It is impossible for one to **directly** visualize, as you may have concluded
already, a structure of the dimension above one's own. What a stack-diagram allows
us to do, however,
is to divide higher-dimensional structures into slices that we *can* visualize
- each slice being the equivalent of a structure of our own dimension.
The result is by no means a perfect copy of the higher-dimensional structure
- but simply a version of the structure that we can comprehend. We will
demonstrate this notion by representing in stack-diagram form a model of a spherical
2-dimensional universe - a universe that happens to exist entirely on the surface of a sphere.
This universe, being a 2-dimensional universe, would be a universe within which, on
perhaps a 2-dimensional planet orbiting some 2-dimensional star, we would find the presence
of **Flatlanders**. A Flatlander, if you don't already know, is an intelligent, thinking
2-dimensional creature who lives in a world limited entirely to a mere *length* and
*width* (the 2 dimensions of the surface of the sphere). Of what importance, you may
ask, would a 'Flatlander' be to us? The fact that we can 'look down' upon a Flatlander, and
consider what he is thinking about the world around him, makes what we call a 'Flatlander'
*invaluable*. It is our knowledge of how a Flatlander experiences **his
own** world, you see, that we can make sense of the world that **we** experience.
A Flatlander,
it would happen, cannot **directly** grasp the concept of a sphere (the structure
representing the shape of his universe) for the very reason that a sphere is 3-dimensional
and the Flatlander's mind is 2-dimensional.

What the Flatlander can in fact grasp, however, is a
sphere as it is presented in
**stack-diagram** form. Having been introduced to the concept of a stack-diagram as
recently as the end of the previous section, it would not appear hard to grasp how the
'stack-diagram' of a sphere (as presented below on the left) would relate
to the sphere as it actually exists. We first divide the sphere into slices. Because the
sphere is a 3-dimensional object, the resulting slices therefore extend into 3-dimensional
directions. It is quite clear, then, and it is agreed, that the Flatlander cannot comprehend
these 3-dimensional directions. What the Flatlander
*can* grasp, however, are the individual 2-dimensional slices themselves, and the fact
that even though they extend into 3-dimensional directions, they are nonetheless arranged in
a **linear** fashion. The Flatlander, in turn, arranges the
slices in a manner that he understands to be linear, and acknowledges that the linear nature
of the slices he has arranged, represents a linear extension into the next dimension. The
process just described, in demonstrating the essence of what a stack-diagram is, shows us
that even though the
Flatlander cannot **directly** visualize a sphere, it is by means of the stack-diagram that
the Flatlander can successfully conceive of the concept of a sphere, regardless of the fact
that a sphere is 3-dimensional and the Flatlander's mind is 2-dimensional. Even the most
involved processes
of the sphere can be expressed to the Flatlander by means of the medium of the
stack-diagram - which you will see as we attempt to stretch the visualization capabilities
of the Flatlander to the very limit.

The first challenge involves expressing to the Flatlander the concept
of a closed 2-dimensional surface (a 2-dimensional surface that curves back
on itself, as a sphere does). Displayed to the left is the model of the spherical
2-dimensional universe
we are working with, in stack-diagram form. The Flatlander's best conception
of a closed 2-dimensional surface is, as you can see, as *an array of closed
1-dimensional surfaces*. To better understand this, imagine moving the slices
on both sides of the central slice to their 3-dimensional positions behind
and in front of the central slice, to form the sphere that the stack-diagram
represents. Once the slices are in their appropriate positions, imagine that
the slices are 'pressed together', leaving no empty space in between them. Upon doing
so, take note of the overlapping circular regions where neighboring slices 'touch'. It is
these 'regions of contact' that 'connect' one slice to another, allowing what we will call
"communication" to occur across the slices. Given that the slices are 'connected', and
given that this allows "communication" to occur across the slices, we find as a very
observable result, that all of the connected 1-dimensional surfaces of the slices "combine"
to form what we know to be the **single** 2-dimensional surface of the sphere. Take
note, though, that when the slices of the sphere are in *stack-diagram form*, it is not
possible to directly observe the 'regions of contact' among the slices, for
the very reason that the slices are individually, separately lined up.
Nonetheless, the concept of "communication" across the slices can be applied
just as effectively. It is this concept of "communication" across the slices of the
stack-diagram that allows the Flatlander to imagine the multiple slices of the sphere to
"combine" to form the **single** 2-dimensional surface of the sphere.

The second challenge involves representing in stack-diagram form a 2-dimensional
object on the surface of our model. We will assume that floating about
in the upper region of our 2-dimensional spherical universe is a 2-dimensional flying
saucer. Our job is to add the flying saucer onto the stack-diagram of the
2-dimensional universe with as little loss of content as possible. Because
the 2-dimensional surface we are working with is represented in the form
of an array of 1-dimensional surfaces, all 2-dimensional objects on that
surface, when represented in stack-diagram form, get "chopped up" into
1-dimensional cross-sections. In order to represent the flying saucer in
the stack-diagram, we must divide it into 1-dimensional cross-sections,
and spread these cross-sections across the appropriate slices - as is done in the
top illustration to the right. The manner in which the cross-sections increase and
then decrease in size is a direct result of the flying saucer being circular in shape.
If we were to plot a point designating the precise location of the saucer
on the sphere shown, it would be at a point located at the exact center of the
saucer (in the center of its central cross-section). Such a point is plotted
in the next stack-diagram to the right. It is by means of this point that
we will refer to the saucer's location in the future. The purpose behind
displaying the saucer in the form of 1-dimensional cross-sections (and
not just as a point) is to emphasize the fact that the saucer, being a
2-dimensional object, possesses a distinct 'width' that spreads it across
a 2-dimensional area of the surface of the sphere - an area that extends
across *multiple slices* of the sphere, that a point by itself cannot
portray.

The third challenge involves expressing to the Flatlander how a sphere closes 2 dimensions. Put simply, this is referring to how an object on the surface of a sphere can travel in a straight line, and end up where it started. To better understand this concept, consider the notion that for any given area on the surface of a sphere there is at most 2 lines that can cross and lie at right angles to each other. These crossed lines represent the 2 dimensions of freedom the flying saucer has with which to move on the surface of the 2-dimensional universe. If we extended both ends of each line outward, they would follow a curved path until meeting on the other side of the sphere at the point exactly opposite to the point at which they originally met. These 2 closed loops represent the paths around the 2-dimensional universe that the flying saucer could take given its 2 dimensions of freedom.

The illustrations among the next several paragraphs portray the flying saucer
travelling along these paths around the 2-dimensional spherical universe we are
working with. The upper sequence
portrays a path around this sphere that is *parallel* to the alignment of
the slices. The lower sequence portrays a path around the sphere
that is *perpendicular* to the alignment of the slices. The path *parallel*
to the slices represents a form of closure that a Flatlander is familiar
with: the curvature of a **circle**. The path *perpendicular* to
the slices represents a form of closure that a Flatlander is not familiar
with: the curvature of a **sphere**. Clearly, a Flatlander can grasp
the closure of a circle. The closure of a sphere, however, will be difficult
to portray given the fact that we are limited to 2 dimensions. The sequences
we have been speaking of, as you will see when you reach them, are arranged in
a sequential, linear fashion.
This format is reflected by the distinct 'frames' the sequences are divided
into - each frame representing a separate instant in time in the execution
of the sequence. Each individual frame itself, as you will see, consists
of the array of slices that we know to make up the sphere.

The presence of the saucer throughout the sequence is made known across
the slices of the frames by means of what can be observed to be a *point*.
In the *parallel* sequence, only one direction
is involved in the execution of the sequence (an order of passage that
is designated by the numbers next to the frames). In the *perpendicular*
sequence, however, the point can be found to leave a slice within a given
frame, and later come back to that frame a second time, returning on a
separate slice. As a means of portraying the distinct 2-dimensional 'width'
that the saucer possesses on the surface of the sphere (the width described
earlier that involves *multiple slices* and that a point by itself
cannot portray), the 1-dimensional *cross-sections* of the saucer
are displayed on the slices of both sequences, appearing on and around
the slice of the point designating the saucer's location on the sphere.

In both sequences, the point representing the saucer's location follows
a circular path around the 2-dimensional spherical universe. In the *parallel*
sequence, displayed to
left, the point completes the entire process without ever leaving the central slice
of the sphere. The *perpendicular* sequence, however, that lies below, is more
complex: though the point makes a trip to the bottom of the central slice and back
as in the *parallel* sequence, the 2 half-circular trips performed
involve a slice-to-slice transfer across the slices that *surround*
the central slice (rather than a trip confined entirely to the central
slice itself). It is extension of the point into the slices that surround the central
slice that represents *curvature* into the **third dimension**. This curvature, it
would happen, is a direct result of the distinct increase and decrease in size that
the inner hollow slices of the sphere undergo, in the process of extending from one small,
solid outer slice to the other. Imagine, if you will, for the sake of study, the slices
of a 'pressed together' sphere to "combine" to form the
**single** 2-dimensional surface of the sphere. It is the *gradual waver* in the
sizes of the inner hollow slices just described that is responsible for the formation of the spherical curvature that we are familiar with.

In order to make the 2 half-circular
trips involved in the *perpendicular* sequence easier to comprehend
(given that we are limited to 2 dimensions), certain 'mental aids' have
been added to the sequence. First of all, the sequence is constructed so
that the point's path across the stack-diagrams used in the sequence forms
a distinct 'circular pattern'. This is an attempt to emphasize the concept
of the curvature into the third dimension that the point engages in when following its
circular path around the sphere - a curvature that, given that we
are limited to 2 dimensions, can be difficult to express. Furthermore, all slices that the
point crosses during the length of the sequence are
*highlighted*. Finally, *arrows* have been added to the sequence
that divide the point's circular trip around the sphere into the 2 distinct
'half-circular trips' described earlier. The arrows are to be used as a guide
by which one envisions the slice-to-slice traversal of the point as it
shifts across the layers of the sequence.

The point's first half-circular trip of the *perpendicular* sequence (displayed
to the left) is from the top to the bottom of the central slice. Imagine that the point
at the top of the central slice
is being 'pushed' downward in the
direction of the arrows (toward the *bottom* of the central slice).
The point "slides" onto the top of the next slice **A-ward** in response to
being 'pushed' into the curvature of the half-circular path it is following.
This process of 'pushing' and "sliding" is a 2-dimensional way to portray
3-dimensional curvature. The point is 'pushed' downward once again and
"slides" onto the "pole" of the sphere that lies **A-ward**. Next the point
is 'pushed' *directly downward* across the flat surface of the small
solid circular slice, and upon reaching the bottom "slides" onto the bottom
of the next slice **B-ward**. Finally, the point is 'pushed' in the direction
of the arrows and "slides" **B-ward**, in effect returning to the central slice
and completing the first 'half-circular trip' along the circular path around the
sphere. For the second half-circular trip we are to picture the point to
be pushed *back upward* (in the direction of the arrows) toward the
area at which it started, making use of the same 'push and slide' process
applied above. If you experience difficulty visualizing the *perpendicular*
sequence, try visualizing it after imagining the slices of the sphere to
move over to the central slice and assume their actual 3-dimensional positions
behind and in front of it. The stack-diagrams in the 2 sequences just presented
represent a Flatlander's basic conception of how a sphere closes 2 dimensions. In
the next section we apply what we've learned in this section to the
**hypersphere**.

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