A Look in Many Dimensions: The Universe of Flatland and Planiverse
Welcome to a universe of infinite dimensions.
Even if we cannot see any
but our own three dimensional universe,
we can explore the realities of living in them here, through a cube.
Inside this cube is an object. But what is it?
Let us first picture ourselves living in a one-dimensional universe.
We have length, but no width or height. Everything in our universe
exists in lines of varying length. However, when we look at the line
ahead of us or behind us, all we can see is a point. We can only see
in zero directions. Aha! we say.
We can see a point. Thus, there is a line, such as ourselves,
inside the box. How long is the line? Why, I have no idea. But it
is a line.
Now, let us travel to a universe of two dimensions. Suddenly, the
cube gets turned on its side. We, the supreme beings of two
dimensions, can see in one dimension. We live with length and height,
and can see the length of the line. Aha! we say. There is a square
inside the box. Or perhaps a triangle. Or perhaps some other shape.
But it is a two dimensional shape like ourselves. I can see the
length and height when I walk around the box. Ah, now I can see.
With light, the segments are of different shades. It is a triangle.
But truly, it is us, the three-dimensional beings which are supreme.
For in our universe, there is not just length and height, but also
width. I can see the trigangle without guessing. For I can
see in two dimensions. But the shape
inside is not a triangle. You see this box?
You must turn it on its side once again. There now. When I
walk around and up and down, with light I can guess at the shape.
It is a triangular prism.
O you poor fools. It is us, the four-dimensional beings which are
supreme. For us, this box holds no mysteries. The entire box must be
turned to show the real shape. For we can see the box as...
the two dimensional beings could see the line
the three dimensional beings could see the plane
Those silly one dimensional beings could only look two ways.
Thus, they thought that their insides were covered, for they could
not conceive of beings looking at the whole line. And then the
two dimensional beings made the same mistake! They too thought that
their insides were covered, that no one could see the red and yellow
shapes inside the triangle. From the four ways they could look,
naught but solid lines. Poor fools, to not realize their whole
universe is revealed to us of higher dimensions. And then the three
dimension beings made the mistake again. For in every one of the
six ways they could look, the box was covered. But we who can see
eight different ways are not deceived. For we can see the inside of
the box as easily as the outside. How sad for everyone else, all
those creatures in the lower dimensions.
O pitiful misguided beings! It is us, the five dimensional
creatures which are truly masters of this universe. Why, we can
see in you as you could see in the three dimensional beings...
The Difficulty in Representing our Universe
As we saw above, although a creature lives in a n-dimensional
universe, it can only see in n-1 dimensions. Take our universe and
this cube as an example.
I have drawn a three dimensional object in two dimensions. Now,
tell me: if this were a real cube, which face of the cube
would be closest to you, the viewer? Is the cube even open or closed?
As you explore the universe of Planiverse, keep in mind the
challenges we face in our three dimensional world. Then imagine
looking at people, artwork, and even computers when you
can only see in one dimension.