GEB, Properties of Natural Numbers and Chamakam

I had written previously about how good I find the book Godel, Escher and Bach: The Eternal Golden Baird. Last week, I read chapters 3 and 4 of the book. In these chapters, Dr. Douglas Hofstadter speaks of Figure-Ground and its prominence in formal systems. Dr. Douglas Hofstadter goes into great detail to explain Figure and Groundcould not only be identified in Escher and others’ paintings but also in music and formal systems. As an example of figure – ground, try this

From the first look, one would see two faces. But a few seconds of careful look would reveal a vase. The Figure and Ground are thus defined as negation of one another. More on figure and ground here. Now, if you take the square in which the whole diagram is present as the universe, then the vase forms the Figure while the faces form the ground. In sketching and painting this is the general way figure and ground are defined. Dr. Douglas Hofstadter takes this concept a little further and tells us about how Bach used the same concept in The Musical Offering.
Dr. Douglas Hofstadter then enters the formal systems domain and tries to explain how figure and ground could be established. More on that later in the post.
In one of the boring moments @ work, I started browsing through the text of Sri Rudram. As I started reading the eleventh anuvaka of Chamakam I found something very intersting. This was not the first time I was reading Chamakam. But somehow, I missed this point till today.

Srirudram, also known as Rudraprasna, is a hymn devoted to lord Shiva. It is part of the Yajur Veda and one of the greatest of the Vedic hymns for all round benefits and to remove all doshas & difficulties. In most of the poojas and homas it recited by the Vedic pundits. Sri Rudram is in two parts. The first part, chapter 16 of the Yajurveda, is known as Namakam because of the repeated use of the word “Namo” in it. The second part, chapter 18 of the Yajurveda, is known as chamakam because of the repeated use of the words “Chame”.
Rudram is divided into 11 sections called Anuvakas. In the first Anuvaka, Rudra is asked to turn away his Ghora rupa (fierce appearance) and to please keep his and his followers’ weapons at bay. Having been pacified, Rudra is requested to destroy the sins of those for whom it is being chanted.
Apart from being a hymn devoted Lord Shiva, Srirudram also contains may hidden secrets in coded format. For example the verses contain coded instructions for preparing various ayurvedic medicines.

In the eleventh Anuvaka of Chamakam, a series of numbers are chanted. It goes like this


Transliteration in english:

Ekaa cha me tisrashcha me pajncha cha me sapta cha me nava cha ma ekadasha cha me trayodasha cha me pamchadasha cha me saptadasha cha me
navadasha cha ma eka vishatishcha me trayovishatishcha me
pamchavishatishcha me saptavishatishcha me navavishatishcha ma

ekatrishachcha me trayastrishachcha me chatasrashcha me
ashhtau cha me dvaadasha cha me shhodasha cha me
vishatishcha me chaturvishatishcha meashhtaavishatishcha me
vaatrishachcha me shhattrishachcha me chatvarishachcha me

chatushchatvaarishachcha meashhtaachatvaarishachcha me
vaajashcha prasavashchaapijashcha kratushcha suvashcha muurdhaa cha
vyashniyashcha antyaayanashcha antyashcha bhauvanashcha

The meaning of this anuvaka is (as translated by the great english mis-translators)

Let me be granted the uneven number one, three, five, seven, Nine, eleven, thirteen, fifteen, seventeen, Nineteen, twenty one, twenty three, twenty five, twenty seven, twenty Nine, and thirty one, thirty three, and even numbers four, eight, twelve, and sixteen, twenty, and twenty four, twenty eight, thirty two thirty six, and forty and forty four, forty eight to ensure food and its production, its continuity, and the urge to enjoy, the origin of all productions, the sun, the heaven, the head of all, the infinite, the all pervading like the sky, time and the like present at the end of total consummation exists at the end of it on the earth as universal form, the Antaryami the immortal, the inner ruler of every thing, the omni present and omni potent.

As a side note, I could never understand how and why Sri Rudram talks about sacrifices to Lord Shiva. I could never find a word meaning sacrifice in the whole of Sri Rudram. May be my Sanskrit knowledge is not as good.

Now a cursory look might suggest that it just as a series of numbers. Lets have a closer look.
The first series of numbers is


All odd numbers.
The second series of numbers is


All multiples of four. So what, you may ask. I am coming to that.
Lets for a moment concentrate on the numbers missing in the two series’. The missing numbers (I go till 34 only) are


Look at these numbers closely. If you divide these numbers by 2, you would get


You get back the odd numbers. Now I dont know and cant see why the first series is quoted til 33 only where as second series is quoted till 44 numbers. This needs to be investigated. But I am certain about one thing. With 2, 4 and series of odd numbers, you can define any number in the natural number system. How, you ask. This is how

If a set, X of all numbers {x}, such that x is divisble by 2 utmost once, forms a set of numbers


If a set, Y of all numbers {y}, such that y is divisible by 4 and y >=4, forms another set of numbers,


union of X and Y i.e., X U Y forms the set of natural numbers.

Dr. Douglas Hofstadter explains in Chapter 3 of GEB that you could construct Figure-Grounds in mathematics. He also explains us that it is even more interesting to construct Figure-Figures. He gives a variety of examples to explain this and finally comes to question “can theorems as well as non-theorems have a form?”. Well my understanding is that if you take this anuvaka as an example, it is certain that theorems and non-theorems can have form and that too a very clearly defined form, just as the case is here. Make a note that X intersection Y is a null set here, which means X union Y covers the complete universe of natural numbers. This is the brilliance of this anuvaka. With two different forms (or Figures) we have a complete picture (natural numbers).
Coming back to Figure and Ground concept, we can identify two clear figures here – odd numbers + odd numbers multipled by 2 and multiples of four in this anuvaka. A clear example of Figure-Figure definition of formal system of natural numbers.
Unfortunately, Dr. Douglas Hofstadter doesnt discuss this anuvaka in GEB. I dont blame him. Obviously, the translations, like the one quoted above, concentrate on Vedas as a part of ritual of offering sacrifices to Lord Shiva but miss the point that after all, Vedas describe the nature. Numbers are as much part of nature as we are. This anuvaka is a clear example of how well Vedas define the nature.
One may disagree with me and say that I am being partial to Vedic knowledge but one cant say that this piece of information had been embedded in the Vedas by coincidence. It is there because it is supposed to be there. Many questions could be asked like
Why 33 and 44?
Why only odd numbers, 2 and multiples of 4?
Why only natural numbers?
and so on
Answers to these questions could be found by investigating into the Vedas with an open mind. A crooked and one sided analysis of Vedas would leave us with sacrificial procedures and pagan gods and godesses, whereas an open ended discussion on Vedas would lead to knowledge hitherto untouched by Science. Just imagine how many secrets of nature could be pried open if
a Sanskrit Scholar, who learnt Sanskrit from Vedic Pundits and not from western anti-Hindu pundits,
a mathematics Scholar, who can put aside the notion that Vedas are holy writ and God’s word and look at Vedas as a mine of knowledge,
a geologist,
a biologist, and others from relevant fields work together, study and decipher the Vedas
The kind of knowledge represented in the Vedas not only gives out great philosophies for life, but also give us great knowledge on the nature of universe itself. If only, India after Independence took this with some pride and enthusiasm, mankind as a whole would have benefited from the study.
Post Script: It occured to me, as I almost finished this post, that to define the numbers in this anuvaka the way they have been, one would need to multiply all natural numbers by 4 (this is the definition of set of numbers represented by Y above). GEB as had been explained here previously, is about self-reference in general, and specifically in formal systems. This anuvaka is a brilliant example of definition of self-referential formal system. There you have. One more special case of Strange Loop. Can we say that this anuvaka is a special case to Godel’s theorem? May be so. May be not. Only pure mathematicians can tell I guess.Update: Dr. Sasidharan Commented ”

The numbers actually refer to mitochondrial DNA and Nuclear DNA and this 11th anuvakam pray for the well being of our DNA which is fundamental for all living thing in this planet. More on Mitochondrial DNA here. How exactly this anuvaka in Chamakam correlates to Mitochondrial DNA is something which would be interesting to look. Please post any interesting links or articles on this link in comments.Update: Thanks again to Dr. Sasidharan for the link explaining how Chamakam describes various aspects of nature. My favorite still is the 11th anuvaka which, as in his earlier comment Dr.Sasidharan wrote, is a representation of Mitochondrial DNA. Heres the article.



  1. Hi nandhu, a very well composed blog ! Yes, indeed a small classic example of what vedas have in them.

    For me this 11th anuvakam is the best I like while chanting Chamakam, as it’s more challenging (I think the biggest tongue-twister of all anuvakams, for me at least) as well as excellent in sounds.

    Until reading this article my understanding of this anuvakam is barely simple – a number other than zero (not created) should either be even or odd. You know Chamakam (in meaning) is for seeking various things mentioned in anuvakams from Lord. So when you ask for whatever is in even quantity and whatever is in odd quantity, you will get full.. 🙂 as you say the whole nature. So I got to the same conclusion. And why precisely till numbers 33 and 44 I think is to match the metrics (the specific chandas involved) as vedas deal both with purport as well as sounds (intonations) and length.

    What I heard is there is a lot of medical science involved in some of the anuvakas. Just hearing them regularly would pacify dreadly diseases. How is that scientifically!(ghanapati garini adigithe clean ga cheppestahru anuko, I understand the fun in reasoning it for ourselves)

    One other thing I am fascinated about is about the discovery of zero from Gayathri Chandas. This is according to what scriptures say. How is that beyond our scope of understanding I think.. at least for now 🙂

    The scholars you mentioned usually revel in Bhakti-yoga (no reasoning anymore) after their courses in Jnana Yoga, they never work for materialistic gains.
    The way to understand them is – a person in the ranks of 80 – 90 % would not be wanting to come down to 10 -20 (which I believe where we are). As per the requirement they do help though.
    People like us want them to contribute in various fields that we like which for them is something lower in value. They don’t want to leave their spiritual path. We can only dream of ! There is a deadlock. 🙂

    Catch you later.


  2. Hey Ravi, thanks for the comment.

    Its brilliant really. Whenever you find something like this, you realize how much knowledge Vedas have to offer from something so abstract as soul to something so real as numbers. I guess you had already come across something called Super Brain Yoga. Manam vinayakudi mundu gunjillu teesthame adi america vallu Super Brain Yoga ani peru petti patent chesaru. Ironic!

    Yes actually we must learn Sanskrit and try to analyse further. Dont know though when I can start that 🙂

  3. namaste,

    i am glad that a young person such as you has started asking the right questions – sri rudram is the common name – it is called sri rudra prashna

    have a look at the mayan calendar

    and could you give your maths insight into why the sequence? remeber that knowledge has to be cryptic – NTK basis (need to know) and hindus (and mayans) are masters of survival.

  4. namaste,

    the need to know sanskrit is also NTK basis – Need to know. If you Indian and can speak any native language (including urdu or for that matter farsi – or mayan – you land up in sanskrit)

    have alook at

  5. @jonn,
    thanks for the comment. It is really pleasing to see people do share the some thoughts I had.

    Learning Sanksrit is something I have kept myself as a Must Do. It is necessary for all Hindus to learn sanskrit, for, only then we will understand the importance of the various scriptures.

    It is my belief that similar to Hindu systems, the other systems like Mayan, Aztec, Sumerian etc have their own understanding of nature and associated phenomena. Thanks for the links. I will go through them when I find some time.

    Thanks again for the comments and your encouragement.

    Just out of curiosity, are you professor or researcher of sanskrit and links to mayan?

  6. The numbers actually refer to mitochondrial DNA and Nuclear DNA and this 11th anuvakam pray for the well being of our DNA which is fundamental for all living thing in this planet.

    Dr S Sasidharan , INFORMM< USM, Penang

  7. @Dr.Sasidharan,

    Thanks for your comment. Your comment leaves the post with a new angle of investigation and a new piece of information. I will dig into it further. However, to start with I will first post your comment to this post.

    This is exactly the reason why I feel if vedas are looked at with a multi-disciplinary approach, a lot of already known and unknown things can be found and most certainly such findings could be turned into scientific experiments and prove in a lab.

  8. Yeah, all the vedas and other sacred book consist of many scientific fact which is unexplored. You can get more information about this by visiting Rudram Chamakam – An original explanation by TBT

    Dr S Sasidharan, Malaysia

  9. @Dr.Sasidharan,

    Thanks for the wonderful link. Made my day!

  10. Friends,

    The whole premise that 11th Anuvaka only goes till 44 is wrong. It goes till 33 for odd numbers and 48 for even numbers.

    Sanjeevi Rao

  11. @sanju, thanks for the comment.

    I agree. As I have mentioned in the post, I could not figure out why odd numbers are quoted till 33 and multiples of 4 are quoted till 48. I might have mentioned “44” but I mean “48”. However, the point is not about how many numbers are quoted. It is more to do with the relationship between these numbers. My premise is not that numbers are quoted till 44 so my basic point still remains.

    I restricted my analysis till 34 as I have mentioned in the article
    “The missing numbers (I go till 34 only)….”

    You cannot create natural number set with
    1) only 2 and odd numbers
    3) only 4 and odd numbers
    You need 2,4 and the set of odd numbers to do it. This is exactly what the 11th anuvaka does. Well at least till number 34 it does. But why till 34 only, is the question. I hope you are getting the crux of the article.

    The discrepancy can only be understood with further analysis. If you have any idea or any relevant concept in your mind, do post a comment about the same.

  12. […] of universal consciousness or nirguna paramatma as explained earlier via an example from Chamakam here. Translators who came from Europe saw Rudram as sacrificial hymns to Shiva but rather these are […]

  13. Namaskara. This blog is really interesting. I’ve been a big fan of GEB it is interesting to see how you “braid” chamaka & formal systems 🙂 I’ve been searching for a good explanation for the presence of numbers and their usage in feminine gender (eka, tisra). Since higher numbers have the same form in all genders, I think it is a fair assumption that all numbers are in feminine gender because of the first two numbers. Sir, any thoughts on the reason for the numbers in feminine gender?

    1. Interesting observation. I didnt notice it. I will need to do some study before I can respond.

  14. Vivek Sarabeswaran · · Reply

    There is an explanation given by Sri Shankar Bhatt in Chapter 2 of Sripada Srivallabha Charithamrutham on the 11th Anuvaka of Chamakam. Sripada Srivallabha was the 1st incarnation of Dattatreya in Kali Yuga around 900-1000 years ago and Sri Shankar Bhatt has written this true story of this great Avatar of Sri Dattatreya.

    ‘Eka Chame’ means ‘one’. Tisra scha me’ means adding one to three, we get four and its
    square root is ‘two’. ‘Pancha cha me’ means adding five to four, we get nine and its root value is
    ‘three’. ‘Sapt cha me’ means adding seven to nine, we get sixteen and its square root is ‘four’. ‘Navacha ma’ means adding nine to sixteen, we get twenty five and its square root is ‘five’. ‘Ekadasa chame’ means adding 11 to 25, we get 36 and its square root is 6. ‘Thriyodasa cha me’ means adding 13 to 36, we get 49 and its square root is 7. ‘Pancha dasa cha me’ means adding 15 to 49, we get 64 and its square root is 8. ‘Sapta dasa cha me’ means adding 17 to 64, we get 81 and its square root is 9. ‘Nava dasa cha ma’ means adding 19 to 81, we get 100 and its square root is 10. Ekavigum satih cha me’ means adding 21 to 100, we get 121, and its square root is 11. ‘Thriyovigum satih cha me’ means adding 23 to 121, we get 144 and its square root is 12.‘Panchavigum satih cha me’ means adding 25 to 144, we get 169 and its square root is 13.
    ‘Saptavigum satih cha me’ means adding 27 to 169, we get 196. Its square root is 14. ‘Navavigum
    satih cha ma’ means adding 29 to 196, we get 225 and square root is 15. ‘Ekatrigum satih cha me’ means – adding 31 to 225, we get 256 and its square root is 16. ‘Thiotrigum satih cha me’ means – adding 33 to 256, we get 289 and its square root is 17. ‘Panchatrigum satih cha me’ means adding 35 to 289, we get 324 and its square root is 18. ‘Sapta trigum satis cha me’ means – adding 37 to 324, we get 361 and its root value is 19. ‘Navatrigum satih cha me’ means adding 39 to 361, we get 400 and its root value is 20.”

    My explanation pleased the pundits in the darbar very much. I, myself was surprised at my explanation. Again I said, ‘all this is the secret of the infinitesimal particles of matter in the creation. This is known to ‘Kaanaada maharshi’. Depending on the number of the infinitesimal particles, different ‘elements are formed’.

    1. This is an excellent observation. I would like to add the explanation you have provided to my blog. I will give full references you have provided and name you as having given this link. Please let me know if I can. Also, if you can point me to a digital version of the book, I would appreciate it. Thanks for this.

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