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Authors: Lawrence M. Krauss

Hiding in the Mirror





A L S O   B Y   L A W
R E N C E   M .  K R A U S S

Atom: A Single Oxygen
Atom’s Journey from the Big Bang
to Life on
Earth . . . and Beyond

Quintessence: The Mystery
of the Missing Mass

Beyond Star Trek: From
Alien Invasions to the End of Time
Physics of Star Trek

Fear of Physics: A Guide
for the Perplexed

The Fifth Essence: The
Search for Dark Matter in the Universe


T H E  M Y S T E R I O U S  A L L U R
E O F E X T R A D I M E N S I O N S ,

F R O M   P L AT O   T O   S T R I
N G    T H E O R Y   A N D   B E

Lawrence M. Krauss



Published by the Penguin Group

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80 Strand, London WC2R 0RL, England First
published in 2005 by Viking Penguin, a member of Penguin Group
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Copyright © Lawrence M. Krauss, 2009

All rights reserved

Figure on page 67: Brendan Crill, The Boomerang
Collaboration. All other drawings by the author.

ISBN: 1-4362-9493-2

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For my mother . . . at last!


There is a dimension, beyond that which is
known to man.
It is a dimension as vast as
and as timeless as infinity.

It is the middle ground
between light and shadow,
between science
and superstition,
and it lies between the
pit of man’s fears
and the summit of his
This is the dimension of

—Rod Serling,
The Twilight





Two parents wake in the
middle of the night to sounds of their
daughter’s crying out in the distance. The father rushes
to her bedroom
and finds her missing. He
frantically searches everywhere, slowly com-
ing to the grim realization that she is gone. His wife
runs into the room
soon afterward, overcome
with panic. At his wit’s end, he dashes out
to the living room and picks up the phone and calls a
neighbor. He re-
turns to his wife and, in
words that are probably unique in the history
of television, tells her:

“Bill’s coming over. He’s a
physicist! He ought to be able to help!”

orty-two years ago,
when I was very young, a
Twilight Zone
episode called “Little Lost Girl” scared the living daylights out
of me. Touching on every child’s fear of being separated from the
safety of parents and home, the episode told the story of a little
girl who falls into another dimension.

When I first thought about writing a book that
might focus on our love affair with extra dimensions, “Little Lost
Girl” came immediately to mind, although I confess I had no memory
of the episode’s title or when it aired. After a short bit of
research on the Web, I was able to locate it, and a few days later,
along with forty-two other episodes I had to buy in a
light Zone
boxed set,
it arrived at my door.

That night I placed the DVD into my computer
and relived my childhood trauma. The eerie thing was that I
remembered everything about the episode . . .
except for the physicist
! But suddenly, upon hearing
that line of dialogue, a rush of memories came flooding back. Of
course! The physicist was the hero of the episode. He came over in
the middle of the night, discovered and traced out the “portal” in
the wall through which the small child and her dog had wandered,
guided her father through the gap, and ultimately reached through
and saved the father and the terrified duo moments before this door
to another dimension closed forever.

I now vividly remember (or I think I remember)
being struck by how exotic and powerful Bill the physicist’s
knowledge seemed, and how much respect this knowledge engendered in
his frightened neighbor. I, too, wanted one day to be privy to such
secrets, and to explain them. I wanted to be the one whom people in
distress knew they could count on. In short, the

Alas, I have been a physicist for over twenty
years now, and except for some students every now and then the
night before an exam, no one has sought out my physics expertise
when in distress. Nevertheless, I sometimes wonder if I write books
such as this to fulfill my desire to provide what Bill had offered
his neighbors: insights that physics has revealed about universal
human mysteries, such as from whence we came, and what may lie
beyond the darkness of the night. Some people seek solace through
the spirit, but for others it comes through knowledge. As Rod
Serling, the creator of the
Twilight Zone,
observed in his weekly introductions beginning in 1959, the human
imagination can create whole universes into which we can travel via
the depths of despair or the peaks of ecstasy. Ultimately our
continuing intellectual fascination with extra dimensions may tell
us more about our own human nature than it does about the universe

We all yearn to discover new realities hidden
just out of sight. So much so, that we have continually reinvented
them throughout human history, whenever the world of our experience
has seemed lacking. But this does not necessarily mean that all of
these worlds beyond our direct experience are unreal. There are
scientists today who truly expect to discover the existence of
extra dimensions and perhaps even extra universes in our lifetime.
I originally began this book because I wanted to explore the unique
cultural and scientific legacy that has led to our current
fascination with exotic new realms that may lie hidden in the
mirror. But I never guessed that the voyage could be so personal. I
now realize that with seven words heard in a television program
some forty-two years ago, my own future may have been


C H A P T E R 1

I call our world Flatland,
not because we call it so, but to make its
nature clearer to you, my happy readers, who are
privileged to live in

o begins perhaps the
most famous mathematical romance ever written. Penned in 1884,
twenty-one years before Albert Einstein revolutionized our notions
of space and time, under the pseudonym “A. Square” by the clergyman
and Shakespearean scholar Edwin A. Abbott,
was a poignant tale told by a wistful
two-dimensional being who had just discovered the miraculous
existence of three-dimensional space and longed to enjoy it. The
unhappy hero of this saga urged us lucky Spacelanders to recognize
the beauty of the higher-dimensional universes that he thus

At around the same time that Abbott was writing
a lonely and tragic artist on the
Continent was imagining another universe beyond the realm of our
perception. Vincent Van Gogh was a tortured genius who is said to
have sold but a single painting in his lifetime. Yet you cannot
walk the streets of Amsterdam today without seeing reproductions in
storefront windows of his haunting self-portraits or his landscapes
with yellow skies and blue earth. In 1882, he wrote to his brother,
who was his sole supporter, “I know for certain that I have a
feeling for color, and shall acquire more and more.” Through his
paintings Van Gogh freed our minds from the “tyranny” of color,
daring us to imagine everyday objects in a completely different
way, and thereby demonstrating that exotic realities could be
discovered in even the otherwise most ordinary things. His
paintings are haunting not because they are so bizarre but because
they are just bizarre enough to capture the essence of reality
while at the same time forcing us to reexamine what exactly reality
is. These are the luxuries of art and literature: to create
imaginary worlds that cause us to reconsider our place within our
own world. Science has comparable impact. It, too, unveils
different sorts of hidden worlds, but ones that we hope might also
actually exist and, most importantly, can be measured.
Nevertheless, the net result is the same: In the end we gain new
insights into our own standing in the universe.

All of these creative human activities reflect
the essence of human imagination, the spark that raises our
existence from the mundane to the extraordinary. If we couldn’t
imagine the world as it might be, it is possible that the world of
our experience would become intolerable. Such imagination almost
defines what it means to be human. Fourteen thousand years ago, in
what is now France, a remote Ice Age ancestor took a walk with a
young child into what many of us today would think of as a dark and
forbidding place. Deep in an underground cave the adult held the
child’s hand against a wall and blew pigment over it, leaving a
shadowlike imprint of a tiny hand that remains to this very day. We
will never know the purpose of this adventure. Did it have some
deep spiritual significance, or was it simply play? It certainly was
not an everyday activity, as our Cro-Magnon ancestors did not tend
to live in the deep recesses of caves such as this. Whatever its
purpose, it represents something very special about humans that
clearly differentiates us from our closest relatives on the
evolutionary tree.

I am not speaking here about art per se.
Rather, I am addressing the deeper, symbolic sense of self that art
reflects. The notion that the imprint on a wall might permanently
record the presence of two individuals in the cave that day implies
not only a recognition of their own existence, but also their
desire to preserve some aspect of it against the vicissitudes of a
dangerous world. For with a sense of self comes a sense of
everything that
self, or the “unknown
possibilities of existence,” as the godlike alien Q on
Star Trek
once described it.

That even earlier humans pondered such unknown
possibilities is testified to by the existence of artistic
renderings that predate the French cave art by at least eighteen
thousand years. In a cave at a site called Hohlenstein-Stadel, in
what is now Germany, a foot-tall figure of a standing human was
discovered. No less striking than the skill of the artist who
created it is the subject matter: This figure has the head of a
lion, not a man. Did this early carving represent some primal
notion of a deity? Or did it merely represent the recognition that
if lions existed, and humans existed, then somewhere, some exotic
combination of the two might exist?

Of course, here again we shall probably never
know what motivated our ancestral carver, but whatever its purpose
the figure reflects an artistic imagining of the possibilities
inherent either in this world or in one beyond it. In the three
hundred centuries that have passed since this figure was created,
human civilization, and human imagination, have evolved
considerably. But there remains a fundamental connection between
our modern efforts and these first, tentative steps: When we imagine
the world beyond our experience, we are digging deep into our own
psyches. In the famous
Twilight Zone
with which I began this book, Rod Serling argued that imagination
is the middle ground between science and superstition. With that in
mind, the central question becomes: To what extent do our
imaginings reflect our own predilections, and to what extent might
they actually mirror reality?

If we can directly test our imaginings against
the weight of observation and experiment, then the answer is easy.
But what if we cannot? When certain notions persist, in many
cultures and many times, are they merely hardwired in our brains?
Or perhaps, even if they are, is it because we are the products of
a natural world that incorporates them?

One such notion will be the focus of this book:
the longstanding love affair of the human intellect with the idea
that there is far more “out there” than meets the eye. Science has,
of course, validated this notion. Whole new realms of the physical
world have been exposed by the spectacular scientific developments
of the nineteenth and twentieth centuries. But in the present
context I mean something more literally “out there.”

Could space itself extend beyond the bounds of
our experience, and can there be whole new dimensions of space just
out of reach of our senses?

It is difficult to disagree with Serling that
imagination adds an extra dimension to the human experience. Still,
the question remains: Is a fifth—or even an eleventh, or
twenty-sixth—dimension purely imaginary?

What if extra dimensions exist but they remain
hidden from even the most sophisticated detectors? Can our
imaginations alone enable us to pierce nature’s veil to discover

This very question drove the most famous of all
philosophers in Western history to write a tale about a
two-dimensional world as an allegory for our own limited
understanding of reality. Twenty-five hundred years ago, in his most
famous set of Dialogues,
The Republic,
Plato invented the allegory of a cave to describe his belief in the
possibility of uncovering hidden realities within all of the
objects of our experience.

Plato envisaged our lives as being like those
of individuals confined in shackles within a cave, unable to
directly see the world of light beyond. These prisoners viewed all
objects located outside the mouth of the cave via the shadows they
cast on the cave’s back wall. To the viewers, who had no other
experience, the shadows themselves represented the real objects.
Imagine, says Plato, through his interlocutor, Socrates, what it
would be like to be unchained and dragged out to the light outside.
First, of course, the brilliant glare would be painful, and one
would crave a return to the dark familiarity of the cave.
Ultimately, however, the true wonder of the world would become
intoxicating—so much so that a return to one’s previous state of
ignorant slavery would be unthinkable. And even if one did return,
how would it be possible to communicate the truth without appearing
mad to those who had no idea of it?

Plato argued, however, that this is precisely
the responsibility of a true philosopher. He must be willing to
forsake the comfort of his own safe vision of reality and embark on
travels through frightening new terrains of the mind. But more
important, he must not be content to remain in his ivory tower of
learning, separate from the rest of the human race, but must be
willing to return to the world of men, to attempt to educate those
who govern the affairs of men in the true workings of the universe.
When Socrates was asked, in Plato’s dialogue, how one could
penetrate the fog that shields us from the true workings of
reality, his response was particularly telling, especially in light
of our current scientific perspective. The answer involved the study
of abstractions—in particular, arithmetic, the science of numbers.
Or, as he put it, “Numbers, then, appear to lead towards the

The study of numbers, said Socrates, should be
followed by, in successively lesser importance, the study of
geometry, then astronomy—as far as it concerns the laws of
motion—then perhaps harmony, the study of sound. Only through the
study of abstractions of the mind—as he viewed these
disciplines—could one release oneself from the chains that bind us
all to the rigid world of our senses.

Plato’s entreaties now appear hauntingly
modern. If his own abstraction—via the two-dimensional shadows of
three-dimensional objects—might open the minds of his
contemporaries to the infinite possibilities of existence, what
mysteries might modern mathematical excursions unveil? Perhaps this
spirit supplemented Abbott’s desire to create a piece of social
satire when he penned

Indeed, the twentieth-century British
mathematician and philosopher Bertrand Russell, in his
Study of Mathematics,
echoed almost verbatim Plato’s
idealism about the hidden power of mathematics: Mathematics,
rightly viewed, possesses not only truth, but supreme beauty . . .
a beauty cold and austere, like that of sculpture, without appeal
to any part of our weaker nature, without the gorgeous trappings of
painting or music, yet sublimely pure, and capable of a stern
perfection such as only the greatest art can show. The true spirit
of delight, the exaltation, the sense of being more than Man, which
is the touchstone of the highest excellence, is to be found in
mathematics as surely as in poetry.

 More recently we have become so
accustomed to the superb predictive power of our mathematical
descriptions of reality that it is easy to take this unexpected
connection between human abstraction and the actual workings of the
natural world for granted. Yet the mathematical physicist and Nobel
laureate Eugene Wigner wrote a famous essay in 1960 entitled

“The Unreasonable Effectiveness of Mathematics
in the Natural Sciences.”

In it he mused about the remarkable success of
mathematics as a description of natural phenomena, or, as he put
it, “The enormous usefulness of mathematics in the natural sciences
is something bordering on the mysterious and . . . there is no
rational explanation for it.”

It was precisely this latter fact—that the
profound connection between mathematics and the natural world seems
to be “a wonderful gift which we neither understand nor deserve,”
as Wigner put it—that led him to speculate further. Does the
“uncanny usefulness of mathematical concepts” suggest that a
perhaps wholly different mathematics from that we have exploited to
describe nature might perform equally well? Namely, are our
physical theories unique—do they represent some fundamental
underlying reality about nature—or have we just chosen one of many
different, possibly equally viable, mathematical frameworks within
which to pose our questions? In this latter case, would the
apparent underlying physical pictures corresponding to these other
mathematical descriptions each be totally different?

Because we have made huge strides in our
understanding of the nature of scientific theories in the
intervening forty years since Wigner penned his essay, I believe we
can safely say that the question he poses is no longer of any great
concern to scientists. We understand precisely how different
mathematical theories can lead to equivalent predictions of
physical phenomena, because some aspects of the theory will be
mathematically irrelevant at some physical scales and not at
others. Moreover, we now tend to think in terms of “symmetries” of
nature, what are reflected in the underlying mathematics. While this
once again argues for the importance of mathematics in our
understanding of nature, these symmetries themselves seem so
fundamental that we expect that any theory that can produce correct
predictions must reflect them. Thus, seemingly different
mathematical formulations can really be understood to reflect
identical underlying physical pictures. There is also a flip side to
the discussion regarding the unusual effectiveness of mathematics
in describing nature. Not all novel mathematical notions that open
new horizons for our imagination have correlatives in the natural
world. If that were the case, science would be no more than
searching for new mathematics.

The power of mathematics will play a large role
in what follows, but when it comes to the relationship between our
scientific imagination and reality, elegance or mathematical beauty
is by itself not sufficient to generate fruitful science. What
matters are results. That is why science isn’t philosophy, and why
nature holds the upper hand. As Richard Feynman once put it,
science is “imagination in a strait-jacket.” In the end our
theories rise and fall based on their successful ability to
quantitatively predict the future. Imagination truly rises to the
level of beauty in science when it allows one to make predictions
about things that one may never have thought were predictable.

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