## Newton’s cows

Cultures with an Indo-European background have had a long history of symbiosis with the bovine animal since they started herding on the steppes in the Black Sea-Caspian region. Indeed, the very emergence of the modern steppes of Eurasia is likely a result of human-animal action to foster a certain pattern and type of grass growth. Hence, not surprisingly, cows frequently appear even in their mathematical literature, like the Greek problem of cows attributed to Archimedes or Nārāyaṇa’s cow population problem. Even after the destruction of Indo-European tradition by Abrahamism in groups like the English such problems persisted and once such example is Newton’s problem from his Arithmetica universalis. It goes thus:
$a_1$ cows graze $b_1$ fields bare in $c_1$ days,
$a_2$ cows graze $b_2$ fields bare in $c_2$ days,
$a_3$ cows graze $b_3$ fields bare in $c_3$ days,
What relationship exists between the 9 quantities from $a_1, a_2, a_3...c_3$?

To solve this we must make some assumptions that Newton indicates in his work: 1) On an average the cows and fields are equivalent. That would mean that we can take each cow to eat the same amount $w$ daily and each field to have the same type and amount of grass $x$. Grass, ungulates and fungi are in a complex relationship. Grass “hire” fungal symbionts to produce toxins like the ergot alkaloids to deter ungulates. There is some empirical evidence that grazing by ungulates triggers grass growth. So we get the assumption: 2) The grass is not at standstill while being grazed daily but is growing back at a daily rate of $y$.

From the above, for the first set of cows and fields, we have $b_1x$ as the total amount of grass at the start of the grazing. The amount of grass growing on the field in day 1 would be $b_1\times 1 \times y$. The total amount of grass eaten by the $a_1$ cows at the end of the day would be $a_1\times 1 \times w$. Thus, we can calculate the amount of grass at the end of day 1 as: $b_1x +1b_1y - 1a_1w$
At the end of day 2 we get: $b_1x+2b_1y-2a_1w$ and so on.
Hence, when the fields are grazed bare in $c_1$ days we get: $b_1x+c_1b_1y-c_1a_1w=0$. By writing $z=-w$ we get the equation: $b_1x+b_1c_1y+a_1c_1z=0$. Similarly, for the other two sets of cows and fields we get the equations: $b_2x+b_2c_2y+a_2c_2z=0$ and $b_3x+b_3c_3y+c_3a_3z=0$. We thus have as set of 3 simultaneous equations in 3 variables:

$b_1x+b_1c_1y+a_1c_1z=0$
$b_2x+b_2c_2y+a_2c_2z=0$
$b_3x+b_3c_3y+c_3a_3z=0$

We can hence eliminate $x, y, z$ using the determinant of the system $\det A= 0$

$\det A = \begin{vmatrix} b_1 & b_1c_1 & a_1c_1 \\ b_2 & b_2c_2 & a_2c_2 \\ b_3 & b_3c_3 & a_3c_3 \end{vmatrix} =0$

If one wants to avoid $\det A=0$ being obtained for solutions: 1) where the amount of grass eaten by a cow is $w=0$; 2) negative values for $y$ or $w$ we have to set further constraints:
$a_1, a_2, a_3...c_3>0$
$c_2>c_1$
$a_1b_2-a_2b_1>0$
$a_2b_1c_2-a_1b_2c_1>0$

In Newton’s original numerical example the matrix of the 9 values is:

$\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix} = \begin{bmatrix} 36 & 10 & 12 \\ 63 & 30 & 27 \\ 162 & 108 & 54 \end{bmatrix}$

This yields:

$\det A = \begin{vmatrix} 10 & 120 & 432 \\ 30 & 810 & 1701 \\ 108 & 5832 &8748 \end{vmatrix} =0$

There seems to have been a error in Newton’s original copy of this problem which he corrected late in his life. A question that comes to us is: Is there some easy algorithm for generating such valid integer nonads and is there some pattern to them?

Posted in Scientific ramblings |

## A novel discrete map exhibiting chaotic behavior

The map proposed by R. Lozi over 40 years ago is one of the simplest two dimensional maps that exhibits chaotic behavior and generates a wide range of interesting structures. The map may be defined thus:

$x_{n+1}=1-a_1|x_n|+y_n$
$y_{n+1}=a_2x_n$
where $a_1, a_2$ are real parameters.

We have discussed our explorations of this map at some length before. At that point we also discussed our discovery of certain generalizations of this class of maps. In course of the quest for the generalized Lozi-like maps, we discovered another generalization remarkable for the diversity of forms it produced. We held back from presenting it because we wished to investigate its geometry further. As a result we are now able to explain the basic geometric determinants of the map; however, there are other aspects that still remain mysterious to us. Hence, we are presenting this map with whatever understanding we could arrive at along with the issues that are still open questions for us. They are maps of great beauty; hence, they are also worth beholding for the sake of their aesthetics.

This map is defined thus:
$x_{n+1}=a_1+a_2x_n+a_3|x_n|+y_n$
$y_{n+1}= a_4-x_n$

Alternatively, we can also define it as:
$x_{n+1}=a_1+a_2x_n+a_3|x_n|-y_n$
$y_{n+1}= a_4+x_n$

Here $a_1, a_2, a_3, a_4$ are real parameters: $a_2 + a_3$ is in the range $(-2,2)$ (see below for explanation), while we explored $a_1, a_4$ primarily in range $(-1.01, 1.01)$ as this is the range where there is higher tendency for the map to yield chaotic attractors. If we take the first definition then the map is bilaterally symmetric about the axial line $y=-x$. It we take the second definition it is similarly symmetric about the axis $y=x$. For the below discussion we refer only to the first definition because the second one shows comparable behavior with change in signs of some of the parameters.

To explore the basic geometric principles behind the attractors generated by this map we started with a set of 200 equally spaced starting points $x_0, y_0$ all of which lie on the circle $x=\cos(t), y=\sin(t)$. With each starting point the map was iterated 2000 times. Then the evolution of each starting point under the map was plotted in one of 11 different colors. We were able to determine the following rules for the structure of these attractors:

1) The most important determinant of the geometry of the attractor is sum of the parameters $a_2, a_3$. When $a_1, a_4$ are closer to 1 and $a_2+a_3 \rightarrow 2\cos\left(\tfrac{2\pi}{p/q}\right)$ where $p,q$ are mutually prime integers, we get a p-ad structure in the attractor (e.g. Figure 1).

Figure 1. Emergence of a heptad structure for 4 different values of $a_2, a_3$

In Figure 1, $a_1=a_4=1$, and $a_2, a_3$ takes successively, by row, the following 4 pairs of values (-0.5450419, 0.1); (-0.2450419; -0.2); (-0.7450419, 0.3); (-0.9450419; 0.5) . In each case $a_2+a_3 \rightarrow 2\cos\left(\tfrac{2\pi}{7/2}\right)$. This determines the central heptad structure but the rest of the attractor shows considerable variability that cannot be explained in a straightforward way from these parameters.

If $a_2 \rightarrow 0$ and $a_3 \rightarrow 2\cos\left(\tfrac{2\pi}{p/q}\right)$ then there is greater tolerance for the range of values $a_1, a_4$ can take.

Figure 2. $a_1=0.3, a_2=0, a_3=2cos\left(\tfrac{2\pi}{9/2}\right), a_4=0.7$

With $a_3=2\cos\left(\tfrac{2\pi}{9/2}\right)$ we get a nonad structure even though $a_1$ is relatively low.

When $a_1 \rightarrow 0$ and $a_4 \rightarrow 1$ $a_2+a_3 \rightarrow 2\cos\left(\tfrac{2\pi}{p/q}\right)$ the p-ad structure in the attractor tends to be retained (Figure 3).

Figure 3. $a_1=-.001, a_2=-.22, a_3=-.22, a_4=0.8$

In this case $a_2+a_3$ is close to $2\cos\left(\tfrac{2\pi}{7/2}\right)$; thus, we get a core with a heptad structure.

Irrespective of $a_1, a_4$, if $a_2, a_3$ are close to each other in magnitude but opposite in sign then we get square structure in the core of the attractor (Figure 4). This is because $a_2+a+3 \rightarrow 0$, which is $2\cos\left(\tfrac{2\pi}{4}\right)$

Figure 4. $a_1=-0.33, a_2=-0.34, a_3=0.34, a_4=0.8$

2) For low values of both $a_1, a_4$ or at least $a_4$, including when they are below 0 and $a_2+a_3 \rightarrow 2\cos\left(\tfrac{2\pi}{p/q}\right)$ we see a loss of the p-ad structure or a flattening of the attractor along the axis of symmetry from the top left side (Figure 5).

Figure 5. Effect of low $a_1, a_4$

In both panels $a_2=a_3=0.305$; $a_2+a_3 \approx 2\cos\left(\tfrac{2\pi}{5}\right)$. In the left panel $a_1= a_4=1$ and the expected pentad structure is retained. In the right panel $a_1=a_4=0.1$; we notice that the pentad structure has been lost with flattening along the axis of symmetry.

These are the basic rules by which the core structure of the attractors can be accounted for. The parameters $a_1, a_4$ can be seen as scaling parameters whereas $a_2, a_3$ can be seen as rotational parameters. The latter pair sets up the rotational structure based on how close their sum approaches $2\cos\left(\tfrac{2\pi}{p/q}\right)$. This is the cosine principle that occurs as the primary structure-determinant in other chaotic maps, including maps which we discovered and discussed before (1, 2). As in the previously discussed cases, this sets the limits of $a_2+a_3$ as (-2, 2); however, here there is greater scope for diversity because the cosine principle is distributed over the sum of two separate parameters. Beyond the cosine principle,n these attractors exhibit additional features whose origins remain mysterious to us:

1) For example, In Figure 3 in addition to the core heptad one can see additional harmonics like a 11-ad, 18-ad and 29-ad structures. What is the explanation for them? In the previous examples of chaotic maps we could explain higher harmonics based on other $\tfrac{p}{q}$ whose $\cos\left(\tfrac{2\pi}{p/q}\right)$ might lie close to the primary cosine. However, in this case the 11-ad, the 18-ad and 29-ad correspond to no such cosines. Hence, their emergence remains a mystery.

2) Further, there is an effect of the starting $x_0, y_0$, which can result in the appearance of a n-ad structure independently of the cosine principle (Figure 6).

Figure 6. The effect the initial points.

In both the above cases the four parameters $a_1, a_2, a_3, a_4$ are respectively, -0.567115129642189, -0.761931293738961, -0.347582505706949, -0.069076271019876. However, the left panel was initialized with a circle of radius 0.9 and the right panel was initialized with a circle of radius $\tfrac{1}{3}$. Only in the second case we see the emergence of a heptad structure, which cannot be accounted for by the sum of $a_2, a_3$. The role of the starting $x_0, y_0$ in generating n-ad structures is another open question.

To further explore the diversity of chaotic attractors within the above-stated parameter space for the 4 parameters of the map we set up our code to search for such maps thus:
1) We set $x_0=0.3\cos(\pi/4), y_0= 0.3\sin(\pi/4)$ as the starting point.
2) We then randomly generated a set of parameters $a_1, a_2, a_3, a_4$ in the range $(-1.01, 1.01)$.
3) For each such set we allowed $(x_0, y_0)$ to evolve for 1000 iterations under the map. If the evolution resulted in the divergence to $\infty$ or convergence to one or few point attractors, then we discarded those parameters.
4) A heuristic for chaos is that very small differences in the initial conditions can result in very different end results upon evolution under the map after a certain number of iterations. Hence, for those parameters which survived the above filters we tested if a second $x_0, y_0$, which differed from the first by $10^{-7}$ in each coordinate diverged from the evolutionary path of the original $x_0, y_0$ after the same number of iterations. If it did so, the parameter set was retained as it was the sign of being on a chaotic attractor.
5) These surviving parameter sets were then explored more fully by studying the evolution of 200 equally spaced points on the circle $x=0.3\cos(t), y=0.3\sin(t)$ for 2000 iterations of the map.

We present below few examples of attractors emerging from the above procedure with parameters listed in order from $a_1$ to $a_4$.

Figure 7. -0.21947270759847, 0.33858182718046, 0.610763950282708, 0.315013298424892
This attractor assumes of the form reminiscent in some ways of the gingerbread man seen in the classical Lozi attractor. The gingerbread man has a “heart” in the form a five-lobed structure with a ellipse within it.

Figure 8. -0.344555260520428, -0.59013448276557, 0.523313496601768, 0.713451223839074
This attractor has parallels to the previous one in being somewhat like the gingerbread man with 4 hands.

Figure 9. -0.26224758869037, -0.488204386695288, 0.879756956817582, 0.301065913862549
This attractor takes the form of a fish or some crustacean naupilus larva.

Figure 10. -0.309558889106847, -0.797052371953614, -0.980002021430992, 0.34323377257213
This attractor looks somewhat like an echinoderm larva.

Figure 11. -0.482874695337377, -0.819933905000798, -0.381899853958748, 0.100596646997146
This is a representative of a prevalent type of attractor generated by this map that may be termed the “Sombrero hat” type. The central heptad structure in this attractor is a mystery because it is not explicable by the cosine conditions above-presented .

Figure 12. -0.442830848232843, 0.0637841782858595, 0.495140142193995, 0.24925118080806
This form of the attractor resembles the butterfly attractor generated by the square-root modification of the Lozi map that we had described earlier.

Figure 13. -0.170671575525775, -0.671315040569752, 0.563780032042414, 0.24513541710563
Another version of the “Sombrero hat” type.

Figure 14. -0.517300257436, 0.795868384265341, 1.00516145080794, 0.493695715968497
For this attractor the radius of the $x_0, y_0$ circle was changed from 0.3 to 1, though it is stable and similar at the former value too. It vaguely resembles some Cambrian animal.

Figure 15. -0.307423420087434, -0.0870326082082465, -0.337442451966926, 0.766096443301067
This attractor assumes a bun-shaped morph that is seen quite often under these maps. An explanation for the multiple harmonics of this attractor remains as yet mysterious to us.

Figure 16. -0.0823114865785465, -0.766779959597625, 0.868352874149568, -0.0100218075420707
Another Sombrero hat type form. The radius of the $x_0, y_0$ circle was changed from 0.3 to .4 for this attractor. How one accounts for the central pentagonal zone of restriction and the triad of octagonal restriction zones around it remains mysterious.

Figure 17. -0.16997885087505, -0.0931982280220837, -0.631216614013538, -0.248738911962137
Here the radius of the $x_0, y_0$ circle was changed from 0.3 to 0.31. It shows a central pentad surrounded by 9 further pentads. An explanation for the emergence of these pentads and the 9 fold harmonic however remains elusive.

Posted in Scientific ramblings |

## 1859 CE and beyond: Some reflections

The yuga between 1800 CE and 1900 CE saw a remarkable change in our understanding of the world at many levels. It is not that some of these ideas did not exist long before that time but they came together in a world-system of science and philosophy in that period. Part of this change can be traced to multiple disparate events that interestingly happened in the year 1859 CE — a time when our nation had sunk to what was perhaps the lowest points of its existence. Due that our nation could not fully participate in the cataclysmic consequences of those events until some time later and the Hindu elite have still not fully internalized the significance of those events.

The Big
The mathematical foundations of mechanics laid by Newton with deep roots in Euclidean geometry had met with near infallible success. The physicists drunk with confidence of the triumph of Newton believed that they could account for all physical phenomena based on that mathematical formulation. But it was ironically this very mathematical formulation that was to point that something was amiss. The great Carl Causs had shown how with just three observations one could calculate the orbit of a celestial body and used this recover the asteroid Ceres which had been lost after going behind the sun. Gauss’s student was Johann Encke, who a few years after his graduation developed methods to calculate the perturbational effects of various celestial bodies on the orbits of other such bodies. It was these methods that led to his discovering the period of a comet that latter came to be known as the famed Encke’ Comet and his other great works on cometary orbits.

Following on the path of Encke’s methods, the great French mathematician Urbain Le Verrier came to be “the man who discovered a planet with the point of his pen”, as his colleague called him. By performing complex and painstaking calculations of Newtonian mechanics he showed that the orbit of Uranus could not be explained by the known data and proposed a new planet to explain it. Not just that, he predicted where that planet would be in the sky pointing to a location at the boundary of Capricornus and Aquarius. With no Frenchmen showing interest in testing his prediction, he sent his predictions over to Encke at Berlin. But the day Encke received Le Verrier’s letter was his birthday and he had organized big party for the evening rather than an observation session. Moreover, he was much more of a man of mathematics than an observer. By some coincidence, he had recently had his former doctoral student Carl Bremiker make excellent new star maps of the Aquarius-Capricornus region for the observatory. Further, Encke’s assistant and former student Johann Galle had sent his doctoral dissertation to Le Verrier for comments and this letter from Le Verrier contained the comments on that in addition to his note on the prediction of a new planet. Thus, with Encke not observing, Johann Galle and his student Heinrich d’Arrest got to use the telescope that night of receiving the Le Verrier’s letter. They discovered Neptune with $1^o$ degree of the position he had predicted. When we saw Neptune for the first time after quite some difficulty with our small refractor, we were able to appreciate the triumph of Galle. This was the ultimate triumph of Newtonian mechanics and Encke fittingly wrote to Le Verrier: “Your name will be forever linked with the most outstanding conceivable proof of the validity of universal gravitation.” By this Encke meant the Newtonian theory of gravitation.

But there was something ironic about Le Verrier’s life’s work. Before his study of Uranus leading the prediction Neptune, he had worked on the other end of solar system, where the elusive Mercury orbits, which these days the ordinary urban man rarely catches a glimpse of (I remember all the times I’ve seen it). Going through calculations involving hundreds of terms he calculated an excess of precession for the orbit of Mercury which could not be accounted for by Newtonian mechanics (September $12^{th}$ 1859). While the same trick of postulating an additional planet or other anomalies were tried, none of them really worked. Ironically the universality of Newtonian gravitation, which Encke thought Le Verrier had proven, now stood to be surpassed. This of course took a while to happen but seeds had been sowed by the new geometry of Bernhard Riemann. This laid the foundation for Einstein’s theory which came in the next century. At the same time as solving the excess precession problem of Mercury, Einstein as also predicted the existence of gravitational waves. Observations of binary pulsars in 1974 indirectly suggested he was right. With the direct detection of gravitational waves in the current century of the common era this prediction of Einstein has finally received its direct confirmation. Thus, one of the pillars of physics in the realm of the “big” was established.

[As an aside, talking of Riemann 1859 was also the year he published his famous paper establishing the relationship between the zeroes of the $\zeta$ function in the complex plane and the distribution of the prime numbers.]

The Small
The 21 year old Gustav Kirchhoff, while still a student, discovered his famous laws of electrical circuits, which any student who has studied elementary high-school physics would have encountered. This was just the beginning of what was to be a remarkable career spanning multiple branches of science and mathematics. Continuing with his electrical work he showed that in a resistance-less wire the electricity would flow at the speed of light. This important result formed the bed-rock of electromagnetism that was taken to conclusion JC Maxwell. He then worked with Robert Bunsen to develop spectroscopy and it was as part of this work that in October of 1859 CE that he reported his observations on how the D-lines in the solar spectrum are further darkened when passed through a Bunsen burner flame with sodium. These observations culminated in Kirchhoff’s famous spectroscopic laws:
1) An incandescent solid, or a liquid or a gas under high pressure produces a continuous spectrum, i.e. like a rainbow of colors.
2) A gas under low pressure produces an emission spectrum, i.e. one with bright-lines of specific frequency.
3) A continuous spectrum when viewed through a cooler low-density gas produces an absorption spectrum, i.e dark lines are seen superimposed on the continuous spectrum. These correspond to the bright lines produced when the same substance is heated and producing an emission spectrum.

The frequency $\nu$ at which the absorption or emission lines are seen depends on the substances emitting or absorbing light and the temperature to which they are heated. Further, by the end of 1859, using the rather elementary device of the below thought experiment, Kirchhoff arrived at a basic theorem for the continuous spectrum:

Figure 1

Let $P_1$ and $P_2$ be 2 isolated, infinite, opaque (do not allow transmission energy through them) plates (as caricatured in Figure 1) in thermodynamic equilibrium, i.e. they are at the same absolute temperature $T$ and the inflow of energy and outflow of energy into either plate is in balance. Let us even assume they are made of different materials as indicated by the different colors in Figure 1. Let us consider the following for a particular frequency of radiation $\nu$. Let $a_1, a_2$ be the absorptivity of the 2 plates, i.e. the amount of radiant energy absorbed per unit time per unit area. Let $e_1, e_2$ be the emissivity of the 2 plates, i.e. the amount of energy they radiate per unit time per unit area. Now the some of the energy incident on them is absorbed while the rest is reflected. This defines the respective reflectivities as $r_1=1-a_1, r_2=1-a_2$. Now, for $P_1$ the outflow of energy per unit time per unit area is $e_1$. Being in thermodynamic equilibrium it is in balance with its inflow $\iota_1$.

We can analyze $\iota_1$ thus:
1) As the first chain of inflow $P_1$ receives $e_2$ from $P_2$. Of this it absorbs $a_1 e_2$ (the first order absorption) and reflects $r_1 e_2$. This is incident on $P_2$ which reflects back $r_1 e_2 r_2$. Of this $P_1$ absorbs $a_1 e_2 r_1 r_2$ (second order absorption) and reflects back $e_2 r_1^2 r_2$. In turn $P_2$ reflects back $e_2 (r_1 r_2)^2$ of which $P_1$ absorbs $a_1 e_2 (r_1 r_2)^2$. $P_1$ reflects back $e_2 r_1^3r_2^2$ and the chain continues ad infinitum. Thus, we can write the first component of $\iota_1$ as:

$a_1 e_2+a_1 e_2 r_1 r_2+ a_1 e_2 (r_1 r_2)^2+a_1 e_2 (r_1 r_2)^3+...\\[5pt] =a_1 e_2(1+r_1 r_2+ (r_1 r_2)^2+(r_1 r_2)^3+...)\\[5pt] =\dfrac{a_1 e_2}{1-r_1 r_2}$

The last step is obtained via the limit of the infinite sum of a geometric series given that $0 \le r_1 r_2<1$

2) The second chain of inflow goes thus: $P_1$ emits $e_1$ of which $P_2$ reflects back $e_1 r_2$. Of this $P_1$ absorbs $a_1 e_1 r_2$ and reflects back $e_1 r_1 r_2$. Of this $P_2$ reflects back $e_1 r_1 r_2^2$. Of this $P_1$ absorbs $a_1 e_1 r_1 r_2^2$ and reflects back $e_1 (r_1 r_2)^2$. Of this $P_2$ reflects back $e_1 r_1^2 r_2^3$. Thus, we can write the second component of $\iota_1$:

$a_1 e_1 r_2+a_1 e_1 r_1 r_2^2+ a_1 e_1 r_1^2 r_2^3+...\\[5pt] =a_1 e_1 r_2(1+r_1 r_2+ (r_1 r_2)^2+(r_1 r_2)^3+...) \\[5pt] =\dfrac{a_1 e_1 r_2}{1-r_1 r_2}\\ [5pt] \therefore \iota_1 = \dfrac{a_1 e_2}{1-r_1 r_2} +\dfrac{a_1 e_1 r_2}{1-r_1 r_2} = \dfrac{a_1 e_2+a_1 e_1 r_2}{1-r_1 r_2}$

Given the thermodynamic equilibrium $e_1 = \iota_1$; hence,

$e_1= \dfrac{a_1 e_2+a_1 e_1 r_2}{1-r_1 r_2}$

We can rearrange the equation as:
$(1-r_1 r_2-a_1 r_2) e_1=a_1 e_2$

Dividing both sides by $a_1 a_2$ we get:
$\dfrac{1-r_1 r_2 -a_1 r_2}{a_2} \left (\dfrac{e_1}{a_1} \right) = \dfrac{e_2}{a_2}$

Given that $r_1=1-a_1$ and $r_2=1-a_2$ we can hence write the above as:

$\dfrac{1-r_1 r_2 -((1 -r_1) r_2)}{1-r_2} \left (\dfrac{e_1}{a_1} \right) = \dfrac{e_2}{a_2} \\[5pt] \therefore \dfrac{e_1}{a_1} = \dfrac{e_2}{a_2}$

Similarly, from $P_2$ we can write:

$e_2= \dfrac{a_2 e_1+a_2 e_2 r_1}{1-r_1 r_2}\\[5pt] \therefore \dfrac{1-r_1 r_2 -a_2 r_1}{a_1} \left (\dfrac{e_2}{a_2} \right)=\dfrac{e_1}{a_1}\\[5pt] \therefore \dfrac{e_2}{a_2} = \dfrac{e_1}{a_1}$

Thus, irrespective of the material composition of the plates, their ratios of emissivity to absorptivity are the same. Since this analysis was done for a given frequency of radiation $\nu = \tfrac{1}{\lambda}$ (where $\lambda$ is the wavelength of the radiation) at a certain equilibrium temperature $T$, we can say that the above ratios are function of these:
$\dfrac{e_2}{a_2} = \dfrac{e_1}{a_1}= f(\nu, T)$

Now Kirchhoff postulated a theoretical body, termed the black body, that absorbed all energy incident on it, i.e. $a_1=a_2=a_b=1$. Thus, for such a black body the emissivity would be $e_b=f(\nu, T)$, i.e. it would be purely a function of the frequency of the radiation and its temperature. This immediately presented the physicists of the age with two challenges: 1) An experimental one, i.e. to construct a radiating body that approximates the black body as closely as possible and to empirically measure the shape of $f(\nu, T)$. 2) A theoretical one, i.e. to derive from a theoretical model of radiation the shape of $f(\nu, T)$.

These challenges proved more revolutionary than physicists of the time thought. In 1869 CE one of the greatest physical theorists of all times, Ludwig Boltzmann, was appointed full professor of mathematical physics at the age of 25. Starting that year for the next two years he spent some time studying with Bunsen and Kirchhoff whose findings we have just alluded to. Deeply inspired by the discussions with them on thermodynamics, he went on to provide a statistical framework to explain the second law of thermodynamics in 1872 CE. Most European physicists of that time, unlike the chemists, did not consider the atomic theory to be real. Boltzmann not only considered atoms to be real (spherical atoms formed the foundation of his work on the second law) but in this work he introduced the idea of discrete energy levels. Later to the shock of the attending physicists in 1891 CE at a conference in Halle, Boltzmann emphatically stated: “I see no reason why energy shouldn’t also be regarded as divided atomically.” It was these ideas that were to provide the ultimate solution to Kirchhoff’s challenge.

On the experimental side it took about 20 years for the first glimmer of understanding to emerge with regard to $f(\nu, T)$, namely that it has one clear maximum when plotted against $\nu$, which moves to lower $\nu$ with decreasing T. Finally, in 1896 CE, Hermann von Helmholtz’s student, Wilhelm Wien, proposed the first reasonable function to account for this shape. It took the form $f(\nu, T)=a\nu^3e^{-b\nu/T}$, where $a, b$ are constants. In terms of its basic shape it resembled what was empirically known for $f(\nu, T)$ and the experiments by Paschen around that time suggested that indeed Wien had found the right curve for the black body radiation. However, new and more precise experiments at lower frequencies soon poured water on this. These experiments were being done by the groups of Lummer and Pringsheim on one hand and Rubens and Kurlbaum on the other at what where probably the best experimental physics labs in world at the dawn of the 1900s. They indicated that the function of Wien failed at lower frequencies.

The final solution to the problem came from the dark horse among the physicists, Max Planck, who had fittingly taken the professorial chair of Kirchhoff upon his death. This chair at Berlin was first offered to Boltzmann, who declined it; with no one taking it, finally it was given to Planck. He had a solid background having studied physics with Kirchhoff and von Helmholtz and mathematics under Weierstrass, who was second in line of academic descent from Gauss. Till the age of 40 he had done competent work in thermodynamics but was most part ‘scooped’ by the great American mathematician and inventor Josiah Gibbs, whose work in turn paralleled that of Boltzmann to enter the textbooks. Despite all this, Planck had for long set his mind on the bigger goal of deriving the correct shape of the black body radiation curve. Ironically, throughout most of this phase Planck was in the “wrong team”, opposing the atomic theory. As of 1897, Planck was still disputing the statistical framework of Boltzmann based on atomic principles. In response, Boltzmann published a paper showing that Planck’s objections were untenable and wrote that: “It is certainly possible and would be gratifying to derive for radiation phenomena a theorem analogously to the entropy theorem from the general laws for these phenomena using the same principles as in gas theory. Thus, I would be pleased, if the work of Dr. Planck on the scattering of electrical plane waves by very small resonators would become useful in this respect, which by the way are very simple calculations whose correctness I have never put in doubt.”

This rebuttal of Boltzmann brought about the gradual conversion of Planck over the next 3 years. His long-standing wish came to fruition fatefully on a Sunday afternoon in the autumn of 1900 when the family of Rubens who had done the black body radiation experiments visited the family of Planck. Rubens told his host about his latest experimental results with respect to the black body radiation measured for low frequencies and its departure from Wien’s proposal. That very evening Planck drawing on his deep study of Kirchhoff’s fundamental problem arrived at correct formula for the black body radiation function:
$f(\nu, T)=\dfrac{a\nu^3}{e^{b\nu/T}-1}$, where $a,b$ are constants.

Figure 2. The form of Wien and Planck’s curves in arbitrary units of frequency and radiance.

Over the next two months, backed by his profound knowledge of thermodynamics, his strong mathematical capacity, the “conversion” he had undergone due to Boltzmann and the inspiration from those very methods of Boltzmann he arrived at the quantum theory where energy is emitted and absorbed in “atomic” packets or quanta. The energy $E$ of these quanta is described as $E=h\nu$, where $h$ is Planck’s constant. In the following years its power was demonstrated by Einstein who explained the photoelectric effect using the same theory. Thereafter, Niels Bohr used the same in combination with inspiration from Darwin’s grandson’s studies to arrive at the first quantum model of the atom. The rest as they say is history: thus, the second pillar of physics in the realm of the small was established.

When my father first read out a basic version of this story to me when I was a kid I was profoundly inspired to study the quantum theory to the extent my meager mathematics allowed me when I grew older.

Order and disorder
In the late 400s and the beginning of the 500s of the common era the great Hindu scientist Āryabhaṭa-I devised several mechanical devices, which were powered by gravity and/or flowing water. One of these was the svayamvaha-yantra in which the water flowing out of a water clock under gravity caused a sphere to rotate around its axis once in 24 hours. This was meant as a teaching device to illustrate the apparent rotation of the heavenly sphere. Over hundred years later, in 628 CE, Āryabhaṭa’s successor and antagonist Brahmagupta, apparently in a bid to outdo him, claimed to have devised a svayamvaha-yantra which was a perpetual motion machine (ajasra-yantra). It was supposed to operate with gravity acting on mercury and buoyancy alternately working to keep a spoked wheel moving for ever. Evidently, this device did not work as expected prompting his successors, like Lalla and Bhāskara-II, to attempt various modifications and alternate designs to arrive at something which worked. While the real goal was obviously not attained, in India these failed attempts were part of a tradition of constructing genuinely working mechanical devices culminating in king Bhojadeva’s automata — a tradition which did not survive the ravages of Mohammedanism. However, its transmission to West Asia appears to have seeded the quest for perpetual motion machines in Europe upon transmission of these ideas via the Mohammedans.

While from today’s vantage point the quest for these machines might look like a sign of lunacy, the ultimate realization that perpetual motion machines are untenable needed the recognition of the laws of thermodynamics. This had to wait for long time and arose from meditations ensuing from the eventual invention of the steam engine in Europe. First, the Englishman Joule’s recognition that work and heat were manifestations of an equivalent quantity, i.e. energy, led to the first law of thermodynamics. This law is essentially the law of conservation of energy: energy can neither be created nor destroyed but only converted from one form to another. This law negated the possibility of having a perpetual motion machine that did work without an equivalent input of energy.

As the English were taking full advantage of their engines driven by the heat energy from burning coal to run their business, their neighbors the French grew increasingly anxious. This prompted the brilliant military engineer Sadi Carnot from a learned French clan to carry out the first theoretical study of the principles behind the engines. His penetrating investigation completed by the time he was 27 more or less laid the foundations of thermodynamics. Subsequently, he suddenly went mad at the age of 36 and died from cholera shortly thereafter. Due to the contagious nature of the disease many of his works were buried with him but what survived was his famous work on the cycle of an engine. He recognized that for an engine to run it needed both a heat source from which it took heat to perform work and a heat sink at lower temperature than it to dump some of that heat that was not converted to work. While the source was rather obvious in the practical steam engines starting from those devised by James Watt, the sink was not — it was merely the ambient surroundings in which the engine operated. Carnot went on to show the maximal efficiency an engine depended on the temperature of the source and the sink. Let $Q$ be the heat the engine takes from a source at (absolute) temperature $T_{so}$. Let $T_{si}$ be the temperature of the sink. Then the maximal work that can be done in a cycle of the engine is

$W=Q\left(1-\dfrac{T_{si}}{T_{so}}\right)$

This is the famous Carnot equation that indicates that not all heat can be effectively converted to work unless the source was at infinite temperature or the sink at absolute zero, neither of which are feasible options.

It is this inability to get all the heat to perform work which poses an additional constraint that negates even a lower form of perpetual motion machine, namely one which conserves energy but at least keeps running forever by cyclically converting one form energy into another. Two noted scientists of the age, Lord Kelvin and Rudolf Clausius (second in line from Gauss via Dirichlet with whom he studied mathematics) formalized Carnot’s discovery by the middle of the 1800s as a law:
“No cycle in which heat is taken from a hot source and converted completely into work is possible” -Lord Kelvin
“Heat does not flow from a body at low temperature to one at high temperature without an accompanying change elsewhere.” -Rudolf Clausius

These became the basic statements of the second law of thermodynamics. Clausius furthered this to define an entity termed entropy. He defined this such that the change in entropy multiplied by temperature specified that portion of heat which could not be converted to work. This led him to state the second law in a rather different way but ultimately entirely equivalent to the above statements:
The entropy of the universe increases during of any spontaneous change.
Here a spontaneous change is one that occurs automatically, i.e. without needing any external work to be done for it to happen, e.g.: 1) when a compressed gas is released into an empty container of larger volume it spontaneously expands to occupy that volume. 2) A hot metal piece placed at room temperature cools to the same temperature.

This statement of the second law provided a deeper insight into the nature of entropy. For instance, let us consider the above example of the gas expansion: the gas occupying a smaller volume is more ordered. This can be expressed in probabilistic sense: The probability of find a gas molecule in a given unit of volume is higher in this state than when it spontaneously expands on being released into a larger empty container. Here the gas molecule moves over a larger space and the probability of finding the molecule in same unit volume decreases. Thus, the gas gets more disordered. Thus, entropy can be seen as a measure of disorder and the second law stated as: “matter and/or energy tend to get more disordered.” The formal description of this idea had its roots in a discovery of the great James Clerk Maxwell in a distinct investigation, namely the function describing the distribution of the velocities of molecules in an ideal gas, again interestingly published in 1859 CE. In course of developing this abstraction further Boltzmann formulated his celebrated formula for the absolute entropy of a substance in 1877 (in the modern form given by Max Planck):
$S=k_B\log(W)$
Here, $k_B$ is Boltzmann’s constant in energy and inverse temperature units and $W$ is the total number of ways in which atoms, molecules or energy elements can be arranged in the sample such that the total energy remains constant. Each such arrangement that fulfills this condition is termed a “microstate”. Thus, $W$ is the total number of microstates in the sample. This simple formulation has profound implications for it allows one to connect the entropy of a sample to the probability of occurrence of each of the arrangements of the “atomic” entities in the sample:

$S=-k_B \displaystyle \sum_i p_i \log(p_i)$

Here $p_i$ the probability with which microstate $m_i$ occurs in the sample.

While this is a thermodynamic concept, one can now extend it to be the general measure of disorder of any system. This can be illustrated with an often-used example: say, on a national day we have large assembly of people. First, consider state-1: When the national anthem is being recited they all stand up in an erect posture and recite the same. Next state-2: once the anthem is over they might adopt a range of different postures with various conversations between individuals or small groups. Thus for a beholder of this system in state-1 the population is ordered and the information coming out from them is clearly perceived and limited (just the national anthem). In state-2, the population is disordered and information coming out from them is not easily perceived as it has a high degree of complexity being the sum of all the many individual conversations taking place. Thus, the abstract generalization of the thermodynamic entropy concept not only gives a general measure of disorder but also the information content of a system. It was this key generalization that Claude Shannon arrived at almost 70 years after Boltzmann’s initial discovery. His historic formula to quantify information essentially took the same form as Boltzmann’s formulation of thermodynamic entropy, just that Shannon’s is a pure number:

$H=-\displaystyle \sum_i p_i \log_2(p_i)$

Here $p_i$ is the probability of the symbol $s_i$ in a certain symbol set occurring in the string $S$. Thus, Shannon entropy specifies the minimal number of bits per symbol needed to encode the string $S$ in binary form. Hence, $H$ also measures the complexity of a string $S$. As an example let us consider the following strings in the ASCII symbol set:
$S_1=$ rAma rAma hare hare; $H(S_1)=2.7345$
$S_2=$ ugram indraM juhomi; $H(S_2)=3.6163$
One notes that $S_2$ has higher Shannon entropy than $S_2$ and provides quantitative evidence for the intuitive idea that the second string is more complex than the first. This remarkable link between the mathematical formulations of two rather disparate entities, one a description of very palpable quantities like matter and energy and the other an abstract quantity, information, can be summarized by quoting Shannon:

“Quantities of the form $H=-\sum p_i \log_2(p_i)$ (the constant $k_B$ merely amounts to a choice of a unit of measure) play a central role in information theory as measures of information, choice and uncertainty. The form of $H$ will be recognized as that of entropy as defined in certain formulations of statistical mechanics, where $p_i$ is the probability of a system being in cell $i$ of its phase space.”

Shannon’s formulation of entropy as a measure of information has profound implications for understanding the foundations of life. This aspect has been of great importance in our own investigations and will touched upon in the final section. Before heading there we may ask if there is a deeper link between the thermodynamic and informational conceptions of entropy? That is a question which remains more mysterious. However, in that regard we will merely quote a noted scientist of our age, Murray Gell-Mann (compare with the above example of the people assembled for the national day):

“In fact, entropy can be regarded as a measure of ignorance. When it is known only that a system is in a given macrostate [gross state of matter and energy in the sample], the entropy of the macrostate measures the degree of ignorance about which the microstate system is in, counting the number of bits of additional information needed to specify it, with all the microstates in the macrostate treated as equally probable.”

We will conclude this section by merely mentioning some even more mysterious issues pertaining to the two laws of thermodynamics. Emmy Noether, probably greatest female mathematician of all times, proved a remarkable theorem now known as Noether’s theorem. A basic version of this theorem can be relatively easily understood by someone with junior college mathematics. However, in its more complete forms it extends into rarefied heights of mathematics. This basic version depends on the Lagrangian formulation of another great mathematician Joseph-Louis Lagrange to describe a physical system. Simply put the the Lagrangian of a system $L$ is the difference between its kinetic energy $T$ (not to be confused with the symbol for temperature) and potential energy $U$:
$L=T-U$

The Lagrangian $L$ is typically expressed as a function of the position $x$ of a body in the system and its time derivative, i.e. velocity $v=\tfrac{dx}{dt}$. As a simple example, consider the Newtonian system of a body of mass $m$ raised to some height dropping under a uniform gravitation field with acceleration $g$. It would have kinetic energy $T=\tfrac{1}{2}m \cdot v^2$ and potential energy as $m\cdot g \cdot x$. Thus its Lagrangian can be written as:

$L(x,v)=T-U=\tfrac{1}{2}m \cdot v^2-m\cdot g \cdot x$

Now, according to Noether’s theorem if $L(x,v)$ of a physical system remains unaffected upon transformation in the coordinate system used to describe it, i.e. its symmetric under the transformation, then there will be a corresponding conservation law. Now, if a physical system is translated linearly to a different position and there is no other physical influence acting on it, its $L(x,v)$ remains unaffected. This implies that the translational coordinate system is a uniform; thus, translational symmetry of the Lagrangian gives us the law of conservation of momentum. Similarly, that $L(x,v)$ is unaffected if the system is translated in time gives us the law of conservation of energy or the first law of thermodynamics. Thus, remarkably conservation of energy is related to time symmetry, i.e. time being an orderly or uniform coordinate system — time does not flow fast and slow at different points along its flow. If that were to happen then energy would not be conserved.

Most physical laws are agnostic to flipping the time coordinate system, i.e. time reversal. For example, if we flipped time on the flight of an arrow from release to fall, there will be no changes to the laws of motion describing it. Similarly, if the time axis were flipped there would be no difference to the laws describing the revolution of planet around its star or the current in a circuit. As per Noether’s theorem if the Lagrangian of a system were unaffected under time-reversal then entropy would be conserved. But this is not so because it violates the second law of thermodynamics. So, it is the one physical law that has an inbuilt time direction, and is thus the odd third pillar of physics. The measurement of entropy gives us the “arrow of time” to use Eddington’s term. The young universe was very hot in a narrow energy range with energy and matter uniformly distributed. This was a low entropy state. With time the energy and matter become less uniformly distributed with clumping to form galaxies and their constituent star systems. Thus, the increasing entropy resulted in a complexity (if viewed in terms of information) of its structure. Thus, the state of the universe as described by its increasing entropy and hence information might be seen as a reflection of its “unfolding” along the arrow of time.

Lessons from life
The final event pertaining to 1859 CE whose consequences we shall talk about happened on 24th November of that fateful year: Charles Darwin published “On the Origin of Species by Means of Natural Selection, or the Preservation of favoured Races in the Struggle for Life”. One may go as far as to say that modern biology was born with that event. Till then biology was a science without a theoretical scaffold unlike physics or chemistry. The other events described in this note, however profound in their implications, had lesser social and intellectual impact than this book of Darwin. On the social front it shook the foundations of the Abrahamistic religions like nothing else coming from science. On the intellectual front it unsettled more thinkers than any other scientific publication. The implications of this evolutionary theory were quite completely grasped by Darwin; however, even its junior co-discoverer Wallace did not fully grasp all its implications, leave alone several of the other intellectuals at and after that time. Indeed, the situation is rather peculiar: while much of biology done since then can be seen as a footnote to Darwin, a large fraction (at least $50\%$) of the biologists do not fully understand the evolutionary theory and how to use it.

It is not commonly understood that the evolutionary theory has a close relationship to Shannon’s generalization of entropy as a measure of information. Its tremendous predictive power stems from this aspect. The way to understand this can be briefly described thus: the sequence of a biopolymer (nucleic acid or protein) is replicated by a replicator or synthesized by a synthetase using another as a template. This in principle produces identical copies of the biopolymer or a single polymer with repeats of the same sequence. If we thus align the sequence of the identical copies of these biopolymers then the Shannon entropy of a column in the alignment will be zero — i.e. there is no disorder. However, this replication process is not always perfect; hence over time copies would emerge with changes along the sequence (mutations). Thus, the column-wise entropy will keep increasing. What natural selection does is: 1) to prevent increase in entropy of certain columns or 2) be blind to differing degrees to the entropy increase in certain columns or 3) favor increasing entropy of certain columns. These three modes of action are commonly seen as 1) purifying selection; 2) weak selection or neutrality; 3) positive selection.

Now, the one thing in biology that lies beyond the entropy principle but still has some relation to it is the semantic principle: Each of the three above types of action relates a certain biological/biochemical meaning of the given residue in the biopolymer. This semantic aspect of the biopolymer sequence is what is the unique “domain” of biology, even as the semantic aspect of a linguistic sequence is the unique domain of a text. As an illustration let us consider the following peptides from Homo sapiens:
Oyxtocin: CYIQNCPLG; $H=2.9477$
R-Vassopressin: CYFQNCPRG; $H=2.9477$
Both these peptides have the same length of 9 amino acids and the same Shannon entropy. However, the first one is primarily involved in signaling reproductive functions like birthing, bonding and lactation whereas the second is involved in regulating water-balance, primarily as anti-diuretic hormone. Thus, even though they have the same information content their biological semantics have diverged as they diverged from a common ancestor.

Yet, entropy does impinge on semantics in a general sense. To illustrate this let us consider a linguistic example first:
$S_1=$: dhiyo yo naH prachodayAt; $H=3.6258$
$S_2=$ buM buM buddhAya buM buM; $H=2.8675$

The two linguistic strings above are both the same length but without knowing anything else, by just comparing the entropies of the two we can state that by itself $S_1$ is likely to have greater semantic complexity or richness of meaning than $S_2$. By the same token, let us look at two equal-length parts of a protein, the Drosophila Antennapedia, which binds a specific DNA sequence and initiates the development of a specific aspect of the animal body plan along the antero-posterior axis:

TNGQLGVPQQQQQQQQQPSQNQQQQQAQQAPQQLQQQLPQVTQQVTHPQQQQQQPVVYASCKL

$H=2.634$

RKRGRQTYTRYQTLELEKEFHFNRYLTRRRRIEIAHALCLTERQIKIWFQNRRMKWKKENKTK

$H=3.6612$

Now, just comparing the entropies of the two parts of the protein we can say that the first has lower complexity than the second. Thus, the second is more likely to be the functionally more important or involved part of the protein. If you were then asked to guess which part was more likely to perform the specific DNA-binding function then the second part would be the obvious choice. Of course the power of this method increases with an alignment of multiple sequences for here we are exploiting not just the entropy distribution but also the effects of natural selection on it. Thus, natural selection acts on the increasing entropy of the biopolymer alignment to retain some of it and discard the rest based on what its semantics are. Thus, with a single sequence using the entropy measure across the sequence, we can infer which part of it is likely to fold into a globular structure, which part might be unstructured or fibrous or if might be embedded in the membrane. With an alignment of sequences we can additionally tell which protein would be an enzyme, what are its active sites and some aspect of its catalysis if it were an enzyme because the biological semantics are based on the ground of chemistry. However, beyond this one would require empirical approaches.

Thus, the biological semantics place limits on the predictive capacity of its foundational theory. In this sense the situation is quite different from the role of the underlying physics in astronomy (e.g. stellar evolution) or chemistry. Further, say if we take DNA replication, the bacteria have one primary DNA polymerase, archaea and eukaryotes other ones with independent origins from the bacterial one. Why that particular DNA polymerase became the primary enzyme of a particular superkingdom does not currently seem to be accountable purely from theory. It could simply boil down to the microscopic events, i.e. the local dynamics of selection acting at the time of the fixation of the polymerases. One could then call it historical contingency, though we currently do not know that for sure. If this were the case then prediction from the foundational theory in biology has certain limits in terms of what it can do by itself and the rest is contingency. Is this comparable to the questions in physics like why the electron’s mass is what it is? That is also not clear to us.

Nevertheless, the idea of the action of natural selection on the products of entropic diversification leads us to the realization that this principle is more general. Indeed, the evolutionary process of life through natural selection and the emergence of structure in the universe can be compared. With regards to the universe, as noted above, the early phase was low in entropy with more uniform distribution of matter and energy. As the entropy increased various alternative configurations of matter and energy emerged. The physical laws dominant in these new matter-energy regimes now “selected” for certain configurations like first the atoms of hydrogen and then those of heavier elements. Further, the laws in the regime of what had by then emerged as chemistry selected for certain atomic combinations, i.e. molecules. At the macroscopic level, they selected for formation of galaxies and stars. This gave a certain temporal sequence for the evolutionary process. The basic set of galaxies with their globular clusters were produced only once and new ones are not forming like in that initial phase. Instead, the old ones are maintained and what is happening is the stellar evolution within those galaxies which had formed long ago. With regard to the stars themselves, formation of those with very low metallicity happened only at certain early time and could not repeat in the second generation as the stellar nebulae were already seeded with heavy elements.

We believe that a similar entropic process is reflected in the emergence of life as we know it. First, we do not see life forming afresh again and again on earth: all of life we know of had a single origin, which might have not been on earth. In fact, the observation that the archaeal and bacterial lineages, while having a common origin, had a phase of genetic separation where there was no sign of the commonplace lateral transfer of genetic material. Thus, after a common origin on a distant source there was likely a phase of spatial separation followed by independent seedings on earth. Second, the study of the proteins in extant organisms points to a “big bang” when a great section of the stem lineages of all currently present protein domains were produced. We do not see evolutionary “bangs” of that magnitude happening again. Thus, we posit that the entropic diversification resulted in temporal layering — i.e. the early diversity was never reproduced. The diversity that emerged in the early phase was acted upon by natural selection to give rise to the stem lineages of all the major old protein domains. Since then selection has been for a good part playing the role of maintenance of the old lineages with only sporadic completely new innovations. This might be compared to the above-mentioned situation with galaxies and intra-galactic stellar evolution in the universe at large.

Epilogue
It would be almost banal to state that events set rolling in 1859 have had profound influence on our current way of thinking. It is story of towering intellectual heroes who covered themselves in glory one hand and forgotten foot soldiers on the other comparable to any great military endeavor. For some of them the end came before they could see the full glory of their findings: Riemann was dead at 39. JC Maxwell was dead at 48. The great Boltzmann penetrated many realms and caught glimpses of others like the treatment of space-time in special relativity and the use of Riemann’s geometry. But by 1906 madness was gripping him and he committed suicide that year. Planck lived a long life of nearly 90 years but for a good part of his life he had difficulty coming to terms with the theory he had birthed. Four of his children died as adults during his own life. However, it is hard to relate to most of them at a personal level as they come from a very alien culture and religion. In any case one can still relive some of their glory moments by re-tracing their scientific paths.

The aftermath of these upheavals brought a lot of drama to science. There was a golden age of physics which seems to have slowed down closer to our times with the three pillars still standing firm but no underlying unification yet in sight. In biology the situation is more peculiar. On one hand, from the “tradition” of Darwin there arose a certain mathematicization which really did not bring much insight in terms of biology — not surprising given the weakness of the philosophical foundations of this particular direction. On the other hand, many practicing biologists themselves often labor on with a poor understanding of what old Charles had so clearly expounded and blunder into dark ditches. However, those who understand its depths can penetrate the biological science.

Any philosophical system that fails the recognize and engage the consequences of these upheavals is likely to be in deep trouble. For the followers of the sanātana dharma a start in these directions was given long ago by traditions of the great Kaṇāda and Pāṇini but somewhere down the line they chose more sterile paths. In some sense these upheavals may be seen as a return of those principles of the great ancients in a modern guise.

Posted in History, Scientific ramblings |

## Cricket in pictures

We may say that cricket has nearly passed us by. In our youth we played and watched the game quite a bit. While in secondary school we were fairly interested in cricket statistics. The sources for statistics were not easily available those days. We would obtain them bit by bit from the occasional sports magazine our father might purchase, or from a cricket magazine at our local English library, or from sports quizzes on television, or they might stream in from acquaintances and certain relatives. But as we grew older this interest waned. When we were in college, our brother was passing through a similar phase of interest, which re-ignited ours and also now we sought to see it through the lens of statistical distributions and probability. However, those experiments were generally tedious and not the most exciting thing we had on our hands. As we aged into young adulthood and beyond, cricket kept increasingly passing out of our focus. Thus, we had reasonable familiarity with the heroics of Kapil Dev, Gavaskar, Shrikant, Tendulkar, Dravid, Sehwag and Laxman, less-so with Dhoni, and even less so with Kohli. Still, we say it has only nearly passed us by because, though we do not watch it anymore, we still have it streaming to us once in a way from friends and colleagues.

Due to such impingements we recently wondered about re-visiting certain statistical distributions in cricket. This urge finally precipitated due some discussion on Twitter regarding cricket statistics. We would like to acknowledge those discussions for inducing us to write this note that many might perceive as an unnecessary endeavor but a man should always pay attention to distributions and statistical properties for they can might help him in other walks of life. Here we take a look a some such for batting performances in Test cricket for we believe that it is the highest and the most satisfying form of the game. The sources of statistics for the below meanderings are the following: 1) A compilation of 73 top batsmen from the HowzStat Cricket Database; 2) A compilation of 301 batsmen scoring 2000 runs or more from Cricinfo; 3) A large data collection for all batsmen for all international matches, which was prepared by and kindly provided as a convenient csv file by Anupam Singh. The first two were leached from html and converted to csv files. The third file needed some post-processing for removing duplicates and null records. That said let us look at the data.

Basic features of batting in test matches

Figure 1

In the first panel of Figure 1 we see the frequency distribution of runs scored by the bat in an innings of a test match. It is a left truncated, right skewed distribution with a clear central tendency: a modal peak around 220 runs and a median of 239 runs. The second panel shows the frequency distribution of the number of fours in a given test innings. It again shows a similar distribution shape as the total number of runs scored by the bat in an innings. It has a modal peak at around 22-23 fours and a median value of 26 fours. That translates to a median value of 104 runs in a test innings being scored by fours. When we compare this to the first graph we can say that speaking in terms of central tendency about $44\%$ of the innings is scored by fours.

Sixes are much rarer in test cricket and their distribution is shown in panel 3 of Figure 1: the fraction of innings with $n=0, 1, 2, 3...$ sixes. This shows a non-linear decay law which might be approximated by an exponential function of the form $y=ae^{-bx}$. While this works in the range 1..14 sixes, it fails to captures the maximal 0 sixes fraction or the more extreme values. The Poisson distribution and power-law also do not approximate it well.

The strike rate of a batsman is defined as the ratio the number of runs scored to the number of balls faced expressed as as a percentage. In the fourth panel we have the frequency distribution of the mean strike rates for test batsmen computed for all scores greater than 10. It shows a strong central tendency close of $50\%$.

Figure 2

This sets the baseline for some further analysis. In Figure 2 we look at the frequency distribution of the scores for each innings. We did expect to see the median score fall with each innings. However, the non-linear nature of the fall and the changing shape of the distribution is notable. The first two innings have an almost triangular distribution, while the third innings is more of a skewed bell-shaped distribution. Obtaining functions that approximate these will be a problem of interest. The fourth innings has the lowest scores. This is due to two reasons: it is the innings of the final chase to win the match. Now, if the final total needed for a win is small then the forth innings scores would be low. Further, as the pitch deteriorates through the match fourth innings will favor the batsmen the least and the bowlers the most. This was even more in the past when the pitches were not protected overnight. However, this decline does suggests that barring special weather conditions, winning the toss and batting first gives the team which does so an advantage in the match.

Figure 3

In Figure 3 we look at frequency distribution of what fraction of big individual scores of a batsmen, i.e those of 50, 100, 150, 200 or greater are scored by fours and sixes. These are roughly normally distributed with a mean close to 0.5: thus, on an average about half of a big individual score in tests is attained by shots touching or clearing the fence.

Figure 4

We then analyzed the probability of a test inning containing a century. We found that the fraction of test innings containing $n=0, 1, 2, 3...$ innings can be well-described by a Poisson distribution. We empirically determined the $\lambda= 0.485$ for this Poisson distribution. The $p\approx 1$ for Poisson-predicted frequencies matching observed frequencies by the $\chi^2$ test. In figure 3 the hatched bars are the predicted values and the blue bars are the observed values.

Figure 5

In Figure 5 we analyzed the frequency distribution of individual scores of batsmen that are $\ge 100$. It shows a clear-cut decay law. This steep decay is best captured by a power law of the form $y=kx^a$, indicated by the blue line in Figure 5 (goodness of fit deviation fraction: 0.96). However, for scores $\ge 130$ an exponential decay fits the observed distribution at least as well as a power law (Red curve in Figure 5; goodness of fit deviation fraction: 0.94). Overall, it is fair to say that a power-law distribution approximately describes the distribution of scores $\ge 100$.

Apprehending the great batsmen
For this analysis we used a dataset of 301 batsmen who have scored over 2000 runs and have had an average career strike rate of $\ge 20$. This dataset records the number of innings played, number of times the player is unbeaten, total runs scored, highest score, career average, career strike rate, and 100s, 50s and 0s scored. We performed a principle component analysis using a subset of the numerical variables in this dataset for which good records are available (Highest score, number of 100s, number of 50s, career average, and strike rate) after scaling and centering. The first two components which together account for $\approx \tfrac{3}{4}$ of the variation, and their plot is show in Figure 6.

Figure 6

There is not much clumping but a set of “greatest” batsmen can be simply separated by choosing those with a first axis value $\le -2.3$ (red dots in Figure 6). At the extreme end of this this axis lie Tendulkar, Bradman and Lara (circled in green in Figure 6), who have quite unequivocally be mentioned as being among the greatest players. In this set two players are clearly separated from the rest (circled in dark red in Figure 6): Sehwag and Vivian Richards. These are two great batsmen marked by the rapid scoring rates and were a delight for the spectator. The one other Indian player who was notable for his scoring rate when I was young was Kapil Dev, who lies to the right side (circled in blue). While possessing a notable strike rate, as we can see, he was far too inconsistent to make it anywhere close to the great batsmen region. Finally, we may list this set of “greatest” batsmen:
1. V Sehwag (INDIA); 2. IVA Richards (WI); 3. BC Lara (WI); 4. ML Hayden (AUS); 5. GC Smith (SA); 6. RT Ponting (AUS); 7. DG Bradman (AUS); 8. MJ Clarke (AUS); 9. SPD Smith (AUS); 10. AB de Villiers (SA); 11. KC Sangakkara (SL); 12. SR Tendulkar (INDIA); 13. Inzamam-ul-Haq (PAK); 14. GS Sobers (WI); 15. Younis Khan (PAK); 16. GS Chappell (AUS); 17. DPMD Jayawardene (SL); 18. HM Amla (SA); 19. VVS Laxman (INDIA); 20. GA Gooch (ENG); 21. SR Waugh (AUS); 22. AN Cook (ENG); 23. Javed Miandad (PAK); 24. JH Kallis (SA); 25. SM Gavaskar (INDIA); 26. S Chanderpaul (WI); 27. R Dravid (INDIA); 28. KF Barrington (ENG); 29. MA Taylor (AUS); 30. AR Border (AUS); 31. WR Hammond (ENG); 32. L Hutton (ENG)

Figure 7

For this group of test batsmen we can visualize the distribution of their average and strike rate (panel 1 and 2 of Figure 7). We observed that both show a reasonable fit for a normal distribution: Average: Shapiro-Wilk $p=0.5866$; strike rate: Shapiro-Wilk $p=0.1437$. Thus, assuming a normal distribution of averages among top batsmen we can calculate the probability of a batsmen having an average like Bradman by chance alone to be a vanishingly small $p=1.99 \times 10^{-11}$. One may say that Bradman lived in a very different era when Australia’s main opponent was England and the other teams like India were not particularly strong. Yet, his record is unusually deviant and points to some special biological ability in him which might be likened to that possessed by a Ramanujan in mathematics. As evidence one might point to facts such as his surviving serious illness, his long lucid life and success in investing. Thus, he can be seen as the father of Australia itself.

Using the same distributions we can infer that the probability of a batsmen with Sehwag’s career strike-rate emerging by chance alone in this set of top batsmen is $p=0.00015$. When we combine it with his average we can infer that the chance of a Sehwag emerging by chance alone among these top players is a minuscule $p=2.3 \times 10^{-5}$.

From this set of 301 batsmen we can get a smaller set of 73 top batsmen based on the fact that they convert 50s to 100s at a rate of $40\%$ or higher. For this set we find that there is a strong linear correlation between the 100s they score and the number of innings they have played ( $r^2=0.84$; Figure 7, panel 3). As ever, Bradman stands apart from the rest. From this we can calculated that this creamy layer of batsmen score a test century once every 7.8 innings (median value; $\mu=8.27$) or approximately once in every four matches (Figure 7, panel 4).

Figure 8

Finally, for this set of 73 batsmen we can look the distributions of the fraction of the innings in which they have remained not out and their highest test scores (Figure 8). These two metrics bring out two of the greatest men in my cricket-watching days: 1) Kallis, perhaps the second greatest all-rounder to date (the first being Sobers whom we have never watched). He has remained unbeaten unusually high number of times. 2) Lara, the only man who score a 400 in test cricket. He was the last of the great black emperors of the Caribbean: what more needs to be said of him?

Some notable Indian test batsmen
We next took a closer look at some of the notable Indian batsmen whose innings we have watched in our career as a spectator of the game: 1) Sehwag, 2) Tendulkar, 3) Dravid, 4) Laxman, 5) Kohli, 6) Ganguly.

Figure 9

Figure 6 shows the probability density distribution of the scores of these 6 batsmen along with their career average. Kohli is still playing so his result will change in the future. The propensity for low scores is higher in Laxman and Ganguly, whose effect is seen in the form of their lower averages. The most notable features are: 1) Sehwag’s far-out right tail with secondary elevations in that region: he was clearly that man who could reach the big scores which balanced out his low scores. $7.7\%$ of his innings are adorned by scores of 150 or more. In contrast, the right tails of Tendulkar, Kohli, and Ganguly terminate more quickly, and Laxman and Dravid are in between. 2) However, both Tendulkar and Kohli have fat right tails keeping with their tendency to score numerous hundreds. Kohli in particular, displays a secondary peak around 100 which is consistent with the fact that he has 10 scores between 100 and 130. 3) Laxman’s peculiarity is a shoulder between 45 and 75. This comes from the fact in 49 of his 225 innings ( $\approx 22\%$) he has scored runs in this range.

Figure 10

We next look at the strike rates of these same 6 players for all innings where they have not scored a 0. As noted above, the ferocity of Sehwag’s batting stands out in this metric with a mean of 84.5 (Figure 7). On the other end, Dravid’s role as the slow-moving defensive formation in the battle array is displayed by his mean of 40.1, which is way below the average for tests (Figure 1). Most of these players have an approximately normally distributed strike rate. However, Kohli’s profile hints a bimodality which suggests that he has played some defensive innings like Dravid and also more attacking ones. But his central peak indicates that he has one characteristic strike rate (Figure 7).

Figure 11

We devised another way of visualizing the same: a scatter plot of runs scored in an innings versus balls faced. On this scatter plot, using all innings where the player faced 50 or more balls, we plot the minimal and maximal angles corresponding the least and highest strike rate for the innings meeting this criterion. The difference between these two is the characteristic strike rate angle (SRA) of the batsman. We also plot the angle corresponding to the median strike rate for innings meeting the above criterion of balls faced. Sehwag stands out right away: He has the maximum median angle and the narrowest SRA (Figure 8). This means that he was consistently the fastest of these great Indian batsmen in tests. In contrast, Dravid has the lowest median angle and the widest SRA. The former value indicates that he was the slowest of these great batsmen but the wide angle indicates that he was capable of fast innings on occasion. Kohli and Tendulkar have similar median angles that are larger than the remaining batsmen other than Sehwag. However, Kohli has a much narrower SRA closer to Sehwag. This suggests that, while Tendulkar and Kohli score(d) at the same overall rate Kohli is more consistent in scoring at that rate. Tendulkar, however, tended to score at very different rates in different innings. On the whole these features affirm Sehwag’s uniqueness in the constellation of great batsmen.

The laws of distributions hold their strong sway but once in a way a man of superhuman capacity might emerge. However, they used to say: “Cricket is a funny game.” It indeed is. Some might have the potential for greatness but, like the Khans passing away into the grasses of the steppe without history recording ever recording their name, they might stumble as an IR Bell or a DI Gower on the brink of greatness. On the other hand, others like a Gooch or SR Waugh, while lacking genius, might still make it to the club of greatness by their bulldog-like stickiness.

Posted in Cricket, Life | Tagged , , , ,

## The maṅgalācaraṇam of the Mānasollāsa

In PDF format

A conversation with a friend brought my mind back to the Mānasollāsa, the encyclopedia of the great Cālukya emperor Someśvara-deva. Below is the maṅgalācaraṇam of that work. The last verse is where the author announces himself. There are the following caveats: 1) The text as I have it is unclear in parts and this seems to arise both from the printing and the underlying manuscripts that were used. In the unclear parts I have taken the readings most obvious to me. 2) There translation appended to verses is approximate and the reader should look at the original.

abhīṣṭa-phaladaṃ siddhi-siddhamantraṃ gaṇeśvaram ।
karṇa-tālāniloddhūta-vighna-tūla-lavaṃ numaḥ॥
We salute the Gaṇeśvara, the giver of desired fruits and the success-granting mantra, who blows away the cotton-tangles of obstacles with the draft from the beating of his ears.

saṃvit-sakhī jayaty ekā kā ‘pi śuddhā sarasvatī ।
yayā svataḥ prabuddhānāṃ prakāśo’tiprakāśyate ॥
The friend of consciousness conquers, who else is that one but the pure Sarasvatī, by whose self the luster of the enlightened ones is caused to exceedingly shine forth.

vande bhava-latā-bījaṃ liṅgarūpaṃ maheśvaram।
so ‘vyaktam api suvyaktaṃ yasyāntaḥ sacarācaram ॥
I salute Maheśvara, the seed of the creeper of existence, in the form of a liṅga, he who is unmanifest and also well-manifest and the [cause of] the end of [that manifestation] with all living and non-living bodies.

kṛṣṇa kṛṣṇa hare rakṣa māṃ vibho māṃ ramāramaṇa mā vibhuṃ kuru ।
te hare narahare namo’stu me dehi deva padam acyutācyutam ॥
Kṛṣṇa! Kṛṣṇa! Hari! protect me!
Me O mighty one! O delighter of Ramā make me powerful.
O Man-lion! Hari! my obeisance is for you, O god Acyuta!
confer on me the state from which one never falls.

naumi veda-dhvani-varaṃ devaṃ dhatte sadaiva hi ।
I salute the god of the excellent chants of the Veda.
Indeed, I always place myself in
the humming of the buzzing bees,
in the womb of Viṣṇu’s navel-lotus.

taṃ namas kurmahe śakraṃ devānām api daivatam ।
yo locana-sahasreṇa viśvakāryāṇi paśyati ॥
I make my obeisance to him, Śakra who is the god of the very gods,
who with his thousand eyes beholds all the happenings of the universe.

yaḥ santataṃ tat tamaḥ paṭalaṃ vidīrya sāvitrikaṃ karaśatair vahati prkāśam ।
taṃ viśva-rakṣaṇa-paṭuṃ paramekam ādyam ādityam adbhuta-vilāsavidhiṃ namāmi॥
Continually rending that screen of darkness,
the impeller, who with his hundred rays bears light,
I salute him, the foremost one, skilled in protecting the world,
the primal Āditya manifesting in a marvelous manner.

sthāṇur yasyecchayā jātaḥ śarīrārdha-bhṛta-priyaḥ ।
arikta-śaktaye tasmai namaḥ kusuma-dhanvane ॥
By whose wish Sthāṇu came to share half his body with his wife,
Obeisance is for him of no mean power, the wielder of the flowery bow.

cālukya-vaṃśa-tilakaḥ śrī-someśvara-bhūpatiḥ ।
kurute mānasollāsaṃ śāstraṃ viśvopakārakam ॥
The forehead-mark of the Cālukya-s, the king śrī Someshvara, has composed the encyclopedia Mānasollāsa for the benefit of the world.

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

## The second strike

Childhood
Thrice a week, starting Fridays, Lootika’s mother taught her daughters mantra-siddhānta after dinner. She had covered the praxis of the secret mantra-s of the Vīṇāśikhā and had moved on to relevant sections of the Jayadratha-yāmala and the rahasya-s of the Kālī-saṃhitā-s. She mainly focused on the exposition of the mantra-rahasya-s and their vidhi-s leaving the curiosities of Śiva’s bad grammar aside to be covered by their language teacher Shilpika. She merely prefaced it with a brief comment: “Kids, sage Pāṇini is the ideal but not everyone spoke his way and even Rudra and Ṣaṇmukha seem to have approved of that.” That autumn Friday evening, as ever, she was seated in their deva-gṛha performing japa of the appropriate mantra-s with her brood of four. After they concluded the tarpaṇa-s to the deities and the lineage of teachers, she said that since it was a pratipad there were going to be no lessons. Instead she said: “Lootika’s friend Somakhya’s mother has lent me this rather interesting book on Mesopotamian incantations. I thought some topics from it might interest you all, given your disposition. Let us read out and discuss some sections from it.”

With her clump of daughters snuggling close to her like the chicks of a galloanseriform bird, she read out the intended text, prefacing it “Girls, here is an account of persons said to be afflicted by a ghost termed the deputy of the goddess Ishtar ”:
‘If a person experiences pulsating of the temples and his hands and his feet go numb …
“If a person experiences pulsating of the temples and rimūtu paralysis …”
“If a person continually has headaches, his ears roar, (and) his eyes become dimmed, his neck muscles continually hurt him, his arm(s) are continually numb, his kidney gives him a jabbing pain, his heart is troubled, (and) his feet continually have rimūtu-paralysis, a pursuing ghost continually pursues that person.”
“[If a m]an’s temples [afflict him and] his face seems continually to be spinning (and), [he gets up (from the bed)[but then] falls (back down again), ‘hand’ of ghost.]”
“If his face seems continually to be spinning, his ears roar (and) his temples give him jabbing pains and get him wet, ‘hand’ of ghost.”
“[If, as a result of affliction] by ‘hand’ of ghost, a [person]’s neck hurts him, (and) his face seems continually to be spinning …’

She then continued: “As one of the ritual cures they have the following incantation:”
‘I have made you swear (by) heaven and earth, (by)[Anu] and Antu, (by) Ellil and Ninlil, I have made you swear (by) Shin, Shamash and Adad, valiant gods. I have made you swear (by) (tuttubu-style) c[loa]k, (by) incense and ﬂour. Be far away, be] faraway, be distant, be distant!’
Having read it out Lootika’s mother paused and asked her daughters: “Girls, what do you think is going on here?”
Jhileeka: “Mom, that sounds like somebody seized by one of the bhūta-s from Vidrum’s house or the yonder cemetery.”
Her mother sternly looked at her: “Jhilli, I have had enough of Vidrum’s house and the cemetery.”
Jh: “But we have managed to see a bunch of them with the ḍāmara-prayoga and Lootika and Vrishchika tell me that that Vidrum indeed has had some such afflictions, albeit of a milder kind.”
Her mother (LM): “Girls, I think you all are taking the fantasies of your claims of bhūtaḍāmara siddhi-s too far. It is not a good thing for one on the path of the mantra-s.”
Varoli simpered: “But then…” But she and Jhilleeka saw their two older sisters catch their eyes and press their fingers to their lips signaling them to be quiet.
LM then caressingly tousled the long locks of Vrishchika and said: “Alini, what do you think is being described here?”
Vrishchika perked up and said: “Mom, I think these sound like symptoms of someone in need of medical treatment. I would say that in the simplest case the patient is suffering from migraine with some aural effects. However, the remaining symptoms such as paralysis of the extremities and suggests something more serious affecting his nervous system. That kidney symptom strikes me as particularly interesting. Putting all together I am wonder, mom, if we are after all not confronting a bhūta but an early account of multiple sclerosis. However, given the times I would not rule that these are the sequelae of an encephalitis or a meningitis.”
LM: “Dear, that’s not an entirely unreasonable conclusion and in line with what I really wanted you all to think about.”
Lootika put her arms around her mother and said: “Ain’t in also interesting that in this old Eastern Semitic world such neurological conditions were clearly seen as something evil that needed a cure much like in our own ancient world. In contrast, among the Abrahamisms which emerged in the Western Semitic world such afflictions were celebrated as the mark of a prophet.”
LM: “My firstborn, that’s an interesting distinction and the struggle between the two interpretations of the syndromes indeed persisted until the times of the unmatta Mahāmada, whom some correctly diagnosed as suffering from a psychosis. That’s why I repeatedly tell you all, my dears, never even by mistake in the future think of marrying an Abrahamist. It as a good as your funeral without any rites.”

At the cusp of youth
With the monsoons having crept in in the god Parjanya was fertilizing the land. Taking advantage of the brief break in the downpour, Lootika’s mother was out in the garden collecting flowers for the deities. While doing so she saw Vidrum’s aunt standing at the gate. Letting her in: “Come in Vaidoorya, hope things are well with you and the rest?”
Vaidoorya: “Auntie, you must give me a good sweet. I have some great news for your”
LM: “What is it my young lady?”
Vai: “Your daughter Vrishchika has made it to the merit list in the college entrance exams.”
LM: “How do you know that? The marks-card will be given only tomorrow at the college.”
Vai: “No, No. You can find your overall place today itself. It is pasted at the exam office near the fort. On my way back I stopped there to check out some names and your daughter’s name was right there. See, she has surpassed you elder one Lootika who barely missed the merit list.”
LM: “That’s a relief. Like her elder sister, I think she too took the preparation lightly, But all things taken I think she was a little more serious than Lootika, who as you know hardly cares for curricular education.”
Vai: “Yes, but we all would have killed to even get Lootika’s marks. It is so tragic that she opted out of a good degree to pursue an ordinary B.Sc. I guess Vrishchika will be more sensible and now enter medical school.”
LM chuckled to herself: “Yes. That’s what she wants to do. I am glad she can make it without any trouble now. You know how many hurdles are placed these days in our path for this particular educational pursuit.”
Vai: “Ah, that is very true, but even with all this crush you brāhmaṇa-s seem quite unbreakable, I think. But in any case, auntie, a sweet from a brāhmaṇī’s hand would do me some good.”
LM: “Dear, I don’t have any special sweet at home but let me give you some creamed honey.”
As Lootika’s mother brought the honey and was handing it over to Vaidoorya, a large bat dive bombed the latter. Vaidoorya was alarmed. LM brought her to the lit interior and checked her out carefully to see if there was any scratch or saliva. Confirming there was none she gave her a napkin to wipe her face. Calming her down, Lootika’s mother said: “Vaidoorya, I must ask you a favor for tomorrow.”
Vai: “Sure auntie.”
LM: “My youngest, Jhilleeka, is ill with a bad upper respiratory infection. Luckily, I have no teaching at college tomorrow; hence, I will be with her. But I think it is best that Vrishchika go straight to the med-school after collecting her report and pay the fees to claim her seat. My husband will be leaving early and will be dropping off Lootika and Varoli at their college. But someone will need to take Vrishchika later in the day to the med-school. I have reasons for not letting her go alone tomorrow with the large amount of cash needed as fees. Since tomorrow is Wednesday, I believe you will be going to the hospital only around the time she needs to go. So could you please take her along.”
Vai: “I will certainly do so and I could also bring her back if she is willing to wait till the evening.”
LM: “That is excellent. She can spend her time in the anatomy museum and the library till then.”

The next morning Lootika’s mother was faced with cacophony of angry protests from her three older kids, which rose above the wild rain and thunder in the exterior. Lootika and Varoli declared that it was rather strange that just for that day their mother wanted them to be ferried to college by their father. For some reason, LM did not give out her reasons for it and clearly she had her husband in her confidence as he simply took her side and paid no heed to his daughters’ dissent. Lootika offered an alternative: “I could cut college and accompany Vrishchika with the cash. I will have my knife with me. May be I could also convince some of my friends who would certainly be armed with more to give us further company.” Her parents cut short any of those plans and told their daughters that they had to do as they had planned.

That afternoon even as LM was fixing the meal for the evening, she was wondering why her husband had not yet returned with the kids. Just then her phone rang; he husband called her saying: “Have you checked the news? There has been an armed robbery at the bank next to the Mahishahrada near Jhilleeka’s school. One guard has been killed in the shoot out and the ATM machine was blown up with a bomb. I have picked up Lootika and Varoli and have taken them with me to the hospital; we are waiting for the road to be opened for me to return. Don’t worry. It looks as though the core situation has come to an end.

She then tried to reach Vrishchika or Vaidoorya but could not connect. With some tension in her mind she called Somakhya’s mother since their house was located closer to scene of the crime. Somakhya’s mother informed her that Somakhya had seen the mayhem from up close on the way back home from college. Somakhya had watched the events unfold hiding behind a clump of vegetation on the escarpment leading into the Mahishahrada. He suspects that it was not just any robbery but the handiwork of terrorists. There were likely two terrorists and they made away on their bike. He said that the guard confronted the assailants with a primitive rifle; however, he was killed as they possessed superior automatic guns. It was based on the form of the assailant’s guns, apparently of American military make, that Somakhya was convinced that they were not just taskara-s but terrorists. One of them was injured by the guard’s fire but he apparently still made away with his companion. They uttered a slogan while fleeing, which Somakhya only imperfectly heard: to him it seemed more like an uṣṇīṣin’s proclamation than that of a mahāmada-rākṣasa. Somakhya’s mother concluded by adding that the blockade might be cleared in an hour or two and that barring any further incident her husband should be able to make it back smoothly with her daughters.

Just then Lootika’s mother got another call. This time it was Vaidoorya: “I have safely returned home with Vrishchika. Thankfully our way was not blocked due to the bank-robbery. She added that if LM permitted she would let Vrishchika remain at their house for some time because her nephew Vidrum wanted some help from her preternatural daughter for his biochemistry class.” LM: “That is fine. I’ll ask my husband to collect her when he is returning.”
Vai: “Don’t worry auntie. My fiance was to arrive but he too will be delayed; so, I could bring her back too if you think she needs an escort. Also if you do not object we can give her something to eat.”
LM: “No, other than water don’t give her anything. I have made stuff for them which needs to be eaten.”
LM heard her daughter yell out from the background: “Being treated like a baby is most annoying. I surely need no escort to walk back from Vidrum’s house. I can return on my own soon as I am done with whatever he wants for his biochemistry.”

Vidrum wanted something understandable on kinase signaling. Both his teacher and the textbook seemed abominably confusing. He had wanted to discuss it with his friend Somakhya but going against that were: First, the whole incident of the bank robbery which had blocked the road. Second, he knew his friend might not be responsive for perhaps Somakhya had told him of these things earlier and Vidrum had simply not shown any interest then. He knew well that Vrishchika, though his junior, could do as well a job as Somakhya or her elder sister as she had mastered these matters while in school itself. Vrishchika was more than happy to hold forth on the topic delving deep into the intricacies of receptor tyrosine kinases, followed by the signaling systems of the TGF-$\beta$ family ligands, and then JAK-STAT signaling. By then Vidrum had reached the limits of his attention and with his concentration flagging he put down his pen. Vrischika: “Remember we are not done. There is still MAP kinase signaling and kinase-catalyzed AMPylation in the least to cover and finally I will discuss some pharmacology of kinase inhibitors.” Vidrum: “Thank you clever girl. I have a good body of notes that should put me in front of the whole class and probably the professor too now. I know you have much more to say but I’ll get them from you another day: with that I will be king!. But I’ve been studying all day and need a little break at this moment.” Then pointing towards a corner of his desk: “Vrishchika, what do you think is in that box?”

Vrishchika walked up to the table and picked up the little wooden box. She was a bit surprised upon opening it as it contained what were clearly bones from a pair of human hands, somewhat blackened by firing: “I see a pair of lunates, some metacarpals and few phalanges. Well Vidrum, where did you find these?”
Vid: “The past weekend I was quite tired from all the studies and my friends Mahish and Gardabh, who were with me had fit of nostalgia. They suggested that we play marbles for a while. I too felt the yearning for this old game that probably would seem very out of place for a man who was striving to attain the exalted station of physician. In any case we gave into our little fancy and decided to play marbles in my backyard. I began digging a hole close to my back wall when I hit something metallic. This caught my attention and I called my friends over to help dig it out. Ere long we had it out of the ground and it was a nice little metal urn, probably made of some kind of pewter. Evidently it had a lid but that was gone and these bones were lying in the bottom of the said urn.”

Vrishchika looked at them closely: “Perhaps they are from an adult female.” Then looking at them again in her palm: “Where is the urn which contained them?”
Vid: “You know how my aunt has a great fascination for such trinkets. She was so captivated by the product of my excavation that I gave it to her.”
Vri: “Vidrum, you have had a lot of trouble in this house. Ask her to deposit it in the cemetery right away. I also suggest that you get rid of these bones as soon as possible. Do not got to the cemetery tonight with them. It might not be safe. I suggest you do it when the sun is up, ideally with your friend Somakhya to give you company.”
Vid: “You guys always puzzle me. What makes you say all this. Yes, I know I have had a lot of trouble here but what difference would it make since the bones were lying in my backyard all this while. But evidently you see something which I am unable to see.”
Vri: “See, there are certain things which I cannot talk about. Not because I want to hide things from you to appear deliberately obscure but because it is the code of conduct of the practitioner. I cannot say much more other than that your aunt should be careful with such items and you too. Anyhow dispose of them tomorrow and I could tell your aunt if you don’t want too.”
Vid: “No. She will be after me as ever. She can never stop on haranguing me with the shit of having a psychiatric evaluation before my parents. So let it slide.”
Vri: “Whatever. I better head home now or else today my mother could make a fuss. I am sure she would not like me walk back home on my own today for whatever strange fancy…”
Vid: “It must be the robbery. I need to talk to Somakhya to hear more about it.”

Vrishchika had gotten a message from her sister Lootika that the turnpikes were finally lifted and they were headed home. So Vrishchika too started walking back home. As she turned the corner on which Vidrum’s house was located and passed the great wall of the cemetery, she positively felt some presence behind her. Looking back repeatedly she saw no human but there was a stray dog some distance away. She remarked to herself: “It is not that. I see due to my ḍāmara-siddhi that those bones of Vidrum have a genius of locus in the form a rather dreadful brahmarākṣasī.” She quickly deployed a Bhadrakālī mantra and doing its japa proceeded. She could sense that the presence was not repelled but kept a distance from her. As she reached the long road leading to her house she looked around her as man can be more wicked than ghost: even in these familiar parts of the city a young woman in particular could never let down her guard. Knowing this she fingered her garala-śaṅkulā and feeling assured walked on. She passed by the śṛṅgāṭaka cart-man and the smell of the frying pastry and the chickpea sauce sent a signal through her olfactory receptors. It felt nice but she had transcended such temptations for while now. However, her mind was distracted from her japa by that sensory stimulus. As her mind was wandering, she suddenly heard the blare of a car-horn beside her. Greatly startled, she looked to the road and was relieved to see that it was her father with her sisters. Varoli opened the door and she hopped in. The final stretch back home was covered in a couple of minutes during which Vrishchika informed her father that she had formally entered medical school by paying the fees and that classes would begin in a week.

The next evening Lootika and her family were having dinner. Normally, her parents in the manner of the Hindus of old spoke very little at their meals. But she and her sisters were not at all like that: dinner time was always the occasion for raucous conversation on all manner of things, which their parents never succeeded in stopping had become resigned to it — they even saw it as an opportunity to hear in on their daughters to make sure they were up to no mischief. But that day LM with a tinge of anxiety remarked to her husband: “Isn’t is strange that Vaidoorya has some weird problem or the other. Do you have any news on the state of her fiance?”. Lootika’s father looked up from his plate and remarked: “It is indeed sad, dear. He has been taken for an in-depth neurological evaluation. There was no evidence for an ischemic attack or a hemorrhage. It was most sudden and unusual with all his organs otherwise being in good shape.”
Vrishchika excitedly remarked: “What happened?”
Jhilleeka: “Mom will frown as usual, but Vrishchika it sounds like you were right. Mom do you remember the alū demon or gallū demon or the mukīl-rēś lemutti demon. It must one of those or more precisely our equivalent of the same.”
LM: “Listen, dear Jhilli this is no joke. The young lady’s fiance is in bad shape.”
Jh: “I know it is not a joke, mom. If you could somehow figure out that yesterday something untoward was going to happen and ask dad to ferry the agrajā-s, then don’t you think we as your daughters should at least have some of that capacity?”
Vri: “Could you at least tell us what happened.”
LM: “Last night Vaidoorya’s fiance was visiting their home. She is to be married next month. He woke up around 2:00 AM complaining of a loud roaring in his ears. He then felt as though somebody kicked him, and his tongue and extremities have been paralyzed since.”

Bliss at the inception of domesticity
As she awoke that Sunday morning Vrishchika found herself entrapped in the cage of Indrasena’s arms, like an antelope in the grip of a lion. Fair Vrishchika’s dense black locks were all around her like the dark solidified pyroclastic flows from the snowy summit of a great volcano. Indrasena: “Sweet Gautamī, what is that got you all trembling and your heart racing the past night?” Vrishchika kissing Indrasena and stretching out a bit like a caracal: “Why dear Ātreya, I seem to have slept well.” Then after a moment she remarked: “Oh, did you sense me wake up? It must have been that nightmarish renewal of memory in my dream.” Ind: “What was it dear?”. Vri: “Ah, an encounter with a particularly nasty brahmarākṣasī from old Vidrum’s lair returned in my dream!”
Ind: “Vidrum’s old lair seems to have been full of all manner of phantoms. Why, even his parting from that place brought a visitation on our Somakhya and Lootika.”
Vri: “Yes this one was of the same genre as what they battled.”
Ind: “What’s the story?”
Vri: “Let me fix breakfast and as we savor it I can regale you with that tale freshened by my dream and with some consultation of my notes.”

With the morning sun glancing off her ear-studs and casting a rainbow on the wall, Vrishchika narrated the tale to her husband with as much drama as she could muster. Perhaps the events were even more climactic as they actually played out but Vrishchika was not one who could give such an recountal. What follows is the crescendo of Vrischika’s narrative:
“The news reached us that the bank robbers had been apprehended in the neighboring state and that they were indeed terrorists from the Pāñcanada. That at least lifted the restriction on our movements that had come about from their strike on the bank. Lootika and I were being frantically contacted by Vidrum on the matter of the urn and the bones with Vaidoorya’s fiance showing no signs of improvement. We went over to his house in the evening and deploying Bhadrakālī all the while placed the bones back in the urn and deposited it in the cemetery. However, while coming back home I repeatedly felt someone pull my hair. Lootika told me that the brahmarākṣasī might still be after us and suggested that we perform a pratikriyā homa right away. As we were doing the same in our deva-gṛha suddenly I found my prastara bundle missing. Lootika started frantically looking for it when my voice seemed to be arrested by a gruffness and a cough, preventing me from reciting the incantations. Lootika finally found the prastara mysteriously turn up under the skin on which she was seated and then she brought me some water to relieve my throat. Somehow recovering my poise I completed the pratikriyā. Soon thereafter while at dinner I received a message from Vidrum that the victim had miraculously made a complete recovery. Given the glitches, I had doubts if my pratikriyā had really worked but now I felt confident of its success.

But that was hardly the end of the story. That night I awoke from sleep, may be around 3:00 AM, with a choking feeling. I felt a throbbing on one side of my temple alternating with mild a headache. I got up to drink some water when I was aghast to see my face in the mirror. I had developed some mysterious dark spots and some white spots on my arm. I thought I might die soon. I kept thinking of you, my dear, and was paralyzed by the thought that I might expire without ever having the sammelana with you. I wanted to send my parting message to when I regained some composure and woke Lootika up. I was clearly not yet in her league in terms of my mantra-vīrya. My sister quickly realized that more power might be needed for this and right away sent a message to her vīra Somakhya asking him for help in the matter. Then she deployed Koṭarākṣabhairava and kept at the mānasika-prayoga for a couple of hours. She also applied some oṣadhi-s on me. Even as the first rays of the sun where seen, my symptoms abated though I still felt ill probably with the loss sleep playing its part. Later that evening, Lootika and I went with Somakhya to the Sarasvatī temple beside the śmaśāna. There he performed a kamaṇḍalu-prayoga visualizing being in the great cremation ground of Kilikilārava where the Saṃkarṣaṇa had slain the gigantic ape Dvivida with the dhyana of Aparārdhanārīśvara who wears scorpion ornaments. Thus, he was able to imprison the brahmarākṣasī. I instantly became an adept in that mantra at that point. Later that weekend, we made the vile brahmarākṣasī speak her tale.”

The brahmarākṣasī’s tale
What ensues is a summary of the autobiographical bhūtānuvacana: My name is Sarah but I was born Savitri. I had a sister named Gayatri and we were born in the Vaṅga in a family of kulīna brahmins who lived in Vīrabhūmī. My father had educated both me and my sister in Sanskrit and several śāstra-s by the time we were to be married. But just then the great famine struck and our parents perished in it while feeding us with whatever little they could forage. By some luck we were rescued by a kāyastha gentleman of highly enlightened views and taken to the great city of Calcutta. He declared that there was no place for child marriage and sponsored our schooling. At school we came under the kindly tutelage of the Baptist minister from America, Dr. Jedediah Wilder, who took me and my sister under his wings. One day he very sensibly asked as to why it was that we had seen so much tragedy? If we were brahmins and well-educated in Sanskrit and śāstra-s should the gods not have answered our call. We told him that all this was prophesied by Lord Viṣṇu as being a natural situation of the dark Kali age. He then very pertinently asked as to why is it that only the Hindoos were dying in droves despite having Viṣṇu and so many other gods to pray to while the Americans and Europeans were living out their allotted lives in such an upright and useful manner. He then made it clear that all this while we were worshiping false gods and introduced us to the light of Lord Jesus the Savior. I immediately saw the light and took it upon myself to bring that light to my suffering countrymen.

My vile sister revolted against Dr. Jedediah Wilder and called him an enemy of the Hindoo dharma. I tried to reason with her. After all we knew Sanskrit and we could read all the vile things in the false Hindoo scriptures by ourselves. We had read the Veda and Upaniṣad in the original. I asked her how could we accept a scripture which called upon women to engage in coitus with a horse or be immolated upon the corpse of her husband in the a name of being a suttee? I then presented to her that even the so call lofty Upaniṣad provided a prescription for a man to beat his wife with his fists. I adduced more evidence showing how the so called sage of the Hindoos, Āpastamba, had recommended a man to place a centipede in the vulva of his wife to block rival males. Sadly, she still remained adamant and ran away. But I could have nothing to do with Hindooism and its filthy scriptures and stood firm in the path of Jesus.

Saying so the brahmarākṣasī uttered a blood-curdling scream and tried to emerge form the kamaṇḍalu where Somakhya had confined her. Lootika and Vrishchika felt their hair being pulled by some unseen entity. However, all three of them simultaneous deployed the utkīlana of Bagalāmukhī to permanently disable that brahmarākṣasī. Vrishchika’s narrative concluded with Somakhya’s remark on that tense day: “She had been impaled like her preta. However, imagine the unpleasant fate of a man lacking the RAG1 gene. If that were to befall Bhārata-s, as it very well could, then that evil which she embodies could some day destroy our people.”
Indrasena: “So how did that brahmarākṣasī land up far away from the Vaṅga-s in the cemetery near Vidrum?”
Vri: “O Atri-putra, we too were much puzzled by that. We wondered if it was a case like the apparition of the old Scottish surgeon. But we got our answer not long ago when Vidrum’s house was being cleared for sale. Among the stones paved into his bathroom was one which bore the name Dr. J. Wilder. Evidently Wilder had translocated to regions of youth upon the death of this Vaṅga woman with her remains and they were buried close to his. Perhaps he too was one of the phantoms harassing Vidrum and his family.”
Indrasena: “Gautamī, it looks as though a trait segregated in an almost Mendelian fashion among these kulīna sisters. I wonder if there is more to it.”

Posted in art, Life |

## Visualizing the Hindu divisibility test

Prologue
This article continues on the themes covered by the last two (here and here) relating to factorization and the primitive root modulo of a prime number. Early in ones education one learns the divisibility tests for the first few primes: 2, 3, and 5. Of these divisibility by 2 and 5 is trivial. Divisibility by 3 is also easily achieved: e.g. Is 771 divisible by 3? $7+7+1=15; \; 1+5=6$. We see that the successive addition of digits reduces the original number to 6 which is divisible by 3; hence, the original number is divisible by 3. Of course, division by 3 is not a big deal; yet, for big numbers this is an easy method for mentally determining divisibility.

Beyond these three simple cases, in our school days we determined the divisibility by other primes using brute force — by dividing the test number by the given prime. When we reached college, a gentleman of great mathematical ability asked us if we could think of a generalization of the divisibility test for 3 for other primes. Seeing that we were unable to do so, he proceeded to tell us of such a method. He then mentioned that he had learned this via Suryanarayana of Vishakhapatnam who possessed enormous mathematical knowledge and capacity. Suryanarayana had further informed him that it was a folk Hindu method that was known in South India. Shortly thereafter, we saw the same method with some tricks for quick implementation in the pre-computer age described by the late Śaṃkarācarya Bhāratī Kṛṣṇa Tīrtha-jī (hereinafter SBKT) of the Govardhana pīṭha in his curious mathematical work with 16 foundational sūtra-s. He called the method veṣṭana, which he rendered in English as osculation, and presented it as an example of his sūtra: “ekādhikeṇa pūrveṇa |

The Hindu divisibility test
The procedure goes thus:
1) Let $p$ be a prime other than 2, 3, 5 for which divisibility is already accounted for. Let the test number be $n_0$ which is written as $n_0=10a_0+b_0$.
2) Find the first number $(10c-1) \mod p =0$, i.e. the first multiple of $p$ which is of the form $10c-1$. Thus, this number will necessarily be of the form $10d+9$, where $c=d+1$. That is how it relates to SBKT’s “ekādhikeṇa pūrveṇa |
3) This $c$ is termed the veṣṭaka or ‘osculator’ for the procedure. We then compute $n_1=a_0+b_0c$.
4) We again write $n_1=10a_1+b_1$ and repeat this procedure: $n_2= a_1+b_1c$.
5) If $p$ divides $n_0$ then this procedure terminates in a small and easily recognized multiple of $p$. Any further application of the above procedure on that number yields the same number.

Let us illustrate it with an example: Is $n_0=3249$ divisible by $p=19$?
For $p=19$, $2 \times 10 -1=19\; \therefore c=2$. Now we use the osculator $c=2$ for applying the above procedure.
We get $324+9 \times 2=342; \; 34+2\times 2= 38; \; 3+8\times 2=19$
We reach 19 indicating that 3249 is divisible by 19. Applying the above procedure to 19 yields 19 again. Indicating that it is a terminal number of the process. In a practical application from the pre-computer age we don’t have to go all the way: we could stop for e.g. at 38, which we recognize as a multiple of 19. Further, for bigger numbers SBKT gives certain tricks which would be useful in such pre-computer applications.

What if a number is not divisible by $p$? As an example consider the case: Is $n_0=178$ divisible by 19? Of course, in this case the answer is quite obvious; nevertheless, it allows us to illustrate what happens if we apply the above procedure on it. We get the following sequence of $n_j, j=0, 1, 2 ...$:
$178\rightarrow 33 \rightarrow (9 \rightarrow 18 \rightarrow 17 \rightarrow 15 \rightarrow 11\rightarrow 3 \rightarrow 6 \rightarrow 12 \rightarrow 5 \rightarrow 10 \rightarrow 1\rightarrow 2 \rightarrow 4 \rightarrow 8 \rightarrow 16 \rightarrow 13 \rightarrow 7 \rightarrow14 ) \rightarrow9$
We observe that, unlike the divisible $n_0$ which converges to a single number, the non-divisible $n_0$ after the first 2 terms enters a repeating cycle of 18 terms. Again, for practical purposes if one just wanted to test divisibility one obtains the answer much earlier.

The basic divisibility graphs
We can do this divisibility procedure for several successive $n_0$, say for all $n_0=1..200$ and $p=19$, and the illustrate the conglomerate results as an ordered graph (Figure 1). The graph’s nodes are the numbers $n_j$ obtained while applying the above divisibility procedure until we hit a cycle or converge. The edges indicate which number leads to which, with the relative thickness indicating the frequency with which they occur for this range of $n_0$ (scaled by hyperbolic arcsine).

Figure 1

One also notices that all $n_0$ in this range, which are divisible by 19, directly converge to it. The remaining numbers converge directly or indirectly by different paths to a central directed ring $R_1$ with components $r_1, r_2, r_3...$. These are the same as the terms of the cycle obtained in the above example. One also notices that in this example the ring size $S(R_1)=18$, i.e. number of terms in the ring, corresponds to cycle length of the decimal expansion of $\tfrac{1}{19}$ or $j=18$ when $10^j \mod 19=1$ for the first time. This $j$, in this case 18, is known as the multiplicative order of $10 (\mod p)$, which was shown by Carl Gauss to correspond to the cycle of the decimal expansion of $\tfrac{1}{p}$ . The terms of this ring $R_1$ includes all the number from $1..18$. Thus, it is also obvious that the sum of the terms of in ring $R_1$ is divisible by $p$:

$\displaystyle \left(\sum_{j=1}^{S(R_1)} r_j \right) \mod p =0$

To further explore these patterns, we next consider the case of $p=13$ and plot the same for graph for $n_0=1..200$ in Figure 2.

Figure 2

We observe that all $n_0$ divisible by 13 in this range converge to 3 terminal nodes: 13, 26 and 39. The remaining $n_0$ converge directly or indirectly to one of 6 rings with 6 terms each. i.e. $S(R_{1..6})=6$. Here again we notice that the size of these rings is the same as the multiplicative order of $10\mod 13$ ($j=6$ when $10^j \mod 13=1$ for the first time), which is cycle length of the decimal expansion of of $\tfrac{1}{13}$. Further, together the 6 rings include all numbers from 1..38 barring 13 and 26 which are divisible by 13. It is easy to see that $\sum_{j=1}^{38} j-13-26$ is divisible by 13. However, we also notice that the sum of the terms of each of the six rings is also divisible by 13:

$\displaystyle \left(\sum_{j=1}^{S(R_k)} r_{kj} \right) \mod p =0; \; k=1..6$

We next consider the case of $p=7$ and $n_0=1..200$ (Figure 3).

Figure 3

In this graph the numbers divisible by 7 converge to 2 endpoints either 7 itself or 49 (when they are multiples of 49). All the remaining numbers directly or indirectly converge to a large ring with 42 terms, which includes all numbers from 1..48 except for 7, 14, 21, 28, 35, 42. Thus, it is easy to see that the sum of the terms of this ring will be divisible by 7. Further, the multiplicative order of $10 \mod 7$ is 6. Thus, the number of terms in the ring is multiple of that $42=6 \times 7$.

To sum up, we may note the following features of these divisibility graphs:
1) The residue system of $p=19$ for base 10 (till $10^j \mod p$ for $j=1,2,3...$ produces 1 as the residue for the first time, i.e. till $j$ is equal to the multiplicative order) is: 10, 5, 12, 6, 3, 11, 15, 17, 18, 9, 14, 7, 13, 16, 8, 4, 2, 1. The base 10 residue system for $p=13$ is: 10, 9, 12, 3, 4, 1. The base 10 residue system of $p=7$ is: 3, 2, 6, 4, 5, 1. We observe that the penultimate residue in each case is what SBKT calls the “osculator” $c$ used in the divisibility test.

2) Looking the graphs, it becomes clear that the non-divisible $n_0$ converge to one or more directed rings, the sizes of which are either the multiplicative order $m$ for base 10 or a multiple of $m$. A number $n$ is a primitive root of $p$ if the residue system of $n^j, j=1,2,3...p-1$ contains all numbers from $1..p-1$. If 10 is not a primitive root of the $p$ , the divisibility by which we are testing, then we will necessarily have multiple rings. If 10 is a primitive root of $p$ we can in some cases get a single ring with its terms being all numbers from $1..p-1$ (e.g. 19). Alternatively, we can get a large single ring whose size can be some multiple of the multiplicative order (e.g. $p=7$) or multiple rings whose sizes are different multiples of the multiplicative order (e.g. $p=17$).

Figure 4. Divisibility graph for $p=17$

3) The terms included in all the rings of convergence taken together for a given $p$ range from $1..10c-2$, where $c$ is the penultimate residue of the system and the osculator. Of course, all numbers divisible by $p$ in this range are excluded from the rings as they are nodes of separate components of the graph that converge to single root, an integer divisible by $p$. Thus, for $p=13$, the osculator is $c=4$. So, the integers covered in the rings would be from 1 to 38, with 13 and 26 not being in the rings. For $p=7$, the osculator $c=5$; hence, the ring would cover the integers from 1 to 48, with 7, 14, 21, 28, 35, 42 being excluded. When $p=17$, the osculator is $c=12$. Thus, the rings will extend from 1 to 118, with 17, 34, 51, 68, 85, 102 being excluded.

4) From the above it is obvious that the sum of the numbers in all the rings taken together would be divisible by $p$. The less obvious feature is that the sum of the numbers in each individual ring of the convergence graph is also divisible by $p$. Thus, the numbers are placed in each ring in such a way that two constraints are simultaneously met, namely: 1)that of divisibility of the sum of the numbers in each ring by $p$ and 2) that of each ring size being a multiple of the base 10 multiplicative order modulo the given $p$. In the case of $p=13$ placing the numbers in 6 separate rings can satisfy the above constraints for each ring. In the case of $p=17$ placing them in 3 separate rings can satisfy them, while in the case of $p=7$ they need to be in a single large ring to meet the constraints.

5) An obvious but structurally distinctive feature of the divisibility graph is that 1 can never be reached directly from any $n_0$ unless it is a member of the ring to which 1 belongs. From the procedure it is clear that it can be reached only from 10. 10 is the first number in the base 10 residue system for any $p>7$ and will necessarily be part of the ring. Further, in whichever ring 1 occurs, by the nature of the procedure, it would always lead next to the osculator $c$ as the next node in the ring. $c$ always comes just before it in the residue system for base 10 (e.g. for 13: 10, 9, 12, 3, 4, 1): thus $10\rightarrow 1 \rightarrow c$ is fixed a motif in the divisibility ring reverse order of their occurrence in the residue system. The longest path of the divisibility graph ending in a ring always passes through this $c$ (However, there could be graphs with multiple equally long paths passing through other ring nodes in certain cases like $p=17$).

One wonders is an encryption system along the lines of the Diffie-Hellman-Merkle mechanism can be made using these divisibility graphs.

The reduced divisibility graphs
The above-described graphs were based on the simple divisibility method found in folk Hindu mathematics and SBKT’s mathematical exposition. However, for practical purposes SBKT gives an additional step, which he terms the “casting out of the primes”. This goes thus: We again write the test number as $n_0=10a_0+b_0$. We then obtain $n_1= a_0+(b_0c \mod p)$, where $c$ is the osculator as above. Thus, we first reduce $b_0c$ to its residue modulo the $p$, the divisibility by which we are testing, before adding it to $a_0$ to arrive at $n_1$.

As a numerical example let us consider testing the divisibility of 99 by 7. The osculator for 7 is $c=5$. Thus $9 \times 5=45$. We now take $45 \mod 7= 3$ and use it in place of 45. Thus, we get $n_1=12$ which allows us to decide on the matter of divisibility right away. However, we could continue the procedure to convergence, as we did above, in order to compute the convergence graphs for different $p$. In this case we observe the following:
1) If 10 is a primitive root of $p$ then we get a bipartite graph: One component derived from all the non-divisible $n_0$ has a central ring, which has as its terms all numbers from $1..p-1$. The other component is a tree graph converging to $p$ which includes all the $n_0$ divisible by $p$. Examples of this are the graphs for $p=7$ or $p=17$

Figure 5. The convergence graph for $p=7$

Figure 6. The convergence graph for $p=17$

2) If 10 is not a primitive root of $p$ then we get a multipartite graph. One component, as above, is a tree graph converging on $p$ and includes all the $n_0$ divisible by $p$. The remaining components have central rings whose size is the multiplicative order $m$ of $10 \mod p$. The number of these ring components is $\tfrac{p-1}{m}$. Here again, the sum of the numbers in each ring is divisible by $p$. Together they include all integers from $1..p-1$. Examples of this are the graphs for $p=13$ and $p=41$.

Figure 7. The convergence graph for $p=13$. Here $m=6$ for base 10; thus we get $\tfrac{13-1}{6}=2$ components with central rings.

Figure 8. The convergence graph for $p=41$. Here $m=5$ for base 10; thus we get $\tfrac{41-1}{5}=8$ components with central rings.

In these graphs too the longest path terminating in a ring passes through the osculator $c$, which is reached from 1 on the ring. Here again, 1 is reached from 10, the first residue of the residue system for base $10 \mod p$ for all primes $p>7$. However, given that $10 \mod 7 =3$, for $p=7$, 1 is reached from 3 in the ring. In conclusion, this version of the divisibility graph makes apparent the close relationship it has to the period of the decimal form of the fraction $\tfrac{1}{p}$, which we considered in the previous article.

Posted in Scientific ramblings |