Hofstadter and Nārāyaṇa: connections across space and time

The scientist-philosopher Douglas Hofstadter presents an interesting single-seeded sequence H in his book ‘Gödel, Escher, Bach: An Eternal Golden Braid’. It is generated by the recurrence relation,

f[n]=n-f[f[f[n-1]]] where f[0]=0 …(1)

Working it out one can see that it takes the form: 0, 1, 1, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 13, 13, 14, 14, 15, 16, 17, 17…

If we connect f[n] \rightarrow n in (1) then we get a tree structure which simulates a specific pattern of succession and duplication (Figure 1).

Hofstader_H_treeFigure 1

As we mentioned before, we only got to read Hofstadter’s book briefly when we first came across it. Hence, we did not have the chance to take in all that was discussed in it. However, it seeded our own explorations along the lines he has proposed in the book. Thus, in that period we discovered for ourselves a two-seeded sequence generated by the recurrence relation,

f[n]=n-f[f[f[n-2]]], where f[1]=f[2]=1 …(2)

This, while similar in from to the above recurrence relation, produces a different sequence: 1, 1, 2, 3, 4, 5, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 13, 14, 14, 14, 15, 16, 17, 18, 18, 18…

Our sequence (2) is related to Hofstadter’s H sequence in that adjacent duplications in his H are replaced by one singleton and a triplication in ours.

As we have described before, a convenient method for visualizing sequences such as this is to ‘rectify’ them: They increase linearly with a constant slope m. Hence, we find that slope and use f[n]-n\cdot m to render the sequence along the x-axis. For many common sequences m=\tfrac{1}{2} works. However, during our experimentation with the sequence (2) (illustrated in Figure 11 in the previous article) it was obvious that m>.5. Hence, we took the arithmetic mean of \tfrac{f[n]}{n} for large n and obtained m \approx 0.682, which served as the required rectification factor. Notably, the same value of m applied for the H-sequence and we could accordingly rectify it (Figure 2)

Hofstader_HFigure 2

Similarly, during our explorations of two other two-seeded sequences (See previous article) we discovered that their rectification factor m=\tfrac{1}{\phi}, where \phi is the Golden Ratio:

f[n]=n-f[f[n-1]], where f[1]=f[2]=1 …(3)
f[n]=f[f[f[n-1]]]+f[n-f[f[n-1]]], where f[1]=f[2]=1 …(4)

Since, (3) in particular resembles H and our above sequence (2) we wondered if we could similarly get a precise expression for their shared m. We noted that in the case of the doubly nested recurrence relation (3) its rectification factor m=\tfrac{1}{\phi} was the first root of the quadratic x^2+x-1=0. For the triply nested recurrence relation (4) we noted that its rectification factor m=\tfrac{1}{\phi} was the root of the cubic x^3-2x+1. Thus, we realized that a connection exists between the rectification factors and algebraic numbers. Armed with this knowledge searched the roots of cubic polynomials to get the rectification factors for H and our sequence (2). The real root of x^3+x-1=0, x=0.6823278 yielded their required rectification factor. In the case of (3) and (4) the rectification factor is the reciprocal of the Golden Ratio, which is the convergent of the famous Meru sequence (known in the west as Fibonacci),
\displaystyle \lim_{n \to \infty} \dfrac{M[n+1]}{M[n]}=\phi

Thus, the rectification factors of the linearly growing two-seeded sequences (3) and (4) were reciprocals of the convergent of a non-linear sequence M. Notably the terms of M appear at each level of the tree of these linear sequences (3) and (4); see Figures 2 and 14 in the previous article. So question arose as to what is the corresponding non-linear sequence to which the rectification factor m=0.6823278 of H and (2) is similarly reciprocally related? The answer to this remarkably leads us to the original cow sequence of the great medieval Hindu mathematician Nārāyaṇa, son of Narasiṃha.

In his Gaṇita-kaumudi, Nārāyaṇa presents one of the earliest studies to identify a discrete formula for the ideal population dynamics of an organism which continually reproduces upon reaching a certain age. He poses the following problem:
prativarṣaṃ gauḥ sūte varṣa-tritayena tarṇakī tasyāḥ |
vidvan viṃśati-varṣaiḥ gor ekasyāś ca santatiṃ kathaya ||
Every year a cow gives birth, from its 3rd year, [and so also] her calves.
O scholar, tell, in 20 years [of reproduction] what will be the clan size from one cow?

He then provides the answer as the following sum of a series:
abdās tarṇy abd[a+ū]onāḥ pṛthak pṛthak yāvad alpatāṃ yānti |
tāni kramaś c[a+e]aikādika-vārāṇāṃ padāni syuḥ ||
Subtract the number of years (when a calf begins giving birth) successively and separately from the number of years till the remainder becomes less than the subtractive. Those are numbers for repeated addition once etc in order. The sum of the summations along with 1 added to the number of years [is the desired number]. Translation as per Ramasubramanian and Sriram’s interpretation.

Let the sequence dh[n], for dhenu (cow), represent the number of cows in the nth year. While Nārāyaṇa gives the direct formula for the nth term, it can be expressed in modern terms rather simply by the below recurrence relation for a triply seeded sequence,

dh[n]=dh[n-1]+dh[n-3], where dh[0]=dh[1]=dh[2]=1 … (5)

This sequence goes as, 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189, 277, 406, 595, 872, 1278, 1873, 2745, 4023, 5896, 8641… Thus, the answer for Nārāyaṇa’s problem with 20 reproductive years is dh[22]=2745.

We note that, just a \phi for the Meru sequence, this sequence too has a convergent which we will call Nārāyaṇa’s convergent N_c,

\displaystyle \lim_{n \to \infty} \dfrac{dh[n+1]}{dh[n]}= N_c

Investigating this value, we found it to be the only real root of the cubic equation x^3-x^2-1=0, i.e. N_c=1.4655712319. This result yields the relationship between the rectification factor m for the Hofstadter H sequence and our sequence (2) on one hand and the cow sequence of Nārāyaṇa on the other: m=\tfrac{1}{N_c}. Thus, the same reciprocal relationship, which we saw between the Golden ratio, the convergent of the nonlinear Meru sequence, and the rectification factors of the linear sequences (3) and (4), is obtained for N_c, the convergent of the dhenu sequence (5), and the linear sequences H and (2).

Since the Hofstadter H sequence includes all Natural Numbers we ask if there is any pattern to the occurrence of the dhenu numbers in it. This relation turns out to be,

f[dh[n]] = dh[n-1]

The same relation also holds for our sequence (2) except that alternately the values are either dh[n-1] or dh[n-1]+1. A further interesting observation emerges as we examine these sequences more closely. The sequence (3) with rectification factor m=\tfrac{1}{\phi} has a fixed bandwidth oscillation (see figure 5 in previous article). The H sequence and sequence (2) have a similar type of fixed bandwidth oscillation. Consistent with this, their recurrence relation and that of sequence (3) resemble each other with one of the terms being directly n itself. In contrast, sequence (4), which also has the rectification factor m=\tfrac{1}{\phi}, shows increasingly larger loops of the size of the Meru numbers and a fractal structure. Hence, we asked if there is a similar fractal sequence with m same as H and sequence (2). Given that in the recurrence relation (4) the two terms recursively call the sequence with one having a subtraction, n-f[f[n-1]], we looked for similar sequences and found that the sequence discovered by Mallows has a comparable structure,

f[n]=f[f[n-2]]+f[n-f[n-2]], where f[1]=f[2]=1 …(6)

Duly, we found that m=\tfrac{1}{N_c} serves as a rectification factor for it (Figure 3) consistent with what we had experimentally determined for the sequence.

Mallows_sequenceFigure 3

N_c and \tfrac{1}{N_c}, unlike \phi, are not constructible by standard compass and straight-edge construction. Rather, they need a construction of the type used for the Delian altar of Apollo, i.e. doubling the cube. It goes thus (Figure 4):

1) Draw a parabola with focus at (.5, 0) and directrix as y=-\tfrac{1}{2}. This parabola has equation y=x^2-x.
2) Draw a rectangular hyperbola with the x- and y-axes as its asymptotes and (1,1) and (-1,-1) as its vertices. This hyperbola has the equation y=\tfrac{1}{x}.
3) The point of intersection of the two conics, X, gives our desired constants: (N_c, \tfrac{1}{N_c}).
4) Using this and the two conics we can construct a rectangle ABCD comparable to the Golden rectangle.
5) Dissecting it using the square of side 1 allows us to construct two further rectangles: ABCD \sim GCXF \sim BEFX. Together these furnish the various powers of N_c as shown in Figure 4.

nArAyaNa_convergentFigure 4

This finally leads to the question of whether this observation regarding the algebraic number emerging as a convergent of a Nārāyaṇa-type series and it reciprocal as the rectification factor of a Hofstadter-like sequence is a more general one. In course of our explorations of Hofstadter-like sequences we discovered a fractal sequence that we termed the seahorse sequence (Figure 12 in the previous article). This is given by the recurrence relation,

f[n]=f[f[f[n-1]]]+f[n-f[f[n-2]]-1] …(7)

For this sequence we experimentally established a rectification factor m=.45. Using the above-described principle we then identified its proper form as the real root of the cubic equation x^3+2x-1=0. Thus, m=0.4533976515. Then we asked if there was a corresponding Nārāyaṇa-like sequence whose convergent is the reciprocal of m. Our search yielded the following triply seeded Nārāyaṇa-like sequence nl provided by the recurrence relation:

nl[n]=2nl[n-1]+nl[n-3], where nl[0]=nl[1]=nl[2]=1 …(8)

It goes as 1, 1, 1, 3, 7, 15, 33, 73, 161, 355, 783, 1727, 3809, 8401, 18529, 40867, 90135, 198799, 438465, 967065, 2132929, 4704323… We found its convergent,

N_1= \displaystyle \lim_{n \to \infty} \dfrac{nl[n+1]}{nl[n]} \approx 2.205

This allowed us to establish it as the real root of x^3-2x^2-1=0, thus N_1=2.2055694304006. Here again for sequence (7) we get m=\tfrac{1}{N_1}. In addition to these examples there is the trivial case of sequences shown in the previous article where the rectification factor is m=0.5. Its reciprocal c=2 corresponds to the trivial duplication sequence: 1, 2, 4, 8, 16, 32… Together these four show us that the form of the polynomial equations for the convergent and its reciprocal are also notable in their parity:
2: \; 2x^2-3x-2=0
0.5: \;2x^2+3x-2

\phi: \; x^2-x-1=0
\dfrac{1}{\phi}: \; x^2+x-1=0

N_c:\; x^3-x^2-1=0
\dfrac{1}{N_c}: \; x^3+x-1=0

N_1: \; x^3-2x^2-1=0
\dfrac{1}{N_1}: \; x^3+2x-1=0

This leads to a conjecture: The reciprocal of an algebraic number which is the convergent of a non-linear Nārāyaṇa (Meru/dhenu)-like sequence serves as a rectification factor for a Hofstadter-like sequence. We have not attempted to prove this formally but the mathematically minded might be interested in doing so.

As we saw above, for \phi and N_c their reciprocals are the rectification factors for two types of Hofstadter-like sequences, namely one with a fixed bandwidth oscillation and another with fractal loops of increasing size. In the case of N_1 its reciprocal rectifies the seahorse sequence (7) which is a fractal sequence. We have not thus far found a fixed bandwidth sequence rectified by \tfrac{1}{N_1}. If such a sequence exists then more generally we might speculate that for each rectification factor there are both fixed bandwidth and fractal sequences.

In conclusion, we find a remarkable link (to us) between the medieval mathematics of Nārāyaṇa and the modern mathematics of Douglas Hofstadter. This yields some interesting results some of which to our knowledge remains unexplored and unproven in terms of formal proofs. Notably, unlike the Golden ratio which appears in many places in mathematics and even in nature, we have not found equivalent occurrences for N_c, N_1 and their reciprocals. It almost appears as if nature has a predilection for things constructible by compass and straight-edge. However, we may note in passing that while writing this article we saw a recent paper proving a theorem that a number which is related to our N_1, precisely N_1-1, appears in the convergents of the random Fibonacci series of Divakar Viswanath.

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Wisdom from a tag system

The case of the mathematician Emil Post, like that of several others, indicates how the boundary between mania and mathematics can be a thin one. Nevertheless, Post discovered some rather interesting things that were to have fundamental implications the theory of computation. One of his discoveries was an interesting class of systems that have come to be termed as tag systems. In its simplest form such a system might be defined thus: We start with a string of the form gx_1x_2...x_n where g is a specific string. If it is encountered at the beginning of the string then we apply the rule:
gx_1x_2...x_n\rightarrow x_2...x_nx_{n+1}h
What it means is that we cut g and some specified number of elements from the start of the string (in this case 1 element in addition to g) and paste string h at the end of string along with a specified number of elements (again 1) in this case.

As a concrete example let us consider the below tag system with three rules (said to be devised by de Mol):
Here the first three rules specify the h that should be pasted at the end of the string if g is respectively a,b,c. The fourth v=2 specifies that for every g that is cut at the beginning of the string we additionally remove one more element i.e. a total of two elements from the beginning of the string. Moreover, it also indicates that if the string length falls below 2 we can no longer remove 2 elements hence the process halts. As an example we can consider a simple starting string aaa and start applying the above rules:

aaa \rightarrow abc \rightarrow cbc \rightarrow caaa \rightarrow aaaaa \rightarrow \\ aaabc \rightarrow abcbc \rightarrow cbcbc \rightarrow cbcaaa \rightarrow caaaaaa \rightarrow \\ aaaaaaaa \rightarrow aaaaaabc \rightarrow aaaabcbc \rightarrow aabcbcbc \rightarrow bcbcbcbc \rightarrow \\ bcbcbca \rightarrow bcbcaa \rightarrow bcaaa \rightarrow aaaa \rightarrow aabc \rightarrow \\ bcbc \rightarrow bca \rightarrow aa \rightarrow bc \rightarrow a

Thus the system evolves for 25 cycles before coming to a halt as the string length drops to 1 at a. We can further plot this as a graph where each string is presented as a height as the system evolves(Figure 1).


tag_a3_entropyFigure 1 and Figure 2 (lower panel)

The evolution of the system presents two interesting features: 1) a step-wise growth, peaking and decay of the length of the string. 2) The complexity of the string which can be measured by its entropy rises and falls periodically. The entropy of the string is calculated using the famous equation of Claude Shannon:
H=\displaystyle -\sum_{j=1}^n p_i\log_2(p_i),
Where p_i is the probability of the i^{th} character appearing in the string. For each cycle this is plotted in Figure 2.

This plot shows the entropy minimal whenever the string falls to the lowest complexity in the form of all a.

Now let us seed the same system with the starting string aaaaaaaaa i.e. 9 successive a and see it evolve. Here it evolves for 153 cycles before finally halting (Figure 3, 4).

Tag_a9Tag_a9_entropyFigure 3 and Figure 4 (lower panel)

This longer evolution is accompanied by greater number of higher order cycles of rises and falls. Yet the overall structure is similar to the previous case where it evolved for only 25 cycles. Notably, we see a similar pattern of entropy evolution of the strings.The rise to maximal string length in the form of all a string results in an entropy minimum followed by complexification at same length to reach paired maxima separated by a central dip in entropy. This is followed by change in string length with the entropy showing a similar cycle for this new string length. A closer look at the strings in the first example suggests that we reach minimal entropy with all a strings with respectively 3, 5, 8, 4, 2, 1 a-s. In the second example we have all a strings with 9, 14, 7, 17, 26, 13, 20, 10, 5, 8, 4, 2, 1 a-s. Remarkably this pattern of the number of a reveals that the tag system is actually computing the famous Collatz sequence or hailstone sequence. This sequence is defined thus for any integer x_n:

x_{n+1}=\dfrac{x_n}{2}, if x_n is even

x_{n+1}=\dfrac{3x_n+1}{2}, if x_n is odd

Thus far all tested integers which have subjected to the above Collatz map finally reach 1 (the Collatz conjecture), which is essentially equivalent to the above tag system coming to a halt. Thus, there is a certain nesting of structure in the these sequences, which is also evident in the plots of the above tag systems strings. Both the aaa and aaaaaaaaa strings have 5, 8, 4, 2, 1 as the lengths of the set of the last 5 all a strings in their evolution. Thus, part of the evolution of the former is identically recapitulated in the latter.

Interestingly, say we start with a string which is not all a, like say aaaacaaa, then for 23 cycles the system evolves through diversified strings until we hit aaaaaaaaaa which pulls the evolution into the Collatzian process. Thus, there is tendency for channelization into the Collatzian convergence for this tag system even for non-all a strings. Of course certain strings can wander for even longer number of cycle in a high entropy realm before falling into the Collatz trap. For example, a system initiated with the string abcbcaabccababc wanders in a high entropy realm for 94 cycles before being channeled into the Collatz process by a 14-a string appearing at cycle 95 (Figure 5).

tag_cycleFigure 5

Finally, if we initiate evolution with the string abcbcaabc the system does not halt. Rather it settles into a 40 step meta-cycle coming back to the same string at every 41st cycle. In each of the 39 following cycles the string length is longer than the starter string. Thus, this starter string being the minimal string-length in the meta-cycle neatly helps define it. Hence, evolution initiated with this string or any of the other 39 strings which occur in the meta-cycle escapes the standard Collatzian route to extinction. This behavior is more like the extension of the Collatz function to the complex plane where in addition to the convergences to 1 at the integers there are other cyclic traps for negative and complex numbers. Thus, if we write the Collatz function as the following map we get the below Julia-set like fractal upon color-coding by the number of iterations required to escape to infinity (Figure 6).

z_{n+1} \rightarrow \dfrac{1}{4}(1+4z_n-(1+2z_n)\cos(\pi z_n))

Collatz_escape_01Figure 6

Though the Collatz conjecture is simple to describe, mathematicians since Paul Erdős have been saying that “mathematics is not yet ready” for proving it. Thus, along with the Goldbach conjecture it is one of those simple to state but baffling problems that lurk at the foundations of the mathematics. Remarkably, a simple tag system as this one provides a model for how a relatively simple mechanism to perform a computation can be devised. Indeed, it is systems such as this that provide analogies to think about computation achieved in nature by the action of relatively unintelligent systems as long as they can run for a large number of steps. In a more general sense systems such as this that tend halt after a finite number of steps also reminds one of the system of sage PĀṆINI for Sanskrit. Here the process halts when it has formed a valid Sanskrit word.

Finally, this tag system also suggests an analogy for the process of the rise and fall of clades of life. Its three rules can be analogized with the the processes of diversification, local extinction and proliferation. Further, the replacement of the two elements is suggestive of the replacement of older lineages by new ones. Thus, under these reasonable models of low level processes we can see a clade increase in number(string length), diversity (entropy increase), go through ups and downs of these and ultimately become extinct or settle into an endless repeating cycle of the same process. This does provide a way of thinking about the fate of certain lineages like the trilobites. They went through many cycles of rise and fall over 270  \times 10^6 years, remaining a dominant arthropod clade through much of this period before a final decline and complete extinction. This makes one wonder if such final extinctions are a generally unavoidable end for systems evolving as analogs of such tag systems. This might even extend to civilizations in human history much as thinkers like Spengler saw them growing, maturing senescing and dying out.

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A note on the cow, the horse and the chariot in the Ṛgveda

yasmai tvaṃ sukṛte jātaveda
u lokam agne kṛṇavaḥ syonam |
aśvinaṃ sa putriṇaṃ vīravantaṃ
gomantaṃ rayiṃ naśate svasti ||

For whom you will make a pleasant world, O Jātavedas!
as he does correct rituals for you, O Agni!
He endowed with horses, offspring, heroes,
cattle and wealth attains well-being.

It would be an understatement to say in the Ṛgveda the cow, the horse and the chariot mattered a lot to the Ārya-s. Indeed, philogical evidence establishes beyond doubt that they were central to the life of the Indo-Iranians. Their mentions number in the several hundreds whereas houses and gold (hiraṇya) is mentioned far fewer times (171 times). If you sample random blocks of variable size measured in terms of half ṛk-s (50 : 5000) from the Ṛgveda and count the number of occurrences of the common words for cow(go), horse (aśva) and chariot (ratha) per block you find that the number of occurrences of any pair of them are strongly correlated (Pearson’s correlation >.9 for each pair). This suggests that they form a complex that is present throughout the RV. Yet this poses a poorly acknowledged but major paradox for the archaeology, dating and the geography of the text:
1) The RV is considered by white indologists and their fellow travelers to have been composed in Gandhāra and Pāñcanada at its oldest ~3500-3200 years before present. The chariots are said to appear in the archaeological record earliest ~4100 YBP in the Sintashta culture on the Bronze age steppes. Thus the two are said to be comfortably consistent, with the RV being composed after the ārya-s had settled in northern India in the late/post-Harappan landscape. The Andronovo culture, which succeeds the Sinthasta, is seen as the intermediary which expanded and transmitted the steppe Indo-Iranian culture to India. This is seen as being consistent with the absence of horses and chariots in the Harappan culture (though in reality horses are recorded in some Harappan sites though rare in both remains and depictions).

2) In contrast the RV shows clear knowledge of specific geothermal phenomena in the Caspian region. Further, its astronomical references compared with those of the Yajurveda and the Vedāñga Jyotiṣa point to dates certainly certainly earlier to ~3500-3200 YBP and even before the earliest Sintashta chariots, perhaps closer to 5000 YBP than 4000 YBP. These dates, while well-known to Hindu scholars since Tilak, are routinely ignored by white indologists as fantasies of the Hindus despite early acknowledgment of the same in their midst by Jacobi. Further, the study of the ethnogenesis of the Indians suggests that the main Indo-Aryan contribution came from the invasion of a population close to the Yamnaya of the early steppe Bronze age rather than the Sinthasta or its successors Andronovo culture which descended from the Yamnaya. This is also more consistent with the estimates for the dates of expansion of the “Indian” branch of the R1a1 Y-haplogroup which is associated with the invasion of India by the ārya-s. Further, the evidence of Sarasvati river also points to an earlier date for the RV, albeit with much lesser certainty.

We do find the early dates for the RV based on astronomical allusions to be convincing. Further, as discussed earlier it does match with other features like: 1) near lack of rice agriculture and dominance of barley; 2) the rarity of the sword and the gadā (which can be associated with Sintashta/Andronovo) as war weapons; 3) hardly any mention of the complex society with multiple service castes found in the Yajurveda brāhmaṇa-s and ritual. Thus, to us the RV in most part bears all the signs of an early text that is actually not consistent with what we know from archaeology of the period around 3500-3200 YBP in northern India. Hence, we accept the presence of a real paradox that needs wider recognition and study to be resolved suitably.

Irrespective, of how exactly the scenario resolves we can paint some aspect of the lives of the ārya-s based on the RV: They were different from the later horse-centric pastoralists like the steppe Iranians (e.g. śaka-s) and Altaic peoples for whom the horse was primary and the cow marginal. In contrast, for the ārya-s the cow was central to their nutrition and economy and the horse played a central military role. While the ārya-s certainly rode horses, its military role in that period was apparently not so much as direct mount as much as for drawing the war-chariot. Accordingly, the chariot was an important symbol of power. This, expressed itself in the form of the chariot/horse races which were an important form of contest, entertainment and ritual. Further, unlike the later horse-centric steppe pastoralists the ārya-s did practice some agriculture even before they arrived in the Indian subcontinent and seemed to have focused on barley cultivation. Here, again cattle were clearly central; they drew the plow, concomitant with which seeding of the furrows took place. Thus, the economy of the ārya-s was necessarily a mixed one combining both cattle-rearing and agriculture. Other than horse with its special military role, goats and sheep were also reared with the latter supplying wool which is mentioned in the RV. However, notably there is no mention whatsoever of cotton. Camels are mentioned 5 times (4 times in maṇḍala 8) – thus, they were known but do not appear to have been very important, unlike in the case of the Avestan Iranians. This suggests that though the two cultures were related there was a geographic/temporal separation between the Avestan Iranians and RV ārya-s, with the former being obviously younger due the daiva-ahura polarization.

ratha_ashva_goFigure 1. (mean: red, median: blue).

That said let us take a closer look at cows, horses and chariots in the RV. In the first experiment we draw 10,000 blocks of 350 half-ṛk-s each at random from the RV and count the number of times the words aśva (excluding the name of the gods Aśvin-s), ratha and go. A histogram of their occurrences (Figure 1) shows that while the cow is quite normally distributed there is some skew for the chariot and even greater skew for the horse (In a typical run we get skewness(ratha)=0.325; skewness(aśva)=0.472; skewness(go)=0.277). This suggests certain regions with higher than typical mention of these.

To understand the origin of this skewness better we next look at the occurrence of these words per maṇḍala normalized by the number of half-ṛk-s per maṇḍala (Figure 2).

ratha_ashva_go_maNDalaFigure 2. (mean: blue horizontal line)

We notice that the maṇḍala-4 of Vāmadeva Gautama has an above average density of references to the word go. We wonder if this is actually a purposeful encoding of their clan name Gotama (literally meaning he with the best cattle) by the main author of the maṇḍala. Of course the self-reference Gotama occurs here but also the phrase gomad (with cattle) is frequently used. This could be a parokṣa signature of the Gotama-s. Then we observe that in maṇḍala-9, the Soma maṇḍala there is the highest per maṇḍala density of this word for cow. This is because of the frequent metonymic usage of cow for milk which is mixed with the soma (Ephedra species) juice in the preparation of the ritual beverage (also probably the placing of the soma stalks on cattle hide). Notably, aśva and ratha are both clearly over-represented together in maṇḍala-5 of the Atri-s.

To understand this better we take a closer look at the mention of these words across the RV by calculating counts in continuously sliding windows of 100 half-ṛk-s from the beginning to the end of the text Figure 3).

ratha_ashva_go_windowFigure 3. (mean across all window is green line)

We notice that whereas go shows a fairly continuous distribution of fluctuations, ratha less so, and aśva shows few clear standout peaks. The largest of these is seen in maṇḍala-1 which corresponds to the famous aśvastuti of Dirghatamas Aucathya which is deployed in the grand Aśvamedha ritual. Notably, there are two standout peaks for aśva in maṇḍala-5 of the Atri-s and maṇḍala-7 of Vasiṣṭha-s which are correlated with corresponding peaks of ratha. In the case of the Atri-s a part of this again relates to a dhānastuti where an aśvamedha-performing Bhārata monarch is praised. But even beyond this the Atri-s and Vasiṣṭha-s to a lesser degree seem to have a special tendency to mention horses and chariots more than on an average in certain contexts. This does not seem to have geographic factor because the Atri-s show links to Kāṇva-s of maṇḍala-8 who show no such special proclivity. Rather, it does raise the possibility that the Atri-s were either special connoisseurs or breeders of horses.

Thus the great Śyāvāśva the Ātreya praises the Marut-s:

gomad aśvāvad rathavat suvīraṃ
candravad rādho maruto dadā naḥ ।
praśastiṃ naḥ kṛṇuta rudriyāso
bhakṣīya vo ‘vaso daivyasya ॥ RV 5.57.7

Endowed with cows, with horses, with chariots, with good heroes
with gold, wealth you have given us, O Maruts!
Make praise be ours O sons of Rudra!
May I partake a share of your divine aid.

See also:
The maNDala graph for R^iShi-sharing
Some trivia concering the Adi-shruti
The Rāmāyaṇa and a para-Rāmāyaṇa in numbers-II: Evolving early Indo-Aryan warfare

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Median and pedal triangles and derived fractals: an introductory account

It is rather easily seen that joining the midpoints of the sides of a triangle yields four congruent triangles that in turn are similar to the original triangle (Figure 1). This figure might be used to provided a self-evident geometric demonstration of the sum of a series (Figure 1):

1by4_1by3Figure 1

\dfrac{1}{4}+\dfrac{1}{16}+\dfrac{1}{64}... = \displaystyle \sum_{k=1}^{\infty} \dfrac{1}{4^k}=\dfrac{1}{3}

This median triangle is also the source of rather miraculous figure (Figure 2) that has attracted and delighted numerous mathematicians and laymen alike since at least the days of the mathematician Cesaro. The figure in question is obtained rather easily: Remove the median triangle (\triangle EFD) of a starting triangle \triangle ABC. Repeat this procedure on the three remaining congruent triangles. Continue ad infinitum. The figure you get (Figure 2) is famous as the Sierpinski triangle.

sierpinskiFigure 2

While this is the geometrically obvious way of constructing it, the Sierpinski triangle can also be constructed by a slightly less-obvious method. Take a starting triangle and a random initial point. Randomly choose one of the three vertices of the starting triangle and draw the midpoint between the initial point and the chosen vertex. This midpoint becomes the new initial point. Again chose one of the three vertices of the starting triangle randomly and draw the midpoint between the new initial point and the chosen vertex. Repeat this process endlessly. Surprisingly, all the points drawn by the above procedure will settle down into a locus which is the same Sierpinski triangle (Figure 3). One of the very first programs we have memory of writing was to draw this figure by this method.

SierpinskiFigure 3

Notably, this figure also arises rather organically from arithmetic, thereby pointing to a deep connection between simple numbers and geometry. Consider the below matrix M:

\begin{matrix} & & & & 1 & & & & &\\ & & & 1 & & 1 & & & & \\ & & 1 & & 2 & & 1 & & & \\ & 1 & & 3 & & 3 & & 1 & & \\ 1 & & 4 & & 6 & & 4 & & 1 & \\ \end{matrix}

This is the well-known meru-prastāra of the Hindus of yore. The expansion of the binomial (x+y)^n has the coefficients as above. Now, if we apply the modulo operator M \; \mathrm{mod} \; 2 to this matrix it is converted to a matrix of 0s and 1s. If we assign different color values to 0 and 1 and plot the matrix then we get our Sierpinski triangle (Figure 4).

Meru_sierpinskiFigure 4

The Sierpinski triangle is a good way of illustrating fractal dimension to a beginner. Topological dimension is defined as the degrees of freedom of movement one has on a given geometric entity. On a point we cannot move anywhere other than it; hence we have 0 degrees of freedom and its dimension is 0. On a line we can move along the line; hence we have 1 degree of freedom. On plane likewise we have 2 degrees of freedom and so on. This can be objectively measured for a figure based on how many copies k of the original figure we get when we magnify it by a certain factor M. Thus we define the dimension as:

On can see that if we magnify a linear segment by two then we can fit two copies of the original segment in the magnification; thus D=\tfrac{\log 2}{\log 2}=1. If we magnify a square by two we get four copies of of the original square; thus D=\tfrac{\log 2^2}{\log 2}=2. If we magnify a cube by two then we can fit 8 copies of the original cube in the magnification; thus D=\tfrac{\log 2^3}{\log 2}=3. But what about a figure like the Sierpinski triangle? From the above figure we can see that for each doubling of magnification of the Sierpinski triangle we get 3 copies of it. Thus, D=\tfrac{\log 3}{\log 2}=1.584963. This gives us a measure for fractional dimentionality i.e. we can move on parts of the plane but not all of it. Thus, its dimension is neither 1 as a line nor 2 as the complete plane but in between.

This much is known to many moderately educated people. However, a slightly more involved figure is arises from pedal triangles (Figure 5). Here, we draw the altitudes of an acute angled triangle (for a right triangle two of the altitudes are its sides while for an obtuse angled triangle they will lie outside). By joining the feet of the altitudes i.e. the point where the altitude of the triangle intersects the side we dissect the triangle again into four triangles. Here the triangles are generally unequal. However, the three peripheral triangles generated by this construction are similar to each other and the starting triangle (Figure 5).

Pedal_triangle_similarityFigure 5

If we know carry out a Sierpinski-like process described above for this configuration of triangles we get another interesting fractal figure (Figure 6). For an equilateral triangle it becomes same as the Sierpinski triangle but for all other acute angled triangles it assumes a more complex form. It appears that this fractal has an even greater fractal dimension than the typical Sierpinski triangle.

Pedal_triangleFigure 6

Finally, returning to the midpoints of the sides of a triangle, if we join them to the opposite vertices we then get the medians of a triangle. The three medians are concurrent at the centroid of the triangle. Now if we join the centroid to the three vertices of the triangle we dissect the triangle into three triangles. Repeating this procedure for all three newly obtained triangles and iterating it for each of those triangles and so on we get another interesting figure (Figure 7). This is certain fold in origami which in principle can be carried out on paper. In this figure the area of each triangle tends to 0 as as the number of dissected triangles tends to infinity. Thus, it is good example of the concept of the balance of 0 and \infty conceived by āchārya Bhāskara-II in his account of these concepts.

triangle_centroid_dissectionFigure 7

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Mongolica: Chingiz Khan and the rest

As we have remarked many times on these pages there was Chingiz Khan and the rest. No ruler ever achieved his kind of conquest before or after him. Hence, whenever I hear of the latest claim regarding a discovery of the “cause” for the success of the Chingizid Mongols I remain unimpressed. Yes, we do not deny that those causes might have contributed in someway small way – every man needs luck in life to get somewhere. The population of Mongolia at the time of the great Khan is estimated to have been around 700,000 people, who were divided and lost in conflict. So the available manpower that the Khan had at his disposal was to start with rather limited. Further, one must factor in his situation at the start of his career: whatever truth might have existed in the account that his ancestors Qabul, Qutula and Ambaqai were notable local Khans Temüjin was certainly reduced to dire straits upon the assassination of his father. Hence, the Mongol phenomenon was for most part due to the genius of the man himself.

Thus, the Jewish chronicler Rashid ad-Din remarked: “What event or occurrence has been more notable than the beginning of the government of Chingiz Khan, that it should be considered a new era?” translated by M. Biran based on D Morgan.

How do we place his achievements in the broader context of the earlier empires of the steppe pertaining to Mongolic peoples? There were pre-Chingizid Mongols and Turks who achieved many notable things that contributed to the Mongol system:

1) The Hun Khanates starting from Shanyu Motun (probably originally pronounced as Shanyu Bagā’tur) learned the steppe Indo-Iranian style mobile warfare and made it a mechanism that allowed the Altaic people to overthrow the steppe Iranians and seize their steppe lands for themselves. They were able deploy this mode of warfare effectively on the Han Chinese empire and established a system where mobile steppe warriors could defeat much larger armies raised by sedentary populations. From the available record of conflicts in the meticulous cīna records we see that first Hun Khanate (the Xiongnu) defeated the Yueh-chih (including likely precursors of the Kuṣāṇa-s) to the west and the Tung Hu to the east to establish themselves in the Mongolian center. This was accompanied by the recognition of the region around the Ötükän mountains near the Orkhon river in Central Mongolia as a holy territory of the Khans. It is notable that the Chingiz Khan and his successors recognized this region and established their capital of Qara Qorum in this region. This supports the continuity hinted by Chingiz recognizing the Shanyu the Xiongnu as a temporal predecessor in his letter to the cīna sage Chiu Chuji. Thus, the foundational Hun empire defined the religious geography of the later Mongolic homeland. Importantly, these early Huns instituted the system of the imperial guard. This was to play an important role under Chingiz Khan as the Kheshig commanded by Muqali, Boghorju and Chormagan Noyan.

2) The Uighur Turks, who absorbed the earlier Indo-European Tocharian civilization, showed how a genuine steppe potentate, which mixes militarism directed against sedentary populations and trade with them, could emerge. Notably, they laid the foundations for many aspects of the government of a steppe empire, which the later Mongols would adopt. These included the first script of the Chingizid Mongols. Ironically, they were a major bulwark against the spread of Mohammedanism into the eastern steppes; thus, they allowed the survival and subsequent efflorescence of heathen steppe cultures.

3) In their first phase the Khitans restored power of the Mongolic people through the conquest of northern China and the remnants of the Tang dynasty under their resourceful Khan Ambaqai (sinicized as Abaoji). Thus, they brought to the Mongolic world key political, administrative (e.g. the postal system) and knowledge developments that had occurred during the high-point of Chinese civilization. In their second phase, despite their loss of northern China to the Jurchen, they were able to reestablish a strong kingdom in central Asia under Khan Yelü Dashi. This kingdom played an important role in keeping in check Mohammedan terror and also developing an administrative model which could adopted by steppe people for the rule. One aspect of this was the development of a cadre of meritocractic administrators by means of the imperial examination system. Chingiz Khan drew many of his key administrators from this pool. Yelü Dashi himself was appointed general of the Liao kingdom after he passed the highest level imperial exam. After the loss of the Liao kingdom and his capture by the Jurchen he cleverly escaped from prison and managed to rally about 5-6000 men to his cause. He then joined hands with the Uighurs and founded the Qara Khitai kingdom. Yelü Dashi’s high-point came when he defeated Mahmud Qarakhani to take Ferghana and Khujand and then a combined force of Mahmud Qarakhani and Ahmad Sanjar the Seljuk Sultan at Qatwan. This marked the beginning of the unraveling of the Seljuk kingdom.

These achievements of the Altaic peoples of the steppes had built up over the years and were available for a sharp Khan to a bring together. Indeed, we believe that the key Khitan leaders were such Khans. Both their founder Ambaqai the Khitan and Yelü Dashi achieved much with a relatively small start bringing together various useful elements from Chinese civilization and the Uighurs. Ultimately, despite the loss to the Jurchen Yelü Dashi might have been able to save the Liao kingdom had his king listened to him. However, his king’s capture by the Jurchen resulted in that avenue being sterile. So he pursued his own imperial ambitions as an independent Khan in the west uniting other tribes to his cause such as the Uighurs.

We generally accept Owen Lattimore’s model (also more recently reiterated by M. Biran) that Mongolia itself was “prepared” for the Mongols by the Khitan. As per this scenario they followed the Khitan from a center in what is now Manchuria or close to it to move into Mongolia once the Khitan had defeated various Turkic groups that existed before before them. But how does this square with their link to the earlier Xiongu Shanyu and the holy center at Qara Qorum? We hold that after the break-up of the Rouran Khanate one of the tribes descending from their old confederation but distinct from the Mongolic people, the Turks (who to start might have had some Śaka ancestry), took power in the form of the Kök Türük (Gök Türk = Blue Turk) dynasty under Khan Bumin. The Blue Turks followed their predecessors in accepting the Orkhon region as holy because that is where they erected the famous Kül Tegin steles under their great leaders Bilge Kha’khan and his brother Kül Tegin. However, Chingiz does not acknowledge these Türks as his predecessors rather he skips them going back to the earlier Huns. This suggests that at this point the Mongolic tribes were pushed to the East but they did not forget their once glorious past. Following Lattimore’s suggestion, it does appear that the Menggu/Mengwu tribe appearing as a relatively minor power in the Tang annals were these Mongols from among whom the Chingizid arose. The Khitan victory against the Turkic groups allowed the Mongols proper as one of the related tribes associated with them regain their foothold in Mongolia. The Khitans introduced certain innovations to steppe warfare like the establishment of fortified strongholds along the lines of the old Indo-Europeans in certain fertile steppe patches. These served as centers for greater production of materials such as metal for weapons, some cultivation, brick-making and trade. Thus, when the Khitan power in the East and China collapsed the Mongols were able to take control of a much more productive set of networks in the steppe. From the famous rock-carving on the cliff overlooking the Tuula river facing north towards Ulan Bator dated to the 1000-1100s of CE depicting the ancestress doe (Qoa Maral) and the ancestress Alan Qoa suggest that the Mongols had re-established themselves in Mongolia by then.

Despite all this background the best of the earlier Uighur Kha’khans and Khitan Khans were not Chingiz Khan. Thus, the backdrop of the buildup of innovations among the Altaic people, rather than contradicting, actually shows that Chingiz Khan stood apart from the best the steppe had produced. There are two sides to this, military and administrative. The military side is the one which most people have readily grasped. There have been great kings throughout history but in terms of the power of the enemies whom he defeated and the sheer extant of his empire the Khan stands apart. We have described some snippets of that on these pages but we shall again mention two examples of his military brilliance, one strategic and one tactical: First, the strategic aspect is seen in his configuration of his campaigns. The big enemies to the south, the Jin and the Tangut were objects in need of serious action as settled people but like the Khitans before him once engaged in combating settled people he could have lost the base in Mongolia to others who had the advantage like him, namely powerful mounted cavalry enemies like the Turkic groups to the west and Islamized Turks even further west. So he proceed systematically by : (i) defeating and absorbing the Turkic groups to the immediate west; (ii) clearing the eastern flank completely by conquering the Koreans and eastern Khitan holdouts. Similarly, the northern flank was cleared by defeating all tribal groups to the north; (iii) then inflicting a serious blow on the Jin but not engaging them in their own turf in an involved campaign of taking major fortified centers; (iv) Conducting the great outflanking operation to the west by clearing the forces of the Ghāzis and the Rus to a great distance. By this configuration he ensured that he stabilized a well-defined home-base for the Mongols. Then he stung the Jin strongly enough to ensure that they would find it hard to threaten Mongolia while he dealt with the west but did not engage them in depth right then. Then he proceeded west to neutralize the serious threat from all other cavalry powers which could fight by similar means as him. By this he created strategic depth and a cast a pincer-grip on the Tangut to destroy them in his final campaign. Through this period he kept the Jin on low burn after the initial hammering he gave them. Thus he left for his successors the stranglehold on them which they quickly tightened to put an end to the mighty Jin. It was this strategic execution of patiently engaging and eliminating foes that allowed the Mongols to succeed where others got bogged down. Second, while there are many examples of tactical brilliance through all his campaigns, one notable point is his innovative development of military technology like firepower and siege-craft to take fortified centers of settled zones. This was one deficiency which Altaic steppe powers routinely had – the inability to take fortified power-centers settled zones effectively. After Chingiz we even see a relapse to this state among successor states like those of the Oirats. However, his development of innovative siege-craft, firepower, “biological warfare” and hydraulic works to divert rivers in flooding cities allowed them smash these strongholds like no one before them. Other innovations included the use of military booms (chains) on rivers to block naval operations of enemies, like in the campaign to bring down the Mohammedan Turk Temür Maliq.

In tracking these military achievements one often misses the overall vision that the Khan had for the making of a new world. For someone who became an emperor from a minor power center in a desolate land-locked corner of the world this vision for remodeling the whole of the “known world” is nothing short of breathtakingly audacious. While we find it somewhat hard to create a complete description for, a few notable points stemming from this vision can nevertheless be listed:

1) Various steppe powers controlled the border of the steppe-land and the settled zone. When previous steppe powers penetrated deep into settled civilizational centers they either had to give up their steppe centers or at best conduct some raids and retreat to their steppe-land domain. Those groups that did establish themselves in the non-steppe zone essentially gave up their connection to the steppe-land even if they reproduced many aspects of it in their new homelands. This is poignantly seen in the early Indo-European peoples including our own Aryan ancestors who settled in India. The later waves of their successors like the Śaka-s and subsequently groups like the Huns also fall into the above pattern. The only possible exception were the Kuṣāṇa-s who did achieve something like what the Chingizid Mongols did albeit on a clearly smaller scale. Importantly, they could not hold their original steppe centers close to Mongolia, which they lost to the Altaic peoples. Further, they could bridge their successes in the eastern and western steppes and India into a completely stable coherent unit only for a brief period. In contrast, with Chingiz Khan and his great successors we see a program for the complete conquest and control of civilizational centers of the settled world into a system of rule from the steppe. Importantly they were integrated into one common unit from east to west – perhaps the first “world” system after the early Indo-European times.

2) It is clear that at some point while completing the conquest of Mongolia the idea had crystallized in Chingiz that he should completely remodel the relationship with other power centers. In operational terms it is rather remarkable that he managed to get his new Mongol nation to put his plan into practice. A key aspect of the plan was the kind of accommodation he showed for different preexisting power structures: He was rather clear about meritocracy and was willing to take in various Uighur, other Turkic, Khitan, and Aran/Alan talent for his cause. For instance he appointed the Khitans Yelü Ahai and Yelü Tuhua to rule over the lands he took from the Mohammedans in Khwarizm. He was also quite happy to support and facilitate the mercantile class across ethnicities understanding the important economic role they played. However, he was uncompromising towards the Mohammedan power holders and ulema. Likewise, he showed no mercy for any level of the Jurchen or Tangut and their sinitic or sinicized power-holders. These he simply destroyed and replaced with his talent. This was followed by his successors with the southern Han power centers. This was very different in form and action from previous client-patron relationship that both steppe-land and sedentary powers tried to establish. If there was one system that could have eradicated the Jihad it was this one; the reason it did not ultimately happen was because ultimately the Mongols were still operating off a small demographic base – with a certain population of ~700,000 only that much could be achieved in farther settled lands (also see below).

3) The tremendous prestige Chingiz Khan’s clan held long after his death, also seen in the form of their enormous genetic footprint directly stemmed from the charisma of the man. Even the Moslem Mogol tyrants of India merely termed themselves silsilā-i güregen or the “line of the in-laws” [of the Chingizids]. In demolishing all local tribal allegiances and transferring them to him and his clan he created a entirely new kind of state that in someway might be seen as a version of some modern states where the allegiance of the people is to a notional entity termed the state rather than an ethnic or tribal configuration. Chingiz Khan’s original vision was clearly one of the ultimate cakravartin – which literally meant the conquest of the whole world. Hence, no other independent power configuration was tolerated in any form – if they resisted they were destroyed. While this destructive side has been much described and sensationalized, an oft-missed point is the role of Chingiz Khan as the lawgiver of his people. For the newly re-configured Mongol nation he provided a comprehensive collection of laws and ritual practices that were critical for the unity of the Mongol system – the Ikh Yasag (the Great Yasa). It won the wide attention and wonder, even if grudging and antagonistic, of several observers including the Mohammedans Ibn Khaldun, Mirkhond and Juvaini, the Jew Rashid ad-din, the Armenian Vardan, the Byzantine Georgius Pachymeres and the European John of Plano Carpini. More than one of these mention the law: “the Kha’Khan necessarily needs to be elected for the throne from among Chingizids by a grand Quriltai. The one who seizes the throne independently of the Quriltai should be executed.” The idea here was to provide for an institutional structure to choose the best to be leader for a unified Mongol nation. The first violation against it was by Temüge, the brother of the great Khan, who tried to seize the status of Kha’khan bypassing the grand quriltai. He was sentenced to death in accordance with the yasag. It is notable that the yasag had an aspect paralleled the dharma of the Hindus and maat of the Egyptians. Just as dharma has the divine dimension of the laws laid down by the great gods Mitra and Varuṇa (captured in the śruti) at the base of the yasag was the old “divine law” of the Turks and Mongols known as the törü which is said to have been promulgated by the god Köke Möngke Tengri. On this rested the yasag of the great Khan which was like the smṛti brought to humans by Yama and our law-giver father Manu. Among the Mongols it seems to have inspired later reforms of law like the “Code of Altan Khan” promulgated by Altan Khan in the 1500s. The yasag’s influence was so powerful that even after the conversion of the western Mongols to Mohammedanism it exerted its influence against the sharia. For example, Muhammad Shaybani Khan in the 1500s overturned a decision of his Qāzi-s by stating that the correct decision would follow the yasag of Chingiz Khan. Similarly, after the execution of a brāhmaṇa (who had told the marūnmatta-s that their founder was an unmatta) by a shaikh against the Mogol tyrant Akbar’s wishes we see him increasingly remodeling his law as per the yasag of Chingiz Khan and moving away from the sharia. This culminated in Akbar commissioning the production of the illustrated work, the Chingiz Khan Nāmā, in 1596 CE. Not surprisingly we see the Mohammedan Badā’ūnī pour his scorn on the yasag. Finally, we may also mention in passing that it included dimensions, like the tamgha system, which played a key role in driving the economy of the unified “world system” of the Mongols.

4) A part of genius is how you develop what you learn from others. In this the Chingiz Khan set a precedence which allowed his people to clearly exceed the previous steppe powers including the Uighurs who in a sense where their first teachers. Importantly, in the domain of practical knowledge they were good at picking out things which were critical and useful with great avidity – medicine and weaponry are two cases – we see this starting with the great Khan himself taking a great interest in assembling the best of the knowledge of the world at that point. The quick adoption of printing and standardizing of text production went on to allow extensive documentation of the newly acquired knowledge. This trend continued among the successors of the Chingiz Khan with Hülegü having Nasser ad-din al Tusi take the books that mattered most, like those on astronomy and medicine, from the Hāśīśin library upon the sack of their citadel. He also recognized the value of a individual like Nasser for his knowledge while demolishing the rest of the Mohammedan edifice. His brother Qublai Khan’s astronomical observatory employing astronomers across the empire was another example of the interest in knowledge in a more general sense. Finally, Qublai’s chartering of several thousands of schools throughout his realm allowed this knowledge to plowed back into the population.

5) This vision of the of the unified “world” system is encapsulated in the idea of the Chingizids to have a single script to write all the world’s languages, thus moving away from the trend of sinicizing which the Khitans and others had fallen to. For this it is notable that Qubilai chose the Phags-pa script based on Brāhmī family. This vision was captured in the Italian painting of Jerome of Prague where he is shown reading a text in Phags-pa. Along these lines we also see their effort in composing the first world history having comprehensive volumes on each of the major civilizations of their known world. Thus, this helped place their rise documented starting with the Secret History in the larger canvas of other nations.

Finally, other than demography and internecine conflict a key factor in the ultimate unraveling of the Mongol system was their approach to religion. As archetypal heathens they clearly gravitated towards a version of religion closest in spirit to their own namely the Indo-Aryan religion. They received watered down versions of it from the Uighurs first and then the Tibetans. This however, did not translate into a robust religious policy. Like most archetypal heathens their natural tendency was to adopt a “secular” (used in the weird Indian sense of the usage) attitude towards the religion of their conquered peoples. While Qubilai Khan banned certain abhorrent practices such as halal slaughter of animals typical of Abrahamists as being against the yasag, the Mongols mostly left everyone to their own religion, even as some of them dabbled in various religious systems. As we have detailed before, this proved fatal against the aggressive Abrahamisms, primarily in the form of Mohammedanism. In this regard perhaps they were less vigilant than their predecessors, the Qara Khitai who had mullas conducting Dāyi activity in their realm promptly rounded up and executed.

In conclusion, one may strike some parallels with the Hindu potentates of more recent memory. The Vijayanagarans who lead the comeback against the Islamic Jihad had a fairly strong sense of comprehensively clearing out the Mohammedans in their early years but lacked the geographically strategic planning and use of outflanking in their campaigns that Chingiz showed. This was their undoing in the end. Among the Hindus in recent times Chatrapati Śivaji alone comes across as a parallel to Chingiz in his energy and audacity. His geographical strategic planning was impeccable in many ways like the great Khan. His successors showed the same lack of the grasp of the religious dimension as the Chingizids. Importantly, other than Śivajī and perhaps Hambirrāv Mohite and Moro Tryambak Pingle they clearly lacked a pool of talent for independent strategic and tactical virtuosity, which we see among the deputies of the great Khan both from his family and the larger Mongol circle – Qasar, Tolui, Boghorju, Muqali, Jebe, Subedai and Chormagan. Even with Śivajī’s greatest successor Bajirāv-I we see among his deputies a lesser commitment to the great cause – other than Pilājī Jādhav (at some level also a mentor) and Rāṇoji Śinde the best of his other deputies seem to have questionable in terms of loyalty. Thus, the achievements of the Marāṭhā-s, while ensuring our survival to date, still fell short of their intended goal of re-establishing the unified Hindu nation.

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Braided power: a brief note on the last great steppe power: the Mongol-Manchu system

We first read of matters pertaining to this note which some interest in books which had newly arrived at a library in our old city that we mainly visited for Sanskritic literature. We wished to summarize everything we had learned in our own words at some length from material gathered from various sources for it is important to have a Hindu narrative on world history. However, at this moment all we have is this short note.

Chingiz Khan’s brother Qasar was known as the Qabutu, the skillful [bowman]). He had played a key role in the rise of his brother as the great Khan of all Mongols. Due the shaman Teb Tengri poisoning the Khan’s mind against Qasar they had a fallout but their mother Hoe’lun had him restored to favor and the Khan’s wife Borte instigated him to have Teb Tengri assassinated. Qasar passed his archery skills to his son Esunge Mergen. In 1818 CE Russian excavators found a meter high stone with a notable Mongol inscription at Startaul in eastern Turkestan. It that states that the Chingiz Khan had the inscription carved in the honor of Esunge Mergen hitting a target at a distance of 335 alds which the modern Mongols say to be at the distance of 536 meters. This event is said to have happened during the Mongol Nadaam festival organized by the Khan after the destruction of the Mohammedan empire of the Khwarizm Shah and before his last campaign on the Tangut. Duly the Khan bestowed on Qasar the eastern Ulus-es along with those for his other brothers Qachiun and Temüge and his half-brother Belgutei the wrestler. Belgutei lived unusually long (110 years according to the Jewish chronicler Rashid ad-din) but his successors were mostly undistinguished. Qasar seems to have died shortly after the great Khan but his successors were to have much greater prominence in Mongol politics over a wide swath of land across Mongolia, China and Iran. The ulus of Qasar went over to his son Esunge upon his death. Subsequently, Qasarids spread widely but their main power center lay in the midst of the Khingan mountains, which was the old seat of the Khitan, the pre-Chingizid Mongols who had established the Liao dynasty in China. It was from here the main Qasarid clan of the Qorchins expanded. From time to time they continued to collaborate with their Chingizid cousins. During the Red Turban rebellion of the cīna-s a Qasarid prince sacrificed himself to allow the ignominious flight of the last Mongol emperor of China Toghon Temür. For a while in the 1400s they tried to gain the kingdom of Mongolia (e.g. Adai Khan’s effort) and fought the Ming, Chingizids and the Oirat Mongols. Later, alarmed at their dominance in his domain the Oirat emperor Essen Taiji banished a whole tribe of them. In 1510 CE they rejoined their Chingizid cousins under Dayan Khan and Khatun Mandukhai to defeat the people of Jelme and Subedei’s clans who had rebelled, the Mohammedan warlords from Hami who were waging jihad in Mongolia, and launch raids into the Ming territory.

Coming back to the Liao dynasty, they counted among their feudatories a tribe known as the Jurchen. They spoke a language from a distinct branch of the Altaic family, Tungusic. These Jurchen formed an alliance with the ethnic Han Song kingdom of southern China to overthrow their Khitan overlords of the Liao dynasty. They then founded the Jin dynasty in northern China. The word Jin meant “golden” and their rulers were styled the golden khans. The remnants of the Liao were reorganized in central Asia by the energetic Khan Yelü Dashi as the Qara Khitai kingdom. They raised a mighty force which defeated the Qara Khanids and other Mohammedan Turks to establish a sizable and powerful kingdom. This kingdom was usurped from within by the Naiman Turks. Chingiz Khan destroyed the most powerful eastern Turkic confederations of the Kereit and the Naiman and in the process absorbed the Qara Khitai kingdom. The Jin were then overthrown by the brilliant campaigns of Chingiz Khan and his successors. Chingiz Khan had a soft corner for the remnants of the Khitans as co-ethnic Mongols and one of the causes for hostility against the Jin that he cited was their overthrow of the Khitan Liao dynasty. By 1234 CE all vestiges of Jin power in China had been completely smashed by the Mongols and the surviving Jurchen clans returned to their tribal homeland near the Yalu river where they reverted to their old lifestyle of hunting, small farming and pastoralism .

Among these Jurchen clans arose a strongman by the name Möngke Temür who came from near the Sunggari river, tributary of the Amur, to the north of Korea. He moved south and starting in the early 1400s alternatively allied himself to the Ming and the Koreans and kept his chiefdom afloat. He was asked to join the Ming war of extermination against the Mongols and led a campaign against them. But a retaliatory strike by the Mongols forced him to flee from his fiefdom. Nevertheless, he gathered a large force of several thousand Jurchen under him and then on till his death in an inter-Jurchen farcas served as a mercenary leader for both the Ming and the Koreans. This military experience of the Möngke Temür’s clan paid off and despite suffering several reversals in fate in inter-Jurchen conflicts they eventually united the Jurchen tribes 180 years later under their remarkable leader Nurhachi. Recalling the connection to the Jin kingdom of the older Jurchen, the rising new alliance was termed Aisin state, where the word aisin in the Jurchen language meant golden, just as Jin and altan in Mongolian. The old Jin had a special connection to the nāstika deity Mañjuśri, the primary devatā of the Mañjuśriya-mūlakalpa, and he was said to appear in special visions with his upheld sword to his worshipers on the Wutai mountains in China. The Jin had built the Mañjuśri hall on those mountains in the 1100s. That old connection returned as Nurhachi saw himself as the reincarnation of Mañjuśri and renamed his people as the Manjus (or common usage Manchu). Nurhachi and his sons Khong Taiji (some times called Abahai) and Dorgon over the coming years shepherded the meteoric rise of the Manchus and led them to great victories against the Koreans and the Ming. After completing the subjugation of the Koreans taking advantage of peasant rebellions against the Ming in China they conquered Beijing under Dorgon’s leadership in 1644 CE. In consultation with his council of ministers he agreed that his brother Khong Taiji’s young son Fulin should ascend the Chinese throne at Beijing as the Shunzhi emperor. After Dorgon’s death, under the Shunzhi emperor the Manchu continued their expansion snuffing out all Han cīna resistance by crushing the prince of Gui who led the Ming in the south and demolishing the redoubtable half-Japanese seaman “Koxinga”. Thus the Ching empire of the Manchu came into being.

The Mongols played a key role in the rise of the Manchu. First, even as Chingiz Khan had adopted the Uighur script to place Mongolian in writing, Nurhachi employed the Mongols Erdeni Bagshi and Dahai Jarguchi to create a Manchu script from the Mongol script. Second, he and his sons also won over several important Mongol clans to join hands with the Manchus as a combined force for the invasion of China. These clans were to become the main lords of the what is today Inner Mongolia. Third, his son Khong Taiji who lead the Manchu to their early victories married five ladies from Qasar’s clan. Likewise, 5 of Khong Taiji’s sisters were married to Qasar’s descendants, one to a Chingizid of Dayan Khan’s clan, and another to a Mongol of the Ba’arin clan descending as a collateral line from Chingiz Khan’s ancestor. Thus, the fate of Qasar’s clan was intimately tied with that of the Manchu. Khong Taiji while a Bauddha in these sense of worshiping nāstika deities like Mañjuśri and others he was concerned about the version of vajrayāṇa preached by Lamas of Tibet for he feared it was causing the Mongols to give up their own language and shamanism. He tried to call upon the Mongol leaders from giving up their ways but eventually let it lie not wanting stir up religious strife. Khong Taiji noted that the Chingizids lost the empire of China because the Han nationalist fervor turned against them. Hence, he tried to bring about reconciliation between the Han one hand and the Manchus and Mongols by different means but the Han ethnic riots and revolts against them in 1623 CE made him wary of them. On the other hand the closeness to Mongols is clearly laid out by the Khong Taiji’s grandson via a noted Qasarid princess, the Kangxi Emperor (who was the longest ruling emperor of China, 61 years).

He says: “The [Han] Chinese turn of mind is not straight. As to the Manchus and the Mongols, even several tens of thousands of them, are of one mind. In the years I have been on the throne, the reason why I have declared it difficult to rule over the [Han]Chinese, is their not being of one mind [with us].

On the other hand the Kangxi emperor managed to bring the outer Mongolian Chingizid also into an alliance with him. He says regarding the Khalkha Mongols (Chingizid successors of Dayan Khan):
Of old, the Chin dynasty [i.e. Chin-shih Huang-Di] heaped up earth and stones and erected the Great Wall. Our dynasty has extended its mercies to the Khalkha and set them to guard the northern territories. This will be even stronger and firmer than the Great Wall.

This alliance was to also play an important role in the last phase of Manchu power in China when it appeared as though the spirit of Qasar, the skillful warrior, had reincarnated in one of his descendants Sengge Rinchen for one last blaze of the Mongol military fire even as the Manchus were facing some of their most trying challenges. His story relates to the a curious Chinese civil war, which took place in 1850-1860s, whose geopolitical significance is often lost. Hong Xiuquan a failed Han Chinese civil service aspirant heard the sermons of the American protestant preacher Edwin Stevens and studied his pamphlets on Christianity. Inspired by these and the frustrations of his repeated academic failures he started imagining himself as the Chinese brother of Jesus Christ. Further, inspired in his visions by the teachings of another American Southern Baptist missionary he worked to create his own version of the Christian bible – the Taiping bible. Imbibing various Abrahamistic traits he founded his own iconoclastic cult that started destroying images of the bauddha-s and cīna deities. Notably, his flavor of Chinese Abrahamism had the same traits of iconoclasm, communism, sexual equality and utopianism, which independently arose in other secularized Abrahamistic movements within the Christian world itself (This illustrates that Abrahamism has a powerful potential to channelize such delusions. It is also notable how Hong’s fervor later inspired the Han nationalist Sun Yat-sen ). By 1851 CE Hong had organized a large army, which now seeking to establish the utopian “Heavenly Kingdom” initiated a civil war against the Ching dynasty. Hong’s forces killed the Manchu commander sent against the them and he established himself with Nanjing as his capital. Before its end the Taiping civil war had left 30 million dead or more – a true outbreak of an Abrahamistic disease.

Even as Hong’s story was playing out Sengge Rinchen was born 26th in line from Qasar in the early 1800s. He and his father had returned to the old Mongol herding lifestyle but he was noticed by Ching officials for his extraordinary athletic skills. Given the relationship of his family to the Ching emperors he was invited to the court and and educated there. Soon he rose to the rank of Grand Minister of the Ching and member of the inner circle of the emperor. In 1853 as Taiping war was raging as Grand Minister Consultant he was asked to organize the defense of the capital. He smashed Hong’s army in a great encounter in 1855 and captured two key generals of the “heavenly kingdom of the brother of Jesus Christ”. With that he completely neutralized their threat to the capital. He then advanced against another anti-Ching rebel group, the Nian, which inspired by the White Lotus movement; in 1856 he completely wiped out their infantry. Seeing the English threat to the Ching he began fortify several positions with as good artillery as the Ching could mobilize. In June 1859 the combined British and French amphibious force launched a major attack on the port of Tianjin. Sengge led his Mongol forces personally to a brilliant win against the invaders. However, 1860 to avenge their defeat the Europeans returned with a much larger force stiffened by Sikh levies from India, a large body of Chinese who had shifted allegiance to them against the Ching, disparagingly called the “coolies”, and a powerful heavy artillery reinforcement. The Mongols fought them with great valor but the heavy guns proved too much for them and they were vanquished. This was perhaps that landmark moment in history where the long military tradition started by the steppe Indo-Iranians and passed to the Mongols, Turks and Jurchen had finally been completely eclipsed by the changing technology. However, in course of the negotiations that followed Sengge captured the European emissaries and had them killed as they were taken to Beijing. In retaliation the European forces advanced against Beijing and sacked it burning down the famous Yuanming gardens of the Manchu royalty and destroyed their prized new-style buildings. The emperor was displeased by Sengge’s inability to stop this and demoted him to a lower minister rank.

Despite, having lost their infantry the Nian  taking advantage of the European-inflicted defeats on the Ching started making major gains with their quick-moving cavalry now backed by some of the best commanders sent by the Chinese brother of Jesus Christ to stiffen them. Knowing Sengge’s capability the Manchu reinstated him to his old rank and asked him to tackle the new threat. The British, Americans and French realized that if the Chinese Abrahamism triumphed over the Ching then the center of gravity of Abrahamism might shift from their world to Asia. They perceived this as a disaster as it would gnaw into the very scaffold of their identity and sense of self-worth, namely Abrahamism; hence, they decided to tacitly back the Ching government rather than the rebels. Sengge now not having the deal with the Europeans launched a major operation of the Nian and reduced them in several encounters. However, as his Mongol cavalry was seriously diminished by the defeat against the Anglo-French army, his mobility was seriously hampered. Thus, in 1865 CE he died fighting the Nian. The Manchu empress attended his funeral and erected a great monument for him. I am told by my cīna interlocutors that this monument was destroyed by the Maoists. However, apparently the cīna government is now appropriating Sengge’s legacy as part of an attempt to gain support of the Inner Mongolians. Accordingly they have rebuilt a monument for him. The last phase of Qasar’s descendants was seen in their attempt to modernize the Mongols with help from the Japanese during their ascendancy on the mainland. However, all this ended with the Japanese defeat in WWII.

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Means and conics

By the time one reaches high school one learns that: (i) there are four means that one might find some use of in life (I know there are more though they are hardly used) – the arithmetic mean which is most widely used; the geometric mean; the harmonic mean; root mean squared (RMS), which most people would have encountered while studying Maxwell’s brilliant derivation of the ideal gas velocities from theory. (ii) By then one also might have learned the geometric construction of the arithmetic mean and the geometric mean. The later is merely the midpoint of the length of two collinear segments and the latter arises from the famous geometric mean theorem. But fewer learn of or discover a simple geometric construction that gives all these four means (Figure 1).

Figure 1

The arithmetic and geometric means are obvious in Figure 1. If one does some elementary calculations using parallel-lines, similarity of triangles and bhujā-koṭi-karṇa-nyāya one realizes why the other two lengths are respectively the harmonic mean and RMS. Moreover, it gives an immediate proof of the famous statement one might have also studied in high-school:
HM \le GM \le AM \le RMS latex
One also then sees that:

Figure 2

Now perform the following construction (Figure 2):
1) takes a=1 as specified by the point A. b=x varies as point B moves along the horizontal line.
2) Construct the means of (a,b)=(1,x) as shown in Figure 1.
3) Draw a perpendicular at point B to \overleftrightarrow{AB}. Mark off segments of length equal to the means as constructed in the above step (e.g. point F, I). \overline{BF} is the segment corresponding to the harmonic mean and \overline{BI} is that corresponding to the RMS.
4) Determine the locus of these points as point B moves.

One notices that the HM and the RMS give two distinct hyperbolas (Figure 2). The HM hyperbola has asymptotes y=2, \; x=-1. The RMS hyperbola has asymptotes y=-\sqrt{2}x, \; y=\sqrt{2}x. The arithmetic mean by the same construction specifies a straight line and the geometric mean specifies a parabola which corresponds to the lost construction of Apollonius. All four conics are tangent to each other a (1,1) and illustrate how the four means diverge from each other.

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