This note is neither meant to be complete exposition of this matter nor a complete view of all what we have realized in this regard. Nor can it be completely understood by those who are not insiders of the tradition.
Syllable in the primal realm
In below verse, from what is perhaps the most famous sūkta of the Ṛgveda, the Aṅgira Dirghatamas Auchathya, the founder of the Gotama clan says:
ṛco akṣare parame vyoman
yasmin devā adhi viśve niṣeduḥ |
yas tan na veda kim ṛcā kariṣyati
ya it tad vidus ta ime sam āsate || RV 1.164.39
In the syllable of the ṛc set in the highest world,
therein all the deva-s have taken residence,
he who doesn’t know that what can he do with the ṛc?
verily only they who know can sit together [in this ritual session].
gaurīr mimāya salilāni takṣaty
ekapadī dvipadī sā catuṣpadī |
aṣṭāpadī navapadī babhūvuṣī
sahasrākṣarā parame vyoman || RV 1.164.41
The female gaur [Bos gaurus] having measured out fashions the waters,
she is one footed, two-footed, four-footed,
becoming eight-footed and nine-footed,
she is a thousand syllables in the highest realm.
In the Atharvāṅgirasa-śruti Kutsa āṅgirasa says:
ekacakraṃ vartata ekanemi
sahasrākṣaraṃ pra puro ni paścā |
ardhena viśvaṃ bhuvanaṃ jajāna
yad asyārdhaṃ kva tad babhūva || AV-vulgate 10.8.7c
One-wheeled and one-rimmed it spins,
[with a] thousand syllables north, east, south and west,
With half it gave rise to the all the universe,
That which is the other half, what has become of it?
The Atharvaveda has several more mysteries pertain to this primal realm of syllables but we shall only touch upon some here. In the long Rohita recitation we hear:
ekapadī dvipadī sā catuṣpady
aṣṭāpadī navapadī babhūvuṣī |
sahasrākṣarā bhuvanasya paṅktis
tasyāḥ samudrā adhi vi kṣaranti || AV-vulgate 13.1.42
She is one footed, two-footed, four-footed,
becoming eight-footed and nine-footed,
she of thousand syllables [emits] the series of worlds;
from her the oceans flow forth.
The above ṛc-s have much in them that can be expounded but what we wish to stress here is the repeated allusion to the syllables, “akṣara-s” which are:
1) constituents of the spinning wheel from which the universe originated.
2) residents of the “parame vyoman” which can be understood not just as the highest world in some physical sense but also as the highest in terms of “ideals” and also in terms of time as the primordial world from which all has sprung.
3) constituents or the very form of highest realm that is the seat or the dwelling of the gods.
4) associated with counting or numbers in a sequence.
Indeed, this understanding of the syllables or the primal syllable is the secret behind the Vaidika ritualist receiving the teaching of the 4th foot “paro rajase ‘sāvadoṃ” upon having completed his basic initiation into the sāvitrī.
Now moving many centuries down in time we come across the rudra-yāmala tantra, which teaches:
a-mūlā tat kramā jñeyā kṣāntā sṛṣṭir udāhṛtā ।
sarveṣāṃ caiva mantrāṇāṃ vidyānāṃ ca yaśasvini ।
iyaṃ yoniḥ samākhyātā sarvatantreṣu sarvadā ॥
The process of emission [of existence], whose root is ‘a’ [and] in order known to have ‘kṣa’ as the end has been expounded. O glorious one, this is declared in all tantra-s as the source of everything, of mantra-s and vidyā-s, the giver of all.
This persistent idea of the syllables being the root of the universe, even in the physical sense, and the very manifestation of the gods of the Veda or the tantra might seem strange to one who has not grasped its secret. The way it might be understood is by the conception of the Sanskrit language by its tradition of grammarians who were intimately linked to the apprehension and transmission of Vedic knowledge. Pāṇini and his brilliant successors like Kātyāyana and Patañjali see the starting premise as the Sanskrit syllabary which might be arranged in different ways the: the regular periodic-table like structure or as the Māheśvara-sūtrāṇi. On this their rules operate to emit the whole language. Now we hold that Pāṇini did not invent this out of thin air. He was merely following upon a very old tradition that went back to more than 2000 years before him that conceived this process of rules operating on syllables to generate intricate form as manifest by the Vedic incantations. That was the art of the kavi or the vipra by which he crafted his mantra-s – it relates to the old Indo-European tradition of Chandas with a fixed syllable-count. Now these kavi-s extended this logic to the universe. There in the “parame vyoman” there exist these “thousand syllables” from which the universe was emitted by a series of rules (“vrata-s”) even as the Sanskrit language is emitted from its syllabary. This idea persisted with the tāntrika mantra theory which succeeded the Vaidika prototype.
This vision of generating complex structure from a simple syllabary, reflected in the work of Pāṇini, was increasingly important in the evolution of Sanskrit, where rules allowed for unambiguous (or purposely ambiguous) sense while still allowing for enormous structural complexity. This came at what might be called a “marker cost” i.e. marking the elements precisely: nominals precisely inflect – right in the first sūkta of the RV we see in succession inflections of the name of the great god (agnim, agniḥ, agninā, agne) as though to announce the importance of unambiguous markers of sense. Likewise, the verb assumes an enormous range of forms to convey very precise temporal and modal textures. For this very reason, the traditional grammarians split hairs and go into great depths with samāsa-vigraha or dissection of compounds, precisely because the markers are mostly lost in samāsa. For instance, you are left with the puzzle of whether aśvaśiras is the horse’s head or he who has a head like a horse or whether a lokanatha is the lord of the world or one who is lorded over by the whole world. While it makes Sanskrit a difficult language to master and might generate undue pride upon its operational mastery (caricatured by the advaitin-s as the ineffectuality of Pāṇini’s ḍukṛñ), it has had a profound effect on Hindu thought, the modern implications of which have not entirely been fathomed. Curiously, one facet of it which is central to our current discursion is the fact that in Hindu tradition the number is often subservient to the syllable. There are many ways in which numbers might be linguistically represented: the syllabic contraption of the great āryabhaṭa or the #katapayādi system or the system of using markers like: 1= candramas/Indra; 2=pakṣau; 3=vaḥnayaḥ; 4=vedāḥ; 5=bāṇāḥ; 6=ṛtavaḥ/skandaḥ; 7=munyaḥ/abdhayaḥ; 8=nāgāḥ/vasavaḥ; 9=grahāḥ; 10=dik.
“The extraordinary effectiveness of mathematics”
In our times we see a great thinker of the age Roger Penrose talk of three worlds: 1) The “real world” made of particles and energy including forms of matter-energy we do not yet understand. 2) The world of consciousness or first person experience, what in the western philosophical language would be called qualia. 3) The “Platonic world” or ideal mathematical entities both geometric shapes and numbers and the relationships which they contain (what I think mathematicians call theorems). Penrose points out that a great mystery is the fact these objects of the 3rd or the “Platonic” world are the ones that are used to govern the making and the working of the real world. Penrose believes that ultimately there is only one world but it appears three-fold because we do not understand its mysterious unity. That apart, the mathematical world of Penrose is thus comparable to the world of akṣara-s and the vrata-s which operate on them like Pāṇini’s rules in our tradition, which is placed at the root of the world.
Now the existence of such a pure mathematical world is rather easily perceived by anyone who has played with some mathematics, irrespective of whether he really understands its rigorous formulations or not. Its actual manifestation is what Eugene Wigner called the “extraordinary effectiveness of mathematics” or more recently extensively discussed by astronomer Mario Livio as “Is god (in singular emphasizing the certainty of Abrahamistic credo) a mathematician?” (In a sense following a similar Greek statement by Karl Gauss). One can fill a whole volume and more with examples of this mystery of mathematics and we have narrated some of our personal journeys through such on these pages; yet, we shall offer a few here for it is always worth savoring:
1) A trivial case is the Hindu predilection for huge numbers. At the time the Hindus named them people had little reason to count that much. But today we know the various measures at the extremities of the universe fall in the orders magnitude of many of those big numbers of our ancestors. Thus, long after the fact of their discovery in the “mathematical world” we see them as very real entities in the real world. For instance, Avogadro’s number would be approximately ṣaṣṭhi-vṛndāni.
2) This one is more personal: may be a couple of years after we had learned to construct a hyperbola we learned that this curve is describes how enzyme reaction rate changes with respect to substrate concentration. It was the first time we had the personal experience of the mystery of how a curve discovered in a purely mathematical realm in relation to the Delian problem by the yavana-s appeared in the real world of biochemistry.
3) Another personal example: around the 14th year of our life we constructed a curve known as the witch (of Agnesi), which is rather trivial:
* Draw the generating circle which is tangent to the x-axis at origin O and radius .
* Draw the diameter along the y-axis and a tangent to it at point A.
* Let B be a point moving on the generating circle. Draw which cuts the above tangent at point C.
* Draw a line perpendicular at point C and a line parallel to through point B. These two lines intersect at point D.
* The locus of D as a B moves on the generating circle is the witch which has the equation: .
Looking at the witch in those days of our youth we thought that it looked like a good function for a statistical distribution and wondered if anything in nature might be thus distributed. Only several years later we learned that the great mathematician Augustin-Louis Cauchy or the statistician Poisson had discovered a probability distribution of this shape which has the interesting feature of being “fat-tailed”. Closer to our times it emerged mysteriously in the real world of physics as the shape of broadening of certain spectral lines with molecular collisions. When Pierre de Fermat discovered this curve in the 1600s in the purely mathematical world he certainly could not have had an inkling that it would appear in the real world of physics centuries later, just like the conics of the yavana-s. Thus, over and over again this objects discovered in that “Platonic” realm were seen to govern the working of the real world.
This triumph of mathematics in the structure of the laws of physics gives them the feel of being true because mathematical truths are quite unlike scientific models – while much of the science of ancients has been quite superseded by new science their mathematics still stands firms and it was their mathematical teachings that allowed all the new science to emerge. The planetary models of the ancients might have died but their bhūjā-koṭi-karṇa-nyāya remains as true today as when, to their wonder, they first discovered it. This mathematical foundation perhaps gives a sense of inviolability to the laws or in the least it gives them the sense of being enforced (see more on that below). Indeed, the Hindus of yore did record this sense of inviolability in the laws in connection of the gods seated in that same “parame vyoman”. For instance the Arcanānas Ātreya states:
ṛtasya gopāv adhi tiṣṭhatho rathaṃ satyadharmāṇā parame vyomani ।
The two upholders (Mitrā-varuṇā) of the natural law (ṛta), you two of inviolable laws (dharman) stand in your chariot in the primal realm.
An ideal realm with a syllabary?
As noted above this mysterious intrusion of mathematics from its ideal realm into the real world is rather palpable even for someone with relatively limited knowledge but can the same be said for world of syllables postulated by the Hindus? At the face of it, to most it is less apparent – indeed, to some it even seems to bring up images of the well-known mleccha indological trope of “Hindus as idiots”.
When we look at physics we find the mathematics directly present in the structure of its laws – it is very apparent, even if entirely mysterious. Now, when we look at biology we find that same geometry acting as an enforcer that channels all variety by eliminating what falls outside the allowed geometries. This is what is termed natural selection. This is the foundation of the commonly observed phenomenon of convergent evolution – over and over again synapsid carnivores evolved the same saber-toothed morphologies, archosaur carnivores evolved generally similar skulls with serrated teeth (ziphodont morphology), tetrapods returning to the water assumed fish-like morphs like those of their ancestors. One part of this channeling is from the geometric structure resident in the underlying physics – the shapes that are best suited to fly in the sky or swim within the waters. But a part of it stems from deeper phenomenon that becomes apparent only when we descend down to the molecular level. At the microscopic level the three dimensional geometries of the molecules of life are ultimately encoded in one dimensional strings. The information in these strings is read in many ways, the operationally most important aspect of which is the reading of genetic code. That world is where the encodings occurs as letters quite literally akṣara-s and these can specify many different types of words and phrases by a slew of different rules that run the system from which emerges the 3D geometry. These “linguistic” entities are not just the genetic code but also seen in architectures of domains of proteins. Our studies over decades have shown that these have distinct syntaxes, with these syntaxes being the enforcers of what is seen as natural selection. Bad grammar is relentlessly purged like king Bhoja dismissing the bad grammar of his rivals. At the same time “creativity” of domain architectures within the allowed grammar, including the use of rare domains that conform to rules like a Sanskrit versifier’s use of unusual words, is rampant. Different cellular molecular ecologies have allowed for different styles – great creativity in the bacteria, robust simplicity in the archaea, and unexpected reuse of phrases and words in new contexts like a prolix hack in eukarya. Thus, the biological world offers us a glimpse of an ideal realm which contains as the ideals a set of akṣara-s and rules which operate on them – it is indeed quite a menagerie of rules, like the aṣṭādhyāyī.
Are these “akṣara-s” in addition to or in place of the mathematics present in the ideal realm? We have an intuition that it is more fundamental than the mathematics and the mathematics emerges as a limb of it. However, being of limited intelligence and knowledge, we by ourselves currently have no way of establishing this to be the case. However, it appears to us that the view of one of the eminent scientist-mathematicians of our age Stephen Wolfram bears a relationship to such an idea. While it is not clear if he postulates a ideal realm with computer programs, he does propose the existence of simple computational mechanisms comprised of simple rules acting on a limited set of characters as a powerful alternative to purely mathematical mechanisms in generating enormous complexity.
Measuring the real realm with mumbers
While we have spoken above of the privileging of the syllable over the mathematical abstraction of number and geometry in the primal realm in Hindu thought, this should not be construed as an absence of importance for these. Indeed, the main constituents of ordinary modern mathematics can be seen as numbers and geometry, which respectively stand on that bed-rock of mathematical tradition provided by the two great branches of the Indo-European world, the ārya-s and yavana-s. Any discerning student realizes that the ordinary secular mathematics of today is a successor of an ancient tradition of the Indo-European world, which emerged in the context of ritual constructions going back to at least the ancestral Greco-Aryan period. Such a role for mathematics in the religious process might be seen in many other civilizations – with close parallels in Egypt or much later in time in the temple geometry of Nippon. Ultimately all of this might have a direct connection going back to our shared ancestry with other apes like the chimpanzee. The piling of bricks to construct the altar (literally citi or piled in Aryan parlance) may be compared to recently reported ritual of piling of stones in tree hollows by the troglodyte – a operation with allows for counting in the least.
Geometry of the ritual altars was a key feature that can be traced back to the ancestral Greco-Aryan tradition. The cubical altar, which is at the center of the famous yavana Delian problem, can be seen as the cognate of the near cubical śāmitra altar of the animal sacrifice of the ārya-s. The Delian problem itself, which involves doubling of the cubical altar, might be compared to its planar equivalent found in the Yajurvaidika tradition of the eka-śatavidhā ritual where a square altar of Agni is increased from a single unit to 101 units through successive increments without ever changing the square shape. It was this ancestral tradition that divergently evolved to give rise to the Pythagorean and then Platonic world view among the yavana-s which privileged Euclidean geometry and the discovery of conics and other plane curves. In the midst of the ārya-s it evolved in a more numerical and algebraic form which privileged measurement as the primary feature. This feature emerges early in the tradition of our ancestors: one of the primeval preserved memories of our clan is that of our hoary ancestor Cyavāna Bhārgava measuring out the ritual altar with a depth-measuring device: “cyavānaḥ sūdair amimīta vedim |”. Right in the Ṛgveda we hear the great astronomer ṛṣi Atri Bhauma, who saved our ancestor Ṛcīka Aurva, say this as he praises the great Asura:
imām ū ṣv āsurasya śrutasya
mahīm māyāṃ varuṇasya pra vocam ।
māneneva tasthivāṃ antarikṣe
vi yo mame pṛthivīṃ sūryeṇa ॥ (RV 5.085.05)
I proclaim this great māyā
of the famed Asura Varuṇa,
who, standing in the atmosphere as if with a ruler,
measured out the earth with the sun.
Thus, the physical action of that great Asura Varuṇa who upholds the natural laws (ṛta) is carried out by measurement.
The Aṅgira sage Bharadvāja Bārhaspatya elaborates on this theme in that sūkta which can only be apprehended by those who see the rahasya-s of the heavenly Agni Vaiśvānara and not one who mistakes the ritual fire in front of him to be just a fire.
vaiśvānarasya vimitāni cakṣasā
sānūni divo amṛtasya ketunā ।
tasyed u viśvā bhuvanādhi mūrdhani
vayā iva ruruhuḥ sapta visruhaḥ ॥ RV 6.7.6
By the eye of Vaiśvānara heaven’s heights have been measured out,
by the ray of the immortal one.
Indeed on his head are all the worlds;
like branches his seven tongues have grown.
vi yo rajāṃsy amimīta sukratur
vaiśvānaro vi divo rocanā kaviḥ ।
pari yo viśvā bhuvanāni paprathe
‘dabdho gopā amṛtasya rakṣitā ॥ RV 6.7.7
The skillful one, who measured out the atmospheric realms,
is the sage Vaiśvānara who [measured] out the starry heaven
who spread around all the worlds
the irresistible guardian, the protector of immortality.
Notably, that Gauri who was described in the RV and AV (see above) as the embodiment of the syllables in the primal world also needs to be measured out and this can be done using a god as the measuring rod. This is described Kurusuti Kāṇva:
vācam aṣṭāpadīm ahaṃ
navasraktim ṛtaspṛśam ।
indrāt pari tanvam mame ॥ RV 8.76.12
The syllabary, with eight-feet and nine vertices
is embedded in the natural law, (literally in contact with the natural law)
I have measured out its body by means of Indra.
The same Gauri is also embodied in ritual as a cow who represents the measure of heaven. In Taittirīya saṃhitā 7.1 we have:
sahásra-sammitaḥ suvargó lokáḥ|…
The celestial world is measured in thousands…
While the holy cow stands in front of him the ritualist offers with this incantation:
ubhā́ jigyathur ná párā jayethe
ná párā jigye kataráś canáinoḥ |
índraś ca viṣṇo yád ápaspṛdhethāṃ
tredhā́ sahásraṃ ví tád airayethām íti ||
You two have conquered, you two are not conquered;
Neither of the two of them have been defeated;
Indra and Viṣṇu when you two contested,
you had divided the thousand into three. Thus, [he recites].
tredhā-vibhaktáṃ vái trirātré sahásram | sāhasrīm eváināṃ karoti | sahásrasyaiváinām mātrāṃ karoti |
Thousand is indeed divided into three parts at the three-night [ritual]; he makes her [the cow] a [symbol] of thousand. He thus makes her the measure of a thousand.
rūpāṇi juhoti | rūpáir eváināṁ sám ardhayati | tásyā upotthāya kárṇam ā́ japet |
He makes oblations to her forms. He thus furnishes her with her forms. Rising up he mutters in her ear:
íḍe ránté ‘dite sárasvati príye préyasi máhi víśruty etā́ni te aghniye nāmāni |
“Iḍe, Ranti, Aditi, Sarasvati, Priyā, Preyasi, Mahi, Viśruti”, these, O unassailable one, are your names (Thus, the ritual cow is identified with the goddesses among others embodying the syllabary of the primal word).
sukṛtam mā devéṣu brūtād íti | devébhya eváinam ā́ vedayati | ánv enaṃ devā́ budhyante ||
Declare me as a doer of good among the gods. She indeed lets [this] to be known to the gods. The gods take note of this.
The world of the heavens is much vaster than this one hence it is symbolic represented as measuring in a bigger unit, the thousand. For the yajamāna to let the gods know that he is a doer of good, by which he can conquer that celestial realm, he needs a measuring unit of the thousand. For that he makes the ritual cow a representative of the goddesses who embody that thousand. Thus, for the ārya in ritual other than geometry the number played an important role. Indeed, he encompassed the base number of all gods of the śruti numbering 33 (12 Āditya-s, 11 Rudra-s, 8 Vasu-s and 2 Aśvins) by measuring out the altar in the form of successive squares formed of square bricks from 1 to 289. This embodies the relationship:
Thus the above sequence goes from (spelled out the camaka praśna of the Yajurveda) while the sum of sequence goes from . Thus, the first square is one brick. Adding 3 bricks the next in the sequence on its 3 sides give you . Adding 5 bricks to sides of this new square gives you so on till you get the square of 17. Thus, geometry for the ārya was closely linked with number, and measurement with a cord and ruler. It was the philosophical consequence of this that marked a subtle point of departure of the Hindus from the Platonic realm of the yavana-s: one hears of Plato disapproving of the measured constructions – feature that dominated yavana geometry thereafter. Thus, while geometry and other Pythagorean mathematical traditions moved into the realm of the ideals as absolute-measure-free entities among the yavana-s, they were firmly as part of real world in ārya-s emulating the measurements of the universe performed by the great gods. Based on what the Kāṇva says even the primal syllabary has to be measured out for the construction of the real world.
A different way of looking at numbers
We first encountered a new class of systems, the iterative systems, with the 3/2 floor map. This we originally explored, to our great wonder, in the old way with a hand-held calculator and graph paper. The 3/2 floor map works thus:
where the starting value
The 3/2 floor map for two close starting values for 300 iterations
The remarkable feature about it is how we get chaotic behavior from this very simple system. We can see spiking and bursts of spikes as has been observed with the membrane potential of neurons without any differential equations but just a simple iterative system. We realized that the while the behavior is chaotic it is not entirely random in a statistical sense. For different starting values we get a generally similar frequency distribution of the values visited by over iterations though there might be very subtle differences for different starting values (shown in the figure for 50,000 iterations for six different starting values). Over all the values closest to 0 are 3 times more frequent than those closest to 1. Moreover, there are two clear steps in the distribution at 0.5 and 0.75 which are line with how the 3/2 floor map operates.
The distribution of values of visited by the 3/2 floor map for different starting values
Strikingly, even though the bulk distribution is generally similar for all starting values the actual behavior of is very sensitive to the starting value. For example in the figure we illustrate the evolution of 0.5 versus 0.50001 over 300 iterations. One can notice that beyond the 20th iteration the two sequences completely diverge that they have nothing in common despite similar bulk statistics.
This opened our eyes to the fact that numbers can be looked at in way different from how they behave in the system of classical calculus. In the method of calculus, the key aspect is that numbers close to each other behave similarly so that we can smoothly transition from one to another in a continuous sequence of ever close numbers. In iterative systems even a small change in the number can result in a very different end result – something which goes against the classic operation of calculus. Moreover, whether such behavior is produced or not depends on nature of iteration rule or the “map”.
The importance of another such kind of behavior was further driven home to us by the floor-difference sequence. This sequence is generated thus:
where and is a constant fractional number like say or .
The floor-difference sequence as step graph for various numbers
The floor-difference sequence which results from the above rule can be represented as shown in the figure – as a step graph – a sequence of up and down values. This set of up and down values can hence be also represented as a binary string. When is a rational fraction like 3/2 then we get a regularly repetitive graph which translates into a regular binary string like . Now if is instead an irrational number then we do not get a repetitive graph. Instead, there are more complex patterns which have their own structure. For example, Stephen Wolfram, the pioneer in the study of such systems, has shown that if is a root of a quadratic equation then the pattern of the sequence is obtained by an automaton with simple substitution rules. Thus, he has shown the following substitution rules:
These can be easily verified with the figure.
However, for other irrational numbers the rulers are more complex or subtle. The case of or is rather notable. For we see that the step graph looks quasi-regular: it can represented as repeats of the binary string . This goes fine till we reach position 32 at which place a we get an unexpected string . Then it resumes again as usual with the repeat of the earlier string till position 71 where we again get . From this we see that first deviation occurs at 32 and second at 39 from the point of the first deviation. From then on this pattern repeats with successive insertions of the string occurring alternately at 32 and 39 positions from the previous one. Thus, we can predict and see the string at: 32, 71, 103, 142, 174, 213, 245, 284, 316, 355, 387, 426, 458, 497=458+39. But at that point instead of seeing the next occurrence of the string at 529 (497+32) we instead see it at 536 (497+39). Thereafter, from 536 the sequence of the points of insertion of the string are as usual in the 32, 39 alternate pattern: 536, 568, 607, 639, 678, 710, 749, 781, 820, 852, 891, 923, 962, 994, 1033=994+39. But then instead of getting 1065 (1033+32) next we get 1072=1033+39. Thereafter, from 1072 the sequence of appearance of the string continues as usual with the alternating 32, 39 as: 1072, 1104, 1143, 1175 …
To understand the secret behind this interesting pattern we have to turn to the continued fraction representation of , which was developed by the great Leonhard Euler. By working out the fractions to successive terms we get increasingly better approximations of , which are listed below starting from the 0th :
Now the floor-difference sequence for the first fractional approximation produces a repeat of the string ; the next approximation yields the repeats of string ; the next approximation yields the string . Thus, the first three approximations set up the basic pattern which forms the floor-difference sequence of . We then note that the appearance of the deviant pattern , corresponding to positions alternately separated by 32 and 39 (71=32+39), is related to the denominators of the next three approximations. Strikingly, the further points where we observed break of the preexisting pattern occur at or upon crossing of 536 and 1001 which define the denominators of the 9th and 10th cycles of approximation.
Now when we look at the step graph for the floor-difference sequence of we see that it shows a strikingly regular pattern which can represented as repeats of the string . We see 15 such repeats till we reach the position 106 where we see a deviant string . Then it resumes with the usual repeat until position 219 where we again see the deviant string as above. This deviant string then reappears regularly at 332, 445, 558, 671, 784, 897, 1010, 1123… i.e. every 113th position. Now the secret of this pattern becomes clear when we look at the continued fraction derived approximations of starting from the 0th cycle:
The 0th approximation 3 is the ancient one which is already alluded to as *approximation* of even the RV. The 1st approximation is what Archimedes arrived at. Its floor-difference step graph corresponds to a sequence starting with the string followed by never-ending repeats of . This later string sets up the basic pattern of the sequence. Then we see that as with the denominator of the 2nd approximation yields the point where the first deviant string appears (106). Then the next copy of that deviant string appears 113 positions away from that point, 113 being the denominator of the next approximation. The next irregularity is would appear in the vicinity of the denominator of the next approximation, which is way beyond the number of iterations to which we have calculated the sequence. That explains why serves as great approximation for and was so remarkably captured by Srinivasa Ramanujan in his first construction to approximately square the circle.
Thus, here we see that the application of a procedure (the floor-difference operator) converts an irrational number with uniformly distributed set of digits after the decimal point into an orderly structure – a structure which can be generated by relatively simple rules similar to those of Pāṇini operating on a syllabic string.
As the final example in this section we shall consider the remarkable curlicue curves which were first discovered and described by the mathematician Mendes France. Different types of these curves may be generated using two related algorithms and in addition to a deep aesthetic value have rich implications for various natural systems including optics and the folding of polymers. However, for this discussion we shall restrict ourselves to first of these algorithms which generates these curves thus:
1) Let the seed be a fractional or irrational number like .
2) Start with an initial point say .
3) Then for n iterations do the following:
4) Join successive points to get the desired curlicue. The figure shows curlicues generated using various seeds run for 2000 iterations. Top row from left: 1) ; 2) ; 3) ; 4) Bottom row from left: 5) ; 6) ; 7) .
Curlicue curves with various numbers
First, one can see several aesthetically pleasing interesting patterns: curve produces fir-tree like patterns. The curve produces several ‘S’ shaped patterns. The produces a basic pattern of highly folded “beads” connected by unfolded linkers. Together they are organized into the path of multiple spiraling curves that resemble Johann Bernoulli’s clothoid which emerges in optics (where it is called a Cornu spirals). When a beam from a distant source of light encounters a slit of size comparable to its wavelength we get a typical diffraction pattern with a central peak of high intensity followed by increasingly damped peaks of much lower intensity on its flanks. Now, if the light source is close to the slit then the analysis of the diffraction patterns becomes very complicated. It is in such a case, where a nearby linear light source produces a cylindrical wave front, that this spiral curve can be used to predict the diffraction pattern.
Second, nearness of numbers does not necessarily translate into the similar pattern. We know that was used often in early Hindu mathematics as an approximation for correct to one decimal place. However, this number produces a wandering curve that is not at all similar to that of . Then we see the famous approximation used by Ramanujan correct to six decimal places. It being a rational number does not produce a fractal pattern but a linear sequence of highly folded “beads” connected by a linker element. This resembles the unfolded nucleosome-DNA complex in archaea and eukaryotes. The real in contrast has a highly folded bead similar to the former but the linker gradually curls making curve take an overall spiral form.
In conclusion, here we see numbers playing a different role: different from their role as the backdrop of the smooth continuity of curves emerging as solutions to differential equations but as abstractions similar to syllabic strings, which can be acted upon by simple rules to generate complexity. Here, the actual strings of the digits matter. Seeming uniformly distributed sequences of digits can produce complex patterns that are far from random when sent through the appropriate rules. It would almost seem as though they hide something within them which is not at all apparent until they are made to reveal them by procedures like the floor-difference operation or the curlicue algorithm. This complexity can capture the forms and behaviors we see in nature and produce grammar-like features – some of which also emerges from more complex mathematical descriptions of natural systems.
The possibility of a unified vision
Finally, we intuit the possibility of a unified vision – by no means a complete one or rigorously hammered out. First, we do see the presence of the an ideal realm, one might call it “Platonic”. In our view this is how the “parame vyoman” and “śuddha-bhuvanādhvan-s” of the tantra-s should be understood. We also hold that the correct way of understanding the mīmāṃsaka nityatvam of the word is in this sense rather than the physical śruti composed by our ārya ancestors starting from the early Indo-European period. In this realm reside the numbers or syllables, which are comparable entities in that they are operands of a collection of operations or rules, which in themselves might be simple. However, they are capable of generating great all the complexity we see in the world and mathematics. When this ideal world “shapes” the world of matter-energy we see its expression as a different type of number, the number of measurement. It is this number that appears directly in the sciences. The attempt to apprehend the structure residing in these numbers leads to the discovery of mathematics.
In a biographical sense all this unfolded gradually over time – a story which we might narrate at some other point with different illustrations and more detailed acknowledgments of our influences. The first phase of our life was marked by our experiments in Euclidean geometry and plane curves which culminated around our 14th year in the penetration of calculus. It was the high-point of our personal mathematical attainment in that direction which allowed us to advance into the quantum theory and relativity in physics to the extent we could. As we were fighting our way through Dirac’s book on bra and ket vectors in the summer of our 14th year, on one hand we felt great pleasure with what we had managed to understand but on the other the insurmountable mathematical complexities that underpinned an even deeper conquest of physics put us in place. But our experience was one which nevertheless resounded with the triumph of “ordinary” mathematics – a culmination of the mathematical traditions of the ārya-s and yavana-s in one sense. The existence of the ideal realm came into our focus then and it was a very mathematical one filled with curves. It was around this time we came across an article on the work of Benoit Mandelbrot in a science magazine. This immediately opened the doors to a new world, marking the second phase of our life. However, not having easy or continuous access to a computer this exploration began slowly. But it advanced over the subsequent years with the completion of our transition from the hand-held calculator to the computer and the ideas of Turing and Wolfram become a pratyakṣa. By then we saw a new picture, one converging what has been articulated here.