Of lives of men; of times of men-III

Of lives of men; of times of men-II

As they were talking, they saw a sallow-complexioned youth pass by them some distance away carrying a bat on his shoulder. Vidrum waved out to him and he responded similarly and after some delay so did Sharvamanyu.
Lootika: “Who is he?”
Sharvamanyu: “He’s our old schoolmate Mudgar.”
Vidrum: “You don’t remember him? He was the cricket champ.”
Lootika: “As you guys I know I never cared for that most bizarre of games! I believe my parents rightly advised me and my sisters to stay far away from guys who fritter away all their time at the kandūka-krīḍā.”
Sh: “Hey, but I know Somakhya is quite a fan.”
L: “That does not mean I should be a fan too.”
Somakhya: “Did not recognize him. But now I recall seeing Mudgar last on the day the results of our school-leaving certificate were declared. He was in a funereal state having the pulled the plug in more than one subject. Sadly, his prowess with the bat and the ball helped little in that contest. But he still seems to be keeping to his daṇḍa-bhañjana-krīḍā.”
Vi: “Somakhya, evidently Lootika’s company has made you lose your edge with the game. Forget about the school-leaving certificate – he indeed never cleared it – but don’t you know he is a famous man now?”
So: “Famous?”
Sh: “He apparently smashes big sixes in one of the lesser leagues and is slated to make it to the Indian league this summer to play alongside the stars.”
So: “Ah! the gods have been favorable to him.”

Sh: “That’s indeed true. His father was a noted player for the first division in his days. He hoped his time that some day he might be called up for the national trophy matches. But those days they did not play so much cricket. So, like Mudgar, when he pulled plug in his exams he had no real means of employment. Based on his facility with the game, they gave him a job as peon in railways and he continued to play for them. But he never made much money and Mudgar’s youth was one of those with a low economic status. When you come from such a background, even your school curriculum can be a stiff uphill climb.”
L: “That’s why I say that sports should never be taken so seriously as to think that it can be your profession.”
Sh: “Whatever you might think, they put the little money they had into his cricket and it might pay its dividends now. For all you know, in an year or two, he could be earning much more than any of us.”
V: “Lootika, despite your condescending attitude towards the noble game, I am sure even you have heard of our great hero Musal Gandulkar? He is one of the richest men in the country.”
L: “But that is the whole point. For every Gandulkar, I am sure a million other lesser kids, who thought that they too might become Gandulkar-s, ended up like Mudgar’s father. But on the other hand, Vidrum, even if you are far from being the Caraka of our yuga, you will still make a great contribution to society for having put in heroic effort into your MBBS rather than in kandūka-krīḍā. Moreover, think about this – what is so heroic in Gandulkar? His existence or lack thereof would make a negligible difference to the good of the society. On the other hand, upon graduating I am sure you will make a bigger difference to some people’s lives than this Gandulkar.”

Vidrum: “It feels good to hear that from your mouth, paṇḍitā. But to bring the focus back to why we wanted a long chat with you two – can you provide arguments for why sportsmen should not be accorded special respect or admiration. After all, I could point out that scientists and mathematicians pursuing obscure knowledge for knowledge’s sake are probably as useless to the social good, which you seem to take as an important criterion, as a supremely entertaining cricketer as Gandulkar. At least he contributes to the mental health of the masses by way of entertainment. The scientist in the rarefied realms of inquiry cannot be understood by anyone but a minuscule minority.”
L: “Vidrum, while there is something to what you say, I think you have shifted the goal posts in the mean time. Remember, that, while proximal reasons might be difficult to discern, the normal distribution describes quantitative human traits, including success in sports. A Gandulkar is far to the right of the distribution. I need not remind you that the despite many of you guys having great facility with the game in our school circles, Mudgar notwithstanding, you all are still closer to the mean in this regard, where a great mass from our nation is positioned. Hence, my statement that it is not a good idea to invest in it with such seriousness as one might for a proper profession. Coming to science, I agree that science too is not suitable at all as a career for most people but those closer to the right extreme in scientific ability. I even warned my dear sisters of this even though I trained them in science. Indeed, when our junior college physics teacher asked me to speak to the girls in her class to motivate them to consider science as a career I did something she did not expect. I told them just this it is something for the far-extreme and not for everyone. So they should simply focus on being useful to their families and society.”

Sharvamanyu: “Lootika, note that the times have changed. That’s what I was pointing out. In Mudgar’s father’s day, indeed, only the far-right of the distribution could dream of making the cut – after all only 11 people and few extras can play for the country. But what has happened in the mean time is that the game changed and became potently monetized. This allowed a much larger fraction of individuals to be able to make living out of it.”
Somakhya: “While I don’t disagree with that particular point, let us not forget that still only a very small percentage of our vast population can really make the cut in even the current hyper-monetized cricket circuit. In that regard Lootika has a point. On the other hand I don’t think people are exactly massing to consider cricket as a profession and failing in the attempt to lead more useful lives. There may even be some advantage, as Vidrum suggested, to cricket or more generally other spectator sports because they are a palliatives for the masses – it is better that the masses are engaged in cricket rather than films, which are often better vehicles for sneaking in parasitic memes into our unsuspecting population. Finally, I have to say that, while science and mathematics practiced by its cutting edge practitioners might be understood by very few, even of their own ilk, it is not a bad thing for the general population to strive for as deep a knowledge of these fields as they can achieve. Such knowledge in the population will always be more useful to society than cricket. Thus, in the long run, acquiring a few lessons from JC Maxwell are going to be of much greater consequence to a society than those from a WG Grace. At the same time, it is important to distinguish acquisition of such knowledge from the worship of science or mathematics as opposed to its actual acquisition as propagated by the Marxian liberals. That can be devastating to society.”

Lootika: “It is not for nothing that our ancients, while nor prohibiting the entertainers in anyway, placed the likes of a naṭī, nartakī, śailuṣa, māgadha, jhalla and malla at the level of puṃścalī-s rather than at the pinnacle of society. They were employed and experienced adequately-provided lives but these avocations were not at all seen as an aspiration for the central mass of society. It is also notable that other than the boxer/fighter, which are outlets for the natural and ancestral male aggression, none of the many other sports of the ancients were seen as avenues for professional pursuit.”
Sh: “Ha! Vidrum you may recall our classmate Manjukeshi’s vehement calls for the complete ban on violent sports such as boxing. I wonder what she might think of Lootika’s words.”
Vidrum: “She would have termed it biological essentialism. I might have leaned towards such things in the past when spending lot of time with her and Samikaran but I think I am coming around more and more to see the biologically informed points Somakhya and Lootika have made ever since I have known them. However, it is interesting that Lootika mentions the exception made in our old tradition for these sports relating to male aggression. Why would that have emerged when you have the ever-available, universally more honorable profession of the military?”
Sh: “I think the military is serious stuff – a matter of survival of a nation, while the other thing is entertainment, much like our other sports. That’s why the jhalla or the malla is not placed in the league of the rājanya who is at the top of the social ladder. Yet I am sure our friends might have some deeper biological reason for this.”

Somakhya: “The anatomy of the skull of Homo hints that it might have undergone some selection for over-engineering to survive momentum transfer from blows delivered by the hand. It is reasonable to posit that the emergence of bipedalism in the Homo lineage freed the hands for combat. Indeed, this is a common trait that we see convergently evolving in other bipedal species. Many anseriformes (waterfowl) use their arms in combat (mostly male-male) much like Homo. Thus, the legitimization of the jhalla and malla is likely the result of a deep-rooted social role for such face-to-face hand combat. I think we should indeed distinguish that as a remnant of the old intra-group male aggression relating to dominance and mates as opposed to the inter-group aggression which relates to the emergence of the military profession.”
Lootika: “Indeed, their link to the intra-group male is suggested by the fact that humans clearly recognize its counterparts in non-human animals and considered those as equivalent forms of entertainment. For example, watching the combat between human males is comparable to the specter of intra-specific conflict between males of other animal species. Not surprisingly, among the yukti-s of the old Hindus we see alongside the malla and the jhalla, head-butting of rams and goats, and conflict between various male birds from galloanserae as related entertainments.”

Sh: “Our discussion thus far clarifies the role of intra-group male-aggression and the emergence of sports based on that. But the important question of the emergence of the military profession which Vidrum mentioned remains. That also seems have a male bias. What is its relationship to ancient biological struggles?”
S: “First, there needs to be some tendency for sociality – i.e. tendency for aggregation of members of the same species as opposed to the tendency of existing as free-ranging individuals with no interactions with others from the same species beyond mating. Such sociality can emerge convergently across organisms. Take for example the lion, it is a social cat, in which sociality has emerged very recently, given that closely related cats are solitary in their behavior. Such incipient sociality can emerge quite easily from aggregation offering improved predator-survival or simply persistence of birth associations between siblings or sibling clusters at communal nesting sites. Second, once you have incipient sociality, there is good evidence that traits favoring the ability of a social group to capture resources from a competing group are likely to be selected for. This is the origin of inter-group conflict and the emergence of what we term military today. In our cousins, the chimpanzees, such inter-group conflicts are seen. While males already play a dominant part in chimp conflict, the inter-group raids also feature some females. This situation is persists in some human groups too – for example among the Mongols we hear of some female participation in actual combat activity. But we have keep in mind that females are a reproductive bottle-neck for a social mammal like us. Loss of females is a real loss of fitness for the social group, while loss of even a fraction of the males will not change seriously change fitness as long as the number of females are intact. Thus, inter-group conflict is likely to eventually evolve a strong bias towards male only participation, which will be further exacerbated in favor of males when there is strong sexual dimorphism in mean size, speed and strength between the two sexes.
Lootika interjected: “Of course it must be stressed that biology of the social species in consideration plays a big role in the sex bias of combatants in inter-group conflict. As Somakhya clarified, in mammals, the females bearing small number of live young after prolonged gestation and a clear sexual dimorphism in mean size means soldier-formation will be male-biased. However, in insects of the cockroach clade, wherein termites evolved, the sexual dimorphism generally manifests in the form of bigger females. Thus, the soldier caste in termites shows a clear bias towards females. Similarly, the gene dosage from diploidy, underlying size dimorphism, and relationship bias from haplo-diploidy (sisters being more related than brothers) makes the soldier caste of hymenopterans predominantly female only. There are only rare exceptions where both sexes might participate in soldiering like embryonic soldiers of the the encyrtid wasp Copidosoma and we can explain that. This only reinforces the biological foundations of the sex-bias of the primary participants in inter-group conflicts in different species. That’s why I think this whole talk of sending girls to fight in the military clashes with some fundamentals.”

Vidrum: “This brings us back to a topic we had discussed few years ago when I was swayed by Samikaran. Why a specialized caste, like in our tradition, arose to perform the military role. If the whole population fought you have a numerical advantage; so why set aside a caste for that. What are the biological precedents for this?
Lootika: “Vidrum, I guess we have already explained the gist of this to you in the attempt to bring you out of your classmate Samikaran-the-maniac’s māyā.”
Somakhya: “But now that Vidrum has come away from those fancies I think we should try to unpack that again a bit and also in case Sharva is interested. It is indeed true that having the whole population fighting provides numerical advantage. It appears in chimps there is no particular soldier caste. As we saw in them there is even some female participation. Given the bias against female involvement in lethal combat in mammalian societies like ours, which we have just explained, we could still posit an advantage in the involvement of all males. Now certain ‘Kriegstaat’-societies indeed take that route. For example, the Mongols were one such. Chingiz Khan sent out his officials to conduct detailed censuses of all conquered territories so that the males could be recorded and organized into tümen-s which could be called up for wars. Thus, the Mongols saw all males a potential military man power. Now a society which has alternative structure when under stress will transform into this Kriegstaat pattern. We saw that with Mahārāṇa Pratāpa. We saw it on even larger scales with the Marāthā-s. There, in addition to V1 and V2s, traditional castes involved in various activities like agriculture and livestock-rearing, and even fine specializations like making cow-dung-pats and tapping palm-sap transformed into the warrior caste – that’s how we have a Karhāṭaka brāhmaṇa, the Hoḷkar, the Śinde and the Gāikvāḍ becoming Rājā-s (and Rajñī-s). But in the long run it is a more diversified economy that allows the effective conduct of war. Thus, it means other castes performing their specialized roles and channeling the fruits of the diversified economy to the war machine, which itself is primarily performed a dedicated caste. At a basic level this might mean farmers who can produce food, a essential to field large armies but it can involve various other specialized sub-groups. When the Marāthā-s transformed into a Kriegstaat, that seems to have drawn people away from these specialized guilds. That is where the English won. They could still maintain a large body of seemingly ‘useless’ knowledge-producers, like a Darwin and a Maxwell, for each of whom there were lesser tinkerers who could ultimately supply key technological innovations to the system that kept edge on the English war-machine. The success of this type of specialization in nature is simply evident in the world conquest of the ants, bees and termites.”

Sh: “That brings us back to the normal distribution which Lootika mentioned earlier. I guess that would also predicate that castes are likely to form when intra-group specialization enhances survival of the group.”
Vidrum: “That is fine. While I am no longer convinced by Samikaran’s Marxian uniformity I am still bothered by inequality which caste engenders.”
Lootika: “Over the years Somakhya has impressed upon me that 3 distributions more-or-less help us understand much of what we see around us. One is of course the normal distribution. Be it IQ, height, strength all of these are thus distributed and there is bound to be a consequence from that. Then paradoxically there is the power law distribution. This is the pattern seen in as disparate things as the numbers of genes in genome controlled by a given transcription factor, the sizes of human settlements, and the sizes of grains of sands. It is often called Pareto principle after an Italian who saw that in his country 80 percent of the land is owed by 20 percent of the population. Then there are so-called pathological distributions violating the central-limit theorem like the so called Cauchy distribution, the implications of which Somakhya mentioned sometime ago. Now, coming back to the second of these, the power-law, we can crudely state that the ‘rich get richer’. Thus, a lineage which is good at one thing tends to amass that trait in themselves. Hence, you are bound see inequality with a relatively small number who are really good at somethings and a large fraction being close to incapable. I can see why this can cause resentment in the have-nots. I think the way our tradition tried to resolve this is by specialization so that every group has something to be good at, thus limiting competition in a single track. When this system broke down under the assault of modernity we are seeing all these resentments bubble back. Hence, if at all we are going to find some means of mitigating it we need understand the force of these natural distributions rather than deny them. ”

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Posted in Cricket, Life, Politics | Tagged , , ,

Pattern formation in coupled map lattices with the circle map, tanh map, and Chebyshev map

The coupled map lattices (CMLs), first defined by Kunihiko Kaneko around the same time Wolfram was beginning to explore cellular automata, combine features of cellular automata with chaotic maps. The simplest CMLs are defined on a one dimensional lattice with n cells. The value of the j^{th} cell in the lattice generated by a combination of the action of a chaotic mapping function f(x) and coupling of that value with the values of adjacent cells. Imagine a CML where the j^{th} cell is coupled with the two cells on either side j-1 and j+1 with a coupling fraction of \epsilon \in [0,1]. Then, the value of this cell in the next generation (n+1) (indicated as a superscript) of the CML is given by:

x_j^{n+1} = (1-\epsilon)f\left(x_j^n\right)+\dfrac{\epsilon}{2}\left(f\left(x_{j-1}^n\right)+f\left(x_{j+1}^n\right)\right)

Thus, the CML adds a further level of complexity coming from the chaotic behavior of the map defined by f(x) to the basic one dimensional cellular automaton principle.

One map of interest that can be played on a CML is the circle map discovered by the famous Russian mathematician Vladimir Arnold, which he proposed a simple model for oscillations such as the beating of the heart. It essentially performs the operation of mapping a circle onto itself:

x_{n+1}=x_n+\Omega-\dfrac{K}{2\pi}\sin(2\pi \cdot x_n)

Figure 1 shows the the iterates of x_0=\tfrac{1}{3} for K=.9 and \Omega \in [0,1]. We observe that there are several regions where the circle map generates chaotic behavior and other bands where it is mostly non-chaotic. Also visible more subtly are regions of less-preferred values.

Circle_mapFigure 1

We then play the circle map on a CML of 101 cells by keeping K=0.9 and varying \Omega to take multiple values. We take the coupling fraction \epsilon=0.5, which we found experimentally to give interesting results. We initialize the CML by setting the value of cell 51 to 0.5, and setting the 50 flanking cells on either side of it to the value of 0 in generation n=1. We then let it evolve such that if the cell on either edge of the lattice are neighbors of each other — thus the CML here is in reality plays out on a cylinder. Each value of \Omega results in a different kind of behavior of the circle map (The left panels in Figure 2). The corresponding evolution of the CML is shown in Figure 2, for 500 generations going from left to right.

kaneko_circle_1Figure 2

1) In the first case one can see that the map converges to a single value after a brief initial fluctuation. Correspondingly, when played on the CML it results in the seed anisotropy quickly dying off and the CML settles into a constant state.
2) In the second case the circle map shows an oscillation with a gradual concave rise and a sharp fall. While the oscillations are roughly similar in shape they are not identical. This results in the CML rapidly evolving into a complex pattern. The triangular elements seen in the pattern are reminiscent of those which emerge in cellular automata.
3) In this case the circle map generates sharp approximately regular pattern of oscillations, with rapid, abrupt changes in values. The corresponding CML evolves into a basic pattern of waves. Central seed sets up a pattern that develops into a fairly fixed width pattern the keeps propagating independent of the background waves.
4) Here, the circle map generates oscillations similar to above but slightly less-abrupt and has a more convex descending branch. This results in a more complex pattern developing from the central cell that stands out more clearly from the background waves. It gradually grows in width and shows a central band and flanking elements.
5) In this case the circle map generates oscillations with an abrupt rise and gradual, convex fall. This again, like case 2, rapidly generates a complex pattern.

To investigate the effect of other types of chaotic oscillations applied to the CML, we next considered the tanh map which is based in the hyperbolic tangent function. It is defined thus:

If x_n<0, x_{n+1}=\dfrac{2}{\tanh(r)}\tanh(r(x_n+1))-1,

else, x_{n+1}=\dfrac{2}{\tanh(r)}\tanh(-r(x_n-1))-1

This maps x_n \in [-1,1] \rightarrow [-1,1]. Figure 3 shows a plot of iterations of x_n for the parameter r \in (0,10]. For r<1 the distribution of x_n is all over the place. For r>1 the distribution of x_n becomes more and more U-shaped with a preference for values closer to 1 or -1.

tanh_chaosFigure 3

We then play the above tanh map on the CML with 101 cells initialized with a central cell (51) x_{51}^1=.5 for 500 generations. The coupling fraction is chosen as \epsilon=0.1. The experiment is run for different values of the parameter r. These results are shown in figure 4.

Kaneko_tanh2Figure 4

The first two values of r are in that part of the parameter space of the tanh map that produces highly chaotic oscillations. This results in the CML quickly evolving into nearly random continuous variation.

In the next three cases the effects of the U-shaped distribution of the iterates kicks in and we have predominantly abrupt up-down oscillations of the tanh map. However, the subtle difference in the oscillations causes clearly distinct results, albeit with some common features. In the third example, surprisingly, the CML quickly converges and freezes into several tracks of distinct periodic patterns. In the fourth and fifth case, we see tracks with patterns similar to those seen in the above case. However, they do not freeze, at least in the 500 generations we ran them. Rather, the tracks persist for different number of generations and then become extinct or evolve into other patterns after persisting for even 100 or more generations. These more regular patterns play out in a more irregular rapidly changing background.

In the last experiment presented here we consider the effect of the coupling fraction \epsilon on long-term dynamics. For this purpose we use the Chebyshev map, which is related to the eponymous polynomials of that famous Russian mathematician.

x_{n+1}=\cos(a \cdot \arccos(x_n))

This surprisingly simple map produces extreme chaos with a distribution similar to the tanh map for values of the parameter a>1.5. Values of a=1.5:2 produce interesting behavior in CMLs. In our experiment the we keep the Chebyshev map itself the same for all runs with a=1.8. Figure 5 shows the chaotic pulsations produced by this Chebyshev map.

Chebyshev_mapFigure 5

In this case the CML was run for 5000 generations and every 10th generation was plotted. It was initialized with the central cell x_{51}^1=0.1 and each of the flanking 50 cells on either side were set to x_j^1=-0.75. Here, the \epsilon value is varied to establish the effects of coupling on the behavior of the CML. This is shown in figure 6

Kaneko_ChebyshevFigure 6

The behavior is rather interesting:
1) At \epsilon=0.05 we observe that randomness permeates the entire evolution of the CML.
2) At \epsilon=0.075 there is a fall in randomness with repeated emergence of lines of persistence, lasting for several generations before going extinct. Some times they reappear several 100s of generations later. Within, each line, while it lasts, we see some fluctuations in intensity.
3) At \epsilon=0.085, the randomness mostly dies out by one fourth of the total number of generations of the CML’s evolution. Thereafter, barring the fringes, the lines of persistence alone remain over the rest of the evolution, albeit with some fluctuations of intensity within each line.
4) Interestingly, at \epsilon=0.095 we observe the emergence of “wandering” chaotic lines that emerge from old or spawn new lines of persistence.
5) At \epsilon=0.105, 0.115 we observe that most of the CML rapidly settles down in to strong unchanging lines of persistence.
6) The \epsilon=0.13 shows similar behavior to the above, except that certain lines of persistence display a periodic variation within them like a regular wave.
6) At \epsilon=0.16, we interesting see the return of chaos with repeated episodes of chaotic behavior breaking up old lines of persistence followed by emergence of new lines of persistence.
7) Finally, in the last two runs we see a return to the predominantly random pattern. However, this is qualitatively different from the first case in that it shows some short lines of persistence, which establish small domains of local structure.

Thus, the degree of coupling between the cells of the lattice affect the long term evolution of the system for same initial conditions and driving chaotic oscillator. In the range of \epsilon explored above we see an optimal point for freezing of persistent patterns with randomness dominating in the extremes of the range. However, within the more “orderly” zone we may see outbreaks of mixed chaos and pattern-persistence.

Simple CMLs are computational elementary and conceptually easy-to-understand as simple cellular automata. In some ways they captures natural situations more closely than cellular automata. But on the other hand extracting interesting behavior from them appears to be more difficult. Importantly, they are unique in providing a tractable model for how the local chaotic oscillations couples with other such oscillators. This is seen in many biological systems — networks of neurons, interacting bacterial cells in a colony, colonial amoebozoans and heteroloboseans — all are good natural systems for real-life CMLs to play out. We see chaotic oscillatory patterns in individual cells, which if coupled appropriately, can result in regularized patterns after some generations or rounds of interactions. Both nature of the underlying chaotic oscillator and the degree of coupling will determine whether randomness, frozen patterns, or dynamic but not entirely random patterns dominate. This gives an opening for an important force, namely natural selection, that is often neglected in such dynamical systems-based approaches. Selection is required for setting up the oscillator and its parameters as also the coupling fraction. Further, like CAs, CMLs also have potential as historical models, where local oscillations in populations and their interactions could be captured by the coupling of the chaotic oscillators.

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

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.

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

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?”

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

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

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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

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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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