Some novel observations concerning quadratic roots and fractal sequences

Disclaimer: To our knowledge we have not found the material presented here laid out here presented in completeness elsewhere. However, we should state that we do not follow the mathematical literature as a professional and could have missed stuff.

Introduction
\sqrt{2} has captivated human imagination for a long time. Perhaps, its earliest mention is seen in the tradition of the Yajurveda, which provides an approximation for the number in the form of the convergent \tfrac{577}{408} for construction of diagonals of squares in the vedi (altar) for the soma ritual. Yet, it has secrets that continue to reveal themselves over the ages. Here, we shall describe one such, which we stumbled upon in course of our study of sequences inspired by Nārāyaṇa paṇḍita, Douglas Hofstadter and Stephen Wolfram’s work.

A fractional number h lends itself to an interesting operation (the floor-difference sequence; we had earlier described it here; an operation studied by Wolfram),
f_0[n]=\lfloor (n+1) \cdot h \rfloor -\lfloor n\cdot h \rfloor
Here the integer sequence f_0[n] is defined by performing the above operation. If we use h=\sqrt{2} results in the sequence,
1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1…
This pattern of 1 and 2 is not periodic. Nevertheless, it has defined pattern. Wolfram showed that it can be produced by a substitution system entirely independently of \sqrt{2}, namely,
1 \rightarrow 1,2 and 2 \rightarrow 1,2,1
Notably, the ratio of the number of 1s to 2s in the string produced by the floor-difference operation (or equivalently the substitution system) converges to \sqrt{2}. Thus, the numbers of 1s and 2s in the sequence f_0 generated by the above process results in convergents that are like the partial sums of the continued fraction expression of \sqrt{2},

\sqrt{2}= 1+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\dots}}}}

Thus, in f_0[1:70], the number of 1s is 41 and the number of 2s is 29. This gives us a convergent for \sqrt{2} as \tfrac{41}{29}=1.413793 which is the 4th partial sum of the above continued fraction.

Case-1: Summation sequences defined on the floor-difference operation

Next we define a second integer sequence f_1 based f_0 thus,
f_1[n]=\displaystyle \sum_{k=1}^n f_0[k]==1,

i.e. we take the sum of all 1s present till position n in sequence f_0. Thus, for the above 20 terms of f_0 the corresponding terms of f_1 are,
1, 1, 2, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 8, 9, 9, 10, 11, 11, 12…
The basic idea for this procedure is inspired by Hofstadter sequences and the process to generate the tiling fractals described by Rauzy. We notice right away that the value of f_1[n] increases with n in a step-wise fashion along a linear growth line. But what is the constant of this linear growth?

We can derive this thus: Let x be the number of 1s and y be the number of 2s in a sub-sequence of f_0 of length n. From above we know that the ratio \lim_{n \to \infty} \tfrac{x}{y}=\sqrt{2}. Hence, we may write,
y=\dfrac{x}{\sqrt{2}}\\ x+y=n \; \therefore x + \dfrac{x}{\sqrt{2}} =n\\ x=\dfrac{n\sqrt{2}}{1+\sqrt{2}} =(2-\sqrt{2})n

With this constant 2-\sqrt{2}, we can now “rectify” the sequence f_1 i.e. remove its linear growth by straightening it along the x-axis and capture only its true oscillatory variation along the y-axis (see this earlier account for this). Thus, we get the rectified sequence,
f_2[n]=f_1[n]-(2-\sqrt{2})n
Figure 2 shows the first 500 terms of this sequence.

root2_f2Figure 1

We observe that while f_2 takes a wide-range of positive and negative values they are all contained within a fixed bandwidth of 1. However, the values of f_2 are not symmetrically distributed about 0. The highest positive value is 2-\sqrt{2} and the lowest negative value is 1-\sqrt{2}.

We next perform a serial summation operation on f_2 along the sequence. Given the above asymmetry in f_2 with respect to negative and positive value take by it, we again get a sequence oscillating about a linear growth line. This time we can rectify by taking the midpoint of the bandwidth of f_2, i.e.,
\textrm{Midpoint}(2-\sqrt{2}, 1-\sqrt{2})=\dfrac{3-2\sqrt{2}}{2}

Thus, we defined the rectified sequence f_3 as:
f_3[n]=\displaystyle \sum_{k=1}^n f_2[k]-n \left( \dfrac{3-2\sqrt{2}}{2} \right)

Figure 2 shows a plot of f_3[1:n] up to different values of n. Figure 3 shows the same for a large cycle, n=33435 (see below).

root2_towersFigure 2

root2_towers_bigFigure 3

We see that f_3 has an intricate fractal structure resembling rising gopura-s around a central shrine. A closer examination reveals that the fractal structure of f_3 has cycles of increasing lengths, i.e. the same structure re-occurs with greater intricacy at the cycle of the next length (Figure 2, 3). We determined that the lengths of the cycles centered on the highest successive values of f_3 are 27, 167, 983, 5739, 33435… This led us to establish that ratio of successive cycle lengths converges to 3+2 \sqrt{2}. This number is the larger root of the quadratic equation x^2-6x+1=0.

We can do the same thing with the Golden Ratio \phi which has the continued fraction expression,
\phi= 1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\dots}}}}

In this case, the sequence f_0 can be generated by the well-known substitution rule,
1 \rightarrow 2; \; 2 \rightarrow 2,1
Here the ratio of 2s to 1s in f_0 converges to \phi. We can likewise construct f_1 by counting the number of 2s as we walk along f_0 up to a given n. As with \sqrt{2}, we can rectify f_1 to get f_2[n]=f_1[n]-n(\tfrac{1}{\phi}). Here again, the bandwidth of f_2 is 1 but the values it takes are asymmetrically distributed about 0 with a maximum of \phi-1 and minimum of \phi-2. This gives us the rectification to obtain f_3 for \phi,
f_3[n]=\displaystyle \sum_{k=1}^n f_2[k]-n \left( \dfrac{2\phi-3}{2} \right)

Figure 4 shows the fractal structure of f_3 for \phi comparable to that which we obtained for \sqrt{2}

phi_towersFigure 4

We then established that the fractal cycles for f_3 of \phi are of lengths: 32, 87, 231, 608, 1595, 4179, 10944… when defined on the basis of the successive highest values attained by f_3. Thus, the ratio of successive cycle lengths converges to 1+\phi in the case of the Golden Ratio fractal. 1+\phi is the root of the quadratic equation x^2-3x+1=0. This shows a similarity to the above convergent of the cycles of the \sqrt{2} fractal. Further, while that convergent can be expressed as \left (1+\sqrt{2} \right )^2, this one for the Golden Ratio can be similarly expressed as \left (1+\tfrac{1}{\phi} \right)^2

This leads to the conjecture that all such fractals generated from floor-difference-derived sequences of quadratic roots have as convergents such roots of quadratic equations with a relationship like the above ones to the original root.

root2_normal_likeFigure 5

There are some notable features of the distribution of the values of f_3:
1) The number of values >0 is always more than those <0 for a given fractal cycle. This markedly more for the f_3 of \sqrt{2} as opposed to that of \phi.
2) The distribution of the values taken by f_3 is approximately normal (Figure 5; shown for f_3 of \sqrt{2}).
3) Most notably, the f_3 fractal displays structures with quasi-mirror symmetry (figure 2, 3, 4), when we consider the distribution of values around given central points. For the \sqrt{2} case, convenient central points can be easily found in the form of the highest values reached in successive cycles (also the values on which we centered our cycles). To illustrate this quasi-mirror symmetry we show below 10 values on either side of f_3[2869], the central point of the cycle of length 5739:
f_3[2859:2868]: 1.038574, 0.603576, 0.582792, 0.976221, 0.783863, 1.005719, 0.641789, 0.692073, 1.156569, 1.03528
f_3[2869]: 1.328204
f_3[2870:2879]: 1.035341, 1.156693, 0.692257, 0.642036, 1.006027, 0.784233, 0.976652, 0.583284, 0.60413, 1.03919
We notice that the corresponding mirrored values are not equal on either side but very close. Further, the difference is systematic, i.e. the values on one side are consistently higher than their counterparts on the other side. The pair closest to the central point (1.03528, 1.035341) differs by 6.158394 \times 10^5. The next pair by twice that amount, the next by thrice, the next by 4 times and so on. Thus, as one moves away from the center there is a linear increase in the asymmetry by a constant amount until one reaches the ends of the cycle. By the end of a cycle the difference between the quasi-mirror symmetric pairs reaches a maximum of \approx 0.17. Thus, the minimum difference, i.e., the difference between members of the pair closest to the center-point is \approx \tfrac{0.17}{l}, where l is the length of that cycle. Hence, as the cycles get larger the symmetry increases closer to the central point (Can be seen visually in above figures). Similarly, for the f_3 of \phi we can establish the axis of mirror-symmetry as the being the central point of a cycle. Here too, the same dynamics as reported above for \sqrt{2} are observed, but the maximum difference of a pair for a cycle is \approx 0.22 and accordingly for a given cycle of length l the minimum difference of the quasi-mirror symmetric pairs is \tfrac{0.22}{l}. We have not been able to figure out the significance of these maximum difference values for either sequence and remains an open problem. Moreover, this structure of f_3 is of some interest because it seems asymmetry (or randomness) or perfect symmetry are way more common than quasi-symmetry which we encounter here.

Case-2: Product-division floor-difference
Indeed, contrasting real symmetry is obtained in a related class of sequences that we discovered. We shall describe their properties in the final part of this article. Instead of the floor-difference described above, we use a related kind of operation using irrational square roots of integers define the following sequence:
f_0[n]=\left \lfloor n \sqrt{2}\right \rfloor -2 \left \lfloor \dfrac{n}{\sqrt{2}}\right \rfloor

This is a sequence of 0s and 1s: 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0…

Based on f_0 we can define, along the lines of what we did above, another sequence thus:
f_1[n]=\displaystyle \sum_{k=1}^n f_0[k]==1
It is the count of the number of 1 up to the nth term of sequence f_0. It is an integer sequence of the form: 1, 1, 1, 2, 3, 3, 4, 5, 5, 5, 6, 6, 6, 7, 8, 8, 8, 9, 9, 9, 10, 11, 11, 12, 13, 13, 13, 14, 15, 15…

As with the above cases, we observe that the value of f_1[n] grows in a step-wise linear fashion with n. Thus we can again rectify it by determining its constant of this linear growth. We observe that in this case the f_0 as an equal number of 0s and 1s for a given length. Hence, we get the rectification constant as \tfrac{1}{2}. Thus, we can define a further sequence,
f_2[n]=2f_1[n]-n
We multiple by 2 instead of using \tfrac{1}{2}n for rectification because we can that way keep f_2 an integer sequence.

Then, we define the next sequence based on f_2 thus:
f_3[n]=\displaystyle \sum_{k=1}^n f_2[k]

Since the distribution of f_2 in this case is symmetric about 0 we do not need any further rectification in defining f_3 and it remains an integer sequence. In this case the values of f_3 define a symmetric fractal with a bifid peak-like appearance (Figure 6).

sqrt2_double_floor2Figure 6

Here again, the fractal repeats itself at each cycle, with increasing detail as the length of the cycle increases. However, at every cycle the fractal remains perfectly symmetrical unlike the above-discussed cases (Figure 7). We can define the length of each cycle for this fractal based on the palindromic structure of f_3 for each cycle: Each cycle begins and ends in the sub-sequence: 1, 1, 0, 0, 1, 1

sqrt2_double_floorFigure 7

We determined that the cycle-lengths show the progression: 26, 166, 982, 5738, 33458, 195022…
Strikingly, the maximum value reached by f_3 for each of these cycles shows the progression: 6, 35, 204, 1189, 6930, 40391…
Thus, the ratio of both successive cycle-lengths and the maximum height reached in successive cycles, remarkably, converges to 3+2 \sqrt{2} — this is the same as the convergent for the above \sqrt{2} fractal derived from the floor-difference operation.
Notably, the successive partial sums of the continued fraction for 3+2 \sqrt{2} are,

6, \dfrac{29}{5}, \dfrac{35}{6}, \dfrac{169}{29}, \dfrac{204}{35}, \dfrac{985}{169}, \dfrac{1189}{204}, \dfrac{5741}{985}, \dfrac{6930}{1189}, \dfrac{33461}{5741}, \dfrac{40391}{6930}, \dfrac{195025}{33461}

We notice that the maximum value reached in each cycle is captured by the denominator and numerator of every 1, 3, 5, 7… 2n-1^{th} partial sum. The numerator minus 3 of every 2, 4, 6 … 2n^{th} sum captures the cycle-length: the reduction by 3 is evidently because we defined the cycle based on the re-occurrence of the palindrome.

This kind of sequence derived from the product and division by an irrational square root of an integer can be generated from such square roots too. Using \sqrt{3} yields a fractal with a single peak (Figure 8).

sqrt3_double_floor2Figure 8

Here, the cycle-lengths and maximum value attained by the f_3 converges to \left(2+\sqrt{3}\right)^2=7+4\sqrt{3}. In the case of \sqrt{3} we also have minimum values of f_3, which are <0 (Figure 8); interestingly, the ratio of minimum values from successive cycles also converges to 7+4\sqrt{3}. This number is the root of the quadratic equation x^2-14x+1=0

In conclusion, we find that two different operations of the floor function on irrational square roots or roots of quadratic equations yield fractals, whose cycle-lengths are convergents for roots of quadratic equations, which can be constructed based on the original root. The formal proof of this might be of interest to mathematicians. In the second case this also relates to the maximum value attained by the sequence f_3. Finally, it is notable that in the first case the values of f_3 show a certain quasi-mirror symmetry and an approximately normal distribution. Despite this overall distribution, the actually values are arranged in precise manner as to generate a fractal structure. This might yield an analogy to natural situations where a normally distributed population could organize into a highly, structured pattern.

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Of lives of men; of times of men-II

Of lives of men; of times of men-I

Vidrum: “When we attended the discussions at the Right Wing Debate club we heard the president Rammandir Mishra repeatedly emphasize that South Asian civilization was not a ‘history-centric’ civilization and that history-centricism in the form of the urge to fix dates for the veda-s, itihāsa-s and the purāṇa-s is an imitation of Abrahamism among the Indians. Another prolific author and debater Iraamaavadhaaran declared that we had to move away from an outmoded history of kings, generals and dates and talk more about the common people and their folk culture, performing arts and food. He went on to add that this obsession with the former had undermined history’s status as a science and relevance for the people. Clearly you two seem to differ in this regard. You seem to emphasize both history and that pivotal role of special individuals.”

Lootika: “Ah the evil phrase ‘South Asian’ – and right wing they call themselves! You have spent enough time with us by now to realize that we plainly hold the view that a people which ignores the study of history goes down in death unknowingly. If indeed we Hindus have ignored history then it is not something to celebrate but to correct. It is something which must even extend to scripture – be it the śruti or the āgama – you may, hence, term us as aitihāsika-s. That is why, despite all the fundamental flaws of the tāthāgata-matam and the author’s biases, I have respect for the endeavors of the Tibetan lāmā Taranātha. It should be obvious to the beholder that in the piloting of history special individuals matter more than a whole mass of flotsam individuals, as they vulgarly remark: ‘pūrṇa-kara iva kharaviṣṭāḥ’. In understanding this comment there is need for nuance as no notable figure lives in isolation. To give a rough mathematical analogy, real and imaginary number by themselves can be pretty dry but a little bit of both in the form of a complex number gives a lot more interesting stuff. But then I should also state that we need to distinguish the science of history from archaeology. In the latter, data on the mass and their statistics can deeply inform. Thus, archaeology as a statistical study of the bulk is of great significance to provide a backdrop for history, where the case study of the exceptional dominates. Then there is also a real place for the experience of the first person kind. It is something which our ancestors in their study on aesthetics term the sthāyibhāva. This is something only history in its narrative form can produce. Hence, the great historian kavi of Kashmir remarked:

‘saṃkrānta-prāktanānanta-vyvahāraḥ sacetasaḥ |
kasyedṛṣo na saṃdarbho yadi vā hṛdayaṃ gamaḥ ||’

Marching through endless happenings of history, is there a conscious being
whose heart such a narrative would not enter?

Thus, if history is about the lives of people, archaeology is about the times of people.”

Sharvamanyu: “I get the part of the importance of history but on what basis would you place such confidence in the importance of the exceptional individual in the pivoting of events in history.”
Somakhya: “If something is mappable to a mathematical principle it is often difficult to escape its constraints. But of course one has to be very careful in being sure about the applicability and correctness of such a mapping, especially outside the domain of physics. To illustrate the point of the force of mathematical principles imagine a ladder of given length placed against the wall on one end and touching the ground on the other. It is free to slide along the wall and the ground thereby touching the wall and the ground at different heights and distances from the base of the wall. Then we can ask the question that given a certain height of the wall at which the ladder touches it at what distance from the base of the wall will it touch the ground? As you know this is determined by a mathematical principle, a curve known as the astroid. For a ladder of given length, we cannot have a height on the wall or distance on the ground from wall’s base that violates this mathematical principle. Hence, if we map a comparable two-value optimization problem on this principle then we have clear constraints. A more involved example, which you may have studied a bit, is the remarkable central limit theorem. For many distributions, irrespective of the shape of the distribution, if you draw repeated samples and take their means they would be normally distributed around the mean of that distribution. Thus, the central tendency as estimated by mean might be inferred by sampling, even if we have no clue of the shape of the distribution. Now, there are other peculiar distributions which display an unusual membership scenario – a well-known example is the distribution named after the mathematician Cauchy – here the central events are more common and the infrequent events are way more extreme than a ‘regular’ distribution like the normal distribution. Our investigation of historical events and the effects of individuals suggest that they they follow a distribution, like the former. Hence, they are constrained by the properties of such a mathematical principle. A corollary to it is the massive or pivotal role of the rare individuals whose effects are way more extreme than the rare individuals in a normal distributed scenario.”

Sharvamanyu: “OK, that’s an interesting argument. What about the nuance regarding the mass which Lootika mentioned?”
Lootika: “When there is an exceptional figure there needs to be enough of those among the masses who can resonate with and act on behalf of that figure for the exceptionality to shine through. The size of that mass is what depends of the times of men.”
Somakhya: “Think of a Boghorju, a Jelme or a Subedai, Chingiz Khan or the men who rallied around the Mahārāja of the Marāṭha-s. Such exist way more frequently but by themselves they cannot carry the power, but like a conducting metal when there is a source of electricity they can do so. Through the length and breadth of our land there existed local strongmen like a Tānājī or a Sūryājī Mālusare but only under the exceptional Mahārāja they could be fort-conquerors.”

Vidrum: “Regarding the point about sthāyibhāva, I can see the generation of the states of vīrya and vairāgya from historical narratives of the exceptional rājan-s. Which Hindu would not feel that on hearing a narrative like the killing of the Mohammedan Afzal Khan by Śivājī or the heroic struggle of Pratāpa Siṃha against the Mogol tyrant.”
Sharvamanyu: “I felt it several times myself – there is nothing that touches the core than the vīrya-rasa emanating from a well-narrated account of a hero’s exploits.”
Lootika: “It should not stop with just touching the core it should produce that sthāyibhāva upon entering it (hṛdayam gamaḥ). Indeed, among the those of the second varṇa or those performing that function, a major function of itihāsa is the generation of such a state.”
Sh: “Of course – I can say it has not merely touched my core but transformed it. But what are the other rasa-s that might emerge?”
Lootika: “Take hagiographies – at they core they are histories, albeit exaggerated ones. For many people, especially followers of the uttara-mīmāṃsā doctrine, digvijaya-s of foundational teachers, such as Śaṃkara, and others produce not just a romaharṣaṇa but a deep transmutation of the core much as a the digvijaya of a kṣatriya may produce in V2s as well as some brāhmaṇa-s like us. This is an expression of the adbhuta and some times the kāruṇya rasa. While I am not a follower of the uttara-mīmāṃsā doctrines or those schools, I must say certain hagiographies produce some effect, even in me, though not as long- lasting or profound as those experienced by the adherents. For them it lingers truly as sthāyibhāva-s. Thus, a important aspect of history is the account of the lives and deeds of great people, which produce such sthāyibhāva-s in the reader or listener. This was importantly recognized by our kavi-prajāpati-s starting from Kṛṣṇa-dvaipāyana.”

Vidrum: “But then you all have remarked before: ‘na tasya pratimā asti yasya nāma mahad yaśaḥ’ In line with that, many of our key figures have not left behind biographical material or memoirs. Nor are they celebrated in such. Who ever celebrated the Aulikara Yaśodharman despite his most heroic deeds or emperors of the Coḷa-s from the Draṃiḷa country? So, do you think you are creating some new fancy of celebrating the lives of men contrary to tradition, which you as brahmins defend.”
Lootika: “Not at all. While it is not the practice of brāhmaṇa-s to engage in svastuti, as ordained by the law-giver Manu, that statement does not mean a proscription of biographical statements or memoirs. The role of biography is well-recognized by the South Indian kavi Daṇḍin, Bāṇa of Kānyakubja or the Kashmirian kavi Kalhaṇa. Since, the brāhmaṇa was not prone to svastuti one might not see such but they never shied from brief but clear expositions of their biographies including statements on their intellectual prowess. They knew it was their works (even as a kṣatriya’s conquests) which mattered and stood to immortalize them, not an account of what they ate or their sexual exploits. Who gets a biographical magnification has changed over time. Some like emperor Trasadasyu, praised by my ancestors as a half-god in the śruti itself, have passed out of the memory of all but the most conscious practitioners of the śruti. But their successors like Rāmacandra the Ikṣvāku or Kṛṣṇa Devakīputra have got theirs in the age when we celebrated such heroes. Now they have transmogrified into gods. Other heroes of the martial type, like Samudragupta, did not enter the public imagination much but Candragupta-II Vikramāditya entered it with almost a mythology. After this phase, the Hindu consciousness was seen shifting towards hagiographies of religious rather than regal figures. However, Bhoja was one last figure of the great rājarṣi archetype who again nearly entered mythology. The coming of the Meccan demons enshrined the two great Chahamāna-s, Pṛthivirāja and Hammira, in the Hindu mind, as objects of kāvya, for their heroic last stands. But defeat of the Hindu military order before the army of Islam, despite prolonged fight backs, as in Vijayanagara, saw our people look more towards hagiographies and forgot the heroic figures until much closer to our times Śivājī restored the Hindu confidence. Thus, it is the changing landscape of people’s outlooks that has selected for the caritra-s rather than there being any tradition to keep away from them.”

Sh: “Lootika alluded to the brief biographical statements of the brahmins. So do we learn anything of note at all from these brief biographical statements you allude to?”
Somakhya: “While the biographical details might be brief, one important point we learn is that the old Hindu education system clearly had a mechanism to fast-track scientific talent. We can cite examples from all over the country, in different periods, and in different scientific endeavors. Jīvaka a biologist/physician was professor at Ujjaini by 20. Lolimbarāja in Maharashtra was physician who had compiled a new pharmacopoeia by 25. Āryabhaṭa was professor of astronomy and mathematics before 23 at Pataliputra. The Kashmiran astronomer Vaṭeshvara was a professor by 19. Mañjula, the Magadhan astronomer, who was one of the early users of differential calculus in Hindu tradition, was a professor at 20. Jayanta Bhaṭṭa the Kashmirian wrote a grammatical treatise at age 10. The great Nīlakaṇṭha Somayājin of Cerapada was astronomer of note by 23. Gaṇesha daivajña in Maharashtra derived his hyperbolic approximation of the sine function at approximately the age of 14. Raghunātha paṇḍita of nava-nyāya was a paṇḍita by 18. All authors are seen writing mature treatises by the stated ages as we learn mostly from their autobiographical notes. We can also look back at our historical greats and note that Bhāskara-II writing his work at 36 was already quite senior compared to the rest. As we can see from our own curricular educational experience, which thankfully is finally coming to a close, this fast-tracking to make most of people at the height of their intellectual and physical capacity is something the modern system in our nation woefully lacks.”

Sh: “That’s indeed seems to lend support to your hypothesis that genius mostly expresses itself early in life. Returning to narratives. I guess the ‘hagiography’ of a scientist is mostly lacking in our tradition. Perhaps, you might include the accounts on Nīlakaṇṭha Somayājin or Āryabhaṭa but they might be termed by the undiscerning as too sparse to produce sthāyibhāva-s. If you were to produce such a ‘hagiography’ of one, whom would you pick one, say of our times of or close to it, which will have the effects that Lootika quoted Kaḷhaṇa as mentioning – and narrate it to us if you are so inclined.”
Somakhya: “This is the tricky part. I don’t necessarily see a need to produce a complete ‘hagiography’ most of the times. When we are looking at a scientist it is often their own work which speaks. You would need to grasp the science – once you do so, it speaks loudly and clearly – something a hagiography may or may not ever achieve. When we take mleccha scientists/mathematicians, like say a Leonhard Euler, Carl Gauss, Bernhard Riemann, John Herschel or Charles Darwin not much of their the routine caryā and vyavahāra may intersect with us or be worthy of emulation. We truly belong to a different religio-cultural milieu and there is no need to transplant theirs into ours. Indeed, we sometimes see foolish scientific imitators among our people who transplant their caryā in our midst rather than their sattva and think that it is the real thing. But to us is mostly their work which speaks clearly and that is all that matters to and intersects with our own sphere when we try to reproduce or emulate that – it is that which produces a distinct sthāyi-bhāva, which we would definitely place in the domain of the adbhuta.

Yet, since you ask we would pick Śrinivāsa Ramānuja for a special consideration among those of our own people. There is certainly none like him in recent or distant memory and his life needs no special narration to you all. Nevertheless, as you have given me the chance, I will use it to give word to a few thoughts in his regard. The lay man with very limited mathematical education or imagination gets only some vague sense of his greatness, in most part because of the romanticization of his story – the ‘hagiography’ – if you may. But those with a moderate mathematical education, who attempt to even superficially reproduce that part of his work which they can grasp, get a glimpse of a supreme greatness, which can be latent in the human brain, which one cannot but tend to associate with the realm of the highest adbhuta – the daiva. If this is the experience of the moderately educated, then what to say of the gaṇitajña who swims in the ocean of that vidyā. It is clear that Ramānuja himself was aware of his access to a higher channel – that is why, I would say, he termed his vidyā as directly emanating from Śrī, the mistress of all opulence – a connection to something deeply rich.

The once vigorous Hindu tradition of mathematics had lost its subcontinent-wide connectivity with the irruption of the Meccan demons in the late 1200s. But remarkable developments occurred due to a handful of great intellectuals culminating in Nīlakaṇṭha Somayājin in Cerapada. But after that it was almost as if the break was final. There was no Hindu of note even as a Newton, an Euler, a Lagrange, and a Gauss piled on among the mleccha-s. We were at our lowest ebb, when Ramānuja arose. It is not that he came from a lineage of great intellectuals. His line while belonging to a sect of vaiṣṇava V1s, which had a solid tradition of the śastra-s like their other non-vaiṣṇava coethnics, was not particularly accomplished in recent memory. Yet, he arose in their midst like the god Vaiśvānara, slumbering in the logs, suddenly leaping forth in all fury at the commencement of the ritual and establishing that continuity with the earliest fires of Manu, the Bhṛgu-s, the Aṅgiras-es and emperor Bharata. Single-handed, for a good part in isolation, he literally spanned the gap between the brāhmaṇa-s of Cerapada and the ground of Euler, Gauss and Riemann, like a stride of Viṣṇu in the battle against the dānava-s. That is literally so because in school he discovered for himself the infinite series for trigonometric functions, thus rediscovering what the Cera V1s achieved. Before he left for the isle of our erstwhile mleccha conquerors he discovered for himself some version of the magical zeta function in connection with the prime distribution problem, thus nearly reaching Bernhard Riemann, on the way passing through some of Euler’s conquests. Then scaling beyond the heights reached by none other than Jacobi, by the time close to his early death he had reached a rarefied realm that gaṇitajña-s of the highest order could catch up with only much later. And those findings are linked to all manner of deep mysteries, which the lesser mortals, like us, can get the barest shimmer of, like the ketu of Uṣas before the rise of the eye of king Mitra. In his early death, even as he was uncovering those great mysteries, we almost get a reflection of the tale of the god Vāyu’s simian son. He soared too high and was threatening the sun; hence, he had to be felled by Maghavan. Likewise, it almost appears as though Ramānuja had reached a point where he was connecting to the deepest mysteries, which the deva-s keep well-hidden from the martya, and consequently had to die before that. Thus, even if one can only fathom a little glimpse of his mathematics, one gets the barest view of the high mysteries – so, can there be another tale which produces more adbhuta and even some bhayānaka than this?”

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Of lives of men; of times of men-I

Sharvamanyu and Vidrum arrived at the campus where Somakhya and Lootika were in their final days of college. Sharvamanyu had already been working for several months while Vidrum had just completed the last but one of his major exams for the time being and had a couple of days free. Hence, he joined Sharvamanyu to see his old friends whom they had not met in a while. Under their favorite haunt of the elephant apple tree Vidrum brought up the issue of their impending dispersal: “These may be the last few days we ever see each other. Somakhya here has remarked that companions in life are like a vesture, which when old is cast off. He would cite the Sanskrit cliché ‘vāsāṃsi jīrṇāni’. Since, you two would be going away in the near future we though we should have that more serious conversation that in past I did not much appreciate but now yearn for, perhaps for one last time.”
Sharvamanyu: “On a personal note even you two Somakhya and Lootika would be separated. I have been curious about how you would take it? At least I would have Abhirosha with me and Vidrum will give me company.”
Somakhya: “Indeed man comes into being to be alone and leaves it alone. There is a time in life for everything. I’ve enjoyed the blissful company of fair Lootika for a while and thank the gods for that. When the time for the showdown at Kurukṣetra came, Abhimanyu was not thinking of the time he could have spent with Uttarā. There comes a time in life when man has to leave such things as companionship aside and perform ghorakarman in the battle field of existence. He will need to perform them alone – like brave Abhimanyu in the cakravyūha or Yuyudhāna against the Bharadvāja hero.”
Vidrum: “And what do you think Lootika?”
L: “Not too differently. But then remember everything passes along and times change. It is not like we are parting ways for ever. That might happen only if the Hindu belief were true and we were to die and be born again. While a mere martyā cannot have a say on such matters I do think this is not the time of our ends and the last time we meet.”

Sharvamanyu: “You are expansive storytellers and while I do think you two will might meet again in life, we cannot be so sure of that. So before you leave we thought it would be good to hear some stories from you and perhaps note them down for they might have pearls to glean in them.”
Vidrum: “Last evening Sharva and I saw a movie. I featured the ancient Egyptians. I do not know anything substantial of their history. But somehow thinking of them seized us with a sense of deep foreboding. A feeling that I cannot entirely articulate.”
Sharvamanyu: “The Egyptians of old were a sophisticated civilization making objects of great artistic merit and monumental architecture long before anyone else made such. Yet they are gone. A question we wished to ask you was what our ancestors were up to when the Egyptians performed such feats?”
Lootika: “Certainly none of our ancestors had anything like the Egyptian monuments or art to show. One strand of our ancestry which was in the Sindhu-Sarasvati civilization, however, certainly made all kinds of practical things in their highly functional cities like drains, a drill for teeth, and minute metal objects unlike what others elsewhere ever achieved. But other than their genes and a vaguer cultural inheritance of that strand of our ancestry we as yet know not too much. On the other hand, our entire cultural consciousness came from the steppes of Eurasia in the form of the Aryan strand of our ancestry. They had their military monuments, which they likely transmitted to the SSC, but one usually does not compare them to the material achievements of the Egyptian. However, at that time our ancestors were composing a monument of no less magnitude the Ṛgveda and the knowledge-system it engendered. The reason why it is a monument of no less stature is not easily understood by many. Some point to the Egyptian monuments and ask what do you have to show? We simply say the pudding is in the eating – they are gone while we are still around even if it is a precarious existence as you know. In Sāmkhya we talk of three guṇa-s – the Egyptians were heavy on tamas – mind you, this not to be seen as a negative comment but in the pure physical sense – our people were closer to the sattva pole. Thus, our monuments were mostly of a non-material kind, closer to pure information. Perhaps, that’s why we placed sattva at the top the hierarchy and the pure information was even fetishized by our kautsa-mīmāṃsaka-s.”

Vidrum: “How did their glorious civilization end?”
Somakhya: “Like most civilizations of the occident, their ultimate end was due to the Abrahamistic epidemic of the second kind – truly a reminder that memetic epidemics can end civilizations far more effectively than the many plagues of the biological kind. Yet, the end of a civilization might not necessarily be mappable to just one small time slice in history. Egyptian civilization had a long and glorious record. It had met with a strand of our own in the form of people related to us who established the Mitanni kingdom in West Asia. It clashed and survived against military systems like those of our ancestors in the form of the great Hittites. It successfully overcame religious memetic disease erupting from within in the form of Akhenaten and overcame marauders like the Israel. But the decisive point marking the beginning of the end, in our opinion, was the clash with the military system of our cousins, the Iranians. It was downhill thereafter.”

Sharvamanyu: “Give us some background on these Iranians and tell us a bit more of that clash?”
Somakhya: “While much of these events are obscured by the loss of records, the boosterism of our more distant cousins the yavana-s and other apocryphal tales, we can say that the Iranian power in the world system arose somewhat later than ours. Although they were around in west Asia for sometime since 1000 BCE, they were not exactly a mighty force. But in the steppe-borderland there were still mighty clans of the Indo-Iranian continuum which had played a major role in the rise of Aryan power in India. Among them were the Kuru and Kamboja and it was perhaps in a confederation of such tribes that the Hākamanshiya arose. Their king the great Kurush (Cyrus), named after Kuru, established a great world empire stretching from the eastern reaches of Soghdiana to the western end of Asia Minor sometime between 600-530 BCE. He conquered in succession his Iranian cousins the Medians, the Lydians, who were relatives of the Hittites, and the Babylonians and their vassals, besieging and taking fortified strongholds one after the other like no one had ever before done in West Asia. Finally, while fighting the Śaka-Hindu alliance that was arrayed against him with their elephant force during his invasion of Śakastana, a Hindu spearman stabbed Kurush in the liver resulting in his death.

His son was Kambūjiya (Cambyses) named after the old Kamboja mentioned in our national epic. He led a great force for the conquest of Egypt, which was one of the last major powers left in the region that plainly refused to acknowledge the might of the Iranians. Initially, Kambūjiya and the pharaoh Amasis II had attempted to reach a truce. But Amasis himself had come to power in an internal conflict where he had killed another pharaoh Apries. The kinsfolk of that pharaoh and some disgruntled Egyptians appear to have gone over to the Iranian side with key intelligence for Kambūjiya to plan his advance. The yavana-s also fearing the overbearing Egyptians upon their killing of some yavana-s in Egypt appear to have joined hands with the Iranians. This signaled an excellent opportunity for Kambūjiya to launch his attack. He marched through Arabia to enter Palestine where the Egyptian army tried to stave off the Iranian entry into Africa in an encounter at the fort of Gaza. In the mean time, the old pharaoh died and left his son Psamtik to handle the clash of empires. The siege of Gaza ended in a defeat for the Egyptians who now fearing the inevitable invasion of Africa massed their large army near the eastern end of the Nile delta. The clash with the main Iranian army was one of those epic battles of the ancient world with tens of thousands of Egyptians cut down by the kshathiya horsemen. It ended in a total rout for the Egyptians. After this comprehensive victory, the Iranians launched a further wave of attacks driving the Egyptians into the fort of Memphis where they were besieged. Kambūjiya hoped he could get the Egyptians to easily surrender and sent an emissary for the purpose. But they responded by killing his emissary and everyone else who was with him. The Iranians then quickly took the fort and captured the pharaoh Psamtik. His life was initially spared but as he tried to revolt and regain his kingdom Kambūjiya killed him and even carved a seal depicting him spearing the pharaoh.”
bawa_killing_pharaoh
Vidrum: “Interesting: that’s high drama worthy of a movie”
Sharvamanyu: “But no one will tell it from their angle. Is it not quite a turn of history that these world conquerors have been reduced to the enervated Parsis of today, whose extinction is at hand? I guess that was at the heart of inexplicable unease we have been feeling. Could it happen to us too?”
Lootika: “I fear we may come to see it in our lives or it could happen in the lives of our offspring, and the end can come from within and without, in a bang or a whimper. My worst fear is the end in the latter form, even as a whimpering dog or a hedgehog of an Iranian kicked to death by a Mohammedan. Especially when the scaffold of the edifice is shaky, there is also the possibility of a catastrophic end – everything seems to be going well, when suddenly like a poorly built house or a creaky bridge everything comes down crashing. In our own history this has happened more than once – Rāmarāya of the South Indian empire or the marāṭhā-s were in state of considerable power before their catastrophic collapse. I am sure Somakhya can tell us how this was also the case when the Iranians came to an end.”

Somakhya: “That was indeed the case. Shāh Koshrau-II led a brilliant campaign against the Abrahamistic alliance of the Byzantine Christians and the Khazar Turks, winning a string of victories against them. He even managed to get the heathen powers of the Slavs and the Avar Khaghanate over to his side against the Abrahamistic alliance and by 622 CE the it looked as though the Iranian empire was poised at a high-point, recovering the old Hākamanashiya glory. The śula of the preta was brought down in Jerusalem and even Egypt was reconquered by the Iranians. But the empire had serious cracks within. Thus, when the Abrahamistic alliance launched its holy war demolishing key Zoroastrian temples and the epidemic of the third Abrahamism erupted from within it crumbled and Iranian civilization itself became extinct – for what do a band of Parsis matter when your land is gone?”

Of lives of men; of times of men-II

Posted in Heathen thought, History, Life, Politics | Tagged , , , , , , , ,

The incredible beauty of certain Hamiltonian mappings

In our teens we studied Hamiltonian functions a little bit as part of our attempt to understand classical and quantum physics. A byproduct of it was a superficial interest in the geometry of some of the mappings arising from such systems. This led us to the beautiful mapping, known as the Standard Map, emerging from the study of the Russian (Kolmogorov, Arnold, Chirikov) and German (Moser) schools on the Hamiltonian of a dynamical system receiving periodic kicks. Then we chanced upon the work of Scott et al and learned of a Hamiltonian mapping therein, which will be the topic of this note. While the note is a bit about its geometry and mostly a celebration of its aesthetics, for the unfamiliar reader we shall preface it a very basic background of the underlying physics. Rather than go into the details of the calculus of Hamiltonians, this will only repeat very elementary stuff that you would have studied in the first year of an ordinary junior college or can look up from Feynman’s legendary lectures.

Let the position of an object of mass m in one dimension be denoted by x in some distance unit. The first time derivative of this position it the velocity of the object, v=\tfrac{dx}{dt}=\dot{x}. The momentum of the object is defined as the product of its mass and velocity p=m\cdot v=m\cdot \dot{x}. Newton’s second law tells us that: “The acceleration (a) of an object in the direction of a force (F) acting on it is directly proportional to the magnitude of the force and inversely proportional to its mass (m). This acceleration is the second time derivative of position x. Hence, a=\tfrac{d^2x}{dt^2}=\ddot{x}. Thus, we get,
a=\dfrac{F}{m} \\ \\ \therefore F=m\cdot a=m\cdot \ddot{x}
Since, p=m\dot{x} we get \tfrac{dp}{dt}=\tfrac{d}{dt}(m\dot{x}). Thus, F=\tfrac{dp}{dt}=\dot{p}

Now, one of the most important idealized conceptions of motion of an object is simple harmonic motion (SHM). It results from the opposing action of inertia of a body and ‘elasticity’ of the mechanism holding it. Thus, when the body is displaced by a force in a single dimension from its equilibrium position, x=0, the elasticity is that which tries to bring it back to the equilibrium (imagine pulling a spring or a rubber band with a weight). Newton informs us that inertia is the tendency a body to remain at rest or continue in its line of motion unless an unbalanced force acts on it. Thus, due to inertia, the displaced body when pulled back towards the equilibrium point overshoots it and continues its displacement past it, while the elasticity provides the force that tries to restore it. Thus, inertia and elasticity together set up oscillatory motion or SHM. The force displacing the object can be described by Newton’s second law as given above. In contrast the elastic force can be described simply as something which acts opposite to the direction of the displacement and is directly proportional to the amount of displacement (Hooke’s law; again imagine the restoring force generated by pulling a spring/rubber band will be more the greater you stretch it). Thus, F=-k\cdot x. The k is proportionality constant for the elastic force and the negative sign indicates it acting opposite to displacement. Thus, due to the balance of the displacing and elastic force we get:
ma=-kx\\ \therefore m\ddot{x}=-kx;\; \ddot{x}+\dfrac{k}{m}x=0
The above is the famous differential equation for SHM which every educated teenager knows.

When the object is performing oscillation, its kinetic energy T at a given point can be easily described, T=\tfrac{1}{2}mv^2. From above equation for momentum we get v=\tfrac{p}{m} and plugging it into the equation for kinetic energy we get, T=\tfrac{p^2}{2m}

The potential energy of the object in SHM arises from the elasticity. When you do work against the elasticity it gets stored as equivalent potential energy U. We know the force from elasticity due to Hooke’s law is F=-kx. When we do work W against it that work is described as the total amount, i.e. integral, of the product of force (now with a positive sign as it is done against the elastic force) and the infinitesmal displacement dx,
W=\int F\cdot dx= \int kx\;dx=\dfrac{kx^2}{2}
Since this work gets stored as potential energy we get U=\tfrac{kx^2}{2}.

The Hamiltonian H is a function of position and momentum x, p and time t if there is time-dependent evolution which specifies the total energy of the dynamical system. Thus, from the above calculations of kinetic energy T and potential energy U the Hamiltonian H of this oscillator is,
H=T+U= \dfrac{p^2}{2m}+\dfrac{kx^2}{2}
The Hamiltonian function relates to Hamilton’s equations, which specify that: (i) if you take the partial derivative of the Hamiltonian with respect to momentum then you get the first time derivative of position, \dot{x}, i.e. velocity; (ii) if you take the partial derivative of the Hamiltonian with respect to position then you get the first time derivative of momentum, \dot{p}, i.e. force with a negative sign. Thus,
\dfrac{\partial H}{\partial p}=\dfrac{\partial }{\partial p}\left(\dfrac{p^2}{2m}+\dfrac{kx^2}{2}\right)=\dfrac{p}{m}=v=\dot{x}

\dfrac{\partial H}{\partial x}=\dfrac{\partial }{\partial x}\left(\dfrac{p^2}{2m}+\dfrac{kx^2}{2}\right)=kx=-F=-\dot{p}

Since, our SHM is an idealized system with no dissipation of energy due to friction all we have is the kinetic energy converting to potential and back. Thus, total energy is a constant, H=C. Now, if we define \omega=\tfrac{k}{m}, and redefine p in \tfrac{1}{\sqrt{k}}, x in \sqrt{m} units we get H=\tfrac{1}{2}\omega(p^2+x^2). Thus, x^2+p^2=\tfrac{2C}{\omega} and a plot of x,p is a circle of radius \sqrt{\tfrac{2C}{\omega}}.

Now, imagine that such an oscillator performing SHM receives a series of instantaneous kicks that add energy to the system over time t. An instantaneous kick is modeled using Dirac’s \delta(t) distribution. One way to imagine this distribution at instant t=n, i.e. \delta(t-n) is as a couple of up and down step functions forming a rectangle with unit area under it centered at point n whose width tends to the limit 0 (hence, height becomes \infty). Another way is to imagine it as a limiting Gaussian probability distribution centered on mean n such that the whole probability, i.e. 1 is restricted to the mean. When we couple, i.e. multiply, a function to a delta distribution centered at n and evaluate it, we get value of the function at n. This is because everywhere other than at n the area under it is 0 and at n it is 1. If we want to represent a sequence of such kicks at integer instants then we construct a Dirac’s comb which a series of \delta(t-n) between n=-\infty,\infty. A function coupled to this Dirac comb evaluates to the sum of the values of the function at each integer point. The function we are coupling to the Dirac comb in our example is the additional energy delivered at each instant to the oscillator performing SHM. This is specified as directly proportional to the absolute value of the position of the oscillating body |x| with a constant of proportionality or the coupling constant of the kick, \mu. Thus, we get the Hamiltonian of this kicked system of Scott et al as:
H(x,p,t)=\dfrac{1}{2}\omega(p^2+x^2)+\mu |x| \displaystyle \sum_{n=-\infty}^{\infty} \delta(t-n)

Let the position-momentum curves specified by this Hamiltonian by a locus of points z=x+ip in the complex plane. Then following then try to write a z_n \rightarrow z_{n+1} mapping for the above Hamiltonian. If \mu=0 then we get our standard SHM oscillator and the map is z_{n+1}=e^{-i\omega}z_n; it produces our above-stated circle in the x,p plot when we start with some initially value z_0. Thus we can take this \omega to be an angular value between 0, 2\pi. What the kick does is to cause a position-dependent shift in the momentum of -\mu \;\textrm{sign}(Re(z_n)), where the sign function takes the sign of the real part of our complex number z_n, i.e. position. Hence, with the kicks the mapping is written as:
z_{n+1}=e^{-i\omega}(z_{n}-i\mu \; \textrm{sign}(Re(z_n)))

The maps produced by the above have remarkable geometric and aesthetic properties. Strikingly, when \omega=\tfrac{p}{q}2\pi, where p,q \in \mathbb{N} i.e. natural numbers, the map produces a tiling of polygons where a primary polygon in the tiling is a q-gon along with n-gons with q/2 or 2q sides. For example, figure 1 was produced using \mu=0.715; \omega=\tfrac{5}{8}2\pi; starting z_0=x_0+iy_0 with x_0,y_0 \in (-2,2), 2500 iterations for each z_0: we see the primary octagon and squares.

Hamilton_kicker.21Figure 1

When \omega cannot be expressed in the above form it appears that the map produces a structure that appears to be a circle-packing, i.e. filling of the plane with tangent circles. In Figure 2 we show an example of \omega=\tfrac{e}{\pi} with z_0 having real and imaginary parts in the range (-30, 30) at intervals of 5 run for 500 iterations. We see a circle-packing pattern with increasing symmetry while moving away from the origin on the complex plane.

Hamilton_kicker.22Figure 2

The iterates obtained from each z_0 can be given a different color. With this we can distinguish the x,p orbits obtained from each z_0 via our mapping. This depiction of the map is one of incredible beauty in the subsequent figures we show a few of these.

Hamilton_kicker.23Figure 3

Hamilton_kicker.26Figure 4

Hamilton_kicker.27Figure 5

Hamilton_kicker.7Figure 6

Posted in art, Scientific ramblings | Tagged , , , , , , , , ,

Triangles, Hexes and Cubes

One philosophical question which we have often ponder about is: Are numbers “real”? One way to approach this question is via figurate numbers, where numbers directly manifest as very tangible geometry. This idea has deep roots in our tradition: as we have noted before, the square numbers and their link to odd numbers is directly represented in the square vedi-s of the śrauta ritual. Another basic type of figurate numbers, the triangular numbers, are presented in the early yavana philosophical tradition of the Pythagoreans and play an important role in Platonic thought. Thus, we suspect that contemplation on figurate numbers played an important role in the ancestral philosophical tradition on the ārya-s and the yavana-s. Here we will illustrate some well-known and basic features of figurate numbers to show how geometric conceptualization of them allows one to easily understand and derive certain properties of theirs.

triangular_numbersFigure 1

Figure 1 depicts triangular numbers T_n. We see that they simply emerge from appropriately arranging rows of counters amounting to successive Natural numbers i.e. the set \mathbb{N}. Thus, the geometric figure, the triangle, is directly implied by the existence \mathbb{N}. From Figure 1 we also see that the nth T_n itself represents the sum of natural numbers from 1:n. This is the famous sum of the basic arithmetic series which is first indicated in Vedic tradition as derived by the ancient sage Śākapūṇi. It was subsequently expounded by Āryabhaṭa-I in his Āryabhaṭiyam. The medieval scientist Nārāyaṇa paṇḍita gives an old sūtra for it thus:
saika-pada-ghna-padārdhaṃ saṃkalitam |
sa+eka+pada \rightarrow (n+1); ghna: multiply; padārdhaṃ \rightarrow \tfrac{n}{2}; yields saṃkalitam (sum of series). Thus,

T_n=\dfrac{n(n+1)}{2} \rightarrow 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210 ...

sum_of_seriesFigure 2. This demonstration is given by the great scientist-mathematician Nīlakaṇṭha Somayājin and others like the Munīśvara Viśvarūpa.

By looking at figure 2 the proof for the above formula becomes self-evident. By looking at figure 1 and working our way along successive numbers we notice that we can reach any natural number by means of a sum of at most 3 triangular numbers, like the 3 steps of Viṣṇu spanning the universe. This result was formally proved by Carl Gauss at age 19 following up on his auto-discovery of the sum of the simple arithmetic series as a child. Related to this, is the fractal structure that emerges when we represent triangular numbers in binary (base 2) following a device of Stephen Wolfram (Figure 3).

triangular_binaryFigure 3. The triangular numbers up to T_{511}=130816

We had earlier seen how the problem of determining which triangular number is also a square number led to the famous indeterminate equation, which was generally tackled by the old Hindus using the cakravāla algorithm and how it yields approximations for \sqrt{2}. Other relationships between triangular numbers T_n and square numbers S_m can be easily found using the above construction of Nīlakaṇṭha (Figure 4).

triangle_squareFigure 4

From Figure 4 it is self-evident that,

T_n+T_{n+1}=S_{n+1}
8T_n+1=S_{2n+1}

Triangular numbers and related figurate numbers also relate to arithmetic problems which captured the attention of the Hindu mathematicians over the ages. For example, if we pile triangles formed by the triangular numbers one atop the next then we get a 3D figurate number, the tetrahedral number (figure 5). The tetrahedron thus formed had significance in Platonic thought.

tetrahedral_numbersFigure 5

From the above figure it is apparent that the tetrahedral numbers (Te_n) are nth sums of triangular numbers. Āryabhaṭa-I gives the formula for the nth sum of triangular numbers by the sūtra:
ṣaḍbhaktaḥ sa citighanaḥ saikapadaghano vimūlaḥ |
sa citighanaḥ: Sum of sum of natural numbers is called citighana; ṣaḍbhaktaḥ \rightarrow \div 6; sa+eka+pada+ghanaḥ \rightarrow (n+1)^3; vimūlaḥ \rightarrow -(n+1) (literally subtract cube root of previous term). Thus we have,
\displaystyle \sum_{j=1}^n T_n=Te_n=\dfrac{(n+1)^3-(n+1)}{6}=\dfrac{n(n+1)(n+2)}{6}

triangular_number_sumFigure 6. Sum of triangular number demonstrated by Nīlakaṇṭha Somayājin.

Nīlakaṇṭha Somayājin illustrates the geometric proof for this in his commentary on Āryabhaṭa (Figure 6). The term citighana (literally stacked solid) implies that Āryabhaṭa had 3D sense of it similar to constructions with bricks performed during the piling of citi-s or ritual altars in the śrauta ritual. The earliest Hindu temples, known as caitya also stem from the same root implying the piling of bricks. The relationship of these terms to the piling of citi-s was first pointed out by TA Sarasvati Amma but mostly ignored by others.

The ghanacitighana laid out by Āryabhaṭa-I and subsequently commented upon by many mathematicians is computation of the nth sum of cubes of natural numbers. Āryabhaṭa’s sūtra is:
citi-vargo ghanacitighanaḥ |
citi \rightarrow \sum_{j=1}^n j; vargaḥ: squared; yields sum of cubes of natural numbers (ghanacitighanaḥ). We can use triangular numbers to easily derive this formula (Figure 7)

cube_cuttingFigure 7. The sum of cubes of natural numbers.

We construct a square slab of unit cubes with each side of length equal to a triangular number. Thus, the total number of cubes in the slab is T_n^2. We then dissect the slab such that we separate the pieces shown in different colors first. Then we dissect each color piece further along the grey slicers. These divide the pieces into either square slabs or half-square slabs of unit cubes. We then assemble these slabs into cubes of the same color. This demonstrates that,
\displaystyle \sum_{j=1}^n j^3 =T_n^2=\dfrac{n^2(n+1)^2}{4}
Thus, the nth sum of cube numbers is the square of the nth triangular number.

We may next consider the Hex numbers H_n which are depicted in Figure 8. Such a figure plays an important role in the positioning of deities in maṇḍala-s in the āgamika tradition. We also encounter it as a packing principle in nature.

Hex_numbersFigure 8

We see that they are centered hexagons and based on the figure we can infer their general formula and relationship to triangular numbers:
H_n=3n^2-3n+1; n= 1,2,3,4...
H_n=6T_k+1; k=0,1,2,3...
The latter relationship can be seen by dissecting the hexagon into six triangular sectors of dots making the triangular number T_k after leaving out the central dot. Further, we see that a hex number is always of the form 3m+1 where m is 0 or an integer. The hex numbers are the 3D figurate equivalent odd numbers: In the old śrauta altar construction successive odd numbers of bricks form right-angled “shells” that nest into the next such shell to constitute larger and larger squares. We may likewise ask if this can be extended to 3D, such that we have shells which nest into each other to constitute a cube (Figure 9).

Hex_cubesFigure 9

From figure 9 we observe that based on the geometry of the hex number we can arrange in 3D to constitute such a cubic shell. By nesting successive shells of hex numbers we constitute a cube. Thus we get,
\displaystyle \sum_{j=1}^n H_j =n^3
This tells us that just as the difference between successive square numbers is an odd number the difference between successive cube numbers is a hex number.

In conclusion, the study of figurate numbers might be seen as having their origin in old Greco-Aryan religious tradition. In Greece they took an important place in philosophy whereas in India, beyond the ritual tradition, their more general study assumed a form somewhat closer to their modern counterparts. While Hindus were generally less-inclined towards geometry than their Greek counterparts, this is one area where the Hindus developed a unique unbroken tradition of “geometric algebra” that clearly stretches from the Vedic tradition via Āryabhaṭa-I to the later savants. That older root of it is evident from the fact that even in the period of regionalization, the tradition was similarly but independently continued in disparate parts of the Hindu nation. We see demonstrations of geometric algebra in the south in the famous school of Nīlakaṇṭha Somayājin and his successors like Citrabhānu and Śaṃkara Vāriār; in Maharashtra by Gaṇeśa daivajña and his clansmen; in North India even under Islamic tyranny by the Raṅganātha-Munīśvara school.

The study of figurate numbers continues into modern mathematics. We noted above the discovery of Gauss on triangular numbers. Before him the Leonhard Euler generalized the concept of the root to figurate numbers. The square root of a number y=\sqrt{x} is a generalization coming from the n as the root of the square number n^2. Euler showed that similarly y, a general triangular root of x, can be defined as the solution of the quadratic equation y^2+y-2x=0,
y=\dfrac{\sqrt{8x+1}-1}{2}
We get y to be an integer only when x=T_n; then y=n. Thus, only if \sqrt{x} and its triangular root are simultaneously integers it is both a square and a triangular number. In the below table we show the first few numbers which are both triangular and square along with their square root and triangular root.

Table 1.

 sqrt   troot   Sn/Tm
-----  ------  ------
    1       1       1
    6       8      36
   35      49    1225
  204     288   41616

From these numbers we can see that we can compose simultaneously square and triangular numbers by the two seeded series:
S[n]=34\cdot S[n-1]-S[n-2]+2; where S[0]=0, S[1]=1

Similarly, the solution y for the equation 3y^2-3y+1-x=0 provides the hex root of a number,
\dfrac{3+\sqrt{9-12(1-x)}}{6}

Table 2. First few numbers which are both triangular and hex numbers with their triangular and hex roots.

troot   hroot   Tn/Hm
------  ------  ------
     1       1       1
    13       6      91
   133      55    8911

Table 3. First few numbers which are both square and hex numbers and their square and hex roots.

 sqrt   hroot   Sn/Hm
-----  ------  ------
    1       1       1
   13       8     169
  181     105   32761

Whereas a square number is never prime and T_2=3 is the only prime triangular number, the general formula for hex numbers is a fairly rich prime-generating
quadratic. For instance there are 58 H_n| n<10000 of which 28 are prime: pf=0.4827586. For comparison we draw 58 numbers 10000 taking into account the following: 1) hex numbers are always odd; 2) they are not evenly distributed: the first 100 has 6 hex number while the next 100 has only 2. Thus, we make the clustering pattern of our draws identical to that of hex numbers in windows of 100. Doing a simulation with these constraints we get a probability of the prime fraction in draws of 58 numbers being pf=0.4827586 or higher to be of the order of \approx 0.0002 (Figure 10).

hex_primesFigure 10

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The story of the Orissan cycle-vālā

Vidrum had been under considerable pressure. His parents had made it clear that they would auction away his new bike if he was not ranked within the top 5 in his class in the impending mid-semester exams. They had also made it clear that the goal they had set for him that year was to be in the top 3 ranks by the final semester. It had caused him considerable tension. He thought to himself: “How on earth am I going to supersede classmates like Hemling, Gomay, Tumul, Dandadipa, and Jukuta who are so diligent with the books jousting so fiercely with each other. Then we have my friend Somakhya and the unusually wise girl Lootika, both of whom even the teachers secretly fear. While they take their curricular studies very lightly, they are in the least going to be unstoppable for spots four and five. I have often wondered if their parents ever give them the kind of shit I get.” While he had risen early to study, he had instead spent the time with his geometry box neatly drawing some fascinating constructions. But that was going to hardly matter for the exams at hand. So, as the fear got to him, he looked at the sheet before him which gave the prospectus for the impending exam in chemistry.

After plying his books for some time, he felt so unmotivated that he thought it might be a better idea to study with Meghana. She direly needed his help and that would give him the right motivation to get all the correct answers for the impending questions. He ticked off the chapters that needed to be studied and collected his cheat sheets for each of them. Then his eyes fell upon a chapter titled “Naphthalene and Anthracene”. To his horror he found that he had no notes whatsoever on that. There were awful sounding terms under that heading like “preferred nitration product”, “Friedel-Crafts reaction” and the like. At the end he saw some questions based on that chapter. He felt fear deep in his stomach and wondered what to do about it. Just then it hit him that it might be best he left right away to study with Meghana and on the way head to Somakhya’s house and get the material on naphthalene and anthracene from him.

That evening after having covered quite some ground with Meghana and feeling more confident Vidrum decided to return home. But as he just got out of Meghana’s house and headed to where he had chained his bike he saw to his immense horror that the spokes of his front wheel had been smashed. With his mind numb with this disaster he slowly started walking his bike hoping that he might be able to reach the repair shop near his school and get it fixed without his parents knowing. With a down-cast face he was thus trudging along when he crossed the bylane that led to Lootika’s house. There at the corner of the street he caught sight of Lootika standing with Nikhila leaning on their bikes. He waved out them and was about to proceed along when they asked him what had gone wrong with his new bike. He told them how unbeknownst to him his spokes had been smashed when he was studying with Meghana. He then added: “You all are not studying?” Lootika: “I am 3/4ths done. So I decided to take break of an hour for the evening because it is traditional among brāhmaṇa-s not to study during this hour. Instead, since Nikhila needed some help, I was bringing her up to speed. By the way you don’t have to go all the way near school. If you go past my house and turn left in the little lane there there is a new cycle-repair chap from Odisha. He recently fixed my bike very efficiently and economically.” Vidrum: “Thanks Lootika! that’s wonderful. Let me hurry there and return home to continue with the studies. I still have to catch up with phenols.”

Thus, Vidrum headed to the repair stall and got his bike fixed. The price the repair-man quoted for his work was indeed a good one as Lootika had mentioned but then Vidrum realized that all he had was a big note with no change. The repair-man asked him to wait till he got another customer so that he could give him the change. However, Vidrum had no time for that especially on a day like this. After some wrangling he thought the repair-man sounded honest so he decided to accept his offer when he said that he would come to Vidrum’s house and deliver him the cash pretty soon if he left his address. Being in a hurry Vidrum unthinkingly gave him directions to his house and left.

◊◊◊◊

In the heat of the exams he forgot about his change. Upon the conclusion of the last of the exams Vidrum was riding back home with Meghana. He was feeling slightly relaxed that they had gone quite well though he could not be sure still if he would meet his parents’ cut. Just then they came across a clump of classmates frantically discussing a problem involving Hypsicles theorem. Seeing them Meghana triumphantly announced that she had solved the killer problem correctly. They were surprised to hear that she had done so, given that she was not known for any mathematical capacity at all. Vidrum did not want to let her down by revealing that she had merely copied the solution from him. Instead, he said: “you know we had worked it out before and it came in the exam.” The rest of their classmates who had gathered there said: “Hey thus far we only knew of Hemling and also Lootika and Somakhya to have cracked it – but those two are sort of crazy – you can never guess what obscure thing they might know! How come you figured it out?” Vidrum happy to be at the cynosure of his classmates was about to spin a yarn of how he had cracked the problem when he saw a car drive by them. He noticed that it was Lootika’s mother who was driving her four daughters home from school. It reminded him that after all the solution was not his. Hemling was given to boasting about his mathematical abilities; duly, he had found this problem and brought it up to Somakhya and Lootika to proclaim his superiority over them. But not the ones to let such challenges pass, they solved it after some battle, and as Vidrum and Sharvamanyu were hanging out with them they revealed the solution to the two. Feeling somewhat ashamed at the intent to pass a patently false tale he instead discreetly told his assembled classmates that he had indeed learned some tricks regarding how to solve it while hanging out with Somakhya. It also reminded him, much to his horror, that he had not yet collected the change from the bike-repair-shop. His parents would ask him for the accounts of how he spent his pocket money in day or two and he could not let the cash go unaccounted. So he left his classmates and hurried towards the bike-man’s shack.

In the meantime Somakhya mounted his bike and feeling the relief of the exams being behind him he wended his way towards a new rocky outcrop he had discovered among the basalts. While he cursed the basalts mentally, wondering why the gods had not birthed him in a land rich in Mesozoic strata, he still was excited by the prospect of finding some geodes, which he wished to analyze for their chemistry and structure. In particular he had acquired an interest in the chemistry of zeolites that were found among these basalts. He thought to himself: “for biochemists like us the chemistry of silicon puts many things in a distinct perspective.” He wandered in solitude among the basaltic rises from the great eruptions marking the close of the Dinosaur Age collecting chabazite, mica and quartz geodes. He had lost all sense of time in his pursuits and as he climbed towards the plateau above the temple of Caṇḍikā, he spotted a circle of basalt pillows that appear to have been laid deliberately by the human hand. To investigate it further he got close to them and started looking around. After some examination he realized that the stone circle was a likely remnant of the ancient megalithic peoples streaming into the peninsular of the subcontinent. While scanning the rocky debris in side the circle he found a metal ingot with the image of the Vetāla, an agent of Rudra, stamped on to it. It was clearly from an age closer to the current time. He realized that it was likely dropped by the Vetāla-worshiping tribesmen who gathered on the adjacent hillock on certain nights for an animal sacrifice. His excitement knew no bounds at his find, which suddenly made him aware of the time, and he headed home.

◊◊◊◊

The next day school was off much to the relief of Somakhya. Lootika was to visit him for lunch and then hang on. Somakhya was in his home lab when he heard his mother intercept Lootika at the gate. Much to his embarrassment he heard her quizzing Lootika about the questions on the just concluded exams and how she had answered them. Lootika answered that she had already forgotten about them. S.M: “Lootika, but would it not be nice cosmetics if you guys actually translated your intelligence into splendid results on the exams?” Lootika: “Nice cosmetics may be, but for whom to behold? As long as we pass and go to the next class what do these exams matter? And not that we are barely scraping through.” Then Somakhya’s mother proceeded to ask Lootika in regard to some molecular biologist featured in that day’s newspaper: “Would you someday be rich and famous like him”. Somakhya was now positively concerned knowing how brusque Lootika could be. He half got up from the microscope to distract them and bring her in quickly. Thankfully, Lootika responded with a boring answer: “I am not too good with names of people so I will have to read the article to find out what he works on precisely to give you an appropriate answer.” This answer was so confusingly unexciting for Somakhya’s mother that she decided not to pursue that line questioning any further. Somakhya heaved a sigh of relief that unnecessary fires were not lit.

She ushered Lootika taking the box with some food item which Lootika had brought along asking her: “Should I have my husband drop you back home in the evening – I just saw in the news that an animated protest for the aggrandizement of the local apabhraṃśa was to take place. As you know this is often coupled with pratibrāhmaṇatvam and we could be a target. L: “So it is. A member of one of the protest-organizing committees even asked me to join saying that the brāhmaṇa-s proscribed females from using the artificial Sanskrit language. So it was natural for women to stand by those fighting for all education in the state language! In any case thank you but no worries; my father said he’ll pick me up in the evening.”

After lunch Somakhya and Lootika were engrossed in a deep discussion on silicate minerals and Somakhya was showing her the minute crystals of various minerals under his microscope even as they set aside samples for chemical analysis. Lootika produced from her bag a rock which she had found. It had a striking globular crystal growth with the surface of each globule having a smaller crystals – a bit of fractality. At base of the globular growths was another mineral with a smooth more glassy surface. L: “What is this one?” Somakhya closely examined it with a lens: “The globular crystals are a good specimen of chalcedony while the the mineral at the base is moganite. Such appear to have formed in some quantity in our local basalts during the end of the Mesozoic.”

As they they set up some acid treatments for qualitative analysis Lootika remarked: “Deep within the two of us are svābhāvika biochemists and these minerals have a bit of an alien touch! Studying their chemistry makes me even more skeptical of Silicon-based life outside of computers elsewhere in the universe.”
S: “In most part I tend to agree with one possible exception that deserves more discussion. We have been looking at a bunch of zeolite structures and as you would have noted while there is the same tetrahedral geometry there is much less tendency for lability – and tendency to prefer oxygen over hydrogen – in this indeed I would posit lies the difference between the ability to make life or not. The one exception being certain clays, which some have seen as a possible Si based form of life.”
L: “Ah! talking of clays I forgot to show you this.” She pulled out a bottle with an interestingly colored clay in it. “This one is from Sagaradurga.” Somakhya prepared a mount of the clay under the microscope and remarked: “You see those crystals are believed to be part of a templating mechanism that allows clays to be treated as a type of Si-based life.”
Having taken a close look Lootika donned her spectacles again saying: “But is such accurate templating not a general process in any crystal growth? You would recall our freezer experiment to grow snowflakes that our parents would hate us for. While every snowflake is unique, the six sectors of each snow flake are often accurate copies of each other suggesting that templating mechanism around the core starter is transmitted quite faithfully even in water. But such templating hardly gets you to life. Moreover while it might be moderately accurate across the six sectors it is too mutable between snow flakes to represent any faithful information transmission.”
S: “That’s right. The snowflake is a good example for how accurate templating can happen in crystal growth, but the clay as life hypothesis goes beyond: It posits that certain clays are more prone to attracting mineral precursors from solution or sol to their surfaces to incorporate them into the growing crystal lattice. The difference in this capacity is seen as a form of natural selection which allows certain clays to “prosper and grow” while the less-sticky ones which cannot attract mineral precursors do not grow or spread. Like with the snowflake it reproduces the initial configuration or crystal “defects” accurately in the daughter crystals but is seen as having more intrinsic stability of structure – something we can verify with the crystals we have been examining. When drying occurs and the crystal breaks into pieces, those pieces preserve the original crystal configurations. They are then seen as being dispersed widely by natural forces, like say wind. These then nucleate new clays that then grow – thus, the proponents would say there is genetic transmission. New defects are the mutations, which can then be copied by the same process too. Thus, this theory posits clay to be a system like life with capacity for replication, natural selection, mutation.”

Lootika: “Though it is not at all clear if the mutations in clay in anyway are the subject of natural selection?”
S: “Yes; that, as far as I can say, is a leap of faith in this hypothesis.”
L: “But would it have any links to life as we know it at all?”
S: “Its proponents propose something called the “genetic takeover,” where biochemicals somehow use the original replicating clays as a vehicle for their own chemistry and transmission. This is followed by they becoming independent replicators to break off on their own as life as we know it.”
L: “That indeed can be seductive idea: but it still suffers from the problem of “special nature of life” principle we have proposed. Why don’t we see such proto-life repeatedly forming on earth with the clay phenomenon still available today?”

◊◊◊◊

Suddenly Somakhya received a message from Vidrum asking if he could stop by. Somakhya was surprised – he thought Vidrum would be having a party with Meghana and friends. He showed the message to Lootika and asked: “Why would he want to come to see me? We are in the midst of interesting things what should I tell him?” L: “Yes. As I was leaving for your house I saw a knot of our classmates and heard their raucous yells and whistles even as I passed by Meghana’s house. May be just tell him we are busy with our minerals and he might get bored if he came.” Somakhya did so but Vidrum answered that if Lootika was around it was all the better since it was a matter the two of them would be most suited to give some answers for. Somakhya gave in and asked Vidrum to come over.

Even as Vidrum came in he said: “Good to see you guys; I am sure this is an issue only you might be able to provide an answer for. Now I understand why in the past our society valued brāhmaṇa-s so much.”
Somakhya: “Why dear Vidrum? You seem to be somewhat agitated by something beyond the ordinary. I have never heard you call upon our brahminical credentials for the more routine issues of education that we were supposed to be in charge of in the days of yore. Moreover, the exams just got over and I understand they went quite well for you. So what beckons you to come to us for brahminical services.”
Lootika said nearly laughing: “Hey, I caught a glimpse of you having a whale of a time with some of the classmates. It seems to have ended prematurely?”
Vidrum: “Lootika, why rub salt thus. Women are always like that! I know I am disturbing you in whatever interesting stuff you are doing but if you don’t mind please hear me out.”
Lootika: “Why? Would you have thought of us if something had not interrupted the session with your other friends? Never mind, we are all ears. Tell us what happened.”

Vidrum: “Please bear with me. It is a bit of a long and absolutely bizarre story! Lootika knows the beginning of it but I’ll fill you in Somakhya. That day before the chemistry exam after getting the needful from you I went over to study with Meghana. While at her place it seems a tempo filled with heavy iron rods dumped its cargo on the wheel of my new bike which was parked at her gate. As result some of my spokes were smashed. Thankfully, Lootika recommended me a bike-vālā near her house who repaired the damage but did not have change for the note I gave him. It was most of my pocket money. He promised to send the change over to my house as soon as possible but did not do so. I remembered this only yesterday and ran to get the change from him but his shop seemed to have shut down!”
Lootika interjected: “No wonder he did not return your change Vidrum. Tragically, he died that night!”
Vidrum: “What! That explains everything! This is certainly a matter in your domain!”
Somakhya: “What happened? Do you know how he died?”
Lootika: “That night before the chemistry test I became rather overconfident that I had nearly revised everything by evening. I started whiling away my time and helping little Varoli and Jhilleeka with their impending exams until I suddenly realized it had gotten quite late and I had not yet finished the part on anthracene with all that photodimerization and oxidation. So I decided to get up early in the morning and finish it off along with a round of last minute brush up. Even as I woke up there was some commotion going on. My father was called up by a bunch of people and he had to arrange for an ambulance all of a sudden. There are some days my father has to handle medical emergencies but they are rare these days. At breakfast Jhilli and Varoli remarked that they had heard a loud explosion. It is rather amazing that Vrishchika and I had so blissfully slept through it all. My father informed us that the car-vālā whose shop is next to the late cycle-vālā’s had a storage tank of nitrogen that was attached to the thin wall separating him from the latter’s shack. For some reason that tank of N2 exploded and propelled itself through the wall into the cycle-vālā’s shack and shot out through the roof. But the cycle-vālā who was sleeping in the shack seems to sadly have died from asphyxiation though we were not sure if he was rather hit by shrapnel from the explosion.”
Somakhya: “That’s sad indeed! But Vidrum you have not completed your story. Please continue.”

Vidrum: “Ah. In light of what Lootika has just informed us I think I’m making more sense of the bizarre events. Nevertheless, let me continue in order so that you understand my situation. Last evening with the exams behind us I went with several friends to watch a movie with that I had exhausted all the cash I had. Today we watched another movie at Meghana’s house and then our gang suggested that we go to the famous Kūrmakūpa hotel for lunch. Unfortunately, I had no cash. Our friends counted their cash to see if they could subsidize me but all they could muster was less than what would be needed for a plate. Hence, with much sorrow I decently decided to opt out. Cursing the cycle-vālā I came back home in sorrow. I grabbed a quick lunch from what my parents had left in the fridge and went to the bathroom to wash my mouth. To my shock when I spat out the water it was red as though with blood. I examined my mouth and found no cut. You won’t believe it – each time I did it the same thing happened! Unnerved by this I went to my room and sat at my desk. If this was bad, what happened next positively shook me. The fan on the ceiling suddenly started rotating on its own making a weird noise. It was like a periodic whooping noise that I seemed to have heard some years ago one night at my ancestral village. My folks told me it was the dreaded ghost Daṇḍalūma making his rounds! Then from one corner of my room I heard a girl singing in an unknown language. I looked hard but I saw no one. I have told you before that there were some strange incidents pertaining to the cemetery near my house and when I mentioned them to my aunt she wanted to take me for a psychiatric evaluation.

Now I thought I was really losing my wits as my aunt had claimed. The alluring singing continued and I looked again and I saw the faint outline of the face of a girl with straight thick hair. Then that stopped abruptly and a darkish face appeared on my wall-clock. It looked vaguely familiar. I raked my brain as to who it was. But only now, after hearing what Lootika said, I realize that it was the late Orissan cycle-vālā. He kept scowling at me from the clock and it deeply terrified me. I did not catch much of what the face kept jabbering but I did hear one phrase: “kṛtaghna-kukkuron-vālā-kula”. Now I realize he was pissed off because I cursed him so many times for not returning my change not knowing he had already died. But it just did not stop sometimes he would appear on the fan with that whooping noise in the background, and sometimes in the clock and it would swing violently. Then it would stop and the girl’s singing in that strange language would continue. Finally, I was so scared that I sent you the message and came over. I know you guys have some special means of tackling this type of entity from beyond. So tell me what can be done!”

Somakhya and Lootika: “Wow.”
Lootika: “There is precious little we can do without being there on site I believe. We could come over there in the evening.”
Vidrum: “But my folks would be back home and we can hardly do anything suspicious!”
Somakhya: “Vidrum we cannot just come over for such an adventure. We first need perform some rituals so that Śiva is on our side as we step into your spooky lair. To be frank I’ve always been a bit concerned of your place.”
Lootika: “Indeed. His place is positively haunted. May be we should do it tomorrow.”
Vidrum: “Hey. If you guys are so scared then what to say of me. What do I do? I cannot just go back and spend the night with the angry phantoms.”

Lootika: “There is something more in your case. We can see the possible reasons for the late Orissan to pursue you but that girl appears strange – could there be two ghosts here?”
Somakhya: “That’s good point! Since Vidrum’s place was always haunted, and bhūta-s like certain loci, I wonder if that girl was displaced by the Orissan’s ghost who tried to occupy the same locus?”
Lootika: “In practical terms that possibility makes it quite difficult for us to handle this thing. I was thinking that the best way for us to deal with this issue of Vidrum without us going on site to do some karman was to create a second mobile locus to draw the bhūta into which we could then collect and tranquilize. But that would need us to set up a khārkhoḍa, which would itself need some labor in the first place. But now we have two to deal with. What if the girl goes into the khārkhoḍa, while the Orissan continues to haunt Vidrum?”
Somakhya: “Ah! that can happen. Did the girl look like someone familiar?”
Vidrum: “I think she looked East Asian. Very strange, right?”
Lootika: “Indeed!”
Vidrum: “How long might it take you to set up that khārkhoḍa thingie? Can you make two?”
Lootika: “It could easily take Somakhya a day. If I ask Vrishchika’s services, we might be able get two between all of us in a few days, given that neither me nor my sister can be sure of our success in the first try. Further, remember tomorrow is a holiday but day after we are back to the drudgery of school.”

Somakhya: “While making a khārkhoḍa from scratch is some challenge. I have an unusual solution. By some luck I found this vetāla ingot of the hillmen. It can prove to be an effective as a khārkhoḍa with just 20 minutes of ritual. Vidrum we need you to go away – may be circle around on your bike and come back after half an hour. I’ll need to do this ritual in some silence and Lootika alone can be around as an uttarasādhaka.”

Vidrum duly left. Somakhya showed Lootika how the ingot could be set up as a khārkhoḍa using the Vetālabhairava mantra. When Vidrum returned he handed it over to him: “Vidrum, take this home and place it the southern corner of your room or under the wall-clock.”
Lootika: “So what do you intend to this later this evening?”
Vidrum: “The friends are going to be back from lunch soon and gathering at Gardabh’s house. I intended to join them there. Do you all have some plans?”
Lootika: “Good. We are going to play some badminton once we are done with our study of the zeolites.”
Vidrum laughed out aloud: “Lootika: do you ever get at least a point against Somakhya? He’s quite stiff even for Sharvamanyu who is apparently the best in our school.”
Lootika: “Never mind. It gives me some practice so that I can bully Abhirosha and Vrishchika later. You continue with your plans but do not eat any meat and if possible light a lamp at the Śmaśānasarasvatī shrine and whenever possible make a clicking noise slapping your tongue against the palate.”
Vidrum: “That’s strange.”
Somakhya: “Yes, but do as she says. That’s the best we can do for you now. I know this can be a difficult day for you but before doing anything deposit the khārkhoḍa and bring it with you to school day after and hand it over to me.”
Vidrum looked a bit uncertain but decided to leave them to join his other friends.

Later that evening Lootika’s father came to pick his daughter up. As he did so he reminded Somakhya’s parents that his family was conducting a caitya-yāga to Kumāra the next evening as a relative was going to visit Kārttikeyapura in the mountains. He then reminded them that his wife and he had invited them to come along with Somakhya to participate in the ritual as that way they could convey their offerings too.

◊◊◊◊

The next evening Somakhya’s family arrived at Lootika’s home for the said caitya-yāga. Lootika’s mother opened the gate to let them in. Lootika and Vrishchika came out with their mother and Somakhya immediately glimpsed a certain tension and excitement on their faces, especially that of Lootika. Lootika immediately exclaimed: “I believe it has played out to the worst of our fears. The phantom has struck at Vidrum’s place!”.
Lootika’s mother said with a stern face: “Girls go inside. Let them come in and sit down. This is not the first thing you want to tell them!”
Seeing her mother’s reaction, and their parents busy with pleasantries, to maintain social decency, Somakhya avoided any immediate conversation with the caturbhaginī, who sat in a closely knit clump on the couch eyeing each other and Somakhya with an expectant look. Lootika’s father told them that he was nearly done with the preparations of the yāga and that they could start in a few minutes.

In the mean time Lootika’s mother filled them in saying that their schedule was a was disrupted because of an emergency call that Lootika’s father had to attend to around 3:00 AM that morning: “There is this kid Vidrum who is Lootika and Somakhya’s classmate who lives some distance away near the śmaśāna.”
Somakhya’s mother: “Well I saw him visit our kids just yesterday.”
L.M: “His aunt Vaidoorya was apparently running down the stairs from her room on the upper floor in the dark when it seems her dress got caught in the railing and she tripped and fell headlong. She suffered a major head-injury and had to undergo an emergency procedure. They called my husband because that’s often the fastest way to procure care in this kind of a situation. Moreover, she was a student of mine in the course I teach at the med-school. Sadly, her situation is pretty bad.”
Lootika’s and Somakhya’s mothers quickly caught their kids visual communication. L.M: “I am sure you might have heard this from your son. Our kids like to keep saying that Vidrum’s house is haunted and, as you can imagine, with this tragedy, my girls have been spinning tales of the wildest fantasy all day and have been waiting to share those with Somakhya.”
S.M: “Are kids can be strange isn’t it? On one side they show this precociousness in several directions but almost as though the compensate for that they can spout utterly crazy fantasies that makes me feel worried sometimes.”

Their parents then admonished them to focus seriously on the ritual if they wanted any phala and stop being distracted by their fantasies. After the ritual they packed the offerings to be sent the caitya at Kārttikeyapura. Then they sat down for a feast in course of which Somakhya and Lootika could hardly talk anything about the matter that was really on their mind because of the eyes of the elders upon them. Instead, the dinner conversation revolved around a heated debate sparked by Vrishchika concerning issues of the metabolic syndrome, while at one side Jhilleeka and Somakhya’s father kept out of it having their own discourse on fractality and irrational numbers.

◊◊◊◊

With school resuming after the exams Somakhya and Lootika never got to talk about the events at Vidrum’s place in private. A whole fortnight passed thus when one day Lootika and Vrishchika had just gotten on their bikes to ride to school. They heard someone riding hard behind them to catch up; alarmed they turned around only to be relieved at the sight of Vidrum with his face blanched as a ghost itself. As Vidrum caught up with them, without much of a word he handed over the vetāla ingot to Lootika: “Found it in my aunt’s room! I couldn’t wait to get rid of it. So I’m handing it over to you rather than Somakhya who has been wondering what befell it.”
Lootika: “OK. Let me bury it outside my gate and we can retrieve it later.”
Vrishchika: “Has your aunt’s intracranial swelling subsided? How is she now?”
Vidrum: “She’s recovering slowly. Luckily, yesterday’s neurological tests were positive but it seems she will miss the year.”
Lootika: “Hopefully she recovers well. I believe my mother can some how arrange with the authorities that she doesn’t miss the whole year but only this semester. But you look very pale and wan yourself?”
Vidrum: “Today is report day. I have made it to the 6th place, but not within the first 5 so I fear I may lose my bike and will be back to the chore of walking.”
Lootika: “Ah. Yes let’s see what music we all get in front of our parents from the teacher.”

School ended early that day but they had to wait for their parents to come and collect the reports in the order of their rank. The teacher assigned to their class first handed the report to Tumul and conferred fulsome praise on him before his parents commenting about his high intelligence, courteousness to teachers, and good behavior. Puffed up he left with them exultantly looking down on the rest. When Lootika and Somakhya’s turn came the teacher unceremoniously handed their reports to their mothers: “Somakhya and Lootika have a similar temperament and are likewise tied for the 5th place. They are sometimes very aggressive in their questions to the teachers in class. Somakhya began badgering the chemistry teacher and Lootika joined him to almost bring the teacher to tears with claims that she was teaching things wrongly. Hence, we have had to cut their marks or else they would have ranked higher.” Then she turned to Lootika’s mother and remarked: “Your daughter hangs out most of the time with the boys and has learnt mischief from them like bringing a catapult and a bag full of clay balls to class. She might do better if she cut that out. The only saving grace is that she is the girl with the highest marks in maths by a huge margin given how tough this exam was.”

Then came Vidrum’s turn. Handing his report to his father, the teacher remarked: “Vidrum has been very focused on his studies this term and well-behaved. He makes excellent notes in class and has shown a major improvement since last year with a strong all-round performance to rank 6. Keep it up.” With the sheen taken off Somakhya and Lootika’s performance, Vidrum 6th place seemed to shine brighter their own tainted fifth place. That was not lost on his father and he was let to keep his bike with the relatively mild admonition of getting higher in the next exam.

With school over early for the day Vidrum, Somakhya and Lootika hopped on to their bikes and headed to see the displays at the newly reopened museum of the archaeological survey. On the way Vidrum asked his friends, somewhat triumphantly, if they got some music from their mothers. Somakhya: “Every dog has his day and today is yours. Deep within our folks know only too well that the standards we have to measure against are not these reports but those of our ancestors of yore, the Atharvan-s and the Aṅgiras-es. So beyond some words at the spur of the moment, they don’t get too caught up with such.”
Vidrum: “But what about the chemistry class incident for which you all were docked some marks?”
Lootika: “Hey. We did you all a good service by calling her out. A person who does not know the formula of Carbon disulfide should not be a teacher, let alone that of chemistry. Our parents know fully well we were doing our brahminical duty of pāṣaṇḍa-khaṇḍanam.”

As they went through the new displays in the museum Somakhya brought the attention of his friends to an unassuming set of objects in a small display. One was a metal mirror with a handle which was labeled as a toli. Beside it was a small doll-like object which was labeled an ongon. Then there was a yellow paper on which something was written in a curious script. Beside it the exhibit stated that the mysterious document was found among the possessions of Doherty, the secretary to the English tyrant Elgin, who was buried in the city cemetery. It was undecipherable until recently a lama from Mandi had translated it for the Archaeological Survey.
The first line read: “Princess Irinjinbālā”. Somakhya: “Vidrum – that is your east Asian lady!” She has recorded her tragic fate in this document. See – in the next line she quotes her ancestress, the mother of the great Khan:
‘Not misunderstanding ancient words,
Not forgetting old words.’
She records her ancestry as being from none other than illustrious Qasar and Sübedei.”
They read the translation of what she had written. She was captured by Elgin’s troops in in 1860 and this Doherty brought her against her wishes as a slave-concubine to our town and subject her to much suffering. She was foretold by her uncle, a noted lama, that she would meet a dreadful fate but die in the holy land of Jambudvipa on account of which after much suffering she would eventually attain the svarga of Indra and reside there. The same lama told her that her illustrious ancestor himself was reborn as a great paṇḍita-warrior in Jambudvipa and thus attained the realm of Kubera.”
Lootika: “I wonder if it was her suicide note?”
Vidrum: “That’s pretty remarkable if it was her! I never heard or saw anything of her after that day. What English expedition was this? To Tibet or to Mongolia.”
Somakhya: “No, it was the opium war in China, where she was evidently captured in the English raid near Beijing. Lootika, some day you would encounter and release her. She would certainly give you service in return.”
Lootika: “You talk like your ancestor Kabandha Ātharvaṇa.”
As they walked among the exhibits they stopped to take a careful look at a bow which was labeled as that of the great brāhmaṇa warrior among the marātḥā-s. Somakhya: “I had always the wondered if this great warrior was Sübedei and Qasar reborn in our midst. We seem to have obtained our confirmation today.”

After the museum trip they swung by the temple of Caṇḍikā and climbed up the hill to the circle of basaltic pillows. There, in the middle of it they buried the vetāla ingot again, put some of the remaining soil in Vidrum’s head and asked to have bath upon his return. Somakhya: “Within 4 days you will have a dream where the deceased Orissan will appear to tale his tale. Record it, however painful it might be, and send it to us.”

◊◊◊◊

The four sisters were huddled in their home-lab when Jhilleeka and Varoli asked Lootika if there had been a closure in Vidrum’s case. Lootika: “Yes, he sent us an account of his dream. It brings everything to a resolution. It so happened that after the cycle-vālā died there was no one to claim his body or cremate it. So it was chopped up in the anatomy department of the med-school. However, his brother came a few days later and honestly tried to settle any outstanding payments that were there. One of those was Vidrum’s. Hence, he went to Vidrum’s house on the last day of our exams when only Vaidoorya was around. He tried to return her Vidrum’s money by knocking at the door. She did not understand what he was saying and thought he had entered the house and stolen money. She called the police and they arrested him. In the mean time it appears our cycle-vālā had become a bhūta and he vented his ire the next day on Vidrum’s family especially Vaidoorya. That day Vidrum came to us for help upon being attacked. Somakhya prepared a khārkhoḍa that drew the Orissan’s ghost into it. However, that night Vidrum managed to get our classmates Gardabh or Mahish to let him stay at their house to evade the phantom in his. For some reason his aunt had gone to his room and seeing that pretty ingot, which was the khārkhoḍa, took it with her to her room. That was her doom.”

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Godhūmo vijayate

As we have noted several times before, the Ṛgveda represents a very early stage of the evolution of Indo-Aryan. Its agricultural terms relate to a major puzzle in Indo-European history. There is an unmistakable agricultural vocabulary including words such as kṛṣṭi = cultivation, deriving from the root kṛṣ to plow. The furrow sītā and also the eponymous goddess represent agricultural practice in a field often termed kṣetra. The primary grain cultivated in these fields was yava = barley, one of the oldest crops cultivated by man. Another word generally taken to mean a generic grain is dhāna. All these words have a clear Indo-European origin and are part of a small but consistent vocabulary of agricultural terms that can be traced back to a very early stage of IE if not PIE itself. Yet, the archaeological evidence for agriculture among the early Indo-Europeans is murky. This issue has been raised before by Mallory as a key problem in our understanding of early Indo-European archaeology. There is no mention of wheat, and at best 3 uncertain mentions of rice – i.e. the word odana, which could have simply meant a boiled grain.

The Yajurveda and later layers of the Atharvaveda, in contrast, present a different picture. Rice and wheat are both present and the former is already on its way to overtaking barley as the principle grain. The words for rice include vrīhi (regular rice) and nīvāra (a wild rice). The first word is found in Iranian as birinci and has cognates across the Iranian languages. The first word was also either transferred to Dravidian as arici or acquired by it parallely to Indo-Iranian. It is also found in Greek as oruza, Mongolian ür-e, Japanese uruchi. The same applies for wheat, which in Yajurveda is already seen as godhūma. It is also found in the later Avestan layers as gantuma. Dravidian shows a form similar to the Indo-Aryan word. It also occurs in Hittite as gant and in the Egyptian branch of Afro-Asiatic, and Semitic as well, it has cognates with similar sounds. Thus, these two crops most likely spread rapidly and laterally from a focus and were taken up by related and unrelated language families as they spread. This suggests that, while rice is not found in RV, the Iranians and Indo-Aryans acquired at least some knowledge of this crop even as they neared their current homelands. However, we cannot entirely rule out knowledge going back to an earlier period. The Altaic and Japanese cognates for rice suggest an eastern focus while the western cognates for wheat likewise suggest a western focus. Both of these are consistent with the ecology and genetics of the domestication of these crops.

If we look at rice and wheat production today the stamp of these foci and distinct ecology are clearly visible. In figure 1 we plot the mean annual density (tons/hectare of national land; log-log scale) over 17 years of wheat versus rice production for the 72 countries that cultivate both these crops. We find that they are entirely uncorrelated (r^2=0.034). This is notable given that wheat and rice are among the most cultivated crops in the world (wheat most, then corn, then rice).

Rice_Wheat_densityFigure 1

Now if we divide the plot into 4 quadrants based on the median values for rice and wheat cultivation we find that there is a relatively small group of 12 countries (1/6th of total set) that show a high density for both wheat and rice cultivation (Table 1)

Table 1: High density rice and wheat cultivation; Land is national area in hectares.

Country                           Land         Rice        Wheat
--------------------------  ----------  -----------  -----------
Bangladesh                    13017.00   1162382.50     58596.88
China                        932748.77   6876275.00   4541253.88
Egypt                         99545.00     76581.38    201579.00
India                        297319.00   3244250.00   2661475.00
Iran, Islamic Republic of    162855.00     85245.50    713768.25
Iraq                          43737.00     12825.00    385919.62
Italy                         29413.14     42250.00    589500.00
Nepal                         14335.00     80625.00     66600.00
Pakistan                      77088.00    239241.38   1121625.00
Portugal                       9150.00      3750.00     28000.00
Spain                         49900.00     17144.00    471508.38
Uzbekistan                    42540.00     12625.00    196625.00

It is notable that 4 of these make up bulk of the Indian subcontinent while 3 others are part of the old greater Iranian zone. These lie precisely at the junction of the eastern rice focus and the western wheat focus. Indeed, we find low wheat-high rice density zones like Bhutan, Myanmar, Japan, Korea, and Thailand around the eastern periphery of this zone and reverse on the western periphery. This also raises that possibility that combined cultivation of the two, in addition to ecology, is in part the echo of the entry of the Indo-Iranians into this zone who then stabilized both the eastern and western acquisitions across their core domains.

In figure 2 we plot the wheat versus barley production (in tons; log-log scale). A similar picture is obtained if we plot densities as above. The two, in distinct contrast to rice, show a good correlation (r^2=0.644). Further, while just 72 countries grow both rice and wheat, 96 grow both wheat and barley. This suggests that barley, which was the old IE crop, was to a degree ecologically and operationally equivalent to wheat, with steppe-land as opposed wetland preference (rice). Thus, unlike rice, barley has taken little root in non-Mediterranean Africa. Hence, it appears that as wheat was acquired, it ate into the share of barley across the steppe-land but barley still holds on as the fourth most produced crop. Its role in domesticated animal feed and probably its special role in alcoholic beverage production (already described in the Yajurveda) play an important role in its tenacity and even expansion.

Barley_WheatFigure 2.

Posted in Heathen thought, History, Scientific ramblings | Tagged , , , , , ,