How far do the stars stretch out into space?

And what's beyond them?

In modern times, we built giant telescopes that have allowed us to cast our gaze deep

into the universe.

Astronomers have been able to look back to near the time of its birth.

They've reconstructed the course of cosmic history in astonishing detail.

From intensive computer modeling, and myriad close observations, they've uncovered important

clues to its ongoing evolution.

Many now conclude that what we can see, the stars and galaxies that stretch out to the

limits of our vision, represent only a small fraction of all there is.

Does the universe go on forever? Where do we fit within it?

And how would the great thinkers have wrapped their brains around the far-out ideas on today's

cutting edge?

To begin to get a handle on infinity, we're going to need some perspective on the numbers

and scales that define our universe.

One place to start is a narrow side street in Charles Dickens' London.

A Curiosity Shop, fictional to be sure.

Here you can find an unparalleled collection of stuff.

Old shrunken heads, manuscripts, newspapers, books, and rare examples of impressively large

numbers.

From Zimbabwe comes a 100 trillion dollar note. In late 2008, with that nation battered

by hyperinflation, it was worth about a dollar fifty US.

Go up two orders of magnitude to something decidedly more useful. The fastest supercomputer

in history will soon hum along at 20 thousand trillion calculations per second, a twenty

followed by 15 zeroes.

You'll have to run it about a day and a half for your calculations to equal the number

of grains of sand on all the world's beaches. That's around a sextillion, a ten followed

by 22 zeroes.

That's roughly the number of stars in the visible universe.

Atoms in the visible universe? That's upwards of 10 to the 78th power, a 10 with 78 zeroes.

Cubic centimeters? A mere ten to the 84th, a septvigintillion.

To go up from there, we turn to no less a source than the Guinness Book of World Records.

The largest named number in regular decimal notation: the Buddhist time period Asamkhyeya

is ten to the 140th years, or 100 quinto-quadragintillions.

Then there's the largest number ever used. Graham's number is a calculation of angles

in a type of hypercube.

If you divided the visible universe into the smallest units known, called Planck volumes,

the total of those units wouldn't get you anywhere close to Graham's number.

But it's still nowhere close to the ultimate ceiling: infinity.

For those who find infinity hard to grasp, even troubling, you're not alone. It's a concept

that has long tormented even the best minds.

Over two thousand years ago, the Greek mathematician Pythagoras and his followers saw numerical

relationships as the key to understanding the world around them.

But in their investigation of geometric shapes, they discovered that some important ratios

could not be expressed in simple numbers.

Take the circumference of a circle to its diameter, called Pi.

Computer scientists recently calculated Pi to 5 trillion digits, confirming what the

Greeks learned: there are no repeating patterns and no ending in sight.

The discovery of the so-called irrational numbers like Pi was so disturbing, legend

has it, that one member of the Pythagorian cult, Hippassus, was drowned at sea for divulging

their existence.

A century later, the philosopher Zeno brought infinity into the open with a series of paradoxes:

situations that are true, but strongly counter-intuitive.

In this modern update of one of Zeno's paradoxes, say you have arrived at an intersection. But

you are only allowed to cross the street in increments of half the distance to the other

side. So to cross this finite distance, you must take an infinite number of steps.

In math today, it's a given that you can subdivide any length an infinite number of times, or

find an infinity of points along a line.

What made the idea of infinity so troubling to the Greeks is that it clashed with their

goal of using numbers to explain the workings of the real world.

To the philosopher Aristotle, a century after Zeno, infinity evoked the formless chaos from

which the world was thought to have emerged: a primordial state with no natural laws or

limits, devoid of all form and content.

But if the universe is finite, what would happen if a warrior traveled to the edge and

tossed a spear? Where would it go?

It would not fly off on an infinite journey, Aristotle said. Rather, it would join the

motion of the stars in a crystalline sphere that encircled the Earth.

To preserve the idea of a limited universe, Aristotle would craft an historic distinction.

On the one hand, Aristotle pointed to the irrational numbers such as Pi. Each new calculation

results in an additional digit, but the final, final number in the string can never be specified.

So Aristotle called it "potentially" infinite.

Then there's the "actually infinite," like the total number of points or subdivisions

along a line. It's literally uncountable. Aristotle reserved the status of "actually

infinite" for the so-called "prime mover" that created the world and is beyond our capacity

to understand.

This became the basis for what's called the Cosmological, or First Cause, argument for

the existence of God.

Another century later, Archimedes incorporated "actual infinity" into measurements of curved

lines and volumes.

His method boils down to a process of summation. Place a triangle inside a circle. Turn it

into a square, then a pentagon, and so on. As the number of sides increases, to infinity,

their combined lengths equal the circumference of the circle.

By slicing and dicing curves into an infinite number of straight lines, he was able to compare

a variety of curves, areas, and volumes.

Archimedes anticipated techniques developed two thousand years later.

And yet, his ideas on infinity did not carry forward, due to what the author David Foster

Wallace described as a mathematical allergy to the concept that developed in response

to Aristotle's "potential infinity."

It was Aristotle's ideas that passed into the Christian era along with his cosmology,

with Earth seated firmly at the center.

That view was not universal. Islamic, Hindu, and even some western thinkers posed alternate

views that included infinite space.

In European circles, the issue of infinity resurfaced during the Renaissance.

In 1543, the Polish astronomer Nikolas Copernicus argued that Earth orbits the Sun, not the

other way around.

The old Greek spheres began to fall by the wayside when a distant supernova, then a comet,

were spotted by the astronomer Tycho Brahe. These objects seemed to behave independently

of the other stars.

A monk named Giordanno Bruno inflamed the issue by traveling Europe at the height of

the Inquisition to proclaim an infinite universe. In the year 1600, he was burned at the stake

for this and other heresies.

Just nine years later, in 1609, Galileo Galilee used the first astronomical telescope to show

that the universe is much larger than we thought. In later writings, he even sought to discredit

the distinction between potential and actual infinity.

Galileo was forced to recant his views, and the old Aristotelian view held sway. Any attempt

to assign a value to infinity, in numbers or in nature, was doomed, for that was the

unique province of God.

Finally, at the end of the 19th century, the mathematician Georg Cantor sought once and

for all to divorce metaphysics from the abstract pursuit of math.

Infinity, he wrote, had to be studied without "arbitrariness and prejudice."

He became known for folding finite and infinite numbers into a unified theory of number sets,

considered a foundation of modern math.

One of his defenders used a paradox to show how infinite sets are subject to concrete

comparisons.

Say you've come to stay at this grand hotel.

You're in luck, because here there is an infinite number of rooms.

Oddly enough, you learn there are "No Vacancies."

Fortunately, the manager says: I can still check you in. He assigns you to room #1 and

directs you down the corridor. Then, he goes to work, shifting the guest in room 1 to room

2 -- room 2 to 3 -- 3 to 4 -- and so on.

So in this hotel, there's a number set that includes an infinite number of guests and

rooms. Then there's that same set plus you... two infinite sets, yet one is a subset of

the other.

Being able to use infinite sets of different sizes allowed mathematicians to design equations

describing continuous motion and change over time.

Echoing Aristotle, a critic of the new set theory suggested that the end of the corridor

is still only a potential infinity, with God representing the only actual infinity.

For those who pine for humble accommodations, we'll recommend an alternative later on.

Even as mathematicians embraced infinity, astronomers in the early 20th century still

saw a limited universe... centered on the galaxy, a flat disk of stars.

Did the limits of our vision, like the horizon at sea, conceal an infinite universe beyond?

Albert Einstein, for one, believed that if that were true, then the night sky would be

filled with dense starlight shining from every direction. We'd reel from the effects of infinite

gravity.

Arguing for a finite universe, he described a people living on the 2D surface of a sphere.

To them, a beam of light moving through space would appear to go straight, on an infinite

journey. In fact, it follows a path determined by the overall gravity of the universe, and

curves back around.

Like the old Greek spheres, this view of a static and limited universe began to fall

by the wayside in the 1920s.

Edwin Hubble and Milt Humason used the new 100" telescope on Mt. Wilson in California

to look at mysterious fuzzy patches of sky called "nebulae." They found that these patches

were galaxies like our own, and that some were very far away.

What's more, they found that most are moving away from us. In fact, the farther out they

looked, the faster the galaxies are moving.

This fact, known as Hubble's law, led to an inescapable conclusion: that the universe

is expanding. Furthermore, if you run the clock back on this expansion, it appears that

it all began in one singular moment.

That moment has traditionally been described as an explosion... a "Big Bang."

How large the universe has gotten since then depends on how long it's been growing, and

how quickly.

Using an array of modern telescopes, astronomers have recently narrowed the beginning to 13.7

billion years ago. Taking into account the expansion of space ever since, the radius

of the visible universe, the part we can see, has expanded out to 46 billion light years.

These measurements have raised anew the ancient questions: What's beyond our cosmic horizons?

Is there an edge? Or does it somehow go on forever?

A new set of answers has emerged from a theory designed to address questions that arose from

the original model of the Big Bang.

For one, how did the universe get so large? The Hubble Deep Field contains images of infant

galaxies at less than 10% of the age of the universe, near the edge of our cosmic horizons.

By the time one of those galaxies reached maturity, it would have moved far, far beyond

our horizon.

And what of all the galaxies visible at its horizons?

For another, how did the universe get so smooth? In every direction you look, the density of

galaxies is the same on large scales.

Astronomers believe that whatever process flung the universe outward, must have also

blended it in its earliest moments.

The theory that addresses these questions was based on the discovery that energy is

constantly welling up from the vacuum of space in the form of particles of opposite charge,

matter and anti-matter.

The idea is that in primordial times, an energy field embedded in this so-called quantum vacuum

suddenly moved into a higher energy state, causing space and time to literally inflate,

and our universe to burst forth.

If this theory is right, then our universe is incomprehensibly large. Its author, the

scientist Alan Guth, wrote that the universe as a whole would have grown to at least ten

billion trillion times the size of our visible patch. That's a ten followed by 23 zeroes.

If you think that's big.

A variation on the theory describes the origin of our universe as a physical process that

exists far beyond it, out into the seemingly infinite void that had confounded Aristotle

and other Greek thinkers.

In that case, our universe would have inflated like a bubble, and joined a stream of other

bubble universes frothing up and expanding across an endless ocean of time and space.

A related idea theorizes a cosmic landscape unfolding in vast fractal patterns.

These new, more expansive, visions of the cosmos are not without their paradoxes.

Logically speaking, with infinite stars, infinite planets, infinite universes, you will also

have infinite possibilities.

The so-called infinite monkey theorum has its roots in Aristotle's attempts to illustrate

the perils of thinking about infinity.

Ask a monkey to type, or ask an infinite number of monkeys to type, for an infinite amount

of time. You're sure to get a lot of random letters.

But there is a chance, however small, that somewhere, some how, you'll get the full text

of Shakespeare's Hamlet.

It's clearly absurd.

Then again, consider the increasingly strange nature of our universe, as suggested by some

new observations.

This is where we draw your attention from the famous Hotel Infinity -- to a less well-appointed

alternative.

You're sure to get a big welcome at the old Hall of Mirrors.