The model we encountered in the previous section, as you may quite clearly recall,
was of a 2-dimensional
universe that existed on the surface of a **sphere**. In material even a Flatlander
could understand, what we in essence did was demonstrate how and why a
2-dimensional flying saucer on this spherical surface could travel in what
it perceived to be a straight line, and yet end up where it started (as
would happen on the surface of the earth). No matter how far the saucer travels (and
no matter what direction it travels in) it will come across no boundary
of any kind. The saucer, of course, never detects any curvature during
the trip: to the saucer, everything appears *flat* - flat in the same
way that the surface of the earth appears flat. What we learned in the
previous section (which includes the 2 stack-diagram sequences of the flying
saucer's paths around the sphere) is of major importance to us in that
it is the means by which we can apply the topics just mentioned to the
**hypersphere**.

How could we possibly travel in a straight line though space and end
up where we started? Or travel through space as far as we can (and in any
direction) and never reach an "outer edge"? The key to answering these
questions lies in the procedure of bringing what we've learned in the previous section
"up a notch" through the process of **analogy**. How would we envision a closed
3-dimensional surface (a surface that curves back on itself like a sphere, yet
contains within itself 3 dimensions)? Displayed to the right is the model of the
3-dimensional hyperspherical universe we will be working with, in stack-diagram form.
As you can see, our best conception of a closed 3-dimensional surface is as *an array
of closed 2-dimensional surfaces*.

As a means of better understanding this, let us construct once again what
is our best idea as to how we would *directly experience* the hypersphere:
the 'compressed hypersphere'. The role, it would happen, that the 'compressed hypersphere'
is about to take on here is very similar in nature to the role that
the 'pressing together' of the slices of a sphere played in the previous section. 'Pressing
together' slices, as you may
recall, served the purpose of bringing about physical contact among slices separated
by empty space. The purpose of "compressing" the hypersphere is similar: to allow the
2-dimensional spherical surfaces of the slices of the hypersphere to "overlap" and hence
make contact.
The first step, once again, in constructing the 'compressed hypersphere', is to imagine
that each and every slice of the model of the hyperspherical universe is itself assigned
to its own *3-dimensional plane*, each 3-dimensional plane being **spatially
separate**, lined up side by side. The next step, as you may
recall, consists of imagining, as we've done before, these multiple 3-dimensional planes
to occupy the space that we perceive our own 3-dimensional plane to take up.
Because we have "compressed" several 3-dimensional planes worth of the slices
of the hyperspherical universe onto a single
3-dimensional plane (as you may recall to have been performed in a previous section), the
slices all end up "overlapping" on the plane onto
which they were "compressed". This occurs because the slices are assuming the location they
occupied on the plane they existed on before being "compressed" - a location, it would happen,
that is the same for every slice. Upon having assumed the *same location* on the plane of the 'compressed hypersphere', we can further document the extent to which the slices relate
to each other by taking note of the fact that the slices now share the *same
centerpoint*. Now that the slices have been appropriately arranged, we will now be
informed of the details as to how a closed 3-dimensional surface is formed.

How do the slices of a 'compressed hypersphere' "overlap"? Take note, firstly, that the
spherical slices of the hypersphere we are dealing with are in fact *hollow* slices.
Secondly, consider the notion that an "overlapping" of any 2 given slices of the 'compressed
hypersphere' is to be considered valid only if the 2 slices were *neighboring* slices
before being "compressed". The main factor, however, in determining whether any 2 given slices of the 'compressed hypersphere' "overlap", is the
sizes of the slices themselves. This could be considered to take the form of the lengths of
each slice's **radius**. If the radius of one spherical slice is too
small or too large in relation to the radius of a second slice, the slices will not be close
enough to "overlap". As a rule, *neighboring slices will always be found to be able to
overlap*. When the lengths of radius of two 2-dimensional spherical surfaces are
close enough, a common, shared area of intersection can be observed. This intersection exists in the form of a hollow, rounded spherical contour and is the
equivalent of what we call "overlapping". We will call such an area a 'region of contact'.
The 'region of contact' spoken of here is in every way similar to the 'regions of contact' spoken of in the previous section among the slices of the sphere. The 'region of contact', so to speak, that one would find
among 2 neighboring slices of a 'pressed together' **sphere**, would be a hollow, rounded
*circle*. The 'region of contact' that one would find among 2 neighboring slices of a
"compressed" **hypersphere**, in turn, would be a hollow,
rounded *sphere*. Though the elements involved in these relationships are one dimension
apart in terms of hierarchy, in terms of *concept* the idea is exactly the same.

The hypersphere, it can be agreed, exists in our minds as an array of 3-dimensional slices.
When relating those slices to each other, what we call the 'regions of contact' take on a
special purpose. This purpose is to "unite" slices understood to be separate, into a
single, connected *greater structure*. We know as a fact that there are no 2 neighboring slices of the hypersphere that do not "overlap". What this means, then, is that there are
no 2 neighboring
slices of the hypersphere that aren't linked together (by means of a 'region of contact').
In conclusion, we could state that there is *no slice not linked to another slice*. In
effect, **everything is connected**. This allows what was introduced in the previous section
as "communication" to occur across the slices of the hypersphere. Given what we know so far,
we are to picture what we know to be the 2-dimensional spherical surfaces of each and every
slice of the 'compressed hypersphere' to "combine" together into a *single entity*.
Having pictured this, we are no longer to think of the slices of the hypersphere as individual,
separate surfaces: this 'single entity' is in fact the **single** 3-dimensional surface of
the hypersphere. Our best conception of a closed 3-dimensional surface, then, is as a
"combining together" of an array of closed 2-dimensional surfaces.

If you have difficulty grasping the above material, think back to the *previous
section*
and recall the process of how the slices of the sphere "combined" to form the **single**
2-dimensional surface of the sphere. This act of "combining together" the slices of the
sphere happens to be the **very same process** as the "combining together" of the slices of the 'compressed hypersphere' described in this section: all that differs between the 2 processes is their *dimensional hierarchy*. The process spoken of in the previous
section involves the 1-dimensional surfaces of the slices of the sphere "combining" to
form the 2-dimensional surface of the sphere. The process spoken of in this section is the
*same process*, only now the slices "combining" together are the 2-dimensional surfaces
of the slices of the hypersphere, and the surface formed by this union of slices is the
3-dimensional surface of the hypersphere. The concept of surfaces "overlapping" is an idea
with which we have become quite familiar, it would happen, as it is the means by which we
are able to envision a 3-dimensional surface. Take note, though, that
when the slices of the hypersphere 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. It is for this reason that we must
*imagine* the 'regions of contact' to occur amongst the slices of the stack-diagram,
and in doing so allow "communication" across the slices to occur.

How would we represent a 3-dimensional object on the surface of the
hyperspherical universe we are working with? We will assume that floating
about in the upper region of
our 3-dimensional universe is a spaceship. Because the 3-dimensional surface
we are working with is represented in the form of an array of 2-dimensional
surfaces, all 3-dimensional objects on that surface, when represented in
stack-diagram form, get "chopped up" into 2-dimensional cross-sections.
In order to represent the spaceship in the stack-diagram, we must divide
it into 2-dimensional cross-sections, and spread these cross-sections across
the appropriate slices - as is done in the top stack-diagram on the left.
The manner in which the
slices increase and then decrease in size shows that the spaceship is rounded
in shape. If we were to plot a point designating the precise location of
the spaceship on the hyperspherical universe we are working with, it would be at
a point located at the
exact center of the spaceship (in the center of its central cross-section).
Such a point is plotted in the bottom stack-diagram on the left. It is by
means of this point that we will refer to the spaceship's location in the
future. The purpose behind including the spaceship in the form of 2-dimensional
cross-sections (and not just as a point) is to emphasize the fact that
the spaceship, being a 3-dimensional object, possesses a distinct 'width'
that spreads it across a given 3-dimensional region of the surface of the
hyperspherical universe - a region of width that extends across *multiple slices*
of the hypersphere, that a point by itself cannot portray.

How does a hypersphere close 3 dimensions? For any given region on the surface of a hypersphere there is at most 3 lines that can cross each other and lie at right angles to each other. These crossed lines represent the 3 dimensions of freedom that the spaceship has with which to move on the surface of the 3-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 hypersphere at the point exactly opposite to the point at which they originally met. These 3 closed loops represent the paths around the 3-dimensional universe that the spaceship could take given its 3 dimensions of freedom.

The illustrations among the next several paragraphs portray the spaceship
travelling along
these paths around the 3-dimensional hyperspherical universe we are working with.
The 2 upper sequences portray
paths around the hypersphere that are *parallel* to the alignment
of the slices. The lower sequence portrays a path around this hypersphere
that is *perpendicular* to the alignment of the slices. The paths
*parallel* to the slices represent a form of closure that we are familiar
with: the curvature of a **sphere**. The path *perpendicular* to
the slices represents a form of closure that we are not familiar with:
the curvature of a **hypersphere**. Clearly, we can grasp the closure
of a sphere. The closure of a hypersphere, however, will be difficult to
portray given the fact that we are limited to 3 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 makes up the hypersphere.

The presence of the spaceship 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* sequences, only one direction
is involved in the process of execution (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 3-dimensional 'width' that
the spaceship possesses on the surface of the hypersphere (the width described
earlier that involves *multiple slices* and that a point by itself
cannot portray), the 2-dimensional *cross-sections* of the spaceship
are displayed on the slices of all three sequences, appearing on and around
the slice of the point designating the spaceship's location on the hypersphere.

In all three sequences, the point representing the spaceship's location
follows a circular path around the 3-dimensional hyperspherical universe. In the
*parallel* sequences (that begin to be displayed to the right) the point
completes the entire process without ever leaving
the central slice of the hypersphere. The *perpendicular* sequence,
however, that lies off-screen below, is more complex: though the point makes a trip to
the bottom of
the central slice and back as in the *parallel* sequences, the 2 half-circular
trips performed involve a slice-to-slice progression around 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 **fourth 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 hypersphere 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 'compressed
hypersphere' to "combine" to form the **single** 3-dimensional surface of the hypersphere.
It is the *gradual waver* in the sizes of the inner hollow slices just described that is responsible for the formation of the hyperspherical curvature that allows 3 dimensions to be
closed.

In order to make the
2 half-circular trips involved in the *perpendicular* sequence easier
to comprehend, 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 fourth
dimension that the point engages in when following its circular path around the
hypersphere - a curvature that, given that we are limited to 3 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 hypersphere 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 right) 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 3-dimensional
way to portray 4-dimensional curvature. After being 'pushed' downward once
again the point "slides" **A-ward** and is 'pushed' *directly downward*
through the spherical region of the small solid spherical 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 hypersphere. 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. To give added meaning to
the interaction among slices portrayed in the stack-diagrams just
presented, try picturing the interaction among the slices as it
would occur among the overlapping slices of the 'compressed hypersphere'. This
is understandably difficult given that it would deal with slices that "overlap"
(as opposed to the individually, separately lined up slices of the hypersphere
as it exists in stack-diagram form). To perform such a feat, however, would
perhaps be the closest we can come to *directly experiencing* the
interaction that occurs amongst the slices of the actual hypersphere. The
stack-diagrams in the 3 sequences just presented represent our basic conception
of how a hypersphere closes 3 dimensions. In the next section we study how the
structure of the hypersphere relates to how it is perceived.

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