## Syllable, number and rules in the ideal realm

•October 23, 2016 • Leave a Comment

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].

He continues:
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 number 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: in 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 $a$.
* Draw the diameter along the y-axis $\overline{OA}$ and a tangent to it at point A.
* Let B be a point moving on the generating circle. Draw $\overleftrightarrow{OB}$ which cuts the above tangent at point C.
* Draw a line perpendicular $\overleftrightarrow{AC}$ at point C and a line parallel to $\overleftrightarrow{AC}$ 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: $y=\dfrac{8a^3}{4a^2+x^2}$.

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 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:

$\displaystyle \sum_{n=0}^{16} (2n+1) = (n+1)^2= 289$

Thus the above sequence goes from $1,3,5..33$ (spelled out the camaka praśna of the Yajurveda) while the sum of sequence goes from $1^2..17^2$. Thus, the first square is one brick. Adding 3 bricks the next in the sequence on its 3 sides give you $2^2$. Adding 5 bricks to sides of this new square gives you $3^2$ 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.

continued…

## Chaos in the iterative Hindu square root method of the gaṇaka-rāja

•October 21, 2016 • Leave a Comment

For Hindus big numbers always mattered and our mathematics is quite reflection of this fascination. Since the earliest times, Hindus devised various methods to obtain square roots of numbers, especially approximations of irrational roots correct to multiple decimal places. The earliest of these methods involving a series of terms is seen encoded in the altars for the Soma rituals specified in the saṃhitā-s of the Yajurveda and explicitly spelled out in their the śulbasūtra-s. Indeed, we have evidence that development of these methods continued in the Yajurvaidika tradition as indicated by Rāma dīkṣita’s commentary on Kātyāyana where he provides a tradition regarding a further term to the approximation to get $\sqrt{2}$ correct to 7 decimal places. A similar improvement was likely used in the procedure preserved by Sundararāja dīkṣita in the Āpastamba tradition for an approximate squaring of the circle based on $\sqrt{2}$.

By the last few centuries before the common era the Hindus had already discovered a method similar to what is today known in the west as the first term Newton-Raphson approximation. We also see the exact algorithm for both square roots and cube roots of ācārya Āryabhaṭa further explained for the lay by Bhāskara-I. But the high point of the Hindu tradition of iterative methods is seen in the text of the brāḥmaṇa Chajjaka-putra gaṇaka-rāja probably from Mārtikāvati (unfortunately named Bakshali manuscript: BM), which gives a glimpse of just what Hindu knowledge has been lost over the ages. While this method was misunderstood by the earlier white indological translator of the BM, the sophistication of the gaṇaka-rāja’s method has only more recently become clear. This has been explained and commented upon in detail by the computer scientists Bailey and Borwein in their excellent work on the same. We shall here comment upon an interesting aspect we discovered of the functions involved in the method .

While the method has already been discussed in detail by Bailey and Borwein, we shall go over it here for introducing the system. In order the find the square root of a number $q$ the BM suggests the following procedure:
Take some starting number: $x_n = x_0$
$x_{n+1}=\dfrac{q-x_n^2}{2x_n}$
$y_n=x_n+x_{n+1}-\dfrac{x_{n+1}^2}{2(x_n+x_{n+1})}$

Then $y_n \approx \sqrt{q}$. Now if we take $x_n=y_n$ and iterate the above procedure we get increasingly accurate approximations of $\sqrt{q}$.

As a example let us take $q = 5$ and $x_0=0.1$. Then we have the following:
$1)\; 12.6248003992015985\\ 2) \; 3.6392111847990769\\ 3) \; 2.2506636482615887\\ 4) \; 2.2360679780006203\\ 5) \; 2.2360679774997898$
Thus, in iteration 3 the value of $\sqrt{5}$ correct to 1 decimal place, in iteration 4 it is correct to 8 decimal places and in iteration 5 it is correct to at least 16 decimal places, in line with the Hindu love for big numbers.

Now if we instead take $x_0=2$ because we know that $\sqrt{5}$ should lie somewhere in the vicinity of 2 then we get:
$1) \; 2.2361111111111112\\ 2) \; 2.2360679774997898$
Thus, with this close value right in the first iteration we get it correct to 3 decimal places and in the second to at least 16 decimal places! As Bailey and Borwein had shown it quartically converges on the square root. Now if we take a negative number for $x_0$ it then converges similarly to $-\sqrt{q}$.

Figure 1

Now consider the following alternative procedure where instead of plugging $x_n=y_n$ we plug $x_n= x_{n+1}$ and thus generate for each iteration $(x_{n+1},y_n)$. On plotting the map of $(x_{n+1},y_n)$ we see the points fall on an interesting curve (Figure 1). This curve has two boat-shaped branches which are respectively tangential to the lines $y= \pm \sqrt{q}$. The region of tangency is peculiar in that the curve lingers in the proximity of $y= \pm \sqrt{q}$ over a wide x-interval.

Figure 2

The actual map of the points obtained by the above procedure displays an interesting feature: they are spread all over the two branches of the curve above but fall most frequently in the vicinity of the two root lines. They notably decrease in frequency as one moves away from those lines but we do get to see extreme points far away from the two root lines. Thus, $\pm \sqrt{q}$ serve as the peaks (Figure 2) for the distribution of $y_n$ with a clear decline for greater and lesser allowed values respectively. However, the tails of their distribution are prominent enough that we seen multiple extreme values. The median value for the negative side of the distribution is $\approx -2.3$ and for the positive side is $\approx 2.3$, illustrating the dominance of the values close to $\pm \sqrt{q}$. In line with this, for a large enough number of iterations with a given $x_0$ the overall median value of $y_n$ comes out as $\pm \sqrt{q}$. However, below is an examination of the extreme values reached by $y_n$ for a run initiated with $q=5$, $x_0=0.1$ for 2000 iterations:
Minimum: $-438.98149$
Maximum: $133.19996$
This shows that $y_n$ explores values over 50-100 times the median values in course of the iterations.

To understand this map better let us look at it geometrically (Figure 3). The two expressions that are deployed successively by Chajjaka-putra to get the square root represent the below functions:
$f(t)= \dfrac{q-t^2}{2t}\\ g\left(t\right)=t+f\left(t\right)-\dfrac{f\left(t\right)^2}{2\left(t+f\left(t\right)\right)}\\[10pt] \therefore g\left(t\right)=\dfrac{q^2+6qt^2+t^4}{4qt+4t^3}$

Figure 3

We see that $f(t)$ is a hyperbola with the y-axis as one of its asymptotes. $g(t)$ is a quartic curve, which has $y= \pm \sqrt{q}$ as the as its tangents with the points of tangency being $(\sqrt{q},\sqrt{q})$ and $(-\sqrt{q},-\sqrt{q})$. This curve has a very “flat” type of tangency, i.e. it lingers in the proximity of $y= \pm \sqrt{q}$ over an extended x-range. This is the secret of the gaṇaka-rāja’s method firmly “pulling” things to the vicinity of required square root. Thus, the parametric curve $(f(t),g(t))$ is the one on which the points of the above-described map based on gaṇaka-rāja’s two expressions lie (Figure 1, 3). This curve as noted from the above map has $y= \pm \sqrt{q}$ as its tangents with the point of tangency at $(0,0)$, but like $g(t)$ it lingers over a wide x-range close to the point of tangency. This explains why the map tends to concentrate the points in the vicinity of $y=\pm \sqrt{q}$.

Now, if we look at the positions of actual points of the map on $(f(t),g(t))$ an interesting observation becomes apparent: while tending to cluster in the vicinity of $y= \pm \sqrt{q}$, successive points are not necessarily proximal to each other on the curve. Rather they jump about the curve in a chaotic fashion (Figure 3) either on the same branch or between the two branches. This becomes even rather apparent if we plot a run of $y_n$ against iteration number $n=1..2000$ (Figure 4; The gaps in the plot are where the $y_n$ jumped outside the range of $\pm 50$ which we set for good visualization). What is notable is that the rarer values in addition to appearing chaotically are also rather extreme: the sum of positive $y_n$ for 2000 iterations with $x_0=0.1$ is $4578.39$; of this just 12 values add up to $1005.48$, which is $\approx 22$ percent of the total sum. Each of these 12 extreme values is over 20 times the median value of positive $y_n$. The picture is roughly symmetric for the negative values.

Figure 4

Importantly, this chaotic behavior of $y_n$ is very sensitive to the initial values of $x_n$ with which we start the map. This is dramatically illustrated by two points close to root of $q=5$, $2.2$ and $2.21$ (Figure 5): while for the first 9 iterations the evolution of the two initial values is the same in direction though not magnitude, iteration 10 and beyond they go completely out of synchrony in both direction and magnitude.

Figure 5

In conclusion this map provides an analogy to think about certain processes in nature and historical events. First, it provides a potential model for foraging behavior of variety of organisms. In this model the clustering around the root values represents what might be called the base-line or ordinary foraging and the extreme jumps represent the drastic forays away from their local patch to distant locales. Such behavior may be seen in animals among herbivores moving to new feeding grounds far from their usual feeding areas or certain carnivores like sharks seeking new hunting waters far from their their current zone. This kind of behavior is also seen in certain ciliates like Halteria in the microscopic realm. In the world of protein sequences we see a similar tendency to keep to a tightly constrained space of diversity under purifying selection within which there is a low-radius exploration under neutral drift. This is punctuated by huge saltations that result from strong positive selection for new functional niches. This might happen within a family of proteins or in the proteome of an organism with some proteins showing big saltations in sequence space.

The great mathematician Benoit Mandelbrot, the pioneer in the study of chaos, has brought home the importance of distributions with rare events with extreme values. The role of such events in systems like financial markets has been recently explained at length by Nassim Taleb. In this context, the above-described map provides an analogy for one type of historical evolution of systems. Even with same basic parameter ( $\sqrt{q}$), clearly predictable bulk statistics (e.g. median and range of most frequent values) and similar starting values we see: 1) clear differences in long term evolution with non-overlapping chaotic extreme events different in both magnitude and direction. 2) Extreme events that can disproportionately contribute to the total numerical measure of the events in the series. A historical system evolving under such a model shows us how with very similar starting material and bulk behavior we can have a great difference in actual events and outcomes. This might be similar to actually observed phenomena like the fall empires or the sudden extinction of long-lasting lineages. This is a theme which we might explore further with examples of some other such systems which have been studied for their chaos.

## Ramanujan’s second construction for the approximate squaring of a circle

•October 19, 2016 • Leave a Comment

To experience the greatness of great men one has to relive or redo some acts of theirs to the best of ones ability. In ones youth such enactments might inspire one to make a bid for greatness. Whether this happens or not is mostly up to your genetics. Nevertheless, through the enactments one can at least savor the experience of what it takes to get there. If there was one man in our midst who could have lived up to be a Gauss or an Euler it was Srinivasa Ramanujan.

By redoing some of his acts that are within the grasp of our limited intellect we experienced the monument that he was. He gave two constructions for the approximate squaring of a circle using a compass and a straight-edge. We had earlier described the first and more widely known of those. The second appears in his paper titled “Modular equations and approximations to $\pi$“. In this paper in addition to remarkable approximations for the perimeter of the ellipse, which we had also alluded to before, he gives several series for $\pi=3.141592653589793...$. One of these series with just the first term leads to the below approximation:
$\pi \approx \dfrac{99^2}{2*\sqrt{2}*1103} = 3.141592730013305$ which is correct down to 6 decimal places. It is this kind of accuracy he captures in his first construction for the quadrature of the circle. In the midst of the dizzying series he conjures in a very Hindu style of mathematics, he says that he came up with an empirical approximation which leads to the below construction for the approximate squaring of the circle:

1) Draw circle to be squared with center O.
2) Draw its diameter $\overline{AB}$.
3) Trisect its radius $\overline{AO}$ to get a third of it as $\overline{AF}$.
4) Bisect the semicircle AB to get point C.
5) Draw $\overline{BC}$.
6) On $\overline{BC}$ mark $\overline{CG}= \overline{GH}=\overline{AF}$.
7) Join point A to point H to get $\overline{AH}$ and to point G to get $\overline{AG}$
8) With radius as $\overline{AG}$ cut $\overline{AH}$ to get point I.
9) Draw a line parallel to $\overline{GH}$ through point I to cut $\overline{AG}$ at point J.
10) Join points O and J to get $\overline{OJ}$.
11) Draw a line parallel to $\overline{OJ}$ through point F to cut $\overline{AG}$ at point K.
12) Draw the tangent to the circle at point A and cut it with radius as $\overline{AK}$ to get point L.
13) Draw $\overrightarrow{OL}$ to cut circle at point M.
14) Draw semicircle LM and perpendicular from point O to cut this semicircle at point N.
15) Triplicate $\overline{ON}$ to get $\overline{OQ}=3 \times \overline{ON}$.
16) Produce $\overline{OQ}$ in the opposite direction to cut circle at point R.
17) Draw semicircle RQ and a perpendicular from point O to cut it at point S.
18) Thus, we have $\overline{OS}$ as the side of the square OSTU which has approximately the same area the starting circle.

Ramanujan tells us that his earlier construction gave an “ordinary” value $\pi \approx \dfrac{355}{113}=3.141592920353982$, which is correct to six decimal places. This one, however, gives us the value:
$\pi \approx \left (9^2+\dfrac{19^2}{22}\right)^{\frac{1}{4}}= 3.141592652582646$
This is correct to a whopping eight decimal places keeping with the Hindu love for big numbers.

## Some meanderings among golden stuff-2

•October 4, 2016 • Leave a Comment

Related stuff:
Golden Ratio-0
Golden Ratio-1

If the golden ratio can fascinate erudite men of high IQ then what to say of simpletons like us. Hence, we shall here talk about some more trivia in this regard. The golden ratio is associated with a sequence, the method for generating which was provided in our tradition first by Piṅgala (the Meru-prastha) and was spelt out in full by Virahāṅka in his Vṛtta-jāti-samuccaya. Hence, we may call it the Meru-średhī ( $M$ with elements $M(n)$; $n= -\infty ...-2,-1,0,1,2,3... \infty$; among the mleccha-s it is commonly known as Fibonacci’s sequence).
$\lim_{n\to\infty} \dfrac{M(n+1)}{M(n)}=\phi \approx 1.61803398875$, which provides the relationship to the golden ratio.

Figure 1

In modern parlance this sequence can be seen as integer values emerging from the function (Figure 1):
$y=\dfrac{1}{2\phi-1}\left(\phi^x-\dfrac{\cos(\pi x)}{\phi^x}\right)$
This function oscillates with ever-increasing amplitude for negative $x$, hits $0$ at $x=0$, remains stable between $x=1..2$, then almost linearly climbs to $2$ at $x=3$ and then explodes nearly exponentially.

The sequence can be obtained from this function thusly:
$M(n)=\lfloor y \rfloor$ for $x=n=-\infty...-2,-1,0,1,2,3...\infty$.
$M \rightarrow ...34,-21, 13,-8,5,-3,2,-1,1,0,1,1,2,3,5,8,13,21,34...$
Thus having obtained the sequence, a well-known property of it, apparently discovered by Kepler, becomes apparent:
$M(n)^2=M(n-1) \times M(n+1)- (-1)^n$.

Figure 2

Now this relationship might be linked to an interesting geometrical procedure of dissection of a square which can performed only using the golden ratio (Figure 2). The procedure goes thus:
1) Let there be a square of side $a$.
2) The square is first partitioned into two rectangles one of sides $a$ and $\dfrac{a}{x}$ and other of sides $a$ and $a-\dfrac{a}{x}$.
3) Partition the first rectangle into two congruent right-angled trapezia with $base=\dfrac{a}{x}, height=\dfrac{a}{x}, top=a-\dfrac{a}{x}$. Partition the second rectangle along its diagonal into two right triangles with $base=a, height=a-\dfrac{a}{x}$.
4) Rearrange the the four pieces of the square thus obtained such that they reconstitute a rectangle of equal area as the starting square. This is done by taking two opposites sides of the rectangle to be the sides of the two trapezia $\dfrac{a}{x}$ and the other two opposite sides made by laying the shorter edge of the right triangle against the top of the trapezium. Thus, this side is $a+\dfrac{a}{x}$.
5) Thus, we have:
$a^2=\dfrac{a}{x}\left (a+\dfrac{a}{x}\right)\\[10 pt] \therefore 1= \dfrac{1}{x}+\dfrac{1}{x^2}\\ [10 pt] \therefore x^2-x-1=0$

The positive root of above is $x=\phi$. Thus, only by using the golden ratio can we convert a square into a rectangle as described above. For any other ratio you will have a parallelogram area in the middle which is either in excess or less than the area of the reconstituted rectangle.

Figure 3

Now instead of the golden sectioning of the square let us use a square of side $M(n)$ and divide it as above using $M(n-1)$ and $M(n-2)$ to make the four pieces (Figure 3; e.g. $8, 5, 3$). If we then reconstitute them as a rectangle we will get its sides as $M(n-1)$ and $M(n+1)$ (e.g. $5, 13$). Then by the above relationship regarding these elements of series $M$ this reconstituted figure will have a long thin parallelogram of excess or insufficient area relative to the reconstituted rectangle by 1 square unit. Thus, if we make a cardboard cutout of the above one could “hide away” that parallelogram worth of area difference along a roughly cut diagonal of the reconstituted rectangle. This can produce to paradox of square of side $8$ yielding a rectangle of sides $5,13$. Many years ago we had seen precisely such a “puzzle”, may be made of wood or cardboard. We must confess that then it took a little while before we realized that it stemmed from the golden dissection of the square. We recently learned via Mario Livio that this puzzle was invented by a guy called Loyd.

## Knotting a string: line, parabola, conchoid and knot

•October 1, 2016 • Leave a Comment

The basic construction
In course of studying various methods of constructing conics we stumbled upon a means of using the relationship between uniform circular motion (UCM) and simple harmonic motion (SHM) to construct four distinct loci with common procedure. They seem to describe the progression in the process of knotting a string — in a sense the “Platonic” ideal of the making of a knot. The basic construction goes thus:

1) Draw a circle with origin O as center and radius $r$. This is our directing circle.
2) Let A be a point on the circle and $\overline{BC}$ its diameter along the x-axis.
3) Drop perpendicular from A onto $\overline{BC}$ so that it cuts it at point D.
4) As point A performs UCM on the directing circle the point D performs SHM on the diameter $\overline{BC}$. We can locate point D on $\overline{BC}$ with respect the midpoint and and extremity of its oscillation using two segments, namely $\overline{OD}$ and $\overline{DC}$.
5) We thus get two reciprocal ratios: $\dfrac{OD}{DC}$ and $\dfrac{DC}{OD}$.
6) We draw the radial line which passes through points O and A.
7) Along this radial line, we locate two points symmetrically about point A, namely points E and F such that: $\dfrac{AF}{r}=\dfrac{AE}{r}=\dfrac{OD}{DC}$.
8) The loci of points E and F as A performs UCM on the directing circle are a parabola and a knot curve.
9) Likewise, we locate two points symmetrically about point A, namely points G and H such that:
$\dfrac{AH}{r}=\dfrac{AG}{r}=\dfrac{DC}{OD}$
10) The loci of points G and H as A performs UCM on the directing circle are a straight line perpendicular to the x-axis at r and a conchoid of Nicomedes.

When we derive their equations formally we get a pair of each of the four loci, as will be illustrated below.

Formal derivation of the equation of the parabolas

Let us first consider the case in the above construction as typified point F:
1) The coordinates of point F are $(x,y)$, which are the coordinates of the locus whose equation we are seeking to derive.
2) The coordinates of point A on the directing circle are $(x_1=r\cos(t), y_1=r\sin(t))$.
3) From the construction we obtain the following relationships:
$OD=x_1=r\cos(t)\\ DC=r-OD=r-r\cos(t)\\ \dfrac{AF}{r}=\dfrac{OD}{DC}\\[15pt] \therefore AF=\dfrac{OD}{DC}r=r\dfrac{r\cos(t)}{r-r\cos(t)}=\dfrac{r\cos(t)}{1-\cos(t)}\\[15pt] x^2+y^2=(r+AF)^2=\left(r+\dfrac{r\cos(t)}{1-\cos(t)}\right)^2=\dfrac{r^2}{(1-\cos(t))^2}\\[15pt] \dfrac{y}{x}=\tan(t)=\dfrac{\sqrt{1-\cos^2(t)}}{\cos(t)}\\[15pt] \therefore \dfrac{y^2}{x^2}=\dfrac{1-\cos^2(t)}{\cos(t)}\\[15pt] \therefore \cos^2(t)= \dfrac{x^2}{x^2+y^2}\\[15pt] \therefore \cos(t)= \pm \dfrac{x}{\sqrt{x^2+y^2}}$

With these relationships in place we can eliminate $\cos(t)$ to get:
$x^2+y^2=\dfrac{r^2}{\left( 1\pm \dfrac{x}{\sqrt{x^2+y^2}} \right)^2}=\dfrac{r^2(x^2+y^2)}{(\sqrt{x^2+y^2} \pm x)^2}\\[15pt] \therefore (\sqrt{x^2+y^2} \pm x)^2 =r^2\\ \therefore x^2+y^2+x^2 \pm 2x\sqrt{x^2+y^2} =r^2\\ \therefore (2x^2+y^2-r^2)^2=4x^2(x^2+y^2)\\ \therefore 4x^4+y^4+r^4-4r^2x^2-2r^2y^2+4x^2y^2=4x^4+4x^2y^2\\ \therefore y^4-2r^2y^2+r^4=4r^2x^2\\ \therefore (y^2-r^2)^2=(2rx)^2$

The above equation defines a double parabola. By resolving it we get a dyad of parabolas:
$y^2-r^2=2rx$ and $y^2-r^2=-2rx$
In the above construction we considered only one intersection between the radial line and the directing circle; in reality there are two. Hence, if we consider the intersection happens diametrically opposite to point A then we would get a second mirror image curve. This is why in the formal derivation we get a dyad of opposite facing parabolas.

The two parabolas share the x-axis as their common axis and expand in opposite directions. They intersect the x-axis at $\pm .5r$ and the y-axis and the directing circle at $\pm 1$.

Formal derivation of the equations of the knot curves

We next perform the same operation as above on the case typified by point E. Its coordinates are $(x,y)$, which are the coordinates of the locus whose equation we are seeking to derive.The following relationships remain the same for point E as above:
1) The coordinates of point A on the directing circle are $(x_1=r\cos(t), y_1=r\sin(t))$.
2)$\cos(t)= \pm \dfrac{x}{\sqrt{x^2+y^2}}$
3)$OD=x_1=r\cos(t) \; and \; DC=r-OD=r-r\cos(t)$
But here we have to consider everything in terms of point E; hence we have:
$\dfrac{AE}{r}=\dfrac{OD}{DC}\\[15pt] \therefore AE=\dfrac{OD}{DC}r=r\dfrac{r\cos(t)}{r-r\cos(t)}=\dfrac{r\cos(t)}{1-\cos(t)}\\[15pt] x^2+y^2=(r-AE)^2=\left(r-\dfrac{r\cos(t)}{1-\cos(t)}\right)^2\\[15pt] =\left(\dfrac{r-2r\cos(t)}{1-\cos(t)}\right)^2=\left(\dfrac{r-\dfrac{2rx}{\sqrt{x^2+y^2}}}{1-\dfrac{x}{\sqrt{x^2+y^2}}}\right)^2=\left(\dfrac{r\sqrt{x^2+y^2}-2rx}{\sqrt{x^2+y^2}-x}\right)^2\\[15pt] x^2+y^2=\dfrac{r^2(x^2+y^2)+4r^2x^2-4r^2x\sqrt{x^2+y^2}}{2x^2+y^2-2x\sqrt{x^2+y^2}}\\[15pt] 2x^4+y^4+3x^2y^2-5r^2x^2-r^2y^2=\\ 2x^3\sqrt{x^2+y^2}+2xy^2\sqrt{x^2+y^2}-4r^2x\sqrt{x^2+y^2}\\[10pt] 2x^4+y^4+3x^2y^2-5r^2x^2-r^2y^2=\\ (2x^3+2xy^2-4r^2x)\sqrt{x^2+y^2}\\[10pt] \left(2x^4+y^4+3x^2y^2-5r^2x^2-r^2y^2\right)^2=\\ \left(2x^3+2xy^2-4r^2x\right)^2\left(x^2+y^2\right)\\[10pt] 9r^4x^4-6r^4x^2y^2+r^4y^4-4r^2x^6-2r^2x^4y^2\\ -2r^2y^6+x^4y^4+2x^2y^6+y^8=0\\[10pt] (3r^2x^2-r^2y^2-2rx^3-2rxy^2+x^2y^2+y^4)\\ (3r^2x^2-r^2y^2+2rx^3+2rxy^2+x^2y^2+y^4)=0$

The above equation defines a double knot curve. By resolving it we get a dyad of quartic knots:
$3r^2x^2-2rx^3-r^2y^2-2rxy^2+x^2y^2+y^4=0$
and
$3r^2x^2+2rx^3-r^2y^2+2rxy^2+x^2y^2+y^4=0$

Each of the knots has the x-axis as their symmetry axis and respectively cut the x-axis at $\pm 1.5r$ which marks the maximum horizontal extent of the loop of the knot on the side opposite to the knot point. For both knots the points of inflection and knotting is at $(0,0)$. The free ends of the knot with the left-facing inflection cross over the loop of the knot respectively at $(r, \pm r)$. The free ends of knot with the right-facing inflection cross over the loop of the knot respectively at $(-r,\pm r)$. Thereafter the free ends run below the above pair of parabolas and converge towards them as $x \rightarrow \infty$. The maximum horizontal extent of the humps of the right-facing knot on the side of the knot/inflection point is given by the points $((3-2\sqrt{2})r, \pm (\sqrt{8\sqrt{2}-11})r)$. The same for the left-facing knot is $((2\sqrt{2}-3)r,\pm (\sqrt{8\sqrt{2}-11})r$. For the right-facing knot the maximum vertical extant is given by the points $(2\sqrt{3}-3)r, \pm (\sqrt{6\sqrt{3}-9})r)$. For the left-facing knot it is given by the points $((3-2\sqrt{3})r, \pm (\sqrt{6\sqrt{3}-9})r)$.

Formal derivation of the equations of the straight lines

Now we shall consider the cases with the reciprocal ratio. First we shall tackle the case represented by point H in the construction. Its coordinates are $(x,y)$, the coordinates of our desired locus.
We have: $\dfrac{AH}{r}=\dfrac{CD}{OD}; \; \therefore AH=\dfrac{CD}{OD}r$.
As above point A performing UCM on our directing circle has coordinates $(x_1=r\cos(t),y_1=r\sin(t))$ we get: $OD=r\cos(t); CD=r-r\cos(t)$.
From this we get $AH=\dfrac{1-\cos(t)}{cos(t)}r$
Now as above: $\cos(t)=\dfrac{x}{\sqrt{x^2+y^2}}\\[15pt] \therefore AH=\dfrac{\left(1-\dfrac{x}{\sqrt{x^2+y^2}}\right)r}{\dfrac{x}{\sqrt{x^2+y^2}}}=\dfrac{(\sqrt{x^2+y^2}-x)r}{x}$
Now for the point on the locus we have:
$x^2+y^2=(r+AH)^2=(r+\dfrac{(\sqrt{x^2+y^2}-x)r}{x})^2\\[10pt] x^2(x^2+y^2)=(r\sqrt{x^2+y^2}-rx+rx)^2\\ x^2=r^2$

The above equation defines a dyad of lines which can be resolved as:
$x=r$ and $x=-r$; These are a pair of vertical lines that cut the x-axis at $(\pm r)$.

Formal derivation of the equations of conchoid

Finally, we shall consider the point G $(x,y)$ in the construction to derive locus it specifies. We have $AG=AH$; hence from above we get $AG=\dfrac{(\sqrt{x^2+y^2}-x)r}{x}$. However, for point G we have:
$x^2+y^2=(r-AE)^2=\left(\dfrac{rx+rx-r\sqrt{x^2+y^2}}{x}\right)^2\\[15pt] \therefore x^2(x^2+y^2)=(2rx-r\sqrt{x^2+y^2})^2\\ x^4+x^2y^2=5r^2x^2+r^2y^2-4r^2x\sqrt{x^2+y^2}\\ (x^4+x^2y^2-5r^2x^2-r^2y^2)^2=16r^4x^2(x^2+y^2)\\[10pt] (3r^2x^2-r^2y^2-2rx^3-2rxy^2-x^4-x^2y^2)\\ (3r^2x^2-r^2y^2+2rx^3+2rxy^2-x^4-x^2y^2)=0$

The above equation defines a double conchoid and by resolving it we get the two conchoids:
$x^2(r-x)(x+3r)=y^2(x+r)^2$ and $x^2(x+r)(3r-x)=y^2(x-r)^2$

The two conchoids have the respective lines $x=\pm r$ as the asymptotes of their two branches. The bow-like branches of the two conchoids intersect the x-axis at points $\pm 3r$. The loops of the two conchoids cut the x-axis at $\pm r$ and the point of looping is at origin coincident with the point of knotting of the knots. Thus the two asymptotic lines are the tangents to the loops of the opposite conchoid at $x= \pm r$. The loop of the right-looped conchoid bounded by the points $(x \approx 0.587401r,y \approx \pm 0.450196r)$ which mark its maximum vertical extent. For the loop of the left-looped conchoid the corresponding bounds are the points $(x \approx -0.587401r,y \approx \pm 0.450196r)$.

Polar equations of the above curves
From the construction and the above derivations we can also easily derive the polar equations of the four curves of the original construction thus:
1) Right diverging parabola:
$\rho=r\left(1+\dfrac{\cos \left(\theta \right)}{1-\cos \left(\theta \right)}\right)=\dfrac{r}{1-\cos \left(\theta \right)}$
2) Right facing knot:
$\rho=r\left(1-\dfrac{\cos \left(\theta \right)}{1-\cos \left(\theta \right)}\right)=\dfrac{r\left(1-2\cos \left(\theta \right)\right)}{1-\cos \left(\theta \right)}$
3) Right straight line:
$\rho=r\left(1+\dfrac{1-\cos \left(\theta \right)}{\cos \left(\theta \right)}\right)=\dfrac{r}{\cos \left(\theta \right)}$
4) Right-looped conchoid:
$\rho=r\left(1-\dfrac{1-\cos \left(\theta \right)}{\cos \left(\theta \right)}\right)=\dfrac{r\left(2\cos \left(\theta \right)-1\right)}{\cos \left(\theta \right)}$

In addition to having a nice symmetry to them these equations also represent an interesting progression in the process of knotting a linear entity like a rope or a polymeric molecule like the DNA double-helix: the straight line represents the starting state of the linear entity. The parabola represents the next with a bending appearing in it. Third it undergoes more extensive bending and looping as represented by the conchoid. Finally, it undergoes knotting in the form of the knot curve. Alternating, between the loci specified by the two reciprocal ratios we get this progression. Thus, the combination of UCM and SHM can be seen as mechanism to specify a knot through serial transformation of a linear entity. Another notable point is that the pair of loci produced by the same ratio are mutually asymptotic at their free ends.

Points of intersection

Dynamic version:

Finally, we shall consider the non-obvious points of intersection between the above loci which have not been mentioned earlier. There are 4 sets of them in each of the 4 quadrants which are symmetric when we have all 4 pairs of loci. Hence, we provide those only for the first quadrant with those for the remaining quadrants being the same except for sign changes.

Generating circle-conchoid: $\left(\dfrac{1}{3}r,\dfrac{2\sqrt{2}}{3}r\right)$

Generating circle-knot: $\left(\dfrac{2}{3}r,\dfrac{\sqrt{5}}{3}r\right)$

Knot-conchoid inner branch: $\left(\left(1-\dfrac{1}{\phi }\right)r,\left(\sqrt{2-\dfrac{1}{\phi }}\right)r\right)$ where $\phi=1.61803398875$ is the Golden ratio. This intersection produces an angle of $\dfrac{2\pi }{5}=72^o$ with respect to the x-axis thereby defining a regular pentagon

Knot-conchoid outer branch: $\left(\left(1+\phi \right)r,\left(\sqrt{2+\phi }\right)r\right)$

Left-facing parabola-conchoid inner branch: $\left(\left(2-\sqrt{3}\right)r,\ \left(\sqrt{2\sqrt{3}-3}\right)r\right)$

Left-facing parabola-conchoid lobe: $\left(\left(\sqrt{2}-1\right)r,\left(\sqrt{2}-1\right)r\right)$

Left-facing parabola-knot: $\left(\left(1-\dfrac{1}{\phi }\right)r,\left(\sqrt{2\phi -3}\right)r\right)$

Right-facing parabola-conchoid inner branch: $\left(\left(\sqrt{2}-1\right)r,\left(\sqrt{2\sqrt{2}-1}\right)r\right)$

Right-facing parabola-conchoid outer branch: $\left(\left(1+\sqrt{2}\right)r,\left(1+\sqrt{2}\right)r\right)$

## The Apollonian parabola

•September 25, 2016 • Leave a Comment

Some say that Archimedes and Apollonius of Perga (modern Murtina in Turkey; the center of the great yavana temple of the goddess Artemis in the days of Apollonius) were the two great yavana-s who might have rivaled Karl Gauss or Leonhard Euler in their in intellectual achievements. Beyond doubt a whole lot of modern knowledge rests on the foundation of Apollonius and in the west one could perhaps say that a Kepler or a Newton may have never shone forth had that Apollonian foundation not existed. Indeed, even today the study of Apollonius is a profound experience for the man who can do so. As with yavana religion much of their knowledge including the books of Apollonius were lost due to the ravages of pretonmāda in the west. Not surprisingly even at a later age the Lord Protector of England the duke of Somerset was burning books with yavana geometrical diagrams at the Oxford university.

Ironically in the world of its younger sister, the marūnmāda, Apollonius held a great fascination among those influenced by (Neo)-Platonism. Among the marūnmatta-s, their intellectual ibn Sina from Persia was perhaps the most fascinated by Apollonius. This allowed the preservation of much more of his works than in the west. Not surprisingly later marūnmatta-s referring to his work said that while his Qoranic studies were fine, he should be accused of kaffr-hood, and “innovation” derived from yavana and Hindu knowledge. Indeed, some mullahs suggested that he should have been cut pieces and feed to raptors. Nevertheless, his studies on yavana geometry and Hindu trigonometry has resulted in some interesting preservations which would have otherwise been lost. One such we shall discuss here is an Apollonian construction of a parabola using the geometric mean theorem. It goes thus in modern terminology:

1) Let point O be the origin. Take a point A on the x-axis at a given negative coordinate $(-a,0)$.
2) Let B be a point moving on the x-axis from the origin in the positive direction.
3) Draw a circle with diameter as $\overline{AB}$.
4) Draw the tangent to this circle at point B. It will be perpendicular to the x-axis.
5) The above circle cuts the y-axis at points C and D. Draw lines through these points which are parallel to the x-axis.
6) The above pair of lines will intersect the tangent to the circle at point B at points E and F.
7) The locus of points E and F as B moves along the x-axis is the desired parabola.

From the construction we see:
$\overline{OA}=a$; The x-coordinate of the parabola is $\overline{OB}=x$; The y-coordinate of the parabola is $\overline{OC}=\overline{OD}=y$. Given the circle used for the construction we notice that $\overline{OC}=\overline{OD}$ is the geometric mean of $\overline{OA}$ and $\overline{OB}$ because $\overline{AB}$ is the diameter of the circle (geometric mean theorem). Thus we get:
$y=\sqrt{ax}$
$\therefore y^2=ax$
which is the equation of our desired parabola.

## Leaves from the scrapbook

•September 18, 2016 • Leave a Comment

There were extensive memoirs in the form of electronic scrapbooks of Somakhya, Lootika and some members of their circle. Those in the know read the available excerpts due to matters of considerable interest being recorded in them. Other parts were written in a cryptic language that most could not understand what was really in them. These leaves are from the record of Lootika; some of what is contained in them might have a parallel version in Somakhya’s record.

Entry 22; mārgavakra-mīna, Rākṣasa year of the first cycle: These summer days Vrishchika and I sleep on the terrace. We are often up late talking and duly woke up late too much to our parents dislike. We watch with awe the starry vault turn above us, looking out for meteors, Ulka-s, which are pratyakṣa-s of the dreadful elephant-headed Vināyaka deva-s, four in number created by the mighty Skanda. We often try to catch the faintest objects on good nights. Vrishchika fills me in on the work she is  doing on the genetics of subtler visible quantitative traits in humans and we break our heads about the many problems in our investigation before lapsing into a pleasant sleep dreaming of being in the embrace of our kāmin-s against their firm chests. Many aspects of such a study have been surprisingly neglected as it was derided as no better than disreputable things like phrenology, physiognomy, racism and the like. While with my assistance Vrishchika has a good handle of how to study and understand the actual sequence variation at various loci, we are hard-pressed with many aspects of the quantitative genetics. Somakhya has a long distance friend named Indrasena with whom he is close and often in touch though he has seen him only once or twice. He is from the distant city of Udyānapurīṣa where Somakhya’s cousins Mandara and Saumanasa live. This Indrasena is a jack of many trades and I would love to do intellectual battle with him and over all emerge his superior. But he is quite a master of quantitative genetics and Somakhya has introduced Vrishchika to him to get to help with her roadblocks. I suspect Vrishchika has taken quite a fancy for him and might even combine forces with him against me if such a battle were to be joined. In any case they are now able to detect some interesting signals in our data which we could investigate further.

My classmate Sumalla (Sumallā) is considered very beautiful and desired by many males in my class and beyond. However, Vrishchika identified certain subtle physiognomic traits that made her infer based on her current analysis that Sumalla would develop a metabolic condition starting as early as her 25th year that will ruin her good looks and even make her less desirable by the time she nears her 30th year. She volunteered to be a data point and Vrishchika has remarkably detected four informative polymorphisms in the transcription factor-encoding genes NKX2-6, SMAD6, and MEIS1 and the neuropeptide gene NMU, which make it likely indeed that Sumalla would develop the condition Vrishchika has prognosticated. I wondered how the news should be broken to her but Vrishchika seemed quite happy to handle that – after all she’s the one who is going to be a physician. This brings me back to an interesting evolutionary angle which I had discussed at some length with Somakhya: why is it that such variation has survived? Perhaps the earlier age of mating allowed such variation to pass through is one hypothesis. An alternative is that in the past the extant of food availability selected for these traits that are deleterious in today’s dietary landscape. A further alternative is they were protective against infectious disease(s) that are no longer as threatening to fitness as the metabolic condition these variations facilitate. I wonder if Indrasena will have any success at all in obtaining DNA from skeletons he has identified in a cemetery that he believes comes form a time before the coming of the ārya-s to India.

Entry 23; madhu-biḍāla, Rākṣasa year of the first cycle: Vrishchika is making clones for making recombinant Neuromedin (NMU). I have asked her to also make a clone of a CDC123-like ATP-grasp enzyme from Legionella, which I intend using in the ambitious protein “stapling” experiment. For the past several nights we have seen a new star appear in Cygnus between the stars forming base of the neck and the head of the goose. As a result Vrishchika and I had a tangential conversation on what the beings on various planets around stars along the Milky Way stretching above us might be thinking even as they look out into their skies. Whether there would be beings out there wondering, just as we are, if someone is watching their star from a planet around another star. It made me feel that one area where we have a deeply limiting lacuna in our knowledge is the range of forms life and intelligence on other worlds might assume.

Entry 24; cala-saṃkhya, Rākṣasa year of the first cycle: I wished to learn about the new star we had seen appear in Cygnus; so I went to meet Somakhya. He too was excited about it and informed me that that it was a pulsating red giant called $\chi$ Cygni having reached its maximum. It is a Mira-type variable, whose prototypical star Mira in Cetus he had shown me few years earlier. Thereafter, I had such a engrossing session with Somakhya. It all began when I asked Somakhya about the variability of the star $\eta$ Aquilae which we had noted waiting for sleep to overtake us. He informed me that it was a pulsating star called a Cepheid. From there we moved on to a discussion of oscillators other than the $\sin (x)$-like functions. Somakhya had taught me the van der Pol equation a couple of years ago when explaining how one numerically solves differential equations but I had not paid much attention to its specifics. Today he showed me how some forms of the vdP equation produced oscillations which could show a bit of chaotic behavior. Forms of the vdP are good simulators of the curves of (semi)regular pulsating variables and we played with them a little trying to recapitulate what Somakhya explained to me as being the Blazhko phenomenon in Cepheid variability i.e. the periodic modulation of the basic mode of pulsation.

Entry 25; cala-stṛ, Rākṣasa year of the first cycle:

Simple reaction kinetics

The session I had with Somakhya was so exciting that I had to try to relay all I had learned to my sisters. To give them some background I began by explaining simple chemical kinetics using simple differential equations. Even little Jhilli was able grasp much of it suggesting she is coming of age. It was a good revision for me; hence, I am recording the highlights here. After we had discussed the vdP equation Somakhya and I had segued into other types of oscillations in nature. I am quite conversant with the simple prey-predator population dynamics model: the prey is growing at an exponential rate depending on its population size at time t in a food-rich environment. When prey and predator meet the prey is killed, so its population growth rate is negatively affected by this interaction, which scales as the product of the prey and predator population sizes. In the case of the prey its population grows when it gets food so its growth rate scales as the product of the prey and predator populations while it decreases due on death from inter-predator conflict and migration which are proportional to prey population at time t. This gives us the well-known Lotka-Volterra equation.

Lotka-Volterra prey-predator population oscillations

Then Somakhya introduced me to two interesting equation systems that have been discovered by mathematicians [Footnote 1] which can be seen as different variants of the simple LV equation that describes in population oscillations of prey and predator locked in conflict.

Emergence of chaotic population dynamics in a simple system of 3 complexly interacting species

The first of these can be seen as a model describing population dynamics of 3 complexly interacting species X, Y and Z. X, like in the LV system, grows at an exponential rate proportional to its population at time t. Its growth rate is also similarly proportional to another symbiotic species Z, which might be producing metabolites useful for its growth. In contrast, its growth rate is depressed by its encounter with a parasite Y; thus the decrease in growth rate is proportional to the product of the X and Y populations as in the simple LV system. Like in the simple LV system it grows at a rate proportional to encountering the species X and directly deriving nutrition from it; thus its growth rate is proportional to product of the X and Y populations. Additionally, the growth of Y benefits from metabolites which X produces even without Y directly feeding on X. As a result the total positive dependency of the growth rate of Y is proportional $X^2Y$. Like in the simple LV equation Y again loses individuals to death and migration arising from inter-parasite conflict at a rate which is proportional to its own population at time t. Y has no interaction with Z; hence its growth rate shows no direct dependency on it. Now species Z needs to directly interact with X to grow; hence, its growth rate is positively correlated to product of the populations of X and Z. On the other hand the increase of X by itself results in loss of resources for Z’s growth, so its growth rate shows a further negative dependency on population of X by itself and Z too loses a certain part of its population to death and migration at a rate proportional to its current population size. Remarkably, this system has a notable range of parameters where the populations X, Y and Z vary chaotically. As a result their population maps produce strange attractors similar to the famous Rössler or Lorenz attractors.

Chaotic population dynamics of four competing species with a basic logistic growth

The second model which Somakhya introduced me to was one where multiple species are growing in an environment with a fixed carrying capacity, i.e. logistic growth under the Verhulst equation. Additionally, they also negatively interact with each other, i.e. compete with each other as per the standard LV model. In such a model chaotic population dynamics can emerge even with just 4 competing species. While with 4 species the chaotic picture is seen in a relatively small parameter range, with more species chaos becomes much more common. What this tells us is that chaotic dynamics are likely a regular feature of nature. One of the species in this model under certain starting conditions seems almost certainly prone to extinction before it makes a dramatic oscillatory come back which would never have been expected unless you had learned of such dynamics.

Entry 26; ugra, Rākṣasa year of the first cycle: I reflected with my sisters on the general implications of what we had learned about chaotic behavior in nature. It could produce many impressions some simultaneous and some exclusive in the same or different persons. 1) Even among those without much conceptual grasp or generative capability the strange attractors could produce an aesthetic experience. Somakhya conjectured that randomness does not produce an aesthetic experience in most people – indeed few would call a random scatter of points as aesthetic. In completely convergent attractors like an ellipse or the oblongs produced by LV solutions several more people might have an aesthetic experience but still it might not be very widespread. But with the strange attractors certainly many more would have such an experience – something which might be related to their fractal dimension. Perhaps, this is the reason why the devāyatana-s of our ancestors tended towards increasingly fractal states. We will have to explore this more but even this aesthetic experience is perhaps much more in people who can grasp the rahasya-s. 2) In the person characterized by quantitative arrogance, it might produce a belief of being able to prognosticate complex systems. Such a person might think that they could produce models to sufficiently predict high degrees of complexity. 3) In a person who has studied nature it might produce simultaneously a sense of insight and humility. It makes one aware that behind the seemingly disordered state lie laws. Chaos in itself should be seen as a given in nature, so there could be selection for systems that factor in chaos. 4) For the historian trying to prognosticate the fate of nations it produces a strange paradox. The possibility of such chaotic behavior being common place has an almost seductive hold for those who see it in action. Like in the case of one with quantitative arrogance, one might think this is the way going forward with even more complex equation systems being able to ultimately predict history. Yet for all the swirling chaos in its undergirding macro-history does seem far less chaotic – hence one might ask why?

Entry 27; harivāhana, Rākṣasa year of the first cycle: The dreadful box returns: the red ellipse and the red hyperbola. All this began years ago when we were in school. I have hitherto only recorded bits and pieces out of fear of the thing. But now that it thrust itself back into my existence I am moved to record the whole story supplemented by Somakhya’s own records on the mathematical curiosities that surround it.

It was shortly after I and my sisters had transferred to the new school, an event which was to have a momentous impact on my existence. Intentionally or unintentionally it seemed as if I had rather quickly acquired some notoriety in my class and soon I had a lot of classmates giving me company at almost every free moment in school. At lunch break my classmates Vidrum and Hemalinga came to me and said they had a method to trisect any angle with just a compass and a ruler. I had heard from Somakhya that this was not generally possible and told them so. But they showed me a construction which really seemed to trisect the angle when I checked it using my protractor. I asked them as to how they had discovered it. Vidrum said that he had this great geometry box whose instruments would almost magically guide him towards drawing various figures. Curiously, Vidrum nearly lost that box a month before that incident and Vrishchika and I had found it and restored it to him. That morning Vidrum drew one such a figure and was puzzling over it when Hemalinga studied it and realized that it was a solution for the trisection of the angle with just a compass and a straightedge [Footnote 2].

Construction to approximately trisect a given angle

Just then I caught Somakhya going to eat his lunch at a secluded spot away from everyone else. I called him over and soon Vidrum and Hemalinga described their construction with a triumphant laugh. Somakhya half-smiled and said they were certainly wrong. Over the rest of the lunch break he pored over the figure and proved using some geometry that it was not a true trisector but a saw-tooth like function of the form:
$y=\arctan \left(\dfrac{2\sin \left(\frac{x}{2}\right)}{1+2\cos \left(\dfrac{x}{2}\right)}\right)$
He called it the approximate trisector and showed that it rather closely approximated $y=\dfrac{x}{3}$ in the domain $-\dfrac{4\pi}{3} \leq x \leq \dfrac{4\pi}{3}$

The approx-trisector function

Using this function he established that at $x \approx 124.3^o$ the difference between the real and this approximate trisection was $-1^o$. Thus, for the angles we had tried it was close or below the resolution of my protractor. Hemalinga, who was known in the school for his prodigious mathematical capacity, seemed a bit red-faced and Somakhya rubbed it in a bit further after calculating the first term of the Maclaurin series of approx trisector showing it was $y=\dfrac{x}{3}$ thus proving why it seemed to work. Two years ago Hemalinga ran into us, and as though to make a point to Somakhya, showed us that he had integrated by hand the approx trisector function – something which filled a whole page. He felt very pleased when Somakhya remarked that he could never done that without a computer.

That incident aside, coming back to the the matter at hand, by the end of that school day both Somakhya and I were feeling strangely unwell. I remarked that it might because we had not eaten since morning having skipped lunch due to the pursuit of the approx trisector. However, it did not go away and we were mysteriously unwell for 3 days without any apparent cause and during that illness had a disturbed sleep from the repeated apparition of a saw shaped like the approx-trisector function cutting through us. In course of the summer vacations that year Somakhya smashed his own finger due to my closing his eyes shortly after establishing the properties of a geometric figure which Vidrum had drawn. Early the next year a girl in our class to whom Vidrum was close had died mysteriously right in cemetery near his house. I distinctly remember Vidrum picking up his geometry box from her desk the day after the two great kṣatriya-s who respectively delight in svāhā and svadhā had visited her.

Sometime around the middle of that semester with the boring exams temporarily past us, Somakhya pointed to a purple velvet bag lying on the parapet below the window beside which his desk was stationed at school. He used to be curious about its contents. Hence, one day after school he climbed on to the compound wall and jumped forward to hold on to the parapet and clamber on to it. Thus, he reached the bag that intrigued him so much. But even as he picked it up the whole parapet came crashing down. Luckily, other than being dusted up he was not hurt and our land unlike that of the mleccha-s had no surveillance device for anyone other than me and Vrishchika to know what had really happened. From the bag he retrieved a sturdy geometry box which had elegant instruments, a slide rule with fine calibrations and stencils of ellipses, circles and hexagons and a remarkably smooth curve-fitting tape. I immediately recognized this distinctive box as being that of Vidrum and informed Somakhya that it was so. He decided to repatriate it to Vidrum the next day. To my utter surprise Vidrum refused to take it and even denied that the distinctive box was his. I was utterly puzzled by his denial but Somakhya was more than happy to keep it as its finder. There was a śūlapuruṣīya inscription indicating that the box was made in the Śarmaṇyadeśa – indeed modern Hindus never produced anything of that quality and robustness.

Now this box was a strange one indeed. Somakhya remarked that the curve-fitting tape would eerily take the shape of the locus he was setting out to draw. I recalled Vidrum’s weird statement of how it would almost magically draw figures for him. Somakhya had procured a nice drawing board and a roll of math-paper to go with it. It was then that he introduced me to recreational geometry and kindled my interest for the first time in the subject. Notably, Vidrum had entirely lost interest in drawing geometric figures thereafter and for that matter any interest in geometry beyond passing the exams. Unlike in the past he would ask us as to why we found it so interesting. One day Somakhya presented me a simple geometrical challenge: to draw a conic given its eccentricity. He had told me how his father had figured that one for himself when he was 10 years of age but did not give Somakhya the solution urging him to figure it out for himself. Still being daft at geometry, I struggled to figure it out and as I was thinking about it I lifted my hand to my forehead, still holding the compass from that box. Then, I thought I had an idea and at that moment I flicked my hand off my forehead. Somehow the compass had hooked my spectacles and as I did so it pulled them off and hurled them against the wall with great force damaging it. My parents were very angry with me that day as it was the second set of glasses I had wrecked that month, having lost the first while playing table tennis with Abhirosha.

The next day I was still fighting with the construction when Somakhya gave me a hint: “The eccentricity of a conic is the tangent of an angle between 0 to $\dfrac{\pi}{2}$.” This immediately fired the light bulb in my head and I took that geometry box and drew out the construction. As I placed the curve-fitting tape to draw out the ellipse, it neatly folded into an elliptical path almost magically. I was amazed and drew a nice ellipse with the red pen which looked like an egg of the goddess Vinatā bearing the aquiline god. I next drew a parabola with the angle set at $\dfrac{\pi}{4}$ I wanted to show this construction to my family, so I carefully placed it in my file so that it might not get folded. That night I explained the construction to my father and drew out the sheet to show it him when to my anguish and embarrassment the parabola and the ellipse had vanished even though the rest of the construction was intact. The next day I showed it to Somakhya who was also surprised by how cleanly the conics had vanished. He had seen them himself and wondered how that could be. He brought out the red pen and drew them again himself. We ensured they were there and I again put the sheet into my file. That afternoon just before leaving home I checked the sheet again and the conics had vanished again. Startled I just threw sheet away and went home.

Some days later Somakhya had shown me how to genuinely trisect any angle using a hyperbola. He had again constructed a red hyperbola and demonstrated the construction to me. He was then talking of some peculiar properties of a related parabola and ellipse which were approximate dividers of the angle in certain ratios. He then gave the sheet with the construction for me to repeat it with my own substandard instruments. That evening after school Somakhya was approached by a classmate whose name I do not recall clearly. He belonged to a community previously classified as a depressed class and his dark grayish yellow complexion and facial features suggested that most of his ancestry derived from the mysterious tribal peoples who inhabited India before the coming of the later waves of humans. That guy hardly had any intellectual proclivities to be able hold even a limited conversation with Somakhya but he was perhaps the only guy in the class who had a deep fancy for the paper objects that Somakhya made following the Japanese way (Some of our teachers verily hated Somakhya for his love for this art which was compounded by his utter disdain towards some of them). Indeed, Somakhya used to remark that this paper folding technique of the easterner islanders marked them as a people of great creativity. Hence, whenever that guy would come with sheets of paper Somakhya would indulge him and fold those objects, like an eagle, a bear, a starfish, a walrus and boxes of different types. That guy for his part would collect and neatly preserve Somakhya’s creations in a large box. Thus, that day as Somakhya was folding paper, a lout who was also in our class, whom we knew as Sphichmukh, surreptitiously stole his bag with the geometry box and his drawing board and swiftly made away.

Somakhya was utterly disappointed and the next day he asked me for he sheet of paper on which he had drawn the hyperbolic trisection of an angle, perhaps with a longing for the stolen box. I took it out and we looked at each other in horror: the hyperbola had utterly vanished! I could read Somakhya’s mind that he was beginning to have conflicting feelings about the box. However, he could not take the theft lying down. Via several inquiries he obtained evidence that the thief was Sphichmukh. He decided to recover the stuff and said that he was setting out with Sharvamanyu and Vidrum to punish Sphichmukh and seize the stolen stuff. I saw them mount their bikes and tail Sphichmukh after school. Knowing that Sphichmukh was a lout with other hoodlums as friends they had armed themselves for the exploit. Somakhya had a bicycle chain, Vidrum a nunchaku, and Sharvamanyu a knife. Seeing all this I felt a mixture of fear, curiosity, and excitement and followed them on my bike at a safe distance to see what would happen. As they closed in on Sphichmukh he realized that his game was up, but given his usual temperament he furiously rode to the edge of a pond and threw the bag with the geometry box and Somakhya’s drawing board into the murky waters where buffaloes bathed. The three were livid with fury and it seemed they would have give Sphichmukh himself a jalasamādhi; I feared they could land in prison themselves from that. So I pedaled hard to quickly reach the three of them and suggested that rather than smiting Sphichmukh or shoving him into the pond they should complain to the school authorities and I volunteered as an additional witness for Sphichmukh’s crime. For some reason Vidrum then rather passionately asked Somakhya to let go off the box and forget about it.

Perhaps, a bit mollified by my sight they desisted from a direct attack and Somakhya complained as suggested. However, poor Somakhya had little traction with the school authorities; much to his chagrin, they informed him that since the constructions he was drawing and the board were not part of the curriculum, which was being taught at school they would not take any remedial action. They let off Sphichmukh after mildly lecturing him about the impoliteness of taking others things. Somakhya revealed to me then that more than the loss of the box which he had himself obtained for free he had lost something more precious with it. In the same bag he had kept a khārkhoḍā with the yantra of the pūrvāṃnāya with the trident and the three bhairavī-s and the 3 supine bhairava-s. With the loss of the bag he also lost his siddhi of the mantra-s of the pūrvāṃnāya.

Now we come back to the present. Starting this week we have changed our schedule due to much haranguing by our parents. All four of us get up earlier and bike to the pond and run thrice around it. Thereafter Vrishchika and Varoli return home because they spend a couple of hours preparing for their respective entrance exams before continuing with all the fun science we are doing. They seem more sincere in this business than I ever was. However, I and little Jhilli, being more carefree, go to the adjacent hall to play table-tennis for some time along with Abhirosha. Abhirosha is attempting a difficult exam for whatever she wished to do, which was quite removed from my path of life. Nevertheless, that exam had several stiff mathematical tests. I had trained her the previous year for her university entrance exam and knew that she was quite capable of surviving the impending tests in algebra, calculus and numbers. But she does not have much of flourish in geometry so she is back to consult me. Thus, I and Jhilli decided to look at her problems. There was a problem of polygons whose areas and circumferences approximate $\pi$. I remembered that Somakhya’s father had once posed that to me and Vrishchika – he wanted to see for himself if we were really what what people said about us. I let Jhilleeka solve that one. Then she took out a sheet of math-paper and showed a failed attempt of a construction of what should have been a Cartesian oval on it. As she showed it she pointed to some unnecessary lines and remarked: “Hell, where did these vertical lines come from out of the blue! This is is spooky.” Just then I caught sight of her geometry box and my jaw literally dropped: “How on earth did you get that box?” Abhirosha: “Actually, that’s bit strange. I found it by chance on the bank of the pond sometime ago.” I looked at it more closely and asked Abhirosha: “Did it come with a bag?” Abhirosha: “Not it was all by itself dented and worn but the instruments inside were intact. Seems like the famous German engineering.” I noticed that the dent corresponded to it being trampled by the hoof of a buffalo. Now I was not surprised by the strange lines that seemed to have appeared by themselves. More Abhirosha told me of it, it was apparent that our old box had come back into our lives again. I have sent an email to Somakhya detailing these strange events. I also sent one to Vidrum inquiring about the box. He responded rather quickly saying : “You guys dabble with such things anyhow. You can take care of him and please don’t get me into this matter again. I believe he was perhaps a civil engineer from a town among the Karṇāṭa-s known as Hiriyuru.

:::::::::::::::::::::::
Footnote 1: e.g. works of Sprott and the like

Footnote 2: The construction goes thus:
1) To trisect angle A in this case $57^o$ draw circular arc BC cutting the two rays of the angle at B and C.
2) Then bisect angle A to intersect $\widehat{BC}$ at point D. Connect Point D to point B and C to get segments $\overline{DB}$ and $\overline{DC}$.
3) Trisect $\overline{DB}$ to get point H as the beginning of the 3rd segment of trisected $\overline{DB}$.
4) Join point A to H to get $\overrightarrow{AH}$. $\angle{BAH} \approx \dfrac{\angle BAC}{3}$.
Let the approximate trisection of $\angle BAC=x$ be $\angle BAH= \beta$. From construction $\angle BAD =\dfrac {x}{2}$ and $\angle BAD =\dfrac {x}{2}-\beta$. In $\triangle{ABD}$ we get from construction $\angle ABD=\angle BDA$. Using sine rule we get:
$\dfrac{\sin(\beta)}{\overline{BH}}=\dfrac {\sin (ABH)}{\overline{AH}}=\dfrac {\sin(x/2-\beta)}{\overline{DH}}$

From the construction we get $\overline{BH}=2 \overline{DH}$
Thus we have: $\sin (\beta)=2\sin (x/2-\beta)$

$\sin (\beta)=2\left(\sin (x/2)\cos (\beta)-\cos(x/2)\sin(\beta)\right)$

$\tan(\beta)=2\sin(x/2)-2\tan(\beta)\cos(x/2)$

$\tan(\beta)=\dfrac{2\sin(x/2)}{1+2\cos(x/2)}$

Thus we get: $\beta= \arctan \left(\dfrac{2\sin \left(\dfrac{x}{2}\right)}{1+2\cos \left(\dfrac{x}{2}\right)}\right)$