Moksha: In the light of mathematics

Meditative mathematics

The beginning

Before we delve into the concept of “Moksha” and its logical deductions of mathematics, let us first consider figure 1 as our reference and quickly jot down some properties of the given diagram.

Cyclic Trigonometry
Figure 1

“MoM” is the X-axis. We will later consider it as the material plane.
“SoS” is the Y-axis. We will assume it as the spiritual plane.
∠AOF = x, ∠BOE = y, and ∠COD = z.
OA = R1, OB = R2, and OC = R3.

This is not a unit circle because a unit circle restricts the radial value as 1.
So, getting back to the point.

With the help of the angles, radial lengths, and the reference axis, we can deduce the limiting functions of the radial projections on the “MoM” axis (i.e.material plane).

We are taking the MoM as the first projection plane because human life always initiates from the material plane (horizontal plane) and thrives to move towards the spiritual (we can also term this as vertical or pure intellectual) plane.

So the limiting expressions for x,y, and z with the angular degree of 0 and π∕2 are as follows:

Limits of cos0
Limits of cos0
Limits of cos90
Limits of cos90


If A, B, and C are the three points (representing three life forms) on the concentric circles with a common source “O” then moving on the same orbit and merging (read grouping) with some other point on the same place will only increase or decrease the angular measurement but will never going to reduce the displacement from the source.

Understanding the concept

Say, for instance, point C has a radial displacement of R3 and an angle of “z”. Now if the point “C” moves on its orbit and combines with point “C1′ or point “C2” then the angular value will differ but the radial displacement remains the same (R3). But, this will surely affect the projection of the radius on the material plane (MoM). Like if “C” moves towards “C1” it will increase the projection or in our case involvement with the gross mundane world and vice versa if it moves towards “C2”.

Mathematically, R1, R2, and R3 are maximum when x, y, and z are zero or tending to be zero.
If x, y, or z tends towards or becomes equal to π∕2, then the points (C, C1, or C2) get aligned with the spiritual plane keeping the radial distance unchanged. This only helps the above mentioned geometric points (or lives) to rise above the gross material plane and focus towards the source.

This whole journey is from getting horizontal to vertical just like a baby, who crawls with a horizontal spine after sometime stand erect making its spine vertical. This is the first step towards attaining Moksha.


According to Indian scriptures, there are seven major planes of existence from the lower gross plane to the higher plane*.
These planes are termed as “Bhu”, “Bhuvah”, “Swah”, “Maha”, “Janah”, “Tapah”, and “Satyam”. Beyond the “Satyam” plane, the experience of Moksha takes place. 
“Bhu” being the lowest plane or the gross material plane on which we exist. In our diagram, point “C” resides on the lowest plane and here only the lower three planes are depicted. 
We can visualize these major planes as concentric spheres (I depicted the diagrams in  2D for simpler understanding).

Contradiction and Explanation

Here we can debate about the point “E” and “F” as they are also projected on the material plane.
They indeed remain on this plane but the projected distance is much less than point “C”. So, this depicts that even we dwell on the “MoM”, we are moving from gross to subtle existence directly towards the source.

We will deal with this concept again in the latter part of this article.

Limits of sin90
Limits of sin90

On the “SoS” meridian, all get aligned with only one aspect of this co-ordinate system. That’s the Cosine part. But we still have the other one called the “Sine“. So, even we get aligned with the “SoS” and completely renounce the material perspective, we still have the same radial displacement as Sin(π∕2) which keeps the whole limiting equation equal to their respective radii. We can experience the spiritual aspect of the source but staying at the same distance. This is one of the reasons when a spiritual person feels low when they find themselves on the same plane with the materialistic minds. This is the crisis period of spiritual growth.

Hence, only a partial experience of the whole takes place when the angular displacement ranges from 0 to π∕2 and it repeats itself with π, 2π, etc.

So this entire concept reflects the fact that even if we traverse within the plane from gross material to pseudo-spiritual, we technically cannot get near the absolute truth and repeat the process of birth and re-birth. This is the main obstacle to attain moksha or absolute liberation.
Refer to figure 2.

Sine curve
Figure 2

Then how to reduce the radial distance of point C of the lower plane?

The conceptual path to “Moksha”

Let us refer to figure 3. This diagram more or less illustrates the atomic structure of an element (disregarding the eccentricity of the orbits for simplicity) within the periodic table. We will take the analogy of this description and try to ascertain the reduction procedure of the radii (figure 1) concerning the quantum leaps.

Quantum leap
Figure 3

According to quantum physics, when an electron jumps from the higher shell (orbit) to lower shell**, (from B to A), it releases energy (-3.4eV) in the form of quanta’s or pure light energy and moves towards the source.
So, with the help of these two analogous models (figure 1 and figure 3), we can understand that first, we have to increase the value of x, y, or z (angular magnitudes) from 0 to π∕2 for ending the material aspect (in our minds). This involves spiritual practices prescribed in the scripture which include “Yama, asana, pranayama, meditation, self-inquiry, etc.

After this is done, we have to reduce the vertical projection on “SoS” similarly like the quantum leaps of electrons.
Exempting an individual ego in the form of love and compassion is analogous to light.

Releasing this form of refined and morphed ego to the lower plane (remembering figure 1), will let us jump onto the lower shell thereby reducing the radial distance. The reduction of radial magnitude takes us nearer to “Moksha”.

Once we reach the core, we feel ourselves transformed into a super energy reservoir which creates the building blocks of matter. This is the playground of the creator, we call “Ishvara” in our scriptures.

But, this is not all!

The last leap

We are in the core but not fully merged! In Indian scripture, it is known as “Chanaka Rupa” or like chickpea where we find the two parts attached but with a marked division. When we fully merge into the central point which is primordial, we find ourselves in the supervoid called the quantum field. This quantum field is the “Brahman”. It is beyond everything. It does not interfere with its creation or indulges in it but all the creation remains in it.

This is the ultimate source or the absolute Truth.
Reaching and merging with it collapses our plane into the source by making the coordinates (cosθ, sinθ) to (0,0).

Mathematically, this is “Moksha” or complete liberation.

There are minor planes in between the major ones but, they are volatile and play the role of a transitory system from the lower to the higher planes.

**  The reader may get confused here but, in our figure 1, we have the outer shell as the lower plane as we were trying to understand the notion with the help of planer geometry (also geometry mainly deals with the material world) whereas, in this scenario, it’s about the proposed atomic structure where the outermost shell has the higher energy that represents the subtler aspect of our lives.

2 Comments Add yours

  1. saki says:

    this blog you have is absolutely a gem! thank you so much for our great explanations of our hindu scriptures! could you please make more posts on the existence of sri krishna from a scientific point of view?

    1. Deepayan says:

      I will definitely try! Thanks for your suggestion 🙂

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