As we have seen, our endeavors to comprehend the fourth
dimension mirror our timeless struggle with the concepts of spiritualism.
We now understand that by grappling with the former we often advance our
understanding of the latter (and vice versa). Mathematicians, especially
those studying the fourth dimension, have long made use of spiritual training
in their work, while principles of dimensional theory have provided coherent
explanations for unclear passages in many spiritual texts, such as the
Bible.
However, as we continue to ponder the eternal questions that underlie both
of these themes, we would do well to consider those who have aided us in
the spiritual and fourth dimensional journeys many of us have made in our
lives. We who are fortunate enough to have probed some of the complexities
of both spiritualism and the fourth dimension would likely never have done
so without the direction of our teachers. To forge ahead without proper
appreciation of our guides would be to dash blindly onto shaky intellectual
ground without the proper foundation to support us. By contrast, we can
be assured that deliberation under the watchful eyes of our instructors
will afford us progress in spiritual and fourth dimensional enlightenment,
as it were. In addition, it is by recognizing those who have helped us
in our pursuits that we ensure future generations the same grounding in
the principles of spiritualism and the fourth dimension that we have been
privileged to enjoy.
There have been a host of spiritual teachers throughout
the history of time, and it serves our purpose to merely point to a few
specific cases. In chronological order, we can say that Moses, Jesus, and
Mohammed are examples of individuals whose teachings have underpinned a
great deal of spiritual reflection. In his role as teacher, among other
things, Moses provided a spiritual basis for the introspection of his adherents
by teaching them to follow the Ten Commandments. Several thousand years
later, Jesus preached a doctrine of reciprocal forgiveness, and in so doing
gave his disciples a different framework within which they could look for
spiritual satisfaction. In turn, after a period of about six hundred years,
Mohammed bestowed yet another set of spiritual guidelines for his followers.
In all three instances, we see that it was the individual's capacity as
a teacher that promoted spiritual understanding among his students. In
addition, the preservation of both testaments of the Bible and of the Koran
has extended the instructors' spiritual guidance for all ages, thus allowing
for more directed inquiry into questions of religion right through to the
present day.
Given the resonance of fourth dimensional concepts in
spiritual matters, it stands to reason that we should consider the teachers
of the former as well as the latter. One notable figure is Edwin Abbott
Abbott, whose thought provoking book Flatland gives one of the best introductions
ever written to the idea of dimensions. Another is Salvador
Dali, who brought the fourth dimensional "hypercube" a great deal of
prominence in the art world with his painting "Corpus
Hypercubicus" (which, not surprisingly, also features the second of
the spiritual leaders cited in the foregoing paragraph). Perhaps the most
direct example of a "fourth dimensional teacher," however, is none other
than Professor Thomas Banchoff. Although
recognized as an innovator in the mathematical discipline and in computer
graphics, Professor Banchoff is most accomplished as a teacher. He has
received a substantial number of awards for excellence in teaching from
a multitude of institutions, and Banchoff justifiably considers those given
for teaching to be the most important of his accolades. We see in Banchoff's
authorship of myriad articles on methods of classroom instruction his commitment
to helping his students in their attempts to understand the fourth dimension.
Among Banchoff's precepts, we would do well to commit to his assertion
that "Almost anything you can name is related to the fourth dimension"
(lecture, Yale University, 9/24/98). This statement conveys to us the ubiquity
of the fourth dimension in our lives, much the same way that spiritual
concerns influence nearly every aspect of our thinking. Banchoff's incorporation
of subjects ranging from Peter Pan to shower installations into his work
handily demonstrates to us the fundamentally universal accessibility of
fourth dimensional concepts. Professor Banchoff, like his predecessors
Abbott and Dali, has revolutionized our understanding of the fourth dimension,
and it is by recognizing the work of such people that we allow the next
generation of inquisitive minds to benefit from the same kind of guidance
that these teachers have provided to students today.
With our acknowledgment of the contributions of everyone
mentioned above, we students can be satisfied that we are helping to advance
human understanding of spiritualism and the fourth dimension. Moreover,
in light of the connection between the two subjects, we are assured that
when we support those who offer their teachings on the fourth dimension,
for example, we also confer a benefit upon the spiritually unsettled. Ultimately,
we might ask what else we can do to help. Unfortunately, many of the prophets
and teachers mentioned above are no longer living, and so it is solely
through preserving and respecting their memories that we can promote spiritual
and fourth dimensional enlightenment. Professor Banchoff, however, is still
very much with us, and in addition to reading his articles and so forth,
perhaps we could buy or rent one of his instructional videos, such as "Hypercube:
Projections and Slicing" (which can be obtained from the address listed
on his home page), or buy his book, Beyond the Third Dimension, which is
devoted to topics in the fourth dimension. All in all, in the final analysis,
we must agree that if we ever to achieve some kind of gestalt, by which
we have a genuine understanding of how God can exist throughout time, or
where Heaven and Hell are actually located, we can be sure that our grounding
in the principles of the fourth dimension will have been one of the keystones
of our success.