Monday, October 22, 2012

Pi and Sanathana Dharma

Pi and Sanathana Dharma

The following was an interesting aspect in an interview which I read after one of my brother took the pains to tell me to read it , otherwise I have an healthy contempt not to touch Chennai Times as it is full of filmy gossip and glossy photos of actresses with perky breasts and piercing thighs, not necessarily the ectomorphic variety of the west but the bulky bosomed one of the subcontinent. However, rarely such interviews do pop up as the one from Yann Martel (Canadian Novelist), a great admirer of Hinduism and India. He said the following in his interview, “In mathematics, Pi is an irrational number which means it is number that goes on forever. Yet it's constant in science, so we use this irrational number to get to the rational understanding. It's that contradiction that I liked. To me religion is like that - an irrational number, it doesn't make sense on its own, but it helps make sense of the world.”

After I read this I was not only savoring his comparison but I was also thinking that Indians, especially the Sanathana Dharmic practioners, who have always had an unrestrained foray into many domains , especially in their highly logical, scientific , ritualistic [probably due to the inherent contradiction of life and inevitable necessity to be highly demographically inclusive], mathematical and musical aspects of spiritual and cultural expressions must have had something definitely, I was convinced more so because recently I read a wonderful book called ‘THE ADVENTURES IN NUMBERLAND ‘ by a French mathematician wherein he after doing extensive research into the evolution of numbers and all aspects of mathematics writes in that book a lot about the advanced concepts in mathematics and their multiple usages that Indian had in practical application far ahead of Westerners even coming to decipher them, including Fibonacci numbers and Pi . I am not mentioning these as the outpouring of an ethnocentric fanatic soul but as statement of facts.

I also realized that in our over enthusiasm we must not "mathmaticize" Religion, in case of Hindus the Sanatan Dharma. I am also aware that there is this big difference between Pi and Religion in the "progression". In case of Pi it is fixed repeated numbers, whereas Religion, especially Dharma as we know, does not have any "dimension" or can be "quantified". Another difference, from Mr. Martel's philosophy if I am allowed to digress, is that Dharma includes "unknowns" beyond this world (also known as Earth) and our solar Universe, or to be more precise Universes.

Not everything is unreasonable as far as similarity between these two [ 2] topics. For example, mathematics provide a path to expressions & queries from creative and ever-active Human minds, whereas such mental engines are designed & developed by super-Power that also gave us all the Dharma. Not only highly evolved concepts of Mathematics did exist since or even before the time of Ramayana, and surely indications are there in Holy Gita, but there were many practical and superb applications of mathematical concepts in many of the spiritual and cultural expressions of practioners of Sanathana Dharma.

Well as serendipity has this positive prejudice of popping up from unexpected areas to encourage the enquiring mind, enhance and enrich the understanding and enlighten further, it did pop up when I was explaining to someone the melakartha ragas system and then I was lead to these information taken from many sources but vindicating that we Indians had far greater applications of Pi in most of spiritual enquiries and cultural arena.

Katapayadi_system

I have the book by Ramon Campayo Maximize your Memory wherein he uses something similar

The "ka-Ta-pa-ya" scheme
and its application to
mELAkarta raagas of Carnatic music

The ka-Ta-pa-ya scheme:

The "ka-Ta-pa-ya" rule used by ancient Indian mathematicians and grammarians

is a tool to map names to numbers. Writing the consonants of the Sanskrit

alphabet as four groups with "ka, Ta, pa, ya" as the beginning letters of

the groups we get

1 2 3 4 5 6 7 8 9 0

ka kha ga gha ~ma cha Cha ja jha ~na

Ta Tha Da Dha Na ta tha da dha na

pa pha ba bha ma

ya ra la va Sa sha sa ha

Now, each letter of the group is numbered from 1 through 9 and 0 for the tenth

letter. Thus, ka is 1, sa is 7, ma is 5, na is 0 and so on. So to indicate

the number 356 for example one would try and come up with a word involving

the third, fifth and sixth letters of the groups like "gaNitam" or "lESaca".

However, in the Indian tradition, the digits of a number are written left to

right in the increasing order of their place value - exactly opposite the way

we are used to writing in the western way. Therefore 356 would be indicated

using letters in the 6th, 5th, and 3rd positions of the group e.g. "triSUlaM".

There apparently were upto 4 flavors of this scheme in use in ancient

India. These differ in how to interpret the conjoint consonant. The popular

scheme was to use only the last consonant. And any consonant not attached

to a vowel is to be disregarded. These rules should be used while decoding

a phrase in "katapayadi" scheme.

The following phrase found in "sadratnamAla" a treatise on astronomy,

bhadram budhi siddha janma gaNita Sraddha@h mayadbhUpagi@h

when decoded yields

4 2 3 9 7 8 5 3 5 6 2 9 5 1 4 1 3

which when reversed gives

3 1 4 1 5 9 2 6 5 3 5 8 7 9 3 2 4

which is readily recognised as the digits in "pi" (except that the 17th

digit is wrong - it should be 3) :-)!

(source: The article "The Katapayadi Formula and Modern Hashing Technique"

by Anand V Raman, appearing in "Computing Science in Ancient India", edited

by T.R.N.Rao and Subhash Kak, published by the Center for Advanced Computer

Studies, University of South Western Louisiana, Louisiana LA 70504)

mELakarta raagas:

The raagas of Carnatic music are said to be derived from a definite set of

72 ragas known as mELakarta or janaka or sampoorNa raagas.

Before we go any further let me remind you that in an octave there are 12

tones each separated by a half note. These 12 tones are named as the 7 notes

and variations on some of those 7 notes. These names and their western music

equivalents are as follows:

1. shaDjamam S Doh C

2. Suddha rishabham R1 C# or Db

(read as C-sharp or D-flat)

3. chatuSruti rishabham R2

Suddha gAndhAram G1 Re D

4. shaTSruti rishabham R3

sAdhAraNa gAndhAram G2 D# or Eb

5. antara gAndhAram G3 Me E

6. Suddha madhyamam M1 Fa F

7. prati madhyamam M2 F# or Gb

8. pancamam P Sol G

9. Suddha dhaivatam D1 G# or Ab

10. chatuSruti dhaivatam D2

Suddha nishaadam N1 La A

11. shaTSruti dhaivatam D3

kaiSika nishaadam N2 A# or Bb

12. kAkali nishaadam N3 Ti B

The properties of the janaka raagas are :

a) they contain all the 7 notes of an octave (hence the name saMpoorNa)

exactly once in the scale.

b) the tones of the notes must all be in ascending order in the

aarOhaNa. i.e You cannot pick S, R3, G1 ... because the tone of R3

is higher than the tone of G1. Also the ArOhaNa/avarOhaNa cannot have

jumps back and forth like S, G3, R1, ... or S, N2, P, D1 ... etc..

c) the avarOhaNa should contain the same notes as ArOhaNa in the reverse

order.

Given these properties/rules, we can easily surmise that there cannot be

more than 72 sampoorNa raagas. Because of the need for ascending order of

tones the permissible combinations of R,G and D,N are limited to 6 each, viz.

R1G1, R1G2, R1G3, R2G2, R2G3, R3G3 and D1N1, D1N2, D1N3, D2N2, D2N3, D3N3.

There are two varieties of M viz. M1, M2. So the number of possible different

sampoorNa raagas is 6 x 6 x 2 = 72. So if these 72 are arranged in a regular

order, we can figure out the scale of a janaka raaga if its number in the

list is given. Now, from our "kaTapayaadi" scheme if we can name the raaga

in such a way that the name yields the number, then we have further reduced

the memorising!

Application of "kaTapayaadi" to mELakarta raagas

That is exactly what venkaTamakhi of the 18th century is purported to have done.

He applied the "kaTapaya" scheme to name the janaka raagas to fit their place

in the mELakarta list. Some of these already had suitable names and some had

unsuitable names that were in common use. He changed those names a little to

fit the naming scheme. Thus "kalyANi" becomes "mEcha kalyANi","SankarAbharaNam"

becomes "dheera SankarAbharaNam" etc.

Under this naming scheme, the number of a janaka raaga is obtained by decoding

the first two letters using the "kaTapaya" scheme. For the naming scheme used

for the mELakarta raagas, apart from the decoding rules mentioned above for,

conjoint consonants, in case one of the consonants is from the 'ya' group,

the first consonant is to be considered instead of the last. And finally,

to get back to our familiar western notation, reverse the decoded digits.

For example:

"kharaharapriya" : kha = 2 and ra = 2 i.e 22 reversing the digits : 22

"shaNmukhapriya" : sha = 6 and mu = 5 i.e 65 reversing the digits : 56

"naThabhairavi" : na = 0 and Tha = 2 i.e 02 reversing the digits : 20

"divyamaNi" : di = 8 and va = 4 i.e 84 reversing the digits : 48

Once you get the number, figuring out the notes is easy. The 72 raagas are

arranged such that the first 36 raagas contain M1 and the next 36 contain M2.

In each half, the various possible combinations of R,G and D,N

occur cyclically with the R,Gs varying slower than the D,Ns.

i.e: for the first six raagas

R1G1 occurs with each of D1N1, D1N2, D1N3, D2N2, D2N3, D3N3

for the next six raagas

R1G2 occurs with each of D1N1, D1N2, D1N3, D2N2, D2N3, D3N3

and so on.

So given a janaya raaga number you perform the following calculation:

1. if NUM is from 1-36, raaga has M1, from 37-72 raaga has M2.

2. if NUM is greater than 36 subtract 36 from it.

3. divide NUM by 6;

a. if remainder=0

i. the sixth D,N combination occurs.

ii. the quotient gives which of the R,G combinations occurs.

b. if remainder is not zero

i. the remainder gives which of the D,N combinations occurs.

ii. the quotient+1 gives which of the R,G combinations occurs.

Taking the example of "shaNmukhapriya":

From the "kaTapaya" rule its number is 56.

56 is greater than 36. So M2 occurs.

56-36=20.

20 divided by 6 : quotient=3, remainder=2

so 3+1=4th RG combination : R2G2 occurs.

and 2nd DN combination : D1N2 occurs.

So shaNmukhapriya has the notes:

S R2 G2 M2 P D1 N2 S

Another example : "varuNapriya"

From the "kaTapaya" rule its number is 24.

24 is less than 36. So M1 occurs.

24 divided by 6 : quotient=4, remainder=0

so 4th R,G combination: R2G2 occurs.

and 6th D,N combination: D3N3 occurs.

thus varuNapriya has the notes:

S R2 G2 M1 P D3 N3 S

RAAMA AND PANDAVAS- A LESSON IN HUMILITY

Management lessons from Ramayana & Mahabharata
It is the order of the day to look for ‘management lessons’ in any thing , be it “Bhagawath Geetha’ , “Ramayana’ or ‘Maha Bharata’. An attempt here to take a few tips for management from ‘Rama

yana’ and ‘Maha Bharata’. View, enjoy and learning part, we leave it to you!

In the Great Epics of India, Ramayana and Mahabharata, war ends not with celebration of victory but with transmission of knowledge. In the Ramayana, Ravana lies mortally wounded on the battlefield, when Rama turns to his brother Lakshmana, and says, “While Ravana was a brute, he was also a great scholar. Go to him quickly and request him to share whatever knowledge he can.” The obedient Lakshmana rushes to Ravana’s side and whispers in his ears, “Demon King, all your life you have taken not given. Now the noble Rama gives you an opportunity to mend your ways. Share your vast wisdom. Do not let it die with you. For that you will be blessed.” Ravana responds by simply turning away. An angry Lakshmana goes back to Rama and says: “He is as arrogant as he always was, too proud to share anything.” Rama looks at his brother and asks him softly, “Where did you stand while asking him for knowledge?” “Next to his head so that I hear what he had to say clearly” Rama smiles, places his bow on the ground and walks to where Ravana lies. Lakshmana watches in astonishment as his brother kneels at Ravana’s feet. With palms joined, with extreme humility, Rama says, “Lord of Lanka, you abducted my wife, a terrible crime for which I have been forced to punish you. Now, you are no more my enemy. I see you now as you are known across the world, as the wise son of Rishi Vishrava. I bow to you and request you to share your wisdom with me. Please do that for if you die without doing so, all your wisdom will be lost forever to the world.” To Lakshmana’s surprise, Ravana opens his eyes and raises his arms to salute Rama, “If only I had more time as your teacher than as your enemy. Standing at my feet as a student should, you are a worthy recipient of my knowledge. I have very little time so I cannot share much but let me tell you one important lesson I have learnt in my life. Things that are bad for you seduce you easily; you run towards them impatiently. But things that are good for you, fail to attract you. You shun them creatively, finding excuses to justify your procrastination. That is why I was impatient to abduct Sita but procrastinated in meeting you. This is the wisdom of my life, dear Rama. My last words I give to you.” With these words, Ravana dies.

There’s similar knowledge transmission after the Mahabharata war gets over. The Kauravas are all dead. As the victorious Pandavas are about to assume control of Hastinapura, Lord Krishna advises them to talk to Bhisma Pitamaha, their grand uncle, who lies mortally wounded on the battlefield. As a result of a divine blessing, death would elude him for some more time. “Make him talk until his last breath. Ask him questions. He has a lot to tell,” says Krishna. Sure enough, when prompted, the dying Bhisma spends hours discussing various topics: history, geography, politics, economics, management, war, ethics, morality, astronomy, metaphysics and spirituality. Bhisma’s discourse is detailed in the Shanti Parva (discussions on Peace) and Anushasana Parva (discussions on Discipline) that makes up a quarter of the Mahabharata. After listening to their grand uncle, the Pandavas have a better understanding of the world, and this makes them better rulers.

Rama asked Ravana for his wisdom before his death. The Pandavas listened to a lengthy discourse from Bhisma as he lay dying on the battlefield. In the context of commercial organizations, this is Knowledge Management. Both these stories draw attention to the value of knowledge. In triumph, it is easy to claim material possessions of the defeated, but it is not easy to claim their knowledge. Knowledge does not outlive death. It is lost forever with the death of the knowledgeable. Every day, people leave organizations, taking their knowledge with them – knowledge which they acquired because they were part of the organization. They take with them knowledge of clients, markets, business processes, tricks of trade, etc. These may not be confidential information or patented information, but it is information that gives an organization its cutting edge. Over the past decade, a whole new business process known as Knowledge Management has evolved that seeks to harness, store, transmit this knowledge. Every CEO agrees that it is a valuable business process, that investment in it is critical. Policies have been made, people have been hired and systems have been deployed. Unfortunately, for all the initial enthusiasm, implementation has been lacking. Often because they are like Sahadeva, the youngest Pandava. In the Mahabharata, he is described as an expert in many predictive sciences such as astrology, palmistry and face reading. But he is also cursed in a manner that if he ever gave information voluntarily, his head will split and he would die instantaneously. That is why he is silent throughout the epic. He knows every fortune and misfortune that his family will go through, but he can never use his knowledge to forewarn them. When Dharmaraja Yudhishtira finally learns of his brother’s prowess he is furious: “Why did you not tell me all that you knew?” All he gets in response is Sahadeva’s silence. Most employees in an organization are Sahadevas (by choice). They are dumb when it comes to sharing knowledge. Knowledge Management is leadership driven. Only a Rama can do it, not a Laskhman . He must first believe in it. We must respect the fact that everyone in the organization, even those who we do not particularly like, are repositories of great wisdom – not only knowledge of things that work but also knowledge of things that do not work. We must make conscious efforts to capture as much of it as possible. Take a step back. Check if you are creatively shunning this rather tedious matter of Knowledge Management. If you are, then remember that Ramayan and Mahabharata were written at a time when civilization had just set in, probably six or seven thousand years ago and even then our saints and sages knew what will take India forward: Knowledge Management. It will be a pity if we do not take advantage of lessons imparted by these immortal epics. The advanced western world makes full use of these lessons, unfortunately we still don’t. We consider Ramayana and Mahabharata as religious books, which they are NOT just religious.

KOLU [GOLU] IN NAVRATHRI IN INDIA AND HINSHI-MOCHI IN JAPAN

While most of the Hindu world is observing the nine-day Navaratri festival, in many South Indian homes there is also another joyous celebration going on. Known as kolu (pronounced as golu), in this festival the family constructs a seven, nine, or eleven tier step-structure and covers it with a nice cloth. Then on the steps are arranged – often in a particular order – every colorful doll and figurine in the house. By tradition each year one may add a new doll or set.

Auspicious decorations (kólam) adorn the arena where a traditional oil lamp is lit in the evenings. Invited guests gather and sit at the doll-filled steps after admiring them.

An anointed pitcher with a coconut on top is usually placed on the first step, and there is an idol of Ganesha on the highest step. Many of the dolls on the other steps are symbolic of the various avataras of the tradition. In the various other steps there can be dolls and figurines from various cultures, with or without religious significance. There could be a set displaying a wedding ceremony, another set consisting of an elderly couple, yet another with a cart and driver, and so on. It is also common to have traditional doll-pair made with black wood (marappácchi bommais), and nicely clothed in colorful costumes. Some of the dolls come from previous generations: from mother and mothers-in-law.

Though men and women participate in the rejoicings, the golu festival is only in homes with daughters, and is generally hereditary. That is, only families with the golu-tradition continue it.

Given that in a community many homes may set up golu, invitations are often time and day specified. So there is an open house during those days and times during which streams of guests come and go. Some of them sing devotional a song or two, sitting at the golu-steps. Every festival in the Hindu world is colorful and joyous, each associated with special snacks. So during the golu days too there is a treat consisting of some snacks: in some cases this may become sumptuous, and invariably most of it is made of protean rich pulses and cereals and therefore highly carminative resulting in excess biogas in the atmosphere, sometimes they even border on a dinner. When the women guests take leave, they are given a special parting gift.

Golu is one of the most beautiful, aesthetically pleasing and religiously least myth-laden festivities of the tradition. It brings the community together in a very unusual way. It is always a pleasure to visit a golu-home.

It may be mentioned in passing that in Japan there is a somewhat similar one-day festival of dolls known as Hina Matsuri. It is also focused on girls. One of its goals is to ensure a happy and fruitful life for the daughters in the family. On this day, it is customary to drink some sweet sake and a special type of sushi, called chirashi. One also makes a special cake with rice in a geometrical form, called hinshi-mochi, for day, consisting of a few colored layers. Two of these are also placed with the dolls.

As with golu, dolls are displayed in special arrangements, often in different tiers. Since this occurs in the season of peach flowers (March), these are used for ornamentation. One also refers to this festival of Momo-no-Sekko (Peach blossom festival). In the Japanese tradition, peach blossoms represent auspiciousness. They are also regarded as representing the qualities of gentleness, peace, and equanimity. The dolls are nicely left in place for all to appreciate for a whole month.

It is remarkable that two traditions (Tamil and Japanese), quite different in many ways, have festivals which have such parallels.

Tuesday, October 16, 2012

MAN IS...

Yajna -A Scientific Interpretation of Yajna

A Scientific Interpretation of Yajna

Forwarded message from Dwija

[Date - Tuesday Jan 23, 2007]

Scientific Interpretation Of Yajna

Literally Yajna means service and sacrifice and a way of living in
society in harmony. It promotes as well as protects the higher human
values of living style in the society. In other words, it can be said
that it is a real human culture, which is helpful in curing the human
diseases as well as in purifying the atmosphere and environment too.

It has been observed that there are two basic energy systems in the
physical world, which are known as 'heat' and 'sound'. In course of
performing Yajna, these two energies, i.e. 'heat' from Yajna (Fire-
Worship) and 'sound' of Mantras (Gayatri and a like other Mantras)
are combined simultaneously to achieve the desired physical,
psychological and spiritual benefits. It is applicable to quote here
that Dr. Howard Steingull, an American scientist, has established
that recitation of Gayatri Mantra produces 110,000 sound waves per
second. In fact, recitation of Gayatri Mantra during fire worship
(Yajna) act as complementary to each other.

Russian Scientist, Mr. Shirowich has expressed his assertion in these
words:- "After successful experiments, we have collected the
following data about the cow and yajna, which even the people of
India do not know: Cow milk contains the greatest power of protection
from atomic radiation, houses having cow dung covered floors enjoy
complete protection from atomic radiation, if cows ghee is put into
fire, its smoke will lesson the effect of atomic radiation to a great
extent. This process is known as Yajna in Indian languages?

Another Scientist, Dr. Hoffkins, has made his remarks as follows:
"Mixing ghee and sugar and burning theme creates smoke which kills
the germs of certain diseases and secretion takes place from some
glands relates to the wind-pipe which fills our heart and mind with
pleasure.

French Scientist Prof. Tilwans has expressed his remarks, "Burning
sugar and its smoke have great power to purify the atmosphere. It
kills the germs of T. B., Measles, Smallpox and cowpox.?

It is to be noted in this connection that the traditional systems of
treatment of physical diseases employ medicines, which are mostly
administered orally, which produce effects only after they are
digested and absorbed into the systems. Major part of medicines taken
orally is left unutilized by the digestive system. Oral medicines
thus tax the digestive system and sometimes they upset digestion
seriously. The same is more or less true to medicines directly
injected into the blood, which produce results quickly, but their
adverse and side effects are mostly observed and felt. While
corpuscles of the blood resent intrusion of any foreign bodies into
the blood, and sometimes the reaction of the system to the sudden,
massive and direct intrusion of foreign matter into the blood through
injection is most serious and even fatal too.

In a Yajna, medicinal herbs are vapourized by offering them into the
sacrificial fire, which enter into the human system in a gaseous form
through the nose, the lungs and pores of the skin. This might he
proved to be easiest, least taxing, least risky and most effective
method of administering a medicine to reach every single cell of the
body.

The possibilities of curing mental diseases by Yajna are even more
alluring. Diagnosis and treatment of mental diseases is still about
in its infancy stage in the modern systems of treatment. They are
neither well-established diagnostic aids nor any recognized system of
treatment of diseases like Neurosis, Psychosis, Schizophrenia,
Depression, Tension, Melancholia, Mania, Hysteria, etc. On the other
hand, psychosomatic diseases are even more rampant than physical
diseases and they harm the human beings. Manifestation of
psychosomatic diseases except in extreme and advanced cases is not so
apparent and that may explain why enough attention has not been paid
to them. The stress and strain of modern life, degradation of social
behaviour and fall of moral values all around, are contributing to an
alarming increase in psychosomatic diseases. Yajna, might offer a
solution to this serious problem as well.

The techniques of Yajna for the treatment of physical and mental
diseases do not comprise mere vapourization of the medicines or herbs
to be employed, but various Samidhas (special kind of wood piece) are
offered into the sacrificial fire of Yajna to create desired effects.
Yajnopathy may take an honoured as well as leading place with the
other therapies of the world like - Allopathy, Homeopathy,
Chromopathv, Naturopathy, etc. to cure the various ailment and
affliction of mankind including physical and mental diseases.
Yajnopathv may come into being and may be ranked as a unique
achievement of the modern age.

The Brahmavarichas Research Institute, a research unit founded in
1979 at Hardwar, India, by Gayatri Pariwar thought it needed that
science and spirituality should come together and collaborate on the
plane of matter and in the realm of consciousness to enhance the
dignity of the individual and promote a new egalitarian social order.
The efforts with some experiments were made by that institute to
bring science and spirituality, the two super forces, together in
order to work out the stems, which would enable them to complement
each other. The main task undertaken was to present scriptural
writings and suppositions in relation to the scientific disciplines
of today, to study the effects of spiritual practices on human body
in laboratory and to project out an alternative system of medicine
utilizing totally harmless herbal plants after comprehensive and
comparative scientific studies on human subjects.

The investigations, experiments and research, carried out in that
institute (Brahmavarichas, ) were, in fact, strenuous efforts
precisely in that direction, but it is a matter of regret that due to
the internal mishandling that could not stay on continuation and
concluding final or advanced results could not come out.

In the present context, it is expected on the basis of seen
happenings as well as observations around that in absence of
spiritual discipline it is not possible to promote social morality
and individual codes of conduct amongst the people. In the absence of
these qualities of social and individual restraints, human cleverness
and newly mobilized resources may play havoc with social order of
civilization.

It is an established truth that spiritual principles, if adopted in
day to day life, can bring about a change not only in the body
physiology hut also in the thinking and ultimately in the personality
as a whole. Yajna, the process of fumigation with medicinal herbs
under specific procedures influences widely the persons participating
in that as well as the surrounding environment. The ingredients used
are the medicinal plants, which have been using since the Vedic
times, are known to lay beneficial effects on human being.

End of forwarded message from Dwija

GITA-A Meta Shloka of Gita Verses

A Meta Shloka of Gita Verses

Shri R. Visvanatha Sastri (Prof. VK's father) has written a shloka, which refers to 18 verses of Gita. Read more about it here. The shloka itself is this

OmahaM prakR^itiM nainam yogabhaktyA mayA yataH /
amAnitvaM sukhaM kAryaM IshvaraH purushaH sa yat.h // * //

The following is a translation of the 18 verses (source: Swami Gambhirananda), with the five verses 13.(7-11) included.

omityekaaksharaM brahma vyaaharanmaamanusmaran.h .
yaH prayaati tyajandehaM sa yaati paramaa.n gatim.h .. 8.13..

He who departs by leaving the body while uttering the single syllable, viz Om, which is Brahman, and thinking of Me, he attains the supreme Goal. [Emphasis on final thoughts. The implied meaning is the emphasis on thoughts in the entire life.] [In the entire Gita, this is the only shloka that begins with Om.]
ahaM sarvasya prabhavo mattaH sarvaM pravartate .
iti matvaa bhajante maaM budhaa bhaavasamanvitaaH .. 10.8..

I am the origin of all; everything moves on owing to Me. Realizing thus, the wise ones, filled with fervour, adore Me. [The wise know that things happen due to Him and Prakriti and hence adore him, as they see the beauty of everything as it is.] [The verses 9.16 (and hence the series 9.(16-19)), 10.20 and 15.14 also begin with aham. The verses 9.24, 16.18 and 18.53 also begin with aham, though it is less likely that they are being referred to here.]
prakR^itiM purushha.n chaiva vidyanaadi ubhaavapi .
vikaaraa.nshcha guNaa.nshchaiva viddhi prakR^itisaMbhavaan.h ..
13.20..

Know both Nature and also the individual soul to be verily without beginning; know the modifications as also the qualities as born of Nature. [Both of them are ~~timeless~~ without beginning.] [Though 9.8 and 13.1 both begin with the same word, it is less likely that this word refers to either of them.]
naina.n chhindanti shastraaNi naina.n dahati paavakaH .
na chaina.n kledayantyaapo na shoshhayati maarutaH .. 2.23..

Weapons do not cut It, fire does not burn It, water does not moisten It, and air does not dry It. [Though it is more subtle than space, Atman can be thought of as space, with space being one of the five elements. Just as the elements cannot interfere in each other, and hence the other four elements cannot interfere in the most subtle space, they cannot interfere in the Atman.]
yogayukto vishuddhaatmaa vijitaatmaa jitendriyaH .
sarvabhuutaatmabhuutaatmat kurvannapi na lipyate .. 5.7..

Endowed with yoga, pure in mind, controlled in body, a conqueror of the organs, the Self of the selves of all beings-he does not become tainted even while performing actions. [Even by doing, he is not contaminated.] [2.48 is a reasonable contender for this. What about 4.41?]
bhaktyaa maamabhijaanaati yaavaanyashchaasmi tattvataH .
tato maaM tattvato GYaatvaa vishate tadana.ntaram.h .. 18.55..

Through devotion he knows Me in reality, as to what and who I am. Then, having known Me in truth, he enters (into Me) immediately after that (Knowledge). [11.54 is a serious contender for this.]
mayaa tatamidaM sarva.n jagadavyaktamuurtinaa .
matsthaani sarvabhuutaani na chaahaM teshhvavasthitaH .. 9.4..

This whole world is prevaded by Me in My unmanifest form. All beings exist in Me, but I am not contained in them! [This verse, together with the next one (9.5), contains one of the three secrets of Gita. Prof. VK-ji explains it beautifully here.]
yataH pravR^ittirbhuutaanaa.n yena sarvamidaM tatam.h .
svakarmaNaa tamabhyarchya siddhi.n vindati maanavaH .. 18.46..

A human being achieves success by adoring through his own duties Him from whom is the origin of creatures, and by whom is all this pervaded. [The only shloka that begins with the word yataH.]
Ch.13-7: amAnitvaM adambhitvaM ...

amaanitvamadambhitvamahi.nsaa kshaantiraarjavam.h .
aachaaryopaasanaM shauchaM sthairyamaatmavinigrahaH .. 13.7..

(7.) Humility, unpretentiousness, non-injury, for-bearance, sincerity, service of the teacher, cleanliness, steadiness, control of body and organs;
indriyaartheshhu vairaagyamanaha.nkaara eva cha .
janmamR^ityujaraavyaadhiduHkhadoshhaanudarshanam.h .. 13.8..

(8.) Non-attachment with regard to objects of the senses, and also absence of egotism; seeing the evil in birth, death, old age, diseases and miseries;
asaktiranabhishhvaN^gaH putradaaragR^ihaadishhu .
nitya.n cha samachittatvamishhTaanishhTopapattishhu .. 13.9..

(9.) Non-attachment and absence of fondness with regard to sons, wives, homes, etc., and constant equanimity of the mind with regard to the attainment of the desirable and the undesirable;
mayi chaananyayogena bhaktiravyabhichaariNii .
viviktadeshasevitvamaratirjanasa.nsadi .. 13.10..

(10.) And unwavering devotion to Me with single-minded concentration; inclination to repair into a clean place; lack of delight in a crowd of people;
adhyaatmaGYaananityatva.n tattvaGYaanaarthadarshanam.h .
etajGYaanamiti proktamaGYaanaM yadato.anyathaa .. 13.11..

(11.) Steadfastness in the knowledge of the Self, contemplation on the Goal of the knowledge of Reality-this is spoken of as Knowledge. Ignorance is that which is other than this.

[There is no contention for these shlokas. They are the well known GYaana shloka, with GYaana here meaning sAdhana.]

sukhamaatyantikaM yattad.h buddhigraahyamatiindriyam.h .
vetti yatra na chaivaayaM sthitashchalati tattvataH .. 6.21..

When one experienece that absolute Blisss which can be intuited by the intellect and which is beyond the senses, and being established (thus) this person surely does not swerve from Reality; [About bliss of the Atman and the attitude of one who has experienced it.]
kaaryamityeva yatkarma niyataM kriyate.arjuna .
saN^ga.n tyaktvaa phala.n chaiva sa tyaagaH saattviko mataH ..
18.9..

Whatever obligatory duty is performed just because it is a bounden duty, O Arjuna, by giving up attachment and the result as well,-that renunciation is considered to be based on sattva.
iishvaraH sarvabhuutaanaa.n hR^iddeshe.arjuna tishhThati .
bhraamayansarvabhuutaani yantraaruuDhaani maayayaa .. 18.61..

O Arjuna, the Lord resides in the region of the heart of all creatures, revolving through Maya all the creatures (as though) mounted on a machine!
purushhaH sa paraH paartha bhaktyaa labhyastvananyayaa .
yasyaantaHsthaani bhuutaani yena sarvamidaM tatam.h .. 8.22..

O son of Prtha, that supreme Person-in whom are included (all) the beings and by whom all this is pervaded-is, indeed, reached through one-pointed devotion.
yadaadityagataM tejo jagadbhaasayate.akhilam.h .
yachchandramasi yachchaagnau tattejo viddhi maamakam.h .. 15.12..

That light in the sun which illumines the whole world, that which is in the moon, and that which is in fire,-know that light to be Mine.

The shloka and the explanation of ProfVK-ji raises many questions:

Why these verses? Why this order? Is there any implied meaning in this verse? What does Prof VK-ji mean when he says "which makes sense if you look at the content of these slokas.".
Why does Prof Vk-ji say that this shloka is the capsule of whole Gita teaching?
Are 7.16, 7.17, 11.38, 11.18, 9.27, 9.22, 13.2, 18.55, 18.66 hiding somewhere?
Does the second word 'aham' refer to 9.16 and hence to 9.(16-19), rather than just to 10.8? If so, does the number of shlokas then becomes 18+3 = 21?
Why is this shloka written on a page of the 14th chapter?
Is the 9.4 really standing for 9.5 and 9.5 and thus adding one more to the number?
Can we add the 9.(16-19) shlokas, as well as 9.(4-5)? This makes the total number of shlokas as 22. Where are the rest 2 shlokas making up to the (next) magic number 24 (Gayathri)?

This post is one of many with the tag Collections of Gita Verses.

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