Spiritual and cosmological applications of triangles

Anantha Anantha

Anand Bhavan, Kanjampatti P.O, Pollachi Via, Tamil nadu 642003, India

parinirvana.anantha@gmail.com

Abstract

It is well known that the laws of triangles are widely applied in classical mathematical branches,

physics, engineering and almost all the areas of science, technology and other fields.

Trigonometry is the extension of triangles. Cosmology is the application of trigonometry. The

Jews, Vedic Aryans, Islam and Taoism were all well versed with triangles and other geometrical

figures. In this article, the author attempts to unlock the hidden spiritual mysteries of triangles

and also about the part played by triangles in cosmology.

MSC :51 M04

Keywords: Euclidean Triangles, Spiritual Applications; Cosmological Consequences

Introduction

The Star of David, a symbol of Judaism, as a religion, and of the Jewish people as a whole. And

it also thought to be the shield (or at least the emblem on it) of King David.

Jewish lore links the symbol to the "Seal of Solomon", the magical signet ring used by King

Solomon to control demons and spirits. Jewish lore also links the symbol to a magic shield

owned by King David that protected him from enemies. Following Jewish emancipation after

the French revolution, Jewish communities chose the Star of David as their symbol. The star is

found on the Flag of Israel.

The six-pointed Star of David is a common symbol for both Judaism and Israel. Known in

Hebrew as a Magen David (shield of David), geometrically it is two triangles superimposed on

each other, forming the shape of a hexagram.

Though today the symbol popularly communicates Jewishness, its associations with Judaism are

newer than one might think. Some historians trace it to Jewish communities in the Middle Ages,

but these claims are neither fully substantiated nor widely accepted [ See figure 3 ]

Geometry is used throughout the Rig Veda. However, unlike modern geometry, the emphasis

was not on proofs but on usage. Furthermore, the rishis had knowledge of the concept of

precision, and this is evident from various descriptions of applications and use of geometry.

Further, the depth of precision was based on the requirements of the task or study being

described. For example, descriptions of constructions were limited in precision whereas

descriptions of astronomical phenomena were described in greater precision.

The following verse in the Rig Veda deals with the formation of the universe.

Who was the measurer prama? What was the model pratimâ? What were the building materials

for things offered nidânam ãjyam? What is the circumference (of this universe) paridhih? What

are the meters or harmonies behind the Universe chandah? What is the triangle

(yoke) praugam [which connects this universe to the source of driving force, the engine]?[2]

All these Sanskrit words (ie: prama) are geometrical terms which also occur later in the

Sulbasutra (where they are also defined).

Chariots are described in great detail in many different verses in the Rig Veda and Yajur Veda.

Dr. Kulkarni writes:

The proficiency in chariot building presupposes a good deal of knowledge of geometry... The

fixing of spokes of odd or even numbers require knowledge of dividing the area of the circle into

the desired numbers of small parts of equal area, by drawing diameters. This also presupposes

the knowledge of dividing a given angle into equal parts.

The Rig Veda is full of references to words in rituals whose definitions we find in subsequent

Brahmanas and in the Sulba Sutra to be pointing to geometrical figures. For example, three types

of fire altars, garhapatya, ahavaniya and dakshina are mentioned in the Rig Veda but defined in

the Shatapatha Brâhmana as being square, circular and semi circular, respectively, and also

having the same area. Considering that ritualistic fire altar designs were not changed over a

period of time and that the shastras were recited for several millennia before being written down,

the rishis must have had a method to calculate the square root of the number two and the value of

pi, without which they would not be able to determine whether the three altars had the same area

or not.[ http://www.hindupedia.com/en/Mathematics_of_the_Vedas]

The concept of infinity was also known during Vedic times. They were aware of the basic

mathematical properties of infinity and had several words for the concept-chief being ananta,

purnam, aditi, and asamkhyata. Asamkhyata is mentioned in the Yajur Veda[11], and the

Brihadaranyaka Upanishad as describing the number of mysteries of Indra as ananta[12]. These

two statements are elaborated in the opening lines of the Isha Upanishad (Shukla Yajur Veda).

This sholka is as much metaphysical as it is mathematical.

pûrnâsya pûrnamadaya pûrnamevâvasishyate

From infinity is born infinity.

When infinity is taken out of infinity,
only infinity is left over.
The concept of Shunya, or zero void, was originally conceived as the symbol of Brahman,

expressing the sum of all distinct forms. The symbol of zero and the decimal system of notation

is described in the Atharvaveda. it describes how the number increases by 10 by writing zero in

front of it. While there is no explicit mention of zero, it must have been common knowledge

based on how it is used.

In fact, the concept of shunya was not just mathematical or scientific, but is deeply rooted in all

branches of thought - especially metaphysics and cosmology. Shunya is the transition point

between opposites, it symboliss the real balance between divergent tendencies. Most ancient

mathematicians defined zero as the sum of two equal and opposite quantities. Zero produces all

figures, but is itself not limited to a certain value. Zero is the primary or final reservoir of all

single numbers. The symbol of zero and the decimal system of notation is described in

the Atharvaveda. It describes how the number increases by 10 by writing zero in front of it.

Sacred geometry is the basis through which the ―multiverse‖ — as the Ancient One references

— comes into being. Sacred geometry is the mathematical formulas that serve as the foundation

or the springboard upon and through which everything exists. Sacred geometry is the place

where mind and matter, the spiritual and the physical, the manifest and unmanifest, the bound

and boundless meet. If you can imagine an architect‘s blueprint you can capture a snapshot of

just how integral to life sacred geometry is. Just as the blueprint informs the layout, placement,

spacing and more of a building, so too does sacred geometry inform the layout, placement, and

spacing of Earth, the universes, and all of their inhabitants. Sacred geometry is found

everywhere, even your body! At the heart of everything is a mathematical code that inspires,

influences, and informs it and how it appears in physical reality. In short, sacred geometry is the
underlying and organizing principle of Creation.[ https://www.sagegoddess.com/what-is-sacred-

geometry/]

Sacred triangles are applied in Tantra. In this article, section1 deals with the tremendous

mysteries of equilateral triangles in Tantra and section 2 attempts by the applications of isosceles

triangles that the shape of our universe is flat.

SECTION 1

Applications of equilateral triangles in Tantra

Case 1

64 Yogini Vidya is an important mode of worship in Tantra. 64 equilateral triangles should be

inscribed either in a wooden plate or on a cotton piece of cloth. The performer should sit on this

and mediate by reciting the 64 yogini mantras. The practitioner will attain spiritual bliss and

ecstasy. The author reveals this hidden spiritual and triangle mystery for the benefit of

humankind.

See Figure 1

Describe an equilateral triangle ABC. [Elements I, prop.1]. Locate the mid points D. E and F of

sides AB, BC and CA respectively. Join DE, EF and FD.

In triangle ABC, all the sides are equal. Since an equal side subtends equal angles, angles ABC,

BCA and CAB are equal. i. e each of these angle is 60 degrees.

Now by side, angle side correspondence, triangles AED, BEF and CDF are congruent. So, the

base angles AED, ADE, BEF, BFE, CFD and CDF are equal. This shows us that teach of these

angles is equal to 60 degrees. So, sides DE, EF and FD are equal. Consequently we obtain that

triangles AED, BEF, CFD and EDF are all equilateral triangles. There are five equilateral

triangles in figure 1.We can describe a number of equilateral triangles in each triangle AED,

BEF, CDF and DEF.

In figure 1, O is the center of both equilateral triangle ABC and DEF. But it is not the center for

equilateral triangles AED, BEF and CDF. So, this construction is mathematically consistent but

spiritually incorrect. It is very very essential and important that the center/ brahmastan/bindu

should be the same for all 64 yogini triangles. So, such constructions are not acceptable. Some

so called spiritual experts foolishly argue that such equilateral triangles hold good for 64 yogini

triangles.

Let us switch over to the spiritually and mathematically correct description of equilateral

triangles.

See Figure 2

Draw an equilateral triangle ABC. [ Figure 2 ] Let the bisectors of angles ABC and ACB meet at

O. This is the center, brahmastan/bindu of equilateral triangle ABC. On BO and CO make BE

and CF such that BE = CF. Join EF. On CF describe an equilateral triangle DEF. Similarly,

inscribe an another equilateral triangle KLM. So, O is the center, brahmastan/bindu for all the

equilateral triangles ABC, DEF and KLM. This is the [figure 2] spiritually and mathematically

correct and consistent construction. By repeating this process we can construct 64 equilateral
triangles within the biggest equilateral triangle ABC. [Figure 2 ] In figure b2, each equilateral

triangle REPRESENTS a particular deity. There are 64 tantra and 64 yoginis.

The readers may find below the relevant 64 yogini mantras for worship:

Worshipping Kali and Her 64 Yoginis

Kali is in all forms. She is particularly present in the 64 Yoginis, all extensions of her Cosmic

Personality. This 64 Yogini Namavali wins the grace of Maha Kali and her ancient mother

aspects.

First published in 1990, these mantras come directly from Sathguru Babaji. Devotees of the

Mother and her 64 forms will be pleased to know that they have been posted extensively in

multiple sites throughout the internet. All are welcome to use these sacred sounds which will

elevate humanity.

For the absolving of sins, clarity of consciousness, a peaceful heart, massive protection, dynamic

aura, robust health, and the magnificent grace of abundance of every type, one should chant the

sacred names of the sixty-four Yoginis. We should never minimize their potential. They are not

minor deities, but the supreme divinity; none other than emanations of the one cosmic Prakriti

residing over every aspect of the relative plane. They can pull a soul from the lowest depths of

illusion into the most elevated heights of grace- light.

To perform this puja, simply prepare a copper yantra with sixty-four petals and chant the

following mantras with an open heart. If copper is not available, draw the yantra with India ink

using Bhoj Patra. The daily repetition of these mantras with purify all darkness, jealousy and

resentment within. Seek out direct spiritual guidance from a soul animated by the Divine. As the
Siddha Rama Devar said, ―Surrender to the young Shakti, and she will not forget you for the rest

of your life.‖

Kali and Her 64 Yogini Namavali

1. Om Kali Nitya Siddhamata Swaha

Mother of the Siddhas

2. Om Kapalini Nagalakshmi Swaha

Lakshmi of Naga

3. Om Kula Devi Svarnadeha Swaha

Of golden body

4. Om Kurukulla Rasanatha Swaha

Ruler of physical pleasure

5. Om Virodhini Vilasini Swaha

Residing within Self

6. Om Vipracitta Rakta Priya Swaha

Who loves passion

7. Om Ugra Rakta Bhoga Rupa Swaha

Enjoyer of passion in the form

8. Om Ugraprabha Sukranatha Swaha

Ruler of the seminal essence

9. Om Dipa Muktih Rakta Deha Swaha

Liberation through the body of passion

10. Om Nila Bhukti Rakta Sparsha Swaha

Who receives pleasure from the touch of passion

11. Om Ghana MahaJagadamba Swaha

Great Mother of the world

12. Om Balaka Kama Sevita Swaha

Attended by the God of Love

13. Om Matra Devi Atma Vidya Swaha

The Goddess with knowledge of Self

14. Om Mudra Poorna Rajatkripa Swaha

Who completely radiates compassion

15. Om Mita Tantra Kaula Diksha Swaha

Initiatress of the Tantric Kaula Path

16. Om Maha Kali Siddhesvari Swaha

Queen of the Siddhas

17. Om Kameshvari Sarvashakti Swaha

Shakti of All

18. Om Bhagamalini Tarini Swaha

Who delivers from calamity

19. Om Nityaklinna Tantraprita Swaha

Fond of Tantra

20. Om Bherunda Tatva Uttama Swaha

Essence of Sexual Fluid

21. Om Vahnivasini Sasini Swaha

Radiance of the moon

22. Om Mahavajreshvari Rakta Devi Swaha

Sensuous Goddess

23. Om Shivaduti Adi Shakti Swaha

Original Feminine Energy

24. Om Tvarita Urdvaretada Swaha

Giver of the Upward Ecstasy

25. Om Kulasundari Kamini Swaha

Desire Itself

26. Om Nitya Jnana Svarupini Swaha

The Great form of Wisdom

27. Om Nilapataka Siddhida Swaha

Giver of perfection

28. Om Vijaya Devi Vasuda Swaha

Goddess who is Giver of wealth

29. Om Sarvamangala Tantrada Swaha

Who Bestows Tantra

30. Om Jvalamalini Nagini Swaha

Snake Goddess

31. Om Chitra Devi Rakta Puja Swaha

Goddess who is worshipped with passion

32. Om Lalita Kanya Sukrada Swaha

Pure Maiden

33. Om Dakini Madasalini Swaha

Shining with rapture

34. Om Rakini Papa Rasini Swaha

Destroyer of sin

35. Om Lakini Sarvatantresi Swaha

Ruler of all tantras

36. Om Kakini Naganartaki Swaha

Who dances with Nagaraj

37. Om Sakini Mitrarupini Swaha

Friend

38. Om Hakini Manoharini Swaha

Mind Stealer

39. Om Tara Yoga Rakta Poorna Swaha

Who in union bestows complete passion

40. Om Shodashi Latika Devi Swaha

Creeper Goddess

41. Om Bhuvaneshwari Mantrini Swaha

Energy of all mantras

42. Om Chinamasta Yoni Vega Swaha

With dripping yoni

43. Om Bhairavi Satya Sukrini Swaha

Supreme Purity

44. Om Dhumavati Kundalini Swaha

Primordial energy of self

45. Om Bagla Muki Guru Moorthi Swaha

Form of the guru

46. Om Matangi Kanta Yuvati Swaha

Youthful beauty enhanced by Love

47. Om Kamala Sukla Samsthita Swaha

Residing in the Semen

48. Om Prakriti Brahmandri Devi Swaha

Goddess residing in the crown

49. Om Gayatri Nitya Chitrini Swaha

Eternal core of the energy of Self

50. Om Mohini Matta Yogini Swaha

Intoxicated

51. Om Saraswathi Svarga Devi Swaha

Goddess of Heaven

52. Om Annapoorni Shiva Samgi Swaha

With Shiva

53. Om Narasimhi Vamadevi Swaha

Beloved Goddess

54. Om Ganga Yoni Svarupini Swaha

Energy of the Yoni

55. Om Aprajita Samaptida Swaha

Who Bestows Orgasm

56. Om Camunda Parianganatha Swaha

Ruler of sacred erection

57. Om Varahi Satya Ekakini Swaha

Oneness of Truth

58. Om Kaumari Kriya Shaktini Swaha

Energy of dedicated action

59. Om Indrani Mukti Niyantri Swaha

Guide to bliss

60. Om Brahmani Ananda Moorthi Swaha

Image of bliss

61. Om Vaishnavi Satya Rupini Swaha

Form of truth

62. Om Mahesvari Para Shakti Swaha

Transcendental energy

63. Om Lakshmi Monoramayoni Swaha

With beautiful yoni

64. Om Durga Satchitananda Swaha

Truth consciousness and bliss

The Sanskrit sacred mantras Om and Aum are identical. But Tantra asserts that males

should recite Om whereas females MUST avoid Om recitation and SHOULD recite Aum.

Case 2

Look at figure 3

In Christianity the word Amen is very important. Every discourse ends with Amen. In Hindu

philosophy, Om and its equivalent word Aum are sacred. It is stated in Tantra that the TANTRIC

sexual union of male and female is identical with Universal consciousness. The genitals of males

and females are SACRED SPACES in Tantra. The Tantric Monks had formulated a RULE to

experience spiritual bliss/enlightenment and universal consciousness and Oneness for the

partners. The rule is that the couples must perform by sitting or laying on a cot/ bed/cotton cloth

with the sacred equilateral triangles as shown in figure 3. In the sexual equilateral triangles ABC

and DEF O is the center/ brahmastan/bindu. So , if a couple enjoy sex on the bed which

possesses the sacred sexual equilateral triangles as indicated in the figure 3, certainly they will

get climax and orgasm and attain spiritual bliss and universal consciousness and Oneness with

the Almighty.

Before coitus, the couples are requested to meditate at least for five minutes.

The Sanskrit sacred mantras Om and Aum are identical. But Tantra asserts that males

should recite Om whereas females MUST avoid Om recitation and SHOULD recite Aum

[https://www.yoginiashram.com/namavali-of-kalis-64-yoginis/]
SECTION 2

Applications of isosceles triangles in cosmology

According to Freedman equations, the geometry of the universe has three possibilities namely

open, closed and flat. Closed universe obeys the properties of spherical geometry, open universe

has the properties of hyperbolic geometry and flat universe agrees with the concepts of classical

Euclidean geometry. Recent measurements of WMAP and Planck predict that to a margin of

0.02% error, our universe is flat. But there is no mathematical formulation for the shape of our

universe. In this work, the author attempts to show seven mathematical methods for the fate of

our universe. The author‘s findings are just some attempts. The author does not seek any TOP

claims. But the author sincerely believes that these studies may be a small clue for further probes

and research.

Case 1 Look at figure 4 ( Euclidean

Let m, x, y and z denote the sum of the interior angles of cosmic triangles ADE, AFG, AHJ and

ABC respectively. In another words,

ADE = x [I]

AFG = x [II]

AHJ = y [III]

ABC = m [IV]
Results

Let us assume that, xy = a (1)

yz = b (2)

zm = c (3)

Dividing (1) by (2), x / z = a / b (4)

Dividing (2) by (3), y/ m = b/ c (5)

Dividing (6) by (7), xm/ yz = a/ c (6)

Cross multiplying, xmc = yza (7)

Applying (1) in RHS, xmc = yz.xy

i.e mc = y2z

Assuming (3) in LHS, m.zm = y2z

i.e m =y (8)

Comparing [I], [IV] and (8) we obtain that the sum of the interior angles of triangles ABC and

AHJ are equal.

In cosmic triangle ABC the latent sides are equal. And by construction the latent sides AH and

AJ in triangle AHJ are equal.

The summit angle BAC is common to these two triangles.

Consequently we obtain that cosmic triangles ABC and AHJ are similar. (9)
Case 2 Look at figure 4 ( Euclidean)

Let e, g, s and w denote the sum of the interior angles of cosmic triangles ADE, AFG, AHJ and

ABC respectively. In another words,

ADE = e [I]

AFG = g [II]

AHJ = s [III]

ABC = w[IV]

Results

Applying the laws of tensor algebra, we can formulate the following tensors:

Aeg (10)

Bws (11)

Cse (12)

Dws (13)

Eeg (14)

Fgw (15)

(10).(11).(13).(14).= Aeg BwsDws Eeg = Unity

(10).(12).(13).(15).= AegCse Dws Fgw = Unity

Equating the above two relations, (10).(11).(13).(14).= (10).(12).(13).(15).

i.e. BwsEeg = CseFgw

Mwseg = Nsegw (16)

According to the law of equality of tensors, if two tensors of same rank and same type are equal,

then their components are one to one equal, using this law in (16) we get the following relations:

w = s; s = e; e = g; g = w`

From the above relations we obtain that e = g = s = w

From the above relation we obtain , e = g = w=s. (17)

From (17) and from we get that the cosmic triangles ADE, AFG, AHJ and ABC are similar.

(18)

Case 3 Look at figure 4 ( Euclidean)

Let u, x, y and z denote the sum of the interior angles of cosmic triangles ADE, AFG, AHJ and

ABC respectively. In another words,

ADE = v [I]

AFG = x [II]

AHJ = y [III]

ABC = z [ IV]
Results:

The angles BDE and DEC are straight angles and so their measures are equal to180 degrees.

Let V be the value of this 180 degree (1) (20)

Using (1) , X + Y = V + A (2) (21)

i.e. X = {V + A– Y } (2a) (21a)

X + B‘ = A‘ + Z = V + C ‗ (3) (22)

I.e. A = {X + B – Z } (3a) (22a)

Also, A = {V + C – Z } (3b) (22b)

X + Y + Z = 2V + C (4) (23)

I.e. X = {2 V + C – Y – Z} (4a) (23a)

Let us assume that eqns. (21a), (22a), (22b) and (23aa) denote sets.

Intersection of the sets A and B, denoted by A ∩ B, is the set of all objects that are

members of both A and B. The intersection of {1, 2, 3} and {2, 3, 4}

is the set {2, 3} .

Let us assume that equations (21a) , (22b) and (23a) are sets.

Considering equations (21a) a n d (22b) and applying the intersection law of

set theory we obtain that, X ∩ A‘ = { V } (24)

Taking equations (3b) and (4a) and assuming interjection law of

set theory we have that, X ∩ A‘ = { - Z } (25)

Comparing (5) and (6) we get that V = - Z (26)

It is well known that in geometry minus theta refers to the vertically opposite angles.

Since vertically opposite angles are equal, (7) implies that V = Z (27)

A brief analysis of {I], {IV] and (27) clearly indicates that cosmic triangles

ABC and ADE are similar. (28)

Discussion

From the equations (9) of case ONE, (18) of case TWO, and (28 ) of case THREE we obtain

that the cosmic triangles we have studied are similar. The similar triangles are possible only in

classical Euclidean geometry. So, in brief we may conclude that the geometry of our universe is

[ 1-8 ]
flat.

References

[1] Sachs,R. K. and Wolfe,A. M. Perturbations of a cosmological model and angular

variations of the microwave background, Astrophys. J.147, 73–90 ( 1967), 73–90

[2] Weinberg, S., Gravitation and Cosmology ,Wiley & Sons ( New York, 1972)

[3] Hu,W., Sugiyama,N. and Silk,J. The physics of cosmic microwave background
anisotropies. Nature 386 ( 1997) 37–43

[4] Bond,J. R., Efstathiou,G. & Tegmark, M. Forecasting cosmic parameter errors from
microwave background anisotropy experiments. Mon Not. R. Astron. Soc. 291 ( 1997) , L33–
L41
[5] Hinshaw,G. et al. Band power spectra in the COBE-DMR four-year anisotropy
map. Astrophys. J. 464,(1996 ) L17– L20

[6] Scott,P. F. et al. Measurement of structure in the cosmic background radiation with the
Cambridge cosmic anisotropy telescope. Astrophys. J. 461( 1996) L1–L4

[7] Netterfield,C. B. et al. A measurement of the angular power spectrum of the anisotropy in
the cosmic microwave background.Astrophys. J. 474 (1997 ) 47–66

[8] Schmidt,B. P. et al. The high-Z supernova search: measuring cosmic deceleration and
global curvature of the Universe using type Ia supernovae. Astrophys. J. 507( 1998 ) 46–63

Figure 4 ( Euclidean ).