a Mathematical Perspective


The sons of men are one, and I am one with them.
I seek to love, not hate;
I seek to serve and not exact due service;
I seek to heal, not hurt.

Let pain bring due reward of light and love.
Let the soul control the outer form,
And life, and all events,
And bring to light the love
That underlies the happenings of the time.

Let vision come and insight.
Let the future stand revealed.
Let inner union demonstrate and outer cleavages be gone.
Let love prevail.
Let all men love.


Let us define two axes in a plane of Cartesian coordinates: the horizontal axis of Light (K, for knowledge of the "real" world), and the vertical axis of Love (L), which may be also considered the realm of "imaginary" numbers (i).

This equal-armed cross defines four quadrants (K+ L+, K- L+, K- L- and K+ L-) within a

Figure 1

 (Please read quadrants I, II, III and IV counterclockwise in all figures)

Part One: The Unit Square

Let us start our journey with: "The sons of men are ONE, and i am 1 with them." The "imaginary unit" (i) is the square root of negative one, a paradoxical statement, even more so if we place it in the first quadrant (K+ L+) of our unit square. Hopefully, we would be able to resolve this paradox by the end of our journey.

Next, three quests (for love, service and healing) are paired to their antitheses (hate, selfishness and pain).

Figure 2

As hate is mostly driven by ignorance, we placed the phrase "i seek to love, not hate" in the second quadrant (K- L+) meaning unenlightened love. As selfishness lacks in both love and light, we placed "i seek to serve, not exact due service" in the third quadrant (K- L-). As we do hurt, ourselves and others, when knowledge is not wisely applied with love, we placed "i seek to heal, not hurt" in the fourth quadrant (K+ L-).

Then, from the lower triad of pain, outer (material) form and chaos (of "the happenings of the times"), we invoke their respective saving forces: love, soul control and light.

Figure 3

Love  transmutes all hate (quadrant II). The white magic of the soul transforms all selfishness (quadrant III). The light of wisdom transfigures all pain (quadrant IV).

Now we are ready for the final stage which will hopefully solve the initial paradox, that is, i squared equals negative one. In other words, an identity (
i) multiplied by its negative (-i) is ONE. How is the equality of the positive and the negative aspects of an identity possible?

"Let vision come and insight." Both "vision" and "insight" have an "
i" twice (i squared). Is it the vision of the UNIFIED purpose of "good" (+1) and "evil" (-1)? Is it an insight into redemptive power of LOVE? If evil is that good which should have been left behind, then -1 and +1 may represent the TIME axis of the "imaginary" UNI-verse. 

Then we are ready to assert three essential FACTS of revelation, of inner union and of love. 

Figure 4

Lastly, a powerful clarion call:

Let ALL men LOVE!

A synthetic TRIANGLE of pure reason lifting all quadrants with the vacuum generated by the SOUND wave of LOVE.

In the centre of the will of God I stand.
Naught shall deflect my will from His.
I implement that will by love.
I turn towards the field of service.
I, the Triangle divine, work out that will
Within the square and serve my fellowmen.


Part Two: The Unit Circle

We could also inscribe a circle within the unit square to explore the trigonometric relationship between Love, Light and the Power of Unity.

In the unit circle, the sine and cosine functions are defined as Love/One and Light/One, respectively.


In  the "real" world, regardless of the quadrant in the journey, all negative experiences are "squared" (transmuted: raised to the power of 2) so that their sum is always ONE. This is the main lesson of the unit circle in the world of "real" numbers.

But what about Love in the "imaginary" space? That is, the self-sacrifice of an identity (i) fusing and redeeming its negative shadow (-i), the ultimate crucifixion.  Euler's formula reveals this apotheosis.

e^{i \pi} = \cos \pi + i \sin \pi.\,\!

This equation simplifies to the ultimate revelation of identity in the "imaginary" space (also called Euler's identity):

e^{i \pi} +1 = 0.\,\!

Words cannot convey the feeling of this revelation. Mathematicians become mystics when they feel the BEAUTY of this formula. 

BEAUTY of the Identity

A reader poll conducted by Mathematical Intelligencer named the identity as the most beautiful theorem in mathematics. Another reader poll conducted by Physics World in 2004 named Euler's identity the "greatest equation ever", together with Maxwell's equations.

The book Dr. Euler's Fabulous Formula [2006], by Paul Nahin (Professor Emeritus at the University of New Hampshire), is devoted to Euler's identity; it is 400 pages long. The book states that the identity sets "the gold standard for mathematical beauty."

Constance Reid claimed that Euler's identity was "the most famous formula in all mathematics."

Gauss is reported to have commented that if this formula was not immediately apparent to a student on being told it, the student would never be a first-class mathematician.

After proving the identity in a lecture, Benjamin Peirce, a noted nineteenth century mathematician and Harvard professor, said, "It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth."

Stanford mathematics professor Keith Devlin says, "Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence."


When words fail, maybe "tears of joy" can bear witness to the depth of the experience, as we fathom --within--  what Unity is.

I am the ONE,
Begotten by the ZERO,
Differentiated into INFINITY,
Integrated into the conscious SYNTHESIS
Of LIGHT, and LOVE and LIFE,
For the GLORY of the ONE.



Related article: Integration Formulas and Identification Functions

Acknowledgment: This essay is dedicated to the loving memory of Prof. David Pastor, my mathematics mentor, who has recently made his transition to his beloved imaginary space.

Mente expectante y el Corazón sereno,
en perfecta adaptación al Presente eterno.

[An expecting Mind and a serene Heart,
perfectly adapted to the eternal Present]

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