How would we approach an occurrence related to the fourth dimension? Or the fourth
dimension in general? Defining what a 4-dimensional plane is involves studying the
properties and configuration of planes in lower dimensions. Because there is no
empirical evidence supporting spaces higher than our own - higher dimensions being
'hidden' from us in the way that they are - we must do our best with
what is presented to us. What this involves, of course, is rigorous study
of lower dimensions. In the process of doing this we assume that everything
found true for lower dimensions must also be true for our own. As the means of carrying
out this method of study, we
relate a lower-dimensional arrangement to **our** dimension in the same way that the
lower-dimensional arrangement relates to **its own** dimension. It is because of the
very fact that lower dimensions exist, you see, that this approach is possible. This
system of reasoning - the process of **analogy** - is the means by which we can
"fill in the blanks" where there would otherwise be no answers. The use of analogy is the
invaluable "tool" with which we can gain knowledge of the fourth dimension and
of the 4-dimensional occurrences related to it - *knowledge that we could not obtain
from what is presented to us by means of sensory experience*.

As was stated above, the process of analogy in essence consists of relating a
lower-dimensional arrangement to *our* dimension in the same way that the
lower-dimensional arrangement relates to *its own* dimension. Given this, take note
that in future sections, the manner in which a lower-dimensional arrangement is related to
our dimension, may be a manner that requires of us to relate to our dimension in ways
*that we never have before*. Upon such an encounter, the new manner in which we are
required to relate to our dimension may seem to break the "rules" that our minds have set
down dictating to us what is and isn't "spatially possible". These "rules" are the very result of a life confined to the limitations of our own dimension.
Do not be alarmed, then, if you come across a structure or concept
in this section or in a future section of this material that seems impossible to believe.
*Expect*, rather, such an encounter, and remind yourself that relating to one's
dimension *in new ways* can in fact be difficult. In summary, then, we should realize
that if something doesn't seem to make sense in terms of the rules we know this dimension to operate by, it is probably because what we have come across exists **outside** the very
boundaries of our dimension, and cannot be expressed in terms of concepts we can claim to
have experienced.

Having been made familiar with the process of *analogy* and with the ways in which
we will relate to it in future sections, we will now go about addressing what
is next to come in the presentation of this section. Presented below is the
result of an effort to gain the 'knowledge' spoken of earlier in this section - the
knowledge that is made possible by the process of *analogy*. It is done
in the spirit of the
classic sequence (displayed at the top of this page) that begins with a point, and
afterward shows how structures can 'glide' in a direction perpendicular to themselves
and leave behind a 'trail' in the process - a trail that in turn becomes the next
structure. Focus, then, your attention on the sequence presented below. Though similar
in many ways to the 'gliding structure' sequence presented above, there is in the sequence
presented below an aspect not found in the 'gliding structure' sequence - an aspect that
allows the sequence to take on deeper meaning: this aspect is involvement with intelligent,
thinking lower-dimensional creatures.

These sequences differ in terms of the 'final notion' they present. The 'final notion' presented by the 'gliding structure' sequence is a question: the question of how one 'glides' a cube in a direction perpendicular to itself. The final notion presented by the sequence below, however, is that of a solid conclusion, reached fully through what was observed in the lower dimensions. Provided are not questions, but answers. Upon encountering the 'creature' in each example, try to avoid 'debating' whether each creature could theoretically exist in reality - if necessary, they can be thought of more as 'concepts' than as actual 'beings'. Upon defining what a 4-dimensional plane is (the goal of the sequence) we will possess an overall view of the fourth dimension in general.

A zero-dimensional creature can comprehend nothing beyond a point. The creature
is on the central point of an array of points, however, surrounded on both sides by
points identical to his own. These points extend outward from the creature's point
into the A 1-dimensional creature can comprehend nothing beyond a 1-dimensional plane. The creature is on the central plane of an array of 1-dimensional planes, however, surrounded on both sides by planes identical to his own. These planes extend outward from the creature's plane into the A 2-dimensional creature can comprehend nothing beyond a 2-dimensional plane. The creature is on the central plane of an array of 2-dimensional planes, however, surrounded on both sides by planes identical to his own. These planes extend outward from the creature's plane into the We, being 3-dimensional creatures, can comprehend nothing beyond a 3-dimensional plane. If our plane were the central plane of an array of 3-dimensional planes identical to our own, into what directions would the surrounding 3-dimensional planes extend? Much like the creatures in the examples above, we cannot comprehend extension into the |

What, then, is the purpose and meaning behind "compressing" planes?
One by one, the examples in the sequence above introduce a new dimension
of extension in the form of an array of planes. Though the planes extend
into very real directions, the given subject who is located on the central
plane (in the middle of the occurrence of the extension) cannot comprehend
these directions of extension. This is the unavoidable result of the subject's
life of confinement to his own dimension. The method of "compressing" planes
does not attempt to explain these directions to the subject, for the simple
reason that the limitations of the subject's visualization capabilites
make comprehending the directions impossible. In response to the subject's
inability to comprehend these directions, we turn to a concept that the
subject *can* comprehend: the concept of being surrounded by planes
**identical** to his own. Though the subject can't comprehend the location
of these planes, he is nonetheless aware that they exist. The subject overcomes
his inability to comprehend the directions that the surrounding planes
extend into by approaching the situation from a more basic level: because
the planes surrounding the subject are identical to his own plane, the
subject in a sense successfully conceives of them by simply visualizing
the abstract concept of *multiple instances* of his own plane . He does this
by assuming the planes surrounding his own plane to all occupy the space that he
perceives his own plane to take up. Surprisingly, this works. The purpose and
meaning behind "compressing" planes is that if this process presents to each subject
a successful version of each array of planes in all three lower-dimensional
examples, why shouldn't the version that this process presents to our own dimension
be equally as valid?

And hence we see that even though a given subject is not capable of
comprehending how an array of planes of his own dimension extends into the
*next* dimension,
what he can in fact grasp is the array of planes as it exists in its "compressed"
state: the subject visualizes all of the planes of the array to occupy
the space he perceives his own plane to take up. Though not an exact copy
of the original array of planes, this version of the array of planes, as you will
see, can accomplish just as much. The "compressed" array of 3-dimensional planes
(described above as a group of 3-dimensional planes all occupying the same space)
is in fact the '4-dimensional plane' mentioned in the title of this section
- and is the means by which we will construct the hypersphere. It is in
the next section that we cover the details of how this '4-dimensional plane'
relates to the slices of the hypersphere spread across it.

to previous section |
to table of contents |
to next section |

Comments, questions, feedback:
[email protected] |