2Dworld and 2D’ers

Since it is impossible for us to imagine a higher dimension into
which we could curve our 3D world, I will employ a useful trick.
We simply make do without one of our spatial dimensions, say the
dimension of depth, and then we can deal with an imaginary 2D
world (let’s be bold and original and call it 2Dworld). Such flat,
two-dimensional worlds have been discussed by many authors
over the years and have been called everything from Flatland
to the Planiverse. The inhabitants of such a universe are flat,
cardboard cut-out beings who are restricted to moving not ‘on’
but ‘in’ a surface. They can move up/down and left/right but
cannot move out of the surface since that would require motion
into the third dimension which is impossible for them. Now the
illusive fourth dimension that is impossible for us 3D beings to
comprehend (but which we would need in order to visualize the
curvature of our 3D space) is equivalent to a third dimension as far
as the 2D’ers, as I will refer to them, are concerned. We have access
to this third dimension even though the inhabitants of 2Dworld
cannot.
What would such a 2D universe look like? For a start, the
inhabitants would find it just as hard to think about a third
dimension as we do trying to think about a fourth. In figure 1.3 are
two such beings. It is quite interesting to consider how they carry
out basic functions. For instance, their eyes would have to have
the freedom to roam about from side to side so that they can see
in both directions. If this weren’t the case, and the eyes were fixed
on either side of their heads then, although they would have the
advantage of being able to see in both directions at the same time,
they would be missing a vital skill. Being able to look at the same
object with both eyes would enable them, as it does us, to judge how far away that object is. If they did have both eyes on the same
side of their heads, however, they would not be able to look behind
them unless they stood upside down! This is because they would
be unable to swivel their heads around; a skill that requires access
to the third dimension. Both these problems could be overcome
if their eyes are free to roam around as I have depicted. Another
way, of course, is for them to have a pair of eyes on each side of
their heads.Aparticularly interesting aspect of 2Dworld is what 2D’ers can
see when they look at objects in their world. First, let me remind
you of what we see when we look at a solid object like a ball. What
we actually ‘see’ is a 2D image on the retina of each eye, which
is very important for depth perception. Even with one eye closed
we know that what we are looking at is a solid three-dimensional
object rather than a flat two-dimensional one, like a disc, due to
the way light shining on the ball provides shading. Even without
this, we know from experience what a ball looks like and how it
behaves. So, when we watch a football match on television we
know that the circular object being kicked is a three-dimensional
football and not just a flat disc that looks like a ball and is rolling
around on its edge. We know this despite not being able to discern
any shading on the underside of the ball and despite the television
picture itself being a 2D projection of the 3D reality.
When we look at a 3D object we only ever see the twodimensional
surface facing us. Our brains then take into account past experience of such an object plus the way light interacts with
that surface to build up a model in our minds of the whole threedimensional
shape even though we cannot see the back of it. How
does this compare with what the 2D’ers see? Their equivalent of a
sphere is a circle. When a 2D’er looks at a circle she will be looking
at it ‘edge on’ and will therefore only see half of its circumference.
She will see on her ‘retina’ a one-dimensional image: a straight line.
Again, she would have to rely on shading to discern the curvature
of the line and would have to rotate the circle to be convinced that
the line curves all the way round. If the circle is being lit from
above, say from a two-dimensional sun overhead, then the top
section of the line she sees will be lighter than the bottom section
which forms the underside of the circle. Thus, how a circle looks
to 2D’ers is not the same as it does to us because they can never
see inside it. From our privileged vantage point looking down on
2Dworld we can look inside all objects, not just the circle but the
2D’ers’ bodies too. All their internal organs will be visible to us,
giving a new meaning to the term ‘open-heart surgery’. It is just
as impossible for 2D’ers to see inside a closed circle in their world
as it would be for us to see inside a hollow ball.Having spooked the 2D’er into believing in the power of the
paranormal, by causing an object to appear out of nowhere—
an object which just a few seconds earlier was locked inside an
impenetrable square—we decide to show off the wonders of 3D
space by introducing him to a sphere by pushing a small ball into
2Dworld. Of course, it will go right through to the other side
provided there is no 2D object in the way. The 2D’er will first see
a point growing into a line that gets longer then shorter before
disappearing. He concludes from the shading that the line is part
of the circumference of a circle and so knows that he is looking at
a circle that starts off small, gets bigger, reaches some maximum
size (when the ball is half way through) then shrinks again to zero
size as it emerges on the other side of 2Dworld. Thus, at any given
moment the 2D’er will only ever see a cross-section of the ball.