make curvy universes. Quantum theory says that the "true"
shape of the cosmos should be the sum of all these possibilities. By requiring that their model universe should
adhere to strict cause
and effect, the team found that the result has just one dimension of time and exactly
three of space.

There is a twist. At the very smallest scales, the structure of space changes: one dimension of the three melts
away to leave only two dimensions. Perhaps
, if you look closely enough, we live in flatland after all.

Beyond space and time: 4D

Time, the great deceiver

26 August 2009 by
Marcus Chown

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Space consists of three dimen
sions. Time, we are told, is also a dimension. So how come it is so different?

Answer: it isn't. "Space and time are not concepts that can be considered independently of one another," says
Roger Penrose

in his book
. In Einstein's special theory of relativity, they dissolve into one
entity. Two objects that to one person seem separated only in space can to another seem divided by space and
time. S
imilarly, two events that seem to be separated only in time might from another perspective occur in
different places as well.

That does not chime with our everyday experience, but only because we are not up to speed. Discrepancies
between two observers' vi
ews only become obvious when their relative speed is close to the speed of light

cosmic speed limit.

Einstein's physics reveals a deep truth: space and time are just different threads of a single seamless fabric
called space
time. Yet there is still
an obvious difference between the two. We can, in principle, travel in any
direction in the three dimensions of space, but we can plod in one direction only in time

forwards from the past
to the future. How do we explain that anomaly?

Ultimately, says ph
Lawrence Schulman

of Clarkson University in New York, that too is down to the
cosmic speed limit. Imagine, for example, pulling back your curtains at 7 am on a bright, sunny morni
ng. The
sun, though, already doesn't exist. It exploded at 6.55 am; we just don't know it yet, because light from it takes
about 8 ½ minutes to travel to Earth.

In this picture, any event

the exploding sun, us standing at the window

is a point in space
time with two
associated "light cones". One represents light racing towards the event through space and time, and one
represents light racing away (see diagram). To see the sun explode as it happens would mean stepping
outside our light cone and moving fa
ster than the speed of light

something that our universe does not allow.

"It is the cosmic speed limit that makes some parts of space
time inaccessible," says Schulman. It breaks the
symmetry between time and space

and means we see information flowing
smoothly in what we call time from
the past to the future

Beyond space and time: 5D

Into the unseen

26 August 2009 by
Amanda Gefter

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(Image: Ethan Hein)

Seeing time as the fourth dimension made sense of Einstein's special relativity. The German mathematician
Theodor Kaluza had even grander designs. In 1919, he sent a paper to Einstein in which he argued that by
adding a fifth d
imension to space
time, it was possible to show that gravity and electromagnetism were two
aspects of one and the same force.

A few years later a Swedish mathematician, Oskar Klein, took Kaluza's idea and
ran with it
. He countered the
obvious objection to the existence of a fifth dimension

that it is not immediately apparent

by showing it could
be tiny and curled up at every point in our four
dimensional space
Thus began the trend of searching for
the unification of forces in the hidden dimensions of hyperspace that continues today in string theory.

Perhaps, however, the fifth dimension is not as tiny as Klein suggested. Using string theory, Harvard physicist
Lisa Randall

Raman Sundrum

of Johns Hopkins University in Baltimore, Ma
showed in 1999

that a
fifth dimension could explain a vexing mystery: why gravity seems
so much weaker

than nature's other forces.
Their model has our familiar four dimensions floating in an infinitely large, negatively curved fifth dimension.
While the electromagnetic and nuclear forces are stuck inside a "b
rane" made of four dimensions, gravity
out into the fifth

Paul Wesson

of the University of Waterloo in Ontario, Canada, has argued that reality in fact has
five dimensions that can be broken down into our four familiar dimensions plus the mass that populates our
world. This theory not
only rids physics of the problem of why things have mass

it becomes an incarnation of

but also of the big bang singularity, the vexing state of infinite temperature and density from which
our universe sprang and where all current physical theo
ries break down. Seen from the perspective of the full
5D universe, the big bang beginning is no more than an illusion.

The possibility of five dimensions is also raising more subtle questions. In one of the most remarkable results of
string theory, theore
tical physicist
Juan Maldacena

posited in 1997 that some string theories in five large
dimensions that include gravity are equivalent to ordinary quantum
field theories in four dimensions withou
gravity. The former is a holographic projection of the latter

potentially making our everyday world as ethereal
as a
hologram projected from the bo
undary of the universe

That might sound esoteric, but the correspondence has recently been successfully applied to computationally
difficult problems in many areas, including the
physics of high
temperature superconductors
. In Maldacena's
picture, the 4D theory is no "truer" a description of the world than the 5D one. Seen in that light, the question,
"how many dimensions does the universe have?"
does not have a unique answer after all.

The question 'how many dimensions does the universe have?' may not have a unique answer

Beyond space and time: 6D


26 August 2009 by
Marcus Chown

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(Image: FilmNut / Cam Russell)

Whenever physicists devise theories of the universe that invoke extra dimensions, they always seem to mean
the kind in which you could theoretically,
if you could find them, move freely. In other words, hyperspace is just
that: space. When it comes to higher dimensions, time does not get a look in.

There's a good reason for that. If there were more time
like dimensions, things could shuttle between arbi
points in our one
dimensional time by looping through other time dimensions, circumventing the limits imposed
by light's finite speed. In other words, time travel would be possible. In our cosmos at least, that seems not to
be the case.

In 1995, thou
Itzhak Bars

of the University of Southern California in Los Angeles saw a hint of an extra time
dimension in M
theory, the umbrella version of string theory. Investigating further, he used va
rious tricks to
construct a theoretical framework in which a second time dimension could exist, but where time travel was not
allowed. Such a two
timing theory would have attractions, as Bars's
subsequent research has shown
. It might,
for example, iron out some wrinkles in the standard model of particle physics.

The catch is that the scheme only works if there is an extra spatial dimension, too. Yet Bars fo
und that things in
this 6D universe would look pretty much like they do in our 4D universe

with one difference. There will be not
one standard model of particle physics, but many 4D "shadows" of a 6D original.

Beyond space and time: 8D

Surfer's paradis

26 August 2009 by
Anil Ananthaswamy

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The E8 pattern: key to the theory of everything? (Image: Peter McMullen / Claudio R
occhini / Wikimedia
Commons )

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Eight dimensions is a rarefied space that is home to the octonions

"the crazy
old uncle nobody lets out of the
attic", as mathematician
John Baez

of the University of California, Riverside, puts it.

Octonions are indeed odd creatures. They are one of only four number syst
ems in which division is possible,
and so allow the full range of algebraic operations to be performed. The way octonions interact, however, is
peculiarly exasperating and unlike anything we are familiar with from our conventional number system (see

So why bother with octonions at all? It's because, for some problems in theoretical physics, they are an
invaluable tool. Matrices filled with octonions are the building blocks of a bizarre mathematical structure known
as the "
E8 exceptional Lie group
", which sits at the heart of a particular form of string theory.

In 2007, E8 hit the headlines when physicist
Garrett Lisi
, who has no univer
sity affiliation and spends most of
his time surfing in Hawaii,
used the E8 group to seemingly unify gravity

with the three other fundamental
forces without using string theory. The publicity surrounding that work ruffled some feathers. "String theorists
have been working on [E8] since the late 70s," say
Michael Duff

of Imperial College London. "We didn't need
surfer dudes to tell us that it was interesting."

Duff himself is agnostic about the value of octonions, pointing out that no

theory in which they pop up has yet
been tested by experiment. "Whether octonions have anything to do with the real world is still anybody's
guess," he says.

Beyond space and time: 10D

String country

26 August 2009 by
Anil Ananthaswamy

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(Image: Wiki Commons)

Ten dimensions, and we finally reach the fabled land of
string theory
. For all the vitriol that has been thrown at
it, string theory is for the moment the only real game in town when it comes to attempts to bundle up quantum
mechanics and general relativity into a "theory of

everything". It holds that all particles that make up matter or
transmit forces arise from the vibration of tiny strings. Those strings are one
dimensional. The space they
wiggle about in is not. In fact, it has 10 dimensions: nine of space, and one of ti

Why? In a nutshell, because the theory doesn't work with any fewer, as physicists
Michael Green and John
Schwarz showed

in 1984: mathematical anomalies crop up that tra
nslate into violent fluctuations in the fabric of
time at scales smaller than the Planck length of 10


That doesn't necessarily mean that 10 is the magic number. Indeed, one
now unfashionable early variant

string theory had 26 dimensions. There are five broadly defined brands of 10D string theory that compete to
explain the universe, with no indication as to which, if any, is the right one. But these dispara
te theories can be
unified into one overarching theory, known as M
theory. M
theory has 11 dimensions.

It is assumed that the extra dimensions of M
theory must in some way be squashed down to a size that we
can't see. The bad news is that there is an almos
t unlimited number of ways in which this can be done. How to
single out the one way that produces our universe remains a problem. "It divides theorists into two camps,"
Michael Duf

of Imperial College, London. Those who say we'll work out the trick eventually are faced by a
growing band who subscribe to the alternative view of the
. This is perhaps the most outrageous
idea that physicists have brought back from their expeditions into higher dimensions: that all possible universes
do actually exist. The universe we know is as it is because it just happens to be the one we ar
e living in.

Perhaps the most outrageous idea that physicists have brought back from higher dimensions is that all possible
universes exist