Exploring other dimensions - Alex Rosenthal and George Zaidan

We live in a three-dimensional world

where everything has length,

width,

and height.

But what if our world were two-dimensional?

We would be squashed down

to occupy a single plane of existence,

geometrically speaking, of course.

And what would that world look and feel like?

This is the premise

of Edwin Abbott's 1884 novella, Flatland.

Flatland is a fun, mathematical thought experiment

that follows the trials and tribulations of a square

exposed to the third dimension.

But what is a dimension, anyway?

For our purposes, a dimension is a direction,

which we can picture as a line.

For our direction to be a dimension,

it has to be at right angles to all other dimensions.

So, a one-dimensional space is just a line.

A two-dimensional space is defined

by two perpendicular lines,

which describe a flat plane

like a piece of paper.

And a three-dimensional space

adds a third perpendicular line,

which gives us height

and the world we're familiar with.

So, what about four dimensions?

And five?

And eleven?

Where do we put these new perpendicular lines?

This is where Flatland can help us.

Let's look at our square protagonist's world.

Flatland is populated by geometric shapes,

ranging from isosceles trianges

to equilateral triangles

to squares,

pentagons,

hexagons,

all the way up to circles.

These shapes are all scurrying around a flat world,

living their flat lives.

They have a single eye on the front of their faces,

and let's see what the world looks like

from their perspective.

What they see is essentially one dimension,

a line.

But in Abbott's Flatland,

closer objects are brighter,

and that's how they see depth.

So a triangle looks different from a square,

looks different a circle,

and so on.

Their brains cannot comprehend the third dimension.

In fact, they vehemently deny its existence

because it's simply not part of their world

or experience.

But all they need,

as it turns out,

is a little boost.

One day a sphere shows up in Flatland

to visit our square hero.

Here's what it looks like

when the sphere passes through Flatland

from the square's perspective,

and this blows his little square mind.

Then the sphere lifts the square

into the third dimension,

the height direction where no Flatlander has gone before

and shows him his world.

From up here, the square can see everything:

the shapes of buildings,

all the precious gems hidden in the Earth,

and even the insides of his friends,

which is probably pretty awkward.

Once the hapless square

comes to terms with the third dimension,

he begs his host to help him

visit the fourth and higher dimensions,

but the sphere bristles at the mere suggestion

of dimensions higher than three

and exiles the square back to Flatland.

Now, the sphere's indignation is understandable.

A fourth dimension is very difficult

to reconcile with our experience of the world.

Short of being lifted into the fourth dimension

by visiting hypercube,

we can't experience it,

but we can get close.

You'll recall that when the sphere

first visited the second dimension,

he looked like a series of circles

that started as a point

when he touched Flatland,

grew bigger until he was halfway through,

and then shrank smaller again.

We can think of this visit

as a series of 2D cross-sections of a 3D object.

Well, we can do the same thing

in the third dimension with a four-dimensional object.

Let's say that a hypersphere

is the 4D equivalent of a 3D sphere.

When the 4D object passes through the third dimension,

it'll look something like this.

Let's look at one more way

of representing a four-dimensional object.

Let's say we have a point,

a zero-dimensional shape.

Now we extend it out one inch

and we have a one-dimensional line segment.

Extend the whole line segment by an inch,

and we get a 2D square.

Take the whole square and extend it out one inch,

and we get a 3D cube.

You can see where we're going with this.

Take the whole cube

and extend it out one inch,

this time perpendicular to all three existing directions,

and we get a 4D hypercube,

also called a tesseract.

For all we know,

there could be four-dimensional lifeforms

somewhere out there,

occasionally poking their heads

into our bustling 3D world

and wondering what all the fuss is about.

In fact, there could be whole

other four-dimensional worlds

beyond our detection,

hidden from us forever

by the nature of our perception.

Doesn't that blow your little spherical mind?