Consciousness: The Last Frontier of Geometry - Maharishi ...

Consciousness: The Last Frontier of Geometry
Catherine A. Gorini
Introduction. Each discipline of knowledge studies different aspects of life and uses
different methodologies and procedures. Many disciplines, however, find that their
distinct paths reach a common end: questions about consciousness. Thus, consciousness
is regarded as the last frontier in humanity’s quest for knowledge. Here, we will describe
how the discipline of geometry, that part of mathematics that investigates patterns of
space, shape, and form. We will see that geometry can also lead, in a very natural way, to
concerns relevant to the study of consciousness and can give unique insights into the
ultimate nature of consciousness.
Throughout its history, mathematics has been highly regarded as a discipline that can
offer valuable insights outside of its designated domain. Thousands of years ago, Plato [1,
page 69] affirmed that “geometry will draw the soul towards truth.” Descartes, several
hundred years ago, modeled his comprehensive search for truth on the methodology of
geometry. Today, Keith Devlin [2, page 9] recognizes that “as an entirely human
creation, the study of mathematics is ultimately a study of humanity itself.”
In this paper, we will first look at how other disciplines have encountered this last
frontier of consciousness and then focus on several themes of geometry—continuity,
higher dimensions, infinity, symmetry, homogeneity of space, and non-Euclidean
geometries—to see what connections they have with consciousness. Finally, we will
discuss the significance of this view of geometry.
Last Frontiers. Physics, the study of matter, force, and energy, has encountered
Heisenberg’s uncertainty principle, which guarantees that merely observing a system
changes the system in a way that cannot be minimized or avoided. This means that the
role of the observer is an integral part of any system being observed. This has two
important implications. The first is that understanding the process of observation,
whereby an observer is connected to an object of observation, is necessary for any
complete theory of physics. The second is that the observer and the observed,
consciousness and matter, mind and body, are not separate but are intimately connected
at their very basis. Thus, a unified field theory of physics must ultimately account for the
role of consciousness.
Biology, the study of living organisms, has located one fundamental structure at the
basis of life: DNA. Encoded in an organism’s DNA is all of the information about the
growth, development, functioning, and behavior of the organism. Biologists know quite
precisely how certain sequences of DNA codons produce specific proteins, but they know
nothing about how the structure of DNA and the biological systems it controls are
connected to the phenomenon of human awareness or consciousness. Perhaps a complete
and correct understanding of this “blueprint of life” could lead to a deep understanding of
the nature of consciousness.

continuity of the straight line and of angular measure [6, Appendix 5].
Until very recently, the most appropriate model for space, time, matter, and energy
has been the continuum. Calculus was developed to handle changing relationships
between continuous quantities. Beginning with the discovery of molecules and atoms,
however, each continuous model used by physics has been shown to be inaccurate at
sufficiently small time and distance scales. The most recent theories of physics, which
model space and time with space-time foam, show that even space and time cannot, at
their finest level, be modeled by a continuum.
It must be the case, then, that continuity is characteristic of how the observer, as a
mathematician or physicist, interprets or understands space, time, and matter. Continuity
must be more closely associated with the consciousness of the observer than with the
object of observation. The continuum is an infinite, expanding structure, flowing evenly.
It is the basis for measuring or quantifying. We can conclude that these qualities are
necessarily attributes of consciousness.
Higher Dimensions. The three dimensions of space that we live in appear to be
inviolable, yet they can be extended mathematically in a very natural way to higher- and
even infinite-dimensional abstract spaces. Higher-dimensional objects or spaces are
typically used to provide a convenient geometric framework in which to consolidate all
possible members of a certain set. For example, the infinite-dimensional phase spaces of
physics consist of every possible state a specific particle or system could have; the time
evolution of the particle or system is then a trajectory through the phase space. Similarly,
the feasibility region of a linear program is the many-dimensional set of all possible
points satisfying a given collection of inequalities. These higher-dimensional spaces are
used because they organize information in an effective way that allows for further
computation. The physical bounds of three dimensions are not boundaries to our
awareness; it is natural and useful for the mind to operate in an all-inclusive way,
unrestricted by the boundaries of the space we live in. Our consciousness more naturally
functions from a unified level, creating a totality or wholeness out of all possible distinct
parts.
Infinity. The infinite is at the heart of mathematics. Considerations of continuity and
higher-dimensional spaces lead us inevitably to the broader concept of infinity in
geometry. The straight line is infinite in extent and has infinitely many points. There are
infinitely many geometric shapes, classified according to the various attributes (number
of edges, measure of angles, and so on). Fractals show the intricate and beautiful
structure that the infinite can support. What can be said of the human awareness that has
created, studied, and classified the infinities of geometry Certainly the faculty that can
comprehend and manipulate those infinities must itself be even more infinite.
Consciousness is infinite, it is capable of comprehending infinity, and it must be the
holistic value of which all other infinities are only a part.
Symmetry. At the basis of the concept of symmetry is the recognition that two
different objects can have essentially the same structure. For example, the symmetries of

a square depend on the congruence of each side with the other three sides and each angle
with the other three angles. Our innate appreciation of symmetry depends on our ability
to simultaneously discriminate (between the different parts of an object or between
different objects) and to locate sameness or equivalency (of those parts or objects).
Consciousness can locate differences, but more importantly, it can use differences as
the basis for harmonizing or balancing those differences through the location of
symmetries. The natural enjoyment of unity in the presence of diversity explains the
universal use of symmetry in the decorative arts. Thus, we see that consciousness is at
once discriminating and harmonizing, synthesizing parts into ever greater wholes.
Symmetry also depends on the comparing of an object to itself, using an object itself
as its own measure. This is reflective of the self-referral quality of consciousness, the
ability of consciousness to know, observe, and interact with itself.
Homogeneity of Space. The fourth postulate of Euclid says that all right angles are
equal; this means that space is everywhere the same in Euclidean geometry. This is
certainly not how the eye perceives the space around us, since a right angle looks “right”
only when viewed straight on. It is also not the best model of the space we live in, as we
see from Einstein’s General Theory of Relativity. This suggests that homogeneity is more
naturally a characteristic of the consciousness of the mathematician than of the space we
live in.
Different Geometries. For hundreds of years, Euclidean geometry was regarded as
“true,” the only possible geometry and the actual description of the space we live in. The
discovery of non-Euclidean geometries that are provably as consistent as Euclidean
geometry inspired a reexamination of the nature of mathematical truth, forcing
mathematicians to accept mathematical truth as relative truth only. Consciousness is
capable of integrating opposites, harmonizing them into a single viewpoint that
transcends the diversity created by the intellect.
Conclusions. The discussion here of these few geometric topics affirm that the study
of geometry is an important and appropriate study for someone interested in
consciousness and the meaning of life. In geometry, one is studying patterns of
consciousness and gaining knowledge about how consciousness functions within itself.
The unique perspective of geometry, focusing on patterns of shape and form, is
complementary to other approaches to the study of consciousness.
The study of geometry suggests that consciousness is an infinite, unbounded,
homogeneous continuum that can support great diversity. It is more natural and effective
for the mind to construct a whole out of infinitely many possibilities than to deal with
finitely many cases. The intellect is harmonizing and unifying, capable of locating the
common features of different structures. Finally, human consciousness, even when
restricted by the precision and logical correctness of mathematics, is capable of
encompassing different, even contradictory, truths.

These conclusions about connections between geometry and consciousness have
implications for the student of geometry. Students of geometry should consider
themselves to be also students of consciousness. The development and refinement of the
intellect has always been considered to be important for mathematicians, but we see now
that an intellectual understanding of the nature of consciousness can support and enrich
our understanding of geometric concepts.
The integration of the study of consciousness with the study of geometry is part of
the curriculum at Maharishi University of Management. At the beginning of their studies,
all students take a course covering the nature of consciousness, intelligence, and
knowledge led by Maharishi Mahesh Yogi. This course serves as a foundation for their
study of all other disciplines, including geometry. All students also practice the
Transcendental Meditation and TM-Sidhi program as an experiential method for research
into the nature of their own consciousness. This foundation has proved to be successful
and rewarding in the study of geometry.
Finally, I would like to note that Maharishi Mahesh Yogi, the world’s foremost
scientist in the area of consciousness, views mathematics as really nothing other than the
study of consciousness:
Mathematical knowledge deals directly with the functioning of
the field of intelligence—consciousness. The principles of
Mathematics are universally valid principles of knowledge that
describe the dynamics of the field of intelligence—the functioning of
the mathematician’s own consciousness. [5, page 160
From this perspective, geometry is necessarily one aspect of the study of consciousness.
The real fulfillment of mathematics, however, is in what it can achieve for humanity:
Vedic Mathematics provides the steps of invincibility, enlivens
the total potential of Natural Law, establishes mastery over Natural
Law and thereby places life on the invincible level of ‘Victory before
War’—life without problems—invincible defence against anything
that is not useful to life. [4, page 351]
This vision of what mathematics can achieve embodies the goals mathematicians have
held throughout history and is an inspiration to all mathematicians.
References
[1] Ronald Calinger. Classics of Mathematics. Moore Publishing Company, Oak
Park, IL 1982.
[2] Keith Devlin. The Language of Mathematics: Making the Invisible Visible.
Freeman, New York, 1998.

[3] David Hilbert. Foundations of Geometry. Open Court, La Salle, IL, 1971.
[4] Maharishi Mahesh Yogi. Maharishi’s Absolute Theory of Defence. Age of
Enlightenment Publications, India, 1996.
[5] Maharishi Mahesh Yogi. Maharishi Speaks to Students: Mastery Over Natural
Law (Vol. 3). Age of Enlightenment Publications, India, 1997.
[6] James Smart. Modern Geometries. Brooks/Cole, Pacific Grove, CA, 1998.