## The mean hyperbola and other mean functions

Let $a, b$ be two numbers such that,
$0\le a \le b$
We use $a, b$ to construct a specific rectangular hyperbola using one of the following methods:
Method-I (Figure 1: this is based on an approach we described earlier)

Figure 1

1) Mark point $C (-a, a)$, which will be the center of the hyperbola to be constructed.
2) Draw two perpendicular lines through $C$ respectively parallel to the x- and y-axes. These two lines are the asymptotes of the hyperbola.
3) Bisect the angle between these asymptotes to construct a line with slope $m=1$ passing through $C$.
4) On this line mark the two foci of the hyperbola $F_1, F_2$, such that they are are equidistant from $C$ and separated from each other by a distance of $4\sqrt{ab-a^2}$.
5) With $F_1$ as center draw a circle with radius equal to $2\sqrt{2(ab-a^2)}$. This will be equal to the distance between the two vertices of the hyperbola.
6) Let $Q$ be a point on the above circle. Draw lines $\overleftrightarrow{F_1Q}, \overleftrightarrow{F_2Q}$.
7) Draw the perpendicular bisector line of $\overline{QF_2}$. It cuts $\overleftrightarrow{F_1Q}$ at point $P$.
8) The locus of $P$ as $Q$ moves on its circle gives you the required rectangular hyperbola (red in Figure 1).

Method-II (Figure 2)

Figure 2

1) Mark point $C (-a, a)$, which will be the center of the hyperbola to be constructed.
2) Draw two perpendicular lines through $C$ respectively parallel to the x- and y-axes. These two lines are the asymptotes of the hyperbola.
3) Bisect the angle between these asymptotes to construct two perpendicular lines with slope $m= \pm 1$ passing through $C$.
4) On the line with slope $m=1$ mark the two foci of the hyperbola $F_1, F_2$, such that they are are equidistant from $C$ and separated from each other by a distance of $4\sqrt{ab-a^2}$.
5) On the line with slope $m=-1$ mark a sequence of points and draw a sequence of coaxial circles, which will pass through each of those points on this line and also the two foci $F_1, F_2$.
6) Each of these circles will cut the two asymptotes on total of 4 points. On each side separately draw segments joining the respective points of intersection of each circle on the two asymptotes (Figure 2).
7) The envelope of these segments is the desired hyperbola.

The said hyperbola has the equation:
$y=\dfrac{a(x+b)}{x+a}$

Figure 3

One observes that its upper branch cuts the y-axis at $(0,b)$ and by construction it is asymptotic to $x=-a, y=a$ (Green curve in Figure 3). But what else notable about this specific hyperbola? Note the intersections of the following lines with the hyperbola (Figure 3):
The line $x=a$ intersects the hyperbola at $y=\mu_A$, the arithmetic mean of $a,b$.
The line $x=b$ intersects the hyperbola at $y=\mu_H$, the harmonic mean of $a,b$.
The line $x=y$ intersects the hyperbola at $y=\mu_G$, the geometric mean of $a,b$.

Thus, this hyperbola can be considered a mean-generating curve where the means of the two numbers $a,b$ can be easily obtained by plugging certain nice values of $x$. It also furnishes the proof for the fact that the three primary means of two numbers lie on a single hyperbola between the point where the hyperbola cuts the y-axis, point $B(0,b)$ and the asymptotic line passing through $A(0,a)$. One notices that the $\mu_A$ cuts the segment $\overline{AB}$ in half. The geometric mean $\mu_G$ cuts $\overline{AB}$ below $\mu_A$ and the harmonic mean $\mu_H$ cuts it below $\mu_G$. This geometry allows one to define two further means, which are inversions (reflections) of $\mu_G$ and $\mu_H$ respectively on the $\mu_A$ line. We define these inversive means as $\mu_{Gi}$ and $\mu_{Hi}$. We discover that the intersection of the line $x=\tfrac{a^{3/2}}{\sqrt{b}}$ with the hyperbola generates $y= \mu_{Gi} = \tfrac{a^{3/2}+b^{3/2}}{\sqrt{a}+\sqrt{b}}$. Likewise, the intersection of $x=\tfrac{a^2}{b}$ with the hyperbola generates $y=\mu_{Hi}=\tfrac{a^2+b^2}{a+b}$. Thus, we have 5 means with the arithmetic mean as the central mean bisecting the segment $\overline{AB}$ with two means above it and two means below it (Figure 3).
In trying to understand these additional means coming from our hyperbola we learned of the work in this regard by Derrick Lehmer-II. He defined two mean generating functions. The first of them is (orange in Figure 3):

$y=\dfrac{a^{x+1}+b^{x+1}}{a^x+b^x}$

We notice that this function is a sigmoid curve having $y=a$ and $y=b$ as its asymptotes (Figure 3). We realized that this function of Lehmer generates the same means as the above-constructed hyperbola. When $x=0$ we get $\mu_A$; when $x=-\tfrac{1}{2}$ we get $\mu_G$; when $x=-1$ we get $\mu_H$. The two other means $\mu_{Gi}$ and $\mu_{Hi}$ respectively emerge at $x=\tfrac{1}{2}$ and $x=1$. Thus, the same symmetry as the with the hyperbola is recapitulated by this function (Figure 3).

Lehmer’s second mean-generating function is (red curve in Figure 3):

$y= \left(\dfrac{a^x+b^x}{2}\right)^{\frac{1}{x}}$

We observe that this curve is also bounded by the same asymptotes $y=a$ and $y=b$; however, it converges to them at very different rate. At $x=1$ it generates $\mu_A$; at $x=-1$ we get $\mu_H$ and it intersects the first mean-generating curve (Figure 3). At $x=2$ we get the $\mu_Q$, i.e. the quadratic mean or the root mean squared (RMS). This mean cannot be obtained by a nice value of $x$ in either the first mean-generating function or the mean hyperbola. What about the value of the second mean-generating curve at $x=0$? To get that we need to do is to evaluate the below limit:

$\displaystyle \lim_{x \to 0} \left(\dfrac{a^x+b^x}{2}\right)^{\frac{1}{x}}$

At first sight, it would seem that it is indeterminate because we get an ugly division by zero making it impossible to evaluate directly. Hence, this cannot be done and we have to take recourse to Johann Bernoulli’s method (commonly called L’Hospital’s method).

We first take the logarithm of the above expression to get:
$\dfrac{\log\left(\dfrac{ (a^x + b^x)}{2}\right)}{x}$

Then we differentiate the numerator and the denominator. The denominator is reduced as $\tfrac{d}{dx}x=1$. For the numerator we get:

$\dfrac{d}{dx}\left(\log\left(\dfrac{ (a^x + b^x)}{2}\right)\right) = \dfrac{a^x \log(a) + b^x \log(b)}{a^x + b^x}$.

We next take the $\lim {x \to 0}$ to get $\log\left( \sqrt{ab}\right)$. We then reverse the logarithm operation by exponentiation and evaluate the limit of the second mean-generating function at $x=0$ to be $\sqrt{ab}$. Thus, Lehmer’s second mean-generating function yields the geometric mean at $x=0$. Lehmer showed that there are thus two families of means of which one includes the inversions of the geometric and harmonic means on the arithmetic mean and the other which includes the quadratic mean or RMS. The only means that are common to both families are $\mu_A, \mu_G, \mu_H$ and the two mean generating curves intersect at $y=\mu_H$ (Figure 3). Thus, these 3 are the only fundamental means.

While these two families of means do not come together is there an operation where combining a fundamental and family-specific mean gives an interesting result:

Let $a_0=1, b_0=1$ and $n \in \mathbb{N}$,
$a_{i+1}=(\mu_Q(a_i, b_i\cdot \sqrt{n}))^2$
$b_{i+1}=(\mu_G(a_i, b_i))^2$

Then $\dfrac{a_{i+1}}{b_{i+1}} \to \sqrt{n}$
This gives us a means of obtaining rational convergents for a given irrational square root. For example if we use $n=6$ in the above procedure we get:

$\dfrac{7}{2}, \dfrac{73}{28}, \dfrac{10033}{4088}, \dfrac{200931553}{82029808}, \dfrac{2523338293565406}{1030148544608239}$

The last of these yields $\sqrt{6}$ correct to 11 places after the decimal point.

Also see:
1) Means and conics
2) The Hindu square root method

## A Political roundup August 15 2018

As I remarked to a friend, much of the stuff in (geo)politics which is relevant to us is what we have predicted before based on the relatively straightforward model of mleccha-marūnmattābhisaṃdhi i.e. the anti-heathen coalition of those infected by Abrahamistic meme-plexes and their secular mutations. What remains is just too uncertain to predict with our limited knowledge and ability. This might be the more interesting and perhaps the more dangerous part of our existence, but for most part we simply have to watch it unfold. That is part of the reason why we are not too motivated to write a lot about this on these pages. Nevertheless, we felt it might be appropriate to provide review of for another Independence Day has dawned and given that the elections are scheduled for next year.

So let us take a panoramic view by traveling back in time. The great demolition at Ayodhya on 6 December 1992 was a landmark even that hinted that the Hindus were no longer going to take things lying down. hence, we call upon fellow Hindus to observe a moment of remembrance for the men who gave up their lives to bring down that symbol of marūnmatta tyranny. This event and its aftermath created the first notable post-Independence ferment among the Hindus to gain political power and come out of the eclipse of the Gandhi-Nehru dynasty. Overseeing the demolition through benign inaction was the Prime Minister PV Narasimha Rao, an unassuming man of intelligence and a deep and enormous linguistic capacity from the Andhra country (notable was his facility in picking new human and computer languages). But PVNR’s main contribution was his reform of the Indian economy, which for the first time allowed Indians to experience the real middle class life. This boosted the growing confidence of Hindus and after a period of uncertainty culminated in the first NDA government under AB Vajapayee.

While as an adherent of the śruti I found his keeping of the hallowed title of Vajapayee without truly living up to it grating, he was a man of importance for the survival of the Hindu polity. His most important achievement was the conducting of the nuclear weapons tests totally surprising the mleccha and cīna enemies – this was a kind of boldness which had not been exhibited before. Next, he was able shepherd the nation through the economic blockade imposed by the evil mleccha-s and show that the economic platform, which PVNR had laid the foundations for, was indeed robust. In this period he had to handle the jihad of the marūnmatta-s in the Kargil war. The secularized Hindu army carried the day and showed that despite the mleccha attempts to curb them with sanctions they Hindus were not to be taken as walkovers. The mleccha-marūnmattābhisaṃdhi was at this point flummoxed by both the military victories and economic resilience of the Hindu nation and resorted to their usual trick of deploying the internal marūnmatta bomb – the legacy bequeathed on the Hindus by the faux Mahātman and Nehru. The most prominent among these attacks was the arson in Lāṭānarta, where emboldened marūnmatta-s burned down a train carrying Hindus killing many. The man who came to the fore in this dire situation was not ABV but his future successor the Lāṭeśvara from the tailakāra-jāti. It was due to him that the Hindus were able to show to the rākṣasa-like marūnmatta-s that they could not get away with their usual crimes. This was instrumental in bringing peace to the state and teaching a lesson to the marūnmatta in the only language he understands.

But then Hindus are renowned for dropping the axe on their own foot. Thus, lulled by the relatively good times the benign rule of ABV had brought, deciding not to take a long term vision, they instead facilitated the return of the Gandhi-Nehru dynasty via UPA-I. This government was led in proxy by the Helena of India who was a windfall for the mleccha-s, and she ably served as their agent in undermining the Hindu nation. To keep the Hindus in check, the marūnmatta-s were unleashed on a routine basis and this reached a culmination in the invasion of Mumbai on Nov 26 2008 by a mere band of 10 ghazi-s bringing back memories of Ikhtiyar al-Din Bakhtiyar Khalji, the hero of Bangladesh. As the UPA kept facilitating anti-heathen action, the NDA was unable to even put up a proper fight – ABV faded away into a vegetative state and his lieutenant, the aging saindhava LKA lost the plot in many ways (Note his egregious appointment of Kulkarni as an adviser). This only consolidated the grip of the Abrahamistic convergence against the Hindus. To worsen matters, it looked as though the cīna-s might launch a decisive strike on the enfeebled secularized Hindu army – things almost looked lost for the Hindu nation.

It was in such dire circumstances that the Lāṭeśvara became Dillīśvara in 2014. The mleccha-s having known his capacity from his restoring of the Lāṭānarta country greatly feared him. For 12 years they had kept him out their countries because they hated him as the symbol of what they hated most – the unconquered heathens of Hindustan who were the last great bulwark of that old and brilliant Indo-European heathenism, which they had either ground to dust or placed in museum closets. This heathenism was a rival worldview to all that the counter-religions of Abrahamism represented in their essence. Its very presence was a negation of the narratives of Abrahamism and it had to be exterminated. If there was one leader among the Indian masses who could give it a lifeline it was the Lāṭeśvara. Hence, they summoned all they could could in the form of their first responders to prevent his acquisition of power. They raised fake leaders like the broom-wielding koṭvāl of Delhi and the hazardous perpetuator of Gandhian blunders. But none of this worked in the long-run and the Lāṭeśvara took power.

Among the failures of the ABV-LKA duo was their inability to protect the homeland security of Bhārata from marūnmatta-s, śavasādhaka-s and other kaṇṭhaka-s. We had: the hijacking followed by the humiliating release of demonic ghazi-s; the train attack; the ghazwat on the Parliament. Further ABV was hobbled by the machinations of the possibly crypto-preta president KR Narayanan. This is where the Lāṭeśvara NM focused. Despite his inadequacies on the ideological front, his lieutenant Dobhal was able to be quite effective in curbing the violence of the marūnmatta-s. He also dented their ability to finance such operations by the secretive demonetization maneuver. His other lieutenant Rajnath Singh, again, despite his inadequacies, has done a reasonable job in prosecuting a vigorous assault against the secularized mutation of Abrahamism, namely the socialist terrorists of internal India. On the external front the he handled the confrontation with the belligerent Han by tackling the Doklam conflict really well. He also sent a message to the ghazi-s by directing the secularized Hindu army to launch “surgical strikes” against them. This had a psychological effect more than anything else – perhaps the Hindus were doing it for the first time since the Maratha punches delivered in their heydays.

On the economic front things are less clear, in part because I am not qualified to assess the matters too precisely. The general trends point to some success in this regard, especially in terms of curbing and to a degree reversing the damage caused by the UPA misrule. However, the demonetization and the goods and services tax, while done with good intentions and with long-term benefits in mind, might have affected many adversely; however, this is perhaps a matter of perception. On the external front, while NM’s foreign policy has been clearly more robust than that managed by the flaky Jaswant Singh under ABV, its success is not entirely clear. The handling of the Nuclear Suppliers Group matter does look like a failure to us. Finally, like a good Hindu king ought to do, NM has paid much attention to the issue of jana-kṣema and made major strides in this direction. He himself has led the svaccha-Bhārata-abhiyāna from front in attempt the impart the much-need civic sense to the undisciplined Hindus. The ground connectivity progress managed by Gadkari and railways by Prabhu/Goyal seem to be bright spots too. However, specific Hindu issues relating to temple administration, temple rights, animal husbandry and ground level prevention of the infection spread by dayi-s and evangelists have not been satisfactorily addressed.

That last point in the above discussion brings us to proverbial “elephant in the room”. During the UPA rule the mleccha-marūnmattābhisaṃdhi had systematically infiltrated the judiciary and the civil service with pro-Abrahamistic elements. As India is ruled by a democratic form of government, NM and his lieutenants are having a difficult time undoing the evil of this infiltration. This in part is the basis of their inability to do much for certain specific Hindu issues. Going forward, this could result in enormous damage to the Hindus. One only hopes that NM is doing his best to uproot these plants of the enemies completely revamp the affiliation of the administrative machinery. Beyond this, our enemies are certainly rattled by this Julian-like come back of the H under NM. But will he prove to be just that – a Julian?

The mleccha-s and the marūnmatta-s would definitely want it to end that way and NM is going to have to fight hard to keep the saffron flag fluttering. Among other things they would continue their strategy of raising false leaders, like the man of the broom, in various local settings to create a general ferment. The idea here is not to create a rival for NM but to damage the impact of his janakṣema. The śava-s would be used to harry the polity – e.g. the Sterlite Copper agitation in the Tamil country. They would also be tempted to deploy the sword arm of the marūnmatta to attempt something big and bloody. Their new agent in TSP Imran Khan might also be deployed for exerting pressure from western Mohammedan fragment of India. The agent of the Eastern Mohammedan fragment, the white-shrouded evil daughter of Bakhtiyar would be used to expand violence against the Hindus in the east. A political alliance would created around the Kangress and the dredges of the old UPA to attack the NDA like the various enemies of Sudāsa ganging against him. The cīna-s might try to fish in these troubled waters too but we suspect that Trump’s trade-war might come to the aid of the Hindus here. The mleccha deep-state itself is having problems due to the unexpected triumph of Trump. This might keep them busy and Hindus should exploit that. However, the techniques developed against Trump are and will be deployed against NM and the Hindus in India. Finally, the mleccha-s know that if they incite something big and bloody too early or too late it could end up consolidating the Hindus. Hence, we tend to think that the path they would prefer is that of a “million mutinies”. The malcontents from the depressed classes and other groups will be cultivated and deployed. The purchased “news traders”, much like the Piṇḍāri purchase by the English, would be used to amplify these disturbances. Reinforcements would be sent to the socialist terrorists. The idea would be to simultaneously take the sheen of the vikās and break up the unity among the Hindus. If NM returns to power then the stage would be set for bigger and more gory confrontations whose exact unfolding would depend on other geopolitical events of the future.

As a tailpiece we should mention that within hours of completing the body of this text the news reached us that one of the protagonists of the above story, ABV, has passed away after a long life — the last of those were hardly worth living. We have been critical of him on these pages on more than one occasion but now that he belongs the realm of the manes we have to acknowledge his historic role with gratitude. He certainly built the platform that Hindus can use to move forward and the one that the Lāṭeśvara is using. His end brings down the curtain on this turn in the meandering of the Hindu nation. Many Hindus have been spending the last four years doing nothing but bitching about NM. They are nothing but armchair activists who have not participated in electoral contests or organized protective services on the ground. One thing we can tell such folks is do your own little bit for the dharma at the grassroots level. Do not expect the prime minister or mahānta yogī Ādityanātha to be a quantum entity who is at multiple places at the same time performing bābāistic miracles. Also remember that certain parts of the country are seriously compromised with the infections and batting for them would not save them unless they are willing to chip in too. Finally, this period has also seen the death of the Dravidianist leader, the hater of brahma, and the corpulent woman who attempted to destroy the Kumbhaghoṇa maṭha. The Hindu leadership needs to act quickly in this window of opportunity to eradicate the evil of Dravidianism.

Posted in Politics |

## The geometric principles behind discrete dynamical systems based on the generalized Witch of Agnesi

Consider the family of curves defined by the equation following parametric equation

$x=\dfrac{1}{\sqrt{\pi\left(1-a\right)}}\left(\cos\left(\alpha\right)\left(t+1\right)+\dfrac{\sin\left(\alpha\right)\left(1+at+at^2+at^3\right)}{1+t^2}-1\right)$,

$y=\dfrac{1}{\sqrt{\pi\left(1-a\right)}}\left(-\sin\left(\alpha\right)\left(t+1\right)+\dfrac{\cos\left(\alpha\right)\left(1+at+at^2+at^3\right)}{1+t^2}\right)$

where $\alpha= \textrm{atan}(a)$ and $-1\le a \le 1$

It defines a family of probability distribution functions (PDFs): This can be seen from the above equations because

$\displaystyle \int_{-\infty}^\infty y\dfrac{dx}{dt} dt=1$

Figure 1

Examples of these PDFs are illustrated in Figure 1. One can see that when $a=0$ it is symmetric and reduces to a Cauchy distribution, which was discovered by Poisson and Cauchy. This distribution is famous for having “fat tails” and violating the Central Limit Theorem. A simple geometric example of this distribution is the following: Let the point $P=(0, 1)$ be the source of rays that are emanated at an equal angular separation from the adjacent ray. Then the density of the x-intercepts of the rays is equivalent to the PDF corresponding to $a=0$. The family of PDFs in Figure 1 are a generalization of the Cauchy distribution case. When $-1 \le a < 0$ the distribution is more peaked and skewed to the left. Conversely, when $0< a < 1$ the distribution is less-peaked and skewed to the right. When $a\to 1$, the distribution become more and more flat and uniform. They all have fat tails like the Cauchy distribution case. Are these other distributions encountered anywhere? While we have some suspicions that such distributions might occur in the path-lengths of searching behaviors of certain organisms, this is a question we have not been able to definitively answer. Nevertheless, the function behind the above PDFs generates a very interesting class of maps exhibiting chaotic behavior and this is what we shall discuss below.

The function which determines the shapes above PDFs can be written as,

$y=\dfrac{1+ax+ax^2+ax^3}{1+x^2}$

Figure 2

Examples of this curve as illustrated in Figure 2. One notices that it essentially produces unscaled versions of the above PDF curves, which are rotated about the point $(-1,0)$ by the angle $\alpha= \textrm{atan}(a)$. As a result, the x-axis of the PDF curves now becomes the asymptote of these curves which has the equation $y=ax+a$. When $a=0$, again this curve reduces to the shape-determining curve of the Cauchy distribution, i.e. the Witch (of Agnesi). For $a=1$ it becomes identical with the $45^o$ asymptotic line.

We then consider the following mapping procedure:

1) Take (equally-spaced) points on a circle with center at origin and radius $r$. For reasonable aesthetics we commonly take $r=5$

2) Then subject each of these points $P_0=(x_0, y_0)$ to the iterative mapping:

$x_{n+1}=by_n+f(x_n)$

$y_{n+1}=-x_n+f(x_{n+1})$

Here, $b$ is a constant that is taken as 1 or a value close to it; $.99 \le b \le 1.001$ produces the best aesthetics.

$f(x)=\dfrac{1+ax+ax^2+ax^3}{1+x^2}$, i.e the above shape-determining function of the above PDFs.

3) Plot the orbits of all $P_0=(x_0, y_0)$ on the circle under this map.

Figure 3

This mapping procedure is illustrated with an example in Figure 3. We start with a circle shown in red, $r=5$, on which 360 equally spaced starting points $P_0$ lie. We then use $P_0$ to sample the curve $f(x)$ with $a=0.525$ twice, as indicated in the above map, to get the next point. The curve $f(x)$ is shown in green. Its unrotated equivalent, which determines the shape of the corresponding PDF is shown in blue. For this map we take $b=1$ and the iterate it for 250 times. The 250 points comprising the orbit of one $P_0$ are then plotted. This is repeated for all the 360 $P_0$s to get the map show in black in Figure 3. Superimposed on the map is the orbital path of a particular $P_0$ (shown in cyan) for the 250 iterates. The evolution of the orbit is indicated by orange triangles whose three vertices are respectively the two points sampled on $f(x)$ (shown in red and green on the curve) for a given mapping step and the third is the corresponding point on the map. One can see that the map initiated with this particular $P_0$ samples only 4 specific regions of the $f(x)$. Consequently, the orbit results in the six closed loops. Other $P_0$ might sample $f(x)$ more extensively and uniformly, resulting in a more widely dispersed or chaotic orbit. Thus, the union of the orbits of all the initiating $P_0$s on our circle results in a map with fractal structure (Figure 3). The chaotic nature of map becomes immediately apparent from the high degree of sensitivity of the orbits to the starting points.

Having worked out this map, we realized that orbits of single points under above mapping with appropriate change of coordinates are related to the map published in 1980 by Gumowski and Mira (G-M). This was among the first strange attractors for which we, like many others before and after us, had written code for (though we must mention that to date we have not read Gumowski and Mira’s paper). It was then than we became curious about the rotational periodicity exhibited by some examples of the G-M map resulting in rotational quasi-symmetry. We were able to finally provide the rationale for this geometry when we discovered the map which we are currently discussing. The maps discussed here show a strong tendency for rotational periodicity — for example, the above example shows a 6-fold periodicity — what we term the n-ad structure. Indeed, these maps are better for understanding the rotational periodicity than the original G-M maps because all the powers of $x$ in $f(x)$ have the same sign in this map.

In order understand the rotational periodicity in the map we shall have to consider two cases of it separately: first, the case when the radius of the circle on which the starting $P_0$s lie is large; this is empirically taken as $r \ge 4$. The second case is when it is small; this is take $r< 4$. To examine the former case let us consider the cases in the following figures.

Figure 4. Here $b=0.9985$, $r=5$ and $a$ is the sequence of the first few irreducible vulgar fractions, both positive and negative, i.e. the first few fractions of the form $\tfrac{p}{q}$, where $p$ and $q$ are mutually prime. Each $P_0$’s orbit is plotted in 1 of 7 different shades of blue in this figure and all subsequent ones. Thus, we keep cycling through these 7 shades for after every 7th $P_0$.

Figure 5. Here $b=0.9985$, $r=5$ and $a$ is the sequence of further positive irreducible vulgar fractions.

Figure 6. Here $b=0.9985$, $r=5$ and $a$ is the sequence of further irreducible vulgar fractions which are negatives of the above.

Few features become immediately apparent:

1) The maps exhibit strong rotational periodicity. E.g. for $a=0$ we get a tetrad structure.

2) At certain values there is considerable rotational symmetry e.g. $a=\tfrac{1}{6}, a=-\tfrac{3}{4}, -\tfrac{1}{7}$.

3) The extreme cases of this are when the map collapses to a completely symmetric radial structure for $a=\tfrac{1}{2}, -\tfrac {1}{2}, -1$

4) As the value of $a \to \pm 1$ the aspect ratio of the map becomes lower and lower along the y-axis and converges to a straight line-segment at the limiting values of $a$.

5) For values of $a$ close to 0 on either side we get maps with aspect ratio closer to 1.

These observations allowed us to develop a formal description of the geometric principles behind these maps:

1) Since $-1\le a \le 1$ it can be considered the cosine of an angle $\gamma$. Thus, we can write:

$\textrm{acos}(a)=\gamma=\dfrac{2\pi}{n/m}$, where $n,m$ are mutually prime integers.

This fraction $\tfrac{n}{m}$ is the shape-determining fraction of the map (Figure 7). When the radius of our starting circle $r \ge 4$, then the basic rotational periodicity of the map is determined by $n$ and it shows an n-ad structure(Figure 7). Thus, when $\tfrac{n}{m}=3$, $\gamma=\tfrac{2\pi}{3}$. Hence, $a=\cos\left(\tfrac{2\pi}{3}\right)=-\tfrac{1}{2}$. Thus, as we see in Figure 4, $a=-\tfrac{1}{2}$ has a triad structure corresponding to $n=3$. Similarly, when $\tfrac{n}{m}=4$, $a=\cos\left(\tfrac{2\pi}{4}\right)=0$; thus, $a=0$ results in a map with tetrad structure corresponding to $n=4$ (Figure 4).

2) If $\gamma=\textrm{acos}(a)$, then this angle $\gamma$ can be taken to be the interior angle of a polygon inscribed in a circle (Figure 7). Now, flatten this circle into an ellipse such that the eccentricity of the ellipse is $|\tan(\alpha)|=|a|$ (Figure 7). Thus, $0\le |a| \le 1$ describes ellipses within which the map is bounded, which span the entire range from one close to a circle when $|a| \to 0$ to one close to a segment when $|a| \to 1$. The polygon inscribed in the original circle correspondingly gets flattened into a polygon inscribed in the ellipse (Figure 7). This polygon describes both the n-ad structure of the map as described above, whereas the eccentricity of its bounding ellipse describes the aspect ratio of the maps under consideration.

Figure 7. The original generalization of the Witch is in red. The free-rotating version used in the current maps is in green. The polygon inscribed in the circle defined by the $\angle{\gamma}$ is shown in blue dotted segments. The ellipse whose eccentricity is determined by the $|a|=|\tan(\alpha)|$ is shown in red, with the final polygon inscribed in it in violet.

3) In order to illustrate the above principles and describe the finer intricacies of the structure of the map beyond the “coarse features” accounted for by the above rules we need to examine several specific examples. Since cases like $a=\pm \tfrac{1}{2}$ result a “collapse” of the map to a radial structure (Figure 4), to explain in its entirety the role of the fraction $\tfrac{n}{m}$ in determining map shape we add a small positive $\epsilon$ to define $a=\cos\left(\tfrac{2\pi}{n/m}\right)+\epsilon$.

Figure 8. In this set of maps the radius of the starting circle $r=5$, $b=0.9985$, $\epsilon=.0025$. $a$ is expressed as sequence of cosines shown with the corresponding angles and $\tfrac{n}{m}$ above each map in this and the subsequent figures.

In the first 7 cases, $\tfrac{n}{m}=3,4,5,6,7,8,9$. According to the above described geometric rules we find that the map adopts a triad, tetrad, pentad…nonad structure with decreasing aspect ratio (Figure 8).

When $\tfrac{n}{m}=\tfrac{9}{2},\tfrac{11}{2},\tfrac{13}{3},\tfrac{15}{4},\tfrac{17}{4}$, we see an obvious 9-, 11-, 13-, 15-, 17-ad structure but with much higher aspect ratios than what would be expected if the denominator were 1. This is because the $a=\cos\left(\tfrac{2\pi}{n/m}\right)+\epsilon$ results in eccentricities closer to 0. Thus, choosing such $\tfrac{n}{m}$ allows you to obtain n-ad structures at higher aspect ratios (Figure 8).

Further, note the internal structure of the map, i.e. the region close to the center, setting aside the above described patterns for the more peripheral structure described above. The following $\tfrac{n}{m}$ are notable:

For $\tfrac{9}{2}$ we get a tetrad interior;

For $\tfrac{11}{2}$ — pentad interior;

For $\tfrac{13}{3}$ — tetrad interior;

For $\tfrac{15}{4}$ — triad interior;

For $\tfrac{17}{4}$ — tetrad interior (Figure 8).

This indicates that for maps with high aspect ratio, the n-ad structure close to the center is determined by the number $k_1=\left \lfloor \tfrac{n}{m} \right \rfloor$. In some cases, the value of $k_1$ can dominate the shape of the map. For example, when $k_1=\left \lfloor \tfrac{17}{5} \right \rfloor=3, \left \lfloor \tfrac{19}{3} \right \rfloor=6$ we see that $k_1$ is the dominant n-ad of the shape. However, keeping the primary n-ad rule in each of these cases, the periphery shows 17 and 19 spikes respectively.

Next, note the structure of the map where $\tfrac{n}{m} =\tfrac{17}{3}$. At the periphery we see 17 “spikes” keeping with the basic structure determining number $n=17$. Close to the center, we see the pentad structure keeping with the principle described above: $\left \lfloor \tfrac{17}{3} \right \rfloor=5$ However, the overall map is dominated by a hexad structure — i.e., the 17 spikes are arranged into a hexad pattern. This hexad pattern is also seen in the structure of middle layer of the map. This gives us another principle: Especially for larger $n$ if $k_2=\textrm{round}\left(\tfrac{n}{m} \right) > \left \lfloor \tfrac{n}{m} \right \rfloor$, then we can get a further layer of structure with a n-ad structure determined by $k_2$. This can again be seen in the case of $\tfrac{19}{4}$, where $k_2=5$. This results in a pentad dominance with a middle pentad layer between the outer 19 spikes and inner tetrad structure (Figure 8).

4) Finally, we consider the case when the starting circle has $r<4$. In this range we observe that the maps do not entirely obey the above rules. We see a progressive “step-down” in the n-ad structure for values other than $\tfrac{n}{m}=3$. This is illustrated with examples in Figure 9.

Figure 9. Here $r=\tfrac{1}{\phi}$, where $\phi$ is the Golden ratio; $b=0.9985$; $\epsilon=.001$.

We observe right away that for $\tfrac{n}{m}=4$ the structure is only incipiently tetrad. For $\tfrac{n}{m}=5, 6, 7, 8$, on the other hand the n-ad structure of the map is respectively a tetrad, pentad, hexad and heptad. Thus, the rotational periodicity rather being equal to $n$ comes down to $n-1$. For the other fractional $\tfrac{n}{m}$ we see only the $k_1$ structure, where $k_1=\left \lfloor \tfrac{n}{m} \right \rfloor$. We do not see the structure derived from $n$ in $\tfrac{n}{m}$. In the range $\tfrac{1}{\phi} \le r \le 4$ we see a gradual recovery of the structure that clearly manifests at $r\ge 4$. We illustrate this in Figure 10.

Figure 10. Here $r=2$; $b=0.9985$; $\epsilon=.001$.

We observe that for $\tfrac{n}{m}=4, 5, 6, 7, 8, 9, \tfrac{9}{2}$ the typical n-ad structure determined by $n$ as seen in the maps where $r\ge 4$ is restored. However, for the remaining $\tfrac{n}{m}$, $k_1=\left \lfloor \tfrac{n}{m} \right \rfloor$ is the dominant determinant of the map’s rotational periodicity (Figure 10) and we have to wait till $r \ge 4$ for a n-ad structure determined by $n$ to emerge. The emergence of the clear n-ad structure only at higher radius values combined with a more complex chaotic behavior at lower radius values is also seen in the well-known Zaslavsky map based on the Hamiltonian of a kicked rotor. However, it is notable that maps with these values of $\tfrac{n}{m}$ at low $r$ values have some complex internal structures that are difficult to account for by any of these rules. For example, consider $\tfrac{n}{m}=\tfrac{13}{3}$ (Figure 10). In each of the 4 outer projections of the tetrad structure we see an internal hexad attractor. Inside that hexad attractor we see a further pentad attractor. Within the central tetrad element of this map we see a triad excluded space. Finally, in the border of this central tetrad element we see a 7-fold attractor! This is reminiscent of the structures seen in the famous Standard map of Chirikov.

In conclusion, as we saw with an earlier map that we had discovered and the maps derived from certain Hamiltonians of kicked rotors/oscillators, there is a hidden role for the irreducible vulgar fractions in determining the shape of the map. While it is easy to see that the rational numbers are distributed uniformly, that is not the case if they are ordered as per their rank based on their numerators and/or denominators. Examples of such fraction sequences are those described by Moritz Stern, Farey and Brocot. Even as in the fractal structure defined by these fraction sequences, the rank of the fraction is central to the definition of the map’s rotational periodicity in current case. The higher ranked $\tfrac{n}{m}$, i.e. the low magnitude integers like 3, 4, 5, 6 dominate the n-ad shape of the map relative the other rational numbers in their interstices. However, those interstitial values interact with these dominant values to give layers of complex structure to the map.

Posted in Scientific ramblings |

## Reflections on our journey through the aliquot sums and sequences

The numerology of aliquot sums and perfect numbers
The numerology of the Pythagorean sages among the old yavana-s is one of the foundations of science and mathematics as we know it. One remarkable class of numbers which they discovered were the perfect numbers — teleios as they termed it. What are these numbers? Let $n$ be a number and $d_j$ be all its proper divisors, i.e. those divisors of $n$, which are less than $n$. Then we can define an arithmetic function known as the aliquot sum $s(n)$ thus,

$\displaystyle s(n)=\sum_{j=1}^k d_j; d_k

For example, let us consider the number 10. Its divisors are 1, 2,5,10. Its proper divisors are 1, 2, 5. Hence, $s(10)=1+2+5=8$. Now, if $s(n)=n$ then $n$ is called a perfect number (termed pūrṇāṅka in Jagannātha’s Sanskrit edition of Euclid). From Figure 1, we can see that the numbers 6 and 28 are perfect numbers. The Pythagorean interest in them also becomes apparent from the fact that these numbers are associated with certain natural periodicities that have an old Indo-European significance. This becomes clear from their occurrence even in old Hindu tradition. The number 6 is associated with the 6 seasons in brāhmaṇa-s like the Śatapatha-brāhmaṇa and encoded into the śrauta altar. Similarly 28 is also encoded into the altar according to the Śatapatha-brāhmaṇa and corresponds to the count of the nakṣatra-s or days of the lunar month. The number 28 is also used in Vaidika tradition as one of the prescribed counts for the japa of the Savitṛ gāyatrī

Figure 1

Returning to the arithmetic, with our above example of $s(10)$, we can see that $s(n)n$. Such numbers are called abundant numbers. Now, a subset of the proper divisors of 20 add up to 20 (Figure 1). Hence, it is also a semi-perfect or a pseudo-perfect number. There is a rare set of abundant numbers for which no subset of their proper divisors add up to them; they are known as weird numbers. For example, $s(70)=74$; hence, it is an abundant number. However, no subset of its proper divisors, 1, 2, 5, 7, 10, 14, 35 add up to 70. Hence, 70 is a weird number. The next weird number is 836. Thus, due their rarity majority of abundant numbers are semiperfect numbers.

One of the high points of yavana brilliance was the discovery of a general formula for perfect numbers. In Book 9, proposition 36 Euclid states:

“ean apo monados hoposoioun arithmoi hexēs ektethōsin en tē diplasioni analogia, heōs hou ho sumpas suntetheis prōtos genētai, kai ho sumpas epi ton eskhaton pollaplasiastheis poiē tina, ho genomenos teleios estai.”

If as many numbers as we please beginning from an unit be set out continuously in double proportion, until the sum of all becomes prime, and if the sum multiplied into the last make some number, the product will be perfect.

In modern terms we can lay it out thus: If the sum of the series of the powers of 2, where the power are integers $\ge 0$, is a prime number then the product of that prime with the last power of 2 in the series is a perfect number:

$q=\displaystyle \sum_{j=0}^n 2^j$

If $q$ is a prime then $q\cdot 2^n$ is a perfect number. Let us take the first few examples.
$2^0+2^1=3\Rightarrow 2^1\times 3=6$
$2^0+2^1+2^2=7\Rightarrow 2^2\times 7=28$
$2^0+2^1+2^2+2^3=15 \Rightarrow$ Not a prime
$2^0+2^1+2^2+2^3+2^4=31\Rightarrow 2^4\times 31=496$
$2^0+2^1+2^2+2^3+2^4+2^5=63\Rightarrow$ Not a prime
$2^0+2^1+2^2+2^3+2^4+2^5+2^6=127\Rightarrow 2^6\times 127=8128$
After this point the perfect numbers become much less frequent and large.
$2^0...2^{12}=8191 \Rightarrow 33550336$
$2^0...2^{16}=131071 \Rightarrow 8589869056$
$2^0...2^{18}=524287 \Rightarrow 137438691328$

Another way of expressing this is thus: Given a prime number $p$ if $M_p=2^p-1$ is also a prime then $P=2^{p-1}\cdot(2^p-1)$ is a perfect number. The corresponding prime numbers $M_p$ are today famous as the Mersenne primes. For all $p <100$, the following $p$ yield $M_p$: 2, 3, 5, 7, 13, 17, 19, 31, 61, 89
The corresponding $M_p$ are:
3
7
31
127
8191
131071
524287
2147483647
2305843009213693951
618970019642690137449562111

The corresponding perfect numbers are:
6
28
496
8128
33550336
8589869056
137438691328
2305843008139952128
2658455991569831744654692615953842176
191561942608236107294793378084303638130997321548169216

These primes $M_p$ start getting huge and rare rapidly. We computed the above in a few seconds with a modern programming language and computer. However, some of them mark historic feats of arithmetic computation by the unaided human brain in the pre-computer era. For instance, Leonhard Euler computed the 8th $M_p$ (10 digits) and the corresponding perfect number. The Russian village mathematician I.M. Pervushin went further by computing the 9th of these numbers ( $M_p= 19$ digits). With the Great Internet Mersenne Prime Search we now have 50 of them and the corresponding perfect numbers. This also helps us to see why the $M_p$ always end in 1 or 7 and the corresponding perfect numbers end in 6 or 28. That the above are the only even perfect numbers was established by Euler. Unlikely as they seem, there has been no formal proof to show that no odd perfect numbers exist. The modern efforts also show that the perfect numbers are rarer than what the yavana sages thought them to be. The old Pythagorean numerologist Nicomachus states:

“It comes about that even as fair and excellent things are few and easily numerated, while ugly and vile ones are widespread, so also the abundant and deficient numbers are found in great multitude and irregularly placed – for the method of their discovery is irregular – but the perfect numbers are easily enumerated and arranged with suitable order; for only one is found among the units, 6, only one among the tens, 28, and a third in the rank of the hundreds, 496 alone, and a fourth within the limits of the thousands, that is, below ten thousand, 8128.”

The above statement suggests that Nicomachus might have thought that for each decade there is one perfect number. Alternatively, he might have simply stopped counting with the greatest perfect number he knew. That it was the former is strengthened by Platonic siddha (to use Gregory Shaw’s term), Iamblichus, even more explicitly claiming that there is one perfect number per decade. That, however, is clearly wrong as the next perfect number 33550336 is in the crores. In any case the modern confirmation of the real rarity of perfect numbers vindicates the philosophical analogy drawn from these numbers by the yavana siddha — the scarcity of truly perfect things as opposed to the profusion of the supernumerary and the deficient. With respect to the easy enumeration of the perfect numbers Nicomachus and Iamblichus likely meant Euclid’s proposition. This rarity of perfect numbers indicates that the sum of the reciprocals should converge to a constant:

$\displaystyle C_P=\sum_{j=1}^n \dfrac{1}{P_j}= \dfrac{1}{6}+\dfrac{1}{28}+\dfrac{1}{496}+\dfrac{1}{8128}... \approx 0.20452014283893...$

It would be immensely remarkable if this constant turns up somewhere in nature.

Abundant numbers
In any case, this $C_P$ provides a bound for the distribution of abundant numbers and thus, brings us to the point of whether Nicomachus’ statement regarding the abundant numbers is really so? This was what we set out to investigate in our youth armed with a rather meager arithmetic knowledge and a computer. Computing the first few abundant numbers yields a sequence like below:
12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216…

A few rules become immediately apparent:
1) if $P_j$ is a perfect number then $k\cdot P_j$ is an abundant number for all $k \ge 2$. Thus, $P_1=6$ initiates a line of abundant numbers 12, 18, 24, 30…. $P_2=28$ initiates 56, 84, 112…

2) We notice that beyond these a few abundant numbers emerge in the interstices e.g. 20, 88 etc. A notable class of these emergent abundant numbers are of the form $2^k\cdot p_j$, where $p_j$ is the $j^{th}$ odd prime and $k \ge 2$. The first few of these are tabulated below.

These are all shown as red points in panel 1 of Figure 2. One notices that for each jump of $k$ we get a jump in these abundant numbers. Of these, some like 12, 56 and 992 are already accounted for as doubles of the perfect numbers 6, 28 and 496 but the rest are distinct. Their multiples, as with the perfect numbers, are also further abundant numbers.

3) A few further abundant numbers emerge in the interstices, which have a more complex description. They found new lineages of abundant numbers as their multiples continue to be abundant numbers. The first of these is $70=2\times 5 \times 7$. 70 is the product of successive odd primes 5, 7 with 2 leaving out 3, which would yield already accounted-for abundant numbers in a product with 2 via the perfect number 6. Another such is $2002=2\times 7 \times 11 \times 13$. Here 5 is left out because that will again result in already accounted-for abundants in a product with 7. Likewise, leaving out 7, we have $1430=2\times 5\times 11\times 13$. For further numbers in this category we need the next power of 2, e.g. $9724=2^2 \times 11\times 13\times 17$.

4) Till $n=1000$ odd abundant numbers are very rare. The $A[232]=945$ is the first odd abundant number and till $n=200000$ there are only 391 odd abundant numbers as opposed to 49090 even abundant numbers. Of these first 391 odd ones, 387 are divisible by 15. The remaining 4 which are not namely 81081, 153153, 171171, 189189 are divisible by 9009. All odd abundant numbers necessarily need to have the product of at least 3 distinct odd primes as a divisors. As with other abundant numbers, the multiples of odd abundant numbers are also abundant numbers. Based on the separation between the odd abundant numbers, we observed that the first of them 945 is the first of a lineage of odd abundant numbers, which are generated by the formula: $A_o=3 \times (315+210k)$, where $k=0,1,2...51$. Similarly, we observed that a related formula $A_o=11 \times (315+210k)$, where $k=0,1,2...192$ produces a continuous run of 193 odd abundant numbers starting from 3465. After $k=51$ and $k=192$ respectively these formulae do not necessarily produce abundant numbers; nevertheless numbers emerging from these rules continue to remain enriched in odd abundant numbers. Of course, the other emergent odd abundant numbers might have further less-apparent rules. In any case, it appears the difference between two successive odd abundant numbers is most of the times divisible by $2 \times 3 \times 5 =30$ and always divisible by the perfect number 6.

Thus, unlike what the yavana siddha-s thought, the abundant numbers are not entirely unruly. They have complex patterns, but can be described by some rules. The biggest bulk of them are children of perfect numbers — like mortal descendants of the immortal gods. This leads us to the growth of the sequence of abundant numbers, which exhibits striking regularity (Figure 2).

Figure 2.

Given that the sum of the reciprocals of the perfect numbers converges to a constant $C_P$ and given what we have seen above regarding the abundant numbers, they should grow at the upper bound by the equation $y=\tfrac{x}{C_P} \approx 4.89x$ (the dark red line panel 1 of Figure 2). However, since the progeny of perfect numbers are not the only abundant numbers and several emerge by the other rules mentioned above $A[n]$ should grow at a lower rate. During our initial foray into abundant numbers, by empirical examination we thought this growth rate might be exactly 4 (Figure 2, panel 1, light blue line). However, with better computers and computation of 49481 abundant numbers, we later realized that the rate was actually slightly more than 4 (Figure 2, panel 1, dark green line). Another way of visualizing the same is by defining the density of these numbers as the ratio of the number of abundant numbers $\le$ to a certain number $n$ to $n$,

$D= \dfrac{\#A\le n}{n}$

This is shown in Figure 2, panel 2, where we see $D$ converging to a value between 0.248 and 0.247 (red and blue lines). This has been a topic of deep investigation in modern mathematics, starting with the likes of Sarvadaman Chowla and Paul Erdős, and indeed the above has proven to be the approximate bound of the density of abundant numbers. We were quite pleased to have semi-empirically arrived at it for ourselves. Thus, even as the odd and even numbers are distributed in a 1:1 ratio, after initial jitters the ratio of deficient to abundant numbers converges to a constant of $\approx 4.04$ with new lineages of abundant numbers constantly emerging in the interstices of the established ones to maintain a linear growth at this rate.

The basic features of the aliquot sequences
The process of obtaining aliquot sums of a number $n$ can be applied iteratively (the aliquot map): $s(n)=n_1 \rightarrow s(n_1)=n_2 \rightarrow s(n_2)=n_3...$. The sequence $ali(n)=n, n_1, n_2, n_3...$ is called the aliquot sequence of $n$. Thus, if we start with $n= 20$ we get the sequence: 20, 22, 14, 10, 8, 7, 1, 0. More generally, by computing aliquot sequences for all numbers from 1:1000, we observe the following:
1) The number 1 converges in two steps to 0.
2) All primes converge in 3 steps, $p, 1, 0$.
3) Perfect numbers converge in 1 step to themselves.
4) Certain numbers converge to a perfect number. E.g. 25, 95, 119, 143, 417, 445, 565, 675, 685, 783, 909, 913… converge to 6; 608, 650, 652, 790 converge 496. We observe that these numbers have given the appellation “aspiring numbers”.

5) Notably, 220 converges to 284 and 284 converges to 220. $ali(562)=562, 284, 220$. Thus, if from whatever starting point if one reaches 284 or 220 one cycles between them. Such a pair is termed a pair of amicable numbers and was probably known to Platonists as indicated by the heathen Sabian from Harran, Thabit ibn Kurra’s knowledge of these numbers. Thabit discovered a rule to produce a lineage of these numbers, likely drawing on the rule to generate even perfect numbers: Let $a=3\cdot 2^{n-1}-1, b=3\cdot 2^n-1, c=9\cdot 2^{2n-1}-1$ and $n\ge 2$. If $a,b,c$ are primes then we can compose the amicable pair as $a_1=2^n \cdot ab, a_2= 2^n \cdot c$. For $n=2$ we get the pair 220, 284; $n=4$ produces 17296, 18416; $n=7$ produces the pair 9363584, 9437056. However, there are many more amicable numbers between these pairs which cannot be captured by this rule. For example, one can computationally show that 1184, 1210 are an amicable pair. Long after the days of Thabit, starting from Euler down to our times further rules have been found to capture more amicable pairs.

6) If perfect numbers are auto-cycles under the aliquot map, the amicable numbers can be considered 2-cycles. While we do not know if all higher cycles exist under the aliquot map, we can computationally find some of them. For example the pentad 12496, 14288, 15472, 14536, 14264 constitute a 5-cycle. Any number in this pentad will cycle through these values. Even more remarkable is this sequence 14316, 19116, 31704, 47616, 83328, 177792, 295488, 629072, 589786, 294896, 358336, 418904, 366556, 274924, 275444, 243760, 376736, 381028, 285778, 152990, 122410, 97946, 48976, 45946, 22976, 22744, 19916, 17716. These constitute a 28-cycle and any number in this 28-ad will cycle between these values. The lowest member of a 2- and greater cycle is always an abundant number.

7) Still other numbers will eventually converge to an odd prime and via that prime reach 0. There is a remarkable pattern in terms of the preference of the prime via which convergence occurs (Figure 3). 43 is the most preferred prime for convergence for $n=1:1000$ (any links to Heegner numbers or twin primes?). We are not clear as to whether this trend survives with increasing $n$.

Figure 3

8) If we leave out the starting $n$ will every other integer be reached by rest of the aliquot trajectory of some $n$? We can easily provide the answer to this question as no. For example, $2=1+1$. You cannot have two 1s as proper divisors, hence 2 can never be reached from any other number under the aliquot map. Similarly, $5=4+1=2+3$. But each of these sums leading to 5 cannot be a $s(n)$ because we would leave out divisors 2 and 1. Thus, 5 can never be reached from any other number under the aliquot map. On the other hand, a number $n=p+1$ (where $p$ is a prime number) is never untouchable because $s(p^2)=p+1$. Similarly, a number $n=p_o+3$ (where $p_o$ is an odd prime) is never untouchable because $s(2p_o)=p_o+3$. Nevertheless, some numbers are not reached easily until one of the above configurations is encountered for the first time. For example, $ali(1369=37^2)= 1369, 38, 22, 14, 10, 8, 7, 1, 0$ contains 38=37+1 for the first time. Similarly, $ali(2209=47^2)=2209, 48, 76, 64, 63, 41, 1, 0$ contains 48=47+1 for the first time. But the number 52 lies a in sweet spot as as neither 51=52-1 nor 49=52-3 are primes and is untouchable from any other number under the aliquot map. These are some obvious examples of untouchability and touchability; Erdős has shown that ultimately there are an $\infty$ of cases.

Patterns in the convergence length of aliquot sequences
Are these the only convergence patterns seen in the aliquot sequences? Is there any further pattern to the number of iterations needed to converge? To address these questions we can define a further sequence $f[n]$ where each element is the number of iterations taken by the integer $n$ to converge. The plot of $f[n]$ for the first 1000 elements is shown in Figure 4.

Figure 4

The picture is quite remarkable — the majority of values are rather pedestrian but from time to time there are huge eruptive values. The perfect numbers ( $f[n]=1$), 1 and the primitive aspiring numbers, e.g. 25 ( $f[n]=2$), and primes ( $f[n]=3$) form the lowest values of $f$. $f[n]=3$ might also be reached by certain secondary aspiring numbers like 95 or 119. Of these the primes are the most common. It is quite obvious that even numbers have significantly longer convergence paths than odd numbers ( $p=10^{-10}$; Figure 5). We also observed that on an average the abundant numbers have significantly longer convergence paths than the remaining numbers ( $p=5.3 \times 10^{-9}$ for $n=1:1000$). This holds even after the primes are removed from the non-abundant numbers ( $p= 1.9 \times 10^{-8}$ for $n=1:1000$). Similarly, if we compare even abundant numbers and even non-abundant numbers, which occur in the roughly similar counts, the abundant even numbers still have significantly longer trajectories the non-abundant even numbers under the aliquot map ( $p=3.2 \times 10^{-7}$). Due to the relative rarity of odd abundant numbers, these effects are likely to persist beyond $n=1000$. The basic statistics for the convergence trajectory lengths for different types of numbers from $n=1:1000$ are tabulated below and summed up in Figure 5.

$\begin{tabular}{|l|r|r|r|} \hline Number & Median & Mean & Max \\ \hline Even & 12 & 21.7 & 749 \\ \hline Odd & 4 & 4.9 & 17 \\ \hline Abundant & 15 & 34.04 & 749 \\ \hline Non-abundant & 5 & 6.76 & 25 \\ hline Non-abundant Non-p & 7 & 7.84 & 25 \\ \hline Even-abundant & 15 & 34.16 & 749 \\ \hline Even-non-abundant & 10 & 10.36 & 25 \\ \hline \end{tabular}$

Figure 5

During our initial foray into the first 200 terms of this sequence it appeared that certain integers, like 138, never converged under the aliquot map. Going on till $n=1000$ more such terms appeared; however, better computation showed us that the aliquot sequences for some of these $n$ indeed converge after a large number of steps after growing to monstrous values. It was then that we learned that the mathematicians have been studying this problem quite a bit computationally starting with Derrick Lehmer-II (the son of the father-son pair of arithmeticians), one of the doyens of computational mathematics. He was the first who, back in his days after what was a tough fight, showed that $ali(138)$ indeed converges. For us, the hardest nut to crack for $n=1:1000$ was $f[840]$. It simply would not complete in our laptop at home even after running it for over a day with an efficient divisor finding function. Nor did it complete on our primary work station used for most of routine computations. Then we had to bring out our Indrāstra, a 120 core machine that we use for big things. For that we had to first write a multi-threaded version, which was run on 100 cores of this machine and it completed $f[840]$ overnight. We term these large $f[n]$ as the monster numbers. For $n=1:1000$ there are 18 of them, which come in the below-tabulated families, all of which are abundant numbers descending from 6. After the first few terms the members of each family follow the same trajectory to convergence. The final prime via which they converge is termed $p_c$

Figure 6. The evolution of the first members of each family. $f_{840}$: dark red $f_{138}$: cyan; $f_{702}$: dark blue; $f_{720}$: dark green; $f_{858}$: hot pink; $f_{936}$: gray.

One notices that families $f_{138}$ and $f_{858}$ have a higher-order relationship because they converge via 59. However, we keep them as different families for, barring the last 8 elements of $ali(n)$, they followed very different trajectories (Figure 6). Their $p_c$ is the 6th most common $p_c$ for $n=1:1000$, with 45 numbers converging via it (Figure 3). However, 12 of them being monster values it is one $p_c$ over-represented in the aliquot sequences attaining monster values. In contrast, the most common $p_c$ in this range 43 (Figure 3) has only a single monster value (936) in this range. $ali(840)$ shows the most monstrous behavior in this range (Figure 6). It reaches a maximum value of the behemoth number: 3463982260143725017429794136098072146586526240388. I felt please on reaching this number and experiencing the magnificence of $ali(840)$ for myself. After that it remains in the high range for while with two prominent peaks before converging after 749 steps. Such a behavior, with a multiplicity of peaks before convergence, is also seen in the other monstrous families (Figure 6).

We wondered if there was any features of these sequences, which allowed their eruptive growth before finally converging. We observed the following:
1) They start as even abundant numbers, which means that they are statistically in the general group of numbers which are going to have longer trajectories to convergence.
2) Their trajectories tend to remain even, which again increases their probability of having a long convergence path. By specifically studying the first major eruptive case $ali(138)$, we see that it has a convergence trajectory of length 179. It remains even from $ali(138)[1]$ to $ali(138)[175]$. Thus, remaining even is a key factor for a long convergence path.
3) The next key feature observation regarding their growth came from examining the behavior of the first two cases which show above average behavior. These are $f(30)=16$ and $f(102)=19$. First, both are even abundant numbers satisfying the above condition. Then we note their behavior under the aliquot map:
$ali(30)=30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0$
$ali(102)=102, 114, 126, 186, 198, 270, 450, 759, 393, 135, 105, 87, 33, 15, 9, 4, 3, 1, 0$
We see that both of them have a run of 7 continuous abundant numbers, which are multiples of the perfect number 6. This allows a monotonic increase. When they lose this divisor 6 in the 8th iteration they become odd and start falling rapidly and converge. We then looked if this pattern might play out in a truly eruptive example by taking the case of $ali(138)$. We see that from $ali(138)[1]=138$ to $ali(138)[30]=1467588$, then $ali(138)[65]=19406148$ to $ali(138)[72]=348957876$ and again $ali(138)[95]=1013592$ to $ali(138)[117]=100039354704$ each number is a multiple of the perfect number 6. The repeated generation of a multiple of a perfect number under the aliquot map means there can be stretches of unhindered growth with one abundant number succeeding the previous one until that perfect number is lost among the divisors. It is the final loss of 6 and then 2 among the divisors that allows $ali(138)$ to converge. In making these observations we had recapitulated Lehmer’s key findings in the context of the growth of the most mysterious class of aliquot sequences we shall talk about next.

There are 12 $f[n]$, for $n \le 1000$, which could not be determined because the corresponding $ali(n)$ never converged in our computation (the blue dots in Figure 4). A search on the Internet reveals that deep computational efforts by various investigators have not seen an end to these sequences. They belong to 5 families which are known after Lehmer-II who first studied them. Within each family (listed below), after the first few terms the evolution converges to the same trajectory (Figure 7). These families are:
$f_{276}=276, 306, 396, 696$
$f_{552}=552 ,888$
$f_{564}=564,780$
$f_{660}=660,828,996$
$f_{966}=966$

Figure 7. $f_{276}$: dark red; $f_{552}$: dark blue; $f_{564}$: dark green; $f_{660}$: hot pink; $f_{966}$: cyan
All these numbers are abundant numbers descending from 6, which is a tendency they share with the above monster numbers that eventually converge. The frequencies of their largest prime factors are tabulated below:

$\begin{tabular}{|r| r|r|r|r|r|r|r|r|r|} \hline p & 7 & 11 & 13 & 17 & 23 & 29 & 37 & 47 & 83 \\ \hline freq & 1 & 2 & 1 & 1 & 4 & 1 & 1 & 1 & 1 \\ \hline \end{tabular}$

Notably, in this set except $f_{564}$, 4 of the five families have a member with 23 as the highest prime factor. It is now conjectured by mathematicians that these define the founding members of a class of aliquot sequences that would never converge but eventually shoot off to infinity. Examining the eruptive growth of these sequences, we found that the same features that cause the eruptive growth of the converging but monster $n$ hold true here too. Thus, if we consider the first case $ali(276)$, which starts with a double of the first monster $n=178$, we notice that it starts off well with the perfect number 6 as the divisor for the first 5 terms successive terms — $ali(276)[1]=276$ to $ali(276)[5]=1872$. But then the sequence loses 6 in the 6th term a stutters for a few terms even going down from the high point it reached. But then in $ali(276)[9]=2716$ it acquires a new perfect number divisor 28, which drives the abundance of successive terms until the it reaches the giant 28 digit $ali(276)[170]=7165455981799192761913565252$. Lehmer’s study was the first which noted the role of such persistent divisors in driving up the aliquot sequences. Accordingly, the grand old man of arithmeticians, Richard Guy, called these ‘drivers’ and in addition to 2 and the perfect numbers defined a few more of such. What seems to happen is that in these apparently non-converging aliquot sequences the drivers drive the growth to some large number after which they are lost. But before the sequence can crash by attaining an odd number, a new driver emerges and pushes up its growth again. Thus, these sequences might not converge. On the other hand the case of the monstrous $ali(840)$ with $f=749$ tells us that even after many such runs there is some chance of hitting a number, which brings you crashing down though its even almost all the way through. Whatever the case, the aliquot sequence of these non-converging numbers tell us that there are several more numbers in their trajectories, which would similarly not converge. Hence, non-convergence will be encountered frequently as the starting numbers get large.

In conclusion, following along the lines of the ancients, we find that the aliquot sequences offer us some philosophical lessons and mysteries:
1) There is the predictable: The primes and perfects display very predictable and uniform behavior.
2) There is the somewhat predictable: like the amicable numbers, for some of which a rule was found as early as Thabit ibn Kurra. But not all of them can be captured by an easy rule.
3) There is the statistically tendency: We know that an even or abundant number is more likely to have a longer path to convergence though it is not clear at all if any rule can say which of them might be monstrous and which rather common place.
4) There is the unpredictable and mysterious: Why are some numbers “aspiring numbers” which converge to a perfect number? Is there any rule to describe them? Why is 43 the preferred prime for convergence at least for $n=1:1000$. Why do some numbers not converge? Can we predict which numbers will not converge? Do they really not converge at all? As far as I know these remain unanswered. Thus, a very simple arithmetic process gives rise to a whole range of unexpected behavior.

## The ghost in the tattered Gattermann

Vidrum had dropped by to see Somakhya and Lootika when they had just started their household together. They had reconstituted a fairly elaborate lab in the biggest room of their home. They had also completely set up their fire room, which was well-equipped for karman. It had a niche for the images of various deities along with a sacristy. They showed Vidrum around and after uttering some purificatory incantations and, sprinkling water on him from a kamaṇḍalu, led him into the fire-room. There he saw the images of Maghavan along with his parivāra, the six-headed Kumāra, the patron god of Somakhya and Lootika, and of Ucchiṣṭolka and his wife covered in a blue cloth. Thereafter they passed the images of the lord of the yakṣa-s and those of the 7 mothers to finally arrive before the image of the terrifying patron goddess of their ancestors, Atharvaṇa-bhadrakālī. As they stopped before it, Somakhya smeared some vibhūti and kuṃkuma on Vidrum’s forehead from a human calvaria kept before it. There was not much furniture beyond their bike rack and three ample bookshelves for both of them still had their collection of physical books. So they sat on cushions on a floor mat facing those bookshelves.

Vidrum: “I sure you will say that this was bound to happen due to the gods or maybe that it is the way of siddha-s and their kulāṅgaṇā-s or perhaps it might be that two of you were together janman-janman. Whatever the case, I guess you two have to ultimately thank me for having reached this destination in life – I am pretty sure neither of you would have ever spoken to each other had it not been for me…And I hope you will use your mantra-siddhi to aid me to reach a similar destination in life too.”
Somakhya: “Of course Vidrum, we certainly have thank to you.”
Lootika: “We still don’t know how best we should repay you. I would still be in some debt for all the bad things I have said about you. In the least, I hope you would forgive me for that.”
Vidrum smiled and said: “You are forgiven.”
Somakhya: “Though you have forgiven her, don’t be sure it will end with that. You could be at the receiving end in the not so distant future.”
Vidrum: “I’m prepared, though it does appear to me that Lootika has become a more of a good girl over time. In any case Lootika, maybe, I should give you many more chances like Śiśupāla since I know that is your nature. After all, I have not forgotten those early days in school when you told me that you were even jealous of Somakhya. You and your sisters had some ferocity atypical for your sex, though your looks do not betray that. But if someone could hold their own in the domains so peculiar to you it would be Somakhya.”
Lootika controlling a chuckle: “Ouch! But as for being jealous of Somakhya, maybe my steroids got better of me shortly after I told you that.”
Vidrum: “Now don’t tell me it was the steroids since I have heard my friend, your dear Somakhya, remark that such things are ephemeral and in the long run don’t mean much like debris floating on a river. There is perhaps something which is indeed in the realm of those āgamika matters you’ll are known to know.”
Lootika: “But you don’t complete the whole train of physiology…I need not tell you now that there are the steroids and then there is oxytocin. A little unusually amidated peptide can go a long way.”

Vidrum then walked over to their shelves to look at the books and noted a tattered book which had been fortified and inscribed by Lootika along with a daub of kuṃkuma on it. Vidrum: “Gattermann, Heidelberg, 1894. Evidently, this old tattered book is of much value to you Lootika. You have even applied kuṃkuma to it…”
Lootika: “That is not kuṃkuma. That happens to be sulfosalicylic acid’s complex with Fe(III)+ in the famous FeCl3 test. That dark streak below it from a similar complex with salicylic acid. I’m pretty sure the ghost of Gattermann would not have approved of such daubing on your laboratory book, but it was in my earlier days.”
Somakhya: “This was one book she was rather possessive about and did not give it to Varoli who made claim for it.”
Vidrum: “Why so Lootika? Have you not been telling me that such material possessions come and go, ever since when my bike was stolen in school?”
Lootika: “This is special, it was a turning point in my career.”
Vidrum: “How so?”
Lootika: “It is a long story.”
Vidrum: “So be it. Somakhya have you heard it?”
Somakhya: “Not the long form she proclaims. Did not seem anything out of the ordinary.”
Vidrum: “Why don’t you tell us then?”
Lootika: “When we were kids, my father seeing my interest in scientific experimentation had just begun getting me to start a little lab at home and obtained a bunch of chemicals for me. These were the pleasures of Bhārata that a future generation might not have. It was then that I and Vrishchika accompanied our mother to help her haul groceries from the market stalls near our house. After the purchases, we were walking back home when we caught sight of a cart-man who selling the roadside śṛṅgāṭaka with the harimanthaka-sūpa.”
Vidrum: “Ah! that famous śṛṅgāṭaka-seller. My mouth is watering even as you mention it. Eating his śṛṅgāṭaka-s with the chick-pea slurry was one of the high-points of my otherwise dismal youth.”
Somakhya: “We do have some lunch for you. You can see if Lootika might come anywhere near your famed śṛṅgāṭaka and bhṛjjika cart-man of whom I have heard more than once from you. In any case, Lootika continue with your tale for I have not heard all these details either.”

Vidrum: “I guess just as with you, Vrishchika too was quite formed right then as though you’ll were remembering things from your past births? No doubt she intimated even her seniors in the first week of joining med school with a knowledge of morbid anatomy that exceeded them.”
Lootika: “Well, she was one among us caturbhaginī who always fascinated by morbid anatomy. Past births or not, I mentioned Vrishchika because my proclivities too lay in the direction of biological exploration but I did not get distracted to go along the paths of my sister at that point and applied myself to a year of unrelenting chemical experimentation closely following many of the detailed explanations of the śūlapuruṣa Herr Gattermann. The first big thing I did of my own was to extract a mixture of alkaloids from peyotes, which were growing in the nearby rock-garden. I first basified them with NaOH and then extracted them into xylene. Thereafter, neutralized them with repeated salting steps using acetic acid to form alkaloid acetates and extracted the salts back to water and allowed them to crystallize. Buoyed up with the confidence of this success I went on to conquer separation with thin layer and paper partition chromatography. Then I moved on to isolate a conessine-like alkaloid, which seemed to give some relief to certain people with some gastric disorders. Then I took on the tropanes, which subsequently two of my sisters took over and continued. Of them, it was Varoli who had real talent in this direction. That was around the time we first made acquaintance with you. At the end of that, I returned to biology, now as a biochemist in the making, but I had been transformed in many dimensions.”

Vidrum: “Ah! I can now see how that book holds a special place for you. So Somakhya that seems to have been at the root of the virtuosity of your wife you used to episodically praise in our youth followed by the phrase ‘don’t tell her that I think so’.”
Somakhya: “If you look at Gattermann that would not be apparent at all. It can only inspire an already prepared mind. A mind which is also coordinated with the hands and possessed of a certain patience and an eye which can quickly catch the subtle. However, it is said to have even inspired the great chemist Woodward to scale heights like never before.”
Lootika: “It probably gave some of that Woodwardian inspiration to Varoli. She, more than me, had that ability in pure chemistry and the capacity to combine it with a knowledge of theory like what Somakhya has. This was clear from her early interest and graduation to spectroscopy. For me, it was more of getting the fundamentals straight and thinking quantitatively while doing experiments, which held me in good stead in the years which followed.”
Somakhya: “Sure. Spidery, I think we should not keep our guest waiting from savoring your experiments in the kitchen.”

As they were having lunch, Vidrum remarked: “This is the first time I am eating food cooked by Lootika. Her wonderful spread with milk precipitated with HCl from a burette and the liquid N2 chilled stuff is certainly delightful to the tongue. Somakhya, the gods have been doubly good to you to join you with a wife who can cause delights to the gustatory system. I again reiterate, you as brahmins should intercede on behalf of me to get the gods to be at least 1/10th as good to me.”
Lootika smiled and said: “Again, we should state that you perhaps greatly over-estimate our capacity as the knowers of brahman and I think I should give you another perspective. Somakhya’s father remarked that a man who gets entangled in the good rasa-s of his wife’s food soon heads towards pāpman. Hence, he eschewed indulgence in such, observing a vrata of eating mostly that which hardly inspires the tongue – bitter, bland, tasteless and the like. It is thus that he attained siddhi-s like a mahāvratin.”
Vidrum: “Well, you all are the eternal pessimists.”
Somakhya: “Since we are well aware that in life many things that are seen as the door of pleasure eventually lead to sorrow.”

After lunch, Vidrum again went up to the tattered Gattermann and picking it up closely looked at it. Sniffing at it he turned to his hosts and remarked: “You guys had the capacity to summon all kinds of beings from the beyond. But, you know, due to my long-suffering stay in a dwelling that was stationed not far from the famous cemetery of our youth haunted by more entities than I would care to know, I have become uncannily attuned to them.”
Somakhya: “Truth to be told you are way more attuned to them than any of us. We are in fact practically blind to them except when unveiling them via prayoga-s.”
Vidrum: “I must say this Gattermann seems positively haunted by something. Lootika mentioned that it was like yokāi of the Japanese. I wonder if she knows more than she let out while telling us its story.”
Lootika: “I only meant it in a very colloquial sense. I really have not had much of sense of any haunting in that book.”
Vidrum: “Then guys we must do something we did in our youth. We should ply the planchette to see if we can get him to speak.”
Somakhya: “We don’t have a planchette with us now.”
Vidrum: “I’m sure you can do more. Could you not summon him by some other means.”
Somakhya: “We could, if you are willing to be a medium, do a bhūtadarśana. But the last time I did it you said you never wanted to be one again.”
Vidrum: “That’s OK. I think I am game for it again for I think there is something sinister about this book.”
Lootika: “OK we shall try a Kapālīśa Bhairava-Raktacāmuṇḍā-prayoga to draw the entity to give you a bhūtadarśana.”
Somakhya: “No Lootika! We might need that prayoga soon for something more serious and we do not want to deploy it right now. Since were are sarvādhikārin-s we shall deploy the prayoga of Sahasrāra and Viṣvaksena along with his Karimukha-s to bring out the resident.”

As the prayoga got underway Vidrum felt himself lapsing into a strange trance. He wondered if it was the good meal that he had had in a long time which was making him sleepy or if it was something else. But soon it became clear he was going into a bhautika trance. He felt as though he was in a pleasant theater with a nice perfume watching a movie but like a Saṃjaya he also started speaking out in precise detail all that was playing out before his eyes, which looked even more real than real life:
“I repeatedly hear and visualize the following syllables each in a svara lower than the previous one: pau ro mo go ṣu.

Having finished his japa he goes out to wander near the environs of the temple. He walks up to nearby tank that was excavated by a Vijayanagaran general to commemorate his victory over a preta-alliance. The vaiṣṇava is seen collecting some of the green water from it in a container. Stopping near a vast bastard poon tree he closely examines some of its fallen pods and collects a couple into a bag. Returning to the courtyard of the temple he seats himself beside one of the low walls and carefully takes out a brass microscope from a box in his tin case. He sets up some slides and examines the green water he has collected. He thinks to himself: ‘Of all these algae which might be close to the ancestor of the modern land plants? Applying the principles of Mr. Darwin, I believe that somewhere within these silky filamentous forms we should see the origin of land plants. While those which I collected from the sea near Madras have considerable complexity, I doubt they were the ones which gave rise to the land plants for after all from the sea the first transition must have been to fresh water like these forms I’m first seeing. So I must look more closely at these freshwater forms to see if any of them share features specifically with the land plants. Then he makes a slide of a fungus from the pods he had collected and after some examination remarks: ‘This looks like an interesting new species. I will have to study it more closely when I’m back in the college.’ Thus, he is engrossed in his observations. Some a kid with his father passes by the vaiṣṇava. He asks his father:’What is this brāhmaṇa doing?’ The father responds: ‘Let him be, he comes here from time to time to dig dirt and pond scum and look at it through that magical yantra. Don’t ask him anything or he might cast a spell on you.’”

Then Vidrum’s transmission went blank for some time. But he seemed to experience a great quiet and peace with occasionally re-emergence of the syllables he had seen and heard earlier. Lootika remarked to Somakhya: “O Bhṛgūdvaha the Ayyangār seems to have been prescient for his times.” S: “True, dear; I am really curious to know how far he got with his objective.” Then Vidrum’s transmission continued. Vidrum: “I see a man screaming: ‘I am the one, I am the one.’. He looks contorted and with pinpoint pupils as though poisoned by an opioid.”
Lootika: “Yes. He will speak. Don’t worry.”
Vidrum noticed the man freeze for a while and then start speaking: “My name is Sadāhāsa. I belonged to the 3rd varṇa and my people originally came from the Lāṭānarta country. My father ran a grocery shop and I was expected to join in that business. But from an early life, I did not have much proclivity in the direction of my family. After I passed the 10th class with reasonable marks, my father realized that I might be able to get some other means of earning by studying a little more rather than manning the shop. Anyhow my two elder brothers were there to do that. Hence, after some deliberation he let me study further in the science stream in the hope I might become a doctor, a dentist or an engineer. I was never really interested in those professions at all. I just drifted away not knowing what was my true calling and joined the university two years later to obtain a B.Sc. degree in chemistry. My father was unhappy with me continuing with my apparently useless science education and being a drain on his exchequer. I tried to tell him that the degree might give me some knowledge that might help me start a paint shop. It was around that time he was struck down by the rod of the black god Yama. However, my brothers were supportive and as they had opened a new food stall that was meeting with some success; so, they continued to support my education. Thus, I made my way to the M.Sc. program having done tolerably well in the B.Sc course. By some force unknown to me, I became intensely fascinated with organic chemical experimentation in course of this degree and got admission into the Ph.D. program with a stipend in at the university in the dreadful city of Visphoṭaka teeming with all kinds of criminal and debauched elements.

Lootika excitedly: “vallabhatama! hear that! the sesterterpenoid with epitetrathio linkage!” Somakhya: “varārohe! does it not have your mind racing? From whence? from whence?”

Vidrum: “Wow! As in the days of our youth you have managed to make visible a most remarkable phantom!”
Somakhya: “You deserve all the credit for sniffing this one out. Frankly, I did not sense anything there.”
Lootika: “I sort of feel embarrassed that this fellow was lurking all this while much like the hobgoblins in your old house and we could do nothing about it. At least he says he is going to come back to help you.”
Vidrum: “I thought I had seen the last of my goblins but I guess there is more in store.”
Somakhya: “By no means, you have seen the last of them!”

Posted in Heathen thought, Life, Scientific ramblings |

## The hearts and the intrinsic Cassinian curve of an ellipse

Introduction

This investigation began with our exploration of pedal curves during the vacation following our university entrance exams in the days of our youth. It led to us discovering for ourselves certain interesting heart-shaped curves, which are distinct from the well-known limaçon of Etienne Pascal and its special case the cardioid. It also led us to find the intrinsic relationship between the ellipse and the Cassinian curve that is associated with every ellipse. We detail here those observations with the hindsight of multiple decades and the availability of excellent modern geometric visualization software (in this case Geogebra) since the days of our paper and pencil explorations (However, even then we had an excellent set of ellipse and circle templates that our father had gifted us and also an ellipse drawing tool which we had made inspired by the yavanācārya Proclus). We first lay the ground work with some basic results and concepts that provide the necessary background before delving into the heart of the topic under discussion.

The eccentric circles theorem

Given a circle and a point inside it, the locus of the midpoint of the segment joining the said point to a moving point on the circle is another circle with radius half that of the given circle and with its center as midpoint of the given point and the center of the given circle.

Figure 1

Let $(x,y)$ be the coordinates of the point $P$ on the given circle with center at origin $O$ (Figure 1) and radius $2a$. The coordinates of the given point are $F_1=(0, 2c)$. Let the coordinates of the midpoint $M$ of the segment $\overline{F_1P}$ be $(x_1,y_1)$. From Figure 1 it is clear that:

$x_1=\dfrac{x+2c}{2}, y_1=\dfrac{y}{2}$; thus $x=2x_1-2c, y=2y_1$.

By plugging these into the equation of the given circle $x^2+y^2=4a^2$ we get:

$4x_1^2-8cx_1+4c^2+4y_1^2=4a^2$

$x_1^2-2cx+c^2+y_1^2=a^2$

$\therefore (x-c)^2+y^2=a^2 \rightarrow \textrm{Locus}(M)$

Thus, the locus of $M$ is a circle with $A=(0,c)$ as center and radius of $a$: $Q.E.D$.

The ellipse

Given a line $d$ and a point $F$ outside it, what will be the locus of all points such that the ratio of their distances from $F$ and $d$ respectively is a given constant value $e_c$?

Figure 2

From the solution shown in figure we get the equation of this locus to be:

$x^2+(1-e_c^2)y^2-2h(1+e_c^2)y+(1-e_c^2)h^2=0$

Being a quadratic curve it will be a conic. Specifically, when $e_c<1$ it is an ellipse; when $e_c>1$ it is a hyperbola and when $e_c=1$ it is a parabola. This relates to why these curves are called conic sections. We can see that the distance from point $F$ can be represented by the surface of an infinite bicone with vertex at $F$. The distance from line $d$ can be represented by a plane containing $d$. The given ratio $e_c$ specifies the inclination of this plane such that the angle by which the plane is inclined is $\theta=\textrm{arctan}(e_c)$. The intersection of this plane and the bicone generates the conic section, which when projected on the $xy$ plane appears as the conic curve specified by the above equation (Figure 3).

Figure 3

Thus, when the inclination of the plane is less than $\pi/4$ it is an ellipse. When it is exactly $\pi/4$ it is a parabola. When it is between $\pi/4$ and $\pi/2$ we get a hyperbola. Corresponding to this are the Greek terms ellipse: less than; para: equal; hyper: greater than. This number $e_c=\tan(\theta)$ (where $\theta$ is the angle of inclination of the generating plane) is the eccentricity of the conic and $F$ is a focus of the conic.

By definition the bipolar equation of an ellipse is $r_1+ r_2=2a$. Here, $r_1,r_2$ are the distances of a point on the ellipse from the two foci of the ellipse $F_1, F_2$. $a$ is the semimajor axis of the ellipse. $F_1$ is one of the foci of the ellipse ( for instance, as determined by the construction in Figure 2 and 3) then the second focus $F_2$ is at a distance of $2c$ from $F_1$ along the major axis of the ellipse. $c= e_c\cdot a$. Further, it is easy to see that $a^2-c^2=b^2$, where $b$ is the semiminor axis of the ellipse.

The eccentric circles of an ellipse

Given an ellipse and a point $P$ moving on it, 1) what is the locus of the foot of the perpendicular dropped from a focus of the ellipse to the tangent at $P$? (i.e. locus of the intersection of the tangent at $P$ and the line perpendicular to it from one of the foci. 2) What is the locus of the reflection of one of the foci on the tangent drawn to the ellipse at $P$.

Figure 4

From Figure 4 it is clear that the $\triangle F_1QP \cong \triangle PQR$ by the SAS test. Hence, $\overline{F_1P}=\overline{PR}$. By definition of ellipse $\overline{F_1P}+\overline{F_2P}=2a$. Thus, $\overline{PR}+\overline{F_2P}=\overline{F_2R}=2a$. Therefore, the locus of $R$ is a circle $c_1$ with center $F_2$ and radius $2a$.

It is clear from the definition of $R$ that $Q$ is the midpoint (Figure 4) of $\overline{F_1R}$. Therefore, by the eccentric circle theorem applied to the above-defined circle $c_1$ the locus of $Q$ is a circle $c_2$ with radius $a$ and center as the midpoint of $F_1, F_2$, which is the center of the ellipse. Thus, $c_2$ is the solution to the problem of the pedal curve of an ellipse with the pedal point as one of the foci. $c_2$ is also the circumcircle of the given ellipse. These two circles $c_1, c_2$ are the two eccentric circles of an ellipse.

Construction of an ellipse using its eccentric circles

Since the radii of both eccentric circles of an ellipse are defined by only the semimajor axis of the ellipse, the whole family of ellipses with the same semimajor axis will share the radii of the eccentric circles. Hence, we additionally need to define the foci to construct the ellipse given one or both of its eccentric circles.

Figure 5

If we are given just the circumcircle $c_2$ and a focus $F_1$ then we can construct the required ellipse thus (Figure 5): Define focus $F_1=(-c,0)$. Draw a segment connecting $F_1$ to $P$, a point moving on the circle $c_2$. Draw a perpendicular line to $\overline{F_1P}$ at $P$. The envelop of all such lines would be our required ellipse.

Figure 6

If we are given both eccentric circles $c_1$ and $c_2$ then the construction is a little more involved but has interesting consequences (Figure 6). First define the foci $F_1=(-c,0), F_2=(c,0)$. Then draw circle $c_2$ with origin as center and radius $a$. Draw circle $c_1$ with $F_1$ as center and radius $2a$. Let $P$ be a moving point on circle $c_2$. Join $F_1$ to $P$. Draw a line $t$ perpendicular to segment $\overline{F_1P}$ at $P$. With $P$ as center draw a circle which passes through $F_2$. This circle cuts the circle $c_1$ at points $A$ and $B$. Join $F_1$ to both $A$ and $B$. The points where segments $\overline{F_1A}$ and $\overline{F_1B}$ intersect line $t$ are $C$ and $D$ (Figure 6). The locus of points $C$ and $D$ as point $P$ moves on $c_2$ gives us the required ellipse (blue in Figure 6).

The ellipse hearts

Notably the above construction of an ellipse using both the eccentric circles yields a companion curve (purple in Figure 6). It usually assumes a heart-shaped form with a dimple or a cusp that superficially resembles the limaçon of Etienne Pascal. However, a closer examination reveals that it is not that curve and has a distinct shape; we term it the ellipse-heart because every given ellipse will have its unique ellipse-heart. From Figure 6 we can also see that the ellipse-heart can be defined for a given ellipse as the locus of the reflection of the point of tangency on the pedal line from one of the foci. Like the Descartes oval and its dual the limaçon, this ellipse-heart can be seen as the dual of the ellipse. Its shape can be described by the eccentricity $e_c$ of the ellipse. For high eccentricity it shows a prominent dimple that tends towards a cusp as $e_c \to 1$. For $e_c<\tfrac{1}{2\sqrt{3}}$ it becomes a convex oval that becomes a circle identical with the ellipse for $e_c=0$.

In order to derive the equation of this curve, we note that by the above definition of the eccentric circle $c_1$ we have $\overline{AC}=\overline{F_2D}$. We also observe (Figure 6) that the ellipse-heart is obtained by subtracting $\overline{F_2D}$ from the radius vector of the eccentric circle $c_1$. Now, $F_2D$ is itself the radial vector of the ellipse with the focus as the pole. This allows us to define the polar equation of the ellipse using the focus as a pole as is done in celestial mechanics, where one star/planet is at the focus and another star/planet/moon is moving in an elliptical orbit around it. This equation of the ellipse is:

$r=\dfrac{\left(a^2-c^2\right)}{a\pm c\cdot\cos\left(\theta\right)}$

Here the radial $\angle{\theta}$ is known as the “true anomaly”, as in the definition of elliptical orbits in the planetary theories of Nīlakaṇṭha Somayājin and Johannes Kepler. Given that $\overline{AC} \; || \; \overline{F_2D}$ and in the opposite direction we can derive the equation of the ellipse-heart by subtracting the above radial vector from $2a$, the radial vector of the circle $c_1$ with the appropriate signs. Thus, if the ellipse is:

$r=\dfrac{\left(a^2-c^2\right)}{a- c\cdot\cos\left(\theta\right)}$,

then its ellipse-heart is:

$r=2a-\dfrac{\left(a^2-c^2\right)}{a+c\cdot\cos\left(\theta\right)}$

While the square of the above equation has an indefinite integral, evaluating it is a bit complicated. Hence, we resorted to the expediency of numerically integrating it and arrived at the area of the ellipse heart $A_h$ to be:

$A_h=\pi (4a^2-3ab)$, where $a$ and $b$ are respectively the semimajor and semiminor axis of the ellipse.

Thus, $A_h= A_{c_1}-3A_e$, where $A_{c_1}$ is the area of the eccentric circle $c_1$ and $A_e$ that of the ellipse. Further we also get:

$\dfrac{A_h}{A_e}=4\dfrac{a}{b}-3$

Thus, when $\tfrac{a}{b}=\tfrac{5}{4}, \; e_c=\tfrac{3}{5}$, i.e. the three ellipse parameters form a 3-4-5 right triangle then $A_h=2\cdot A_e$. This is a beautiful configuration (Figure 7). Finally, inspired by the above equation for the ellipse-heart we can also define a second ellipse-heart using parametric equations as:

$x=\left(2a-\dfrac{\left(a^2-c^2\right)}{a+c\cos\left(t\right)}\right)\cdot\cos\left(t\right), y=\left(2a-\dfrac{\left(a^2-c^2\right)}{a-c\cos\left(t\right)}\right)\cdot\sin\left(t\right)$

This curve has a classic heart-shape (hotpink in Figure 7) for a part of the range of eccentricities of the ellipse. These curves may be considered bifocal like the ellipse, unlike the limaçons (including the cardioid) derived from the unifocal circle. The ellipse and the ellipse-hearts touch each other at the vertices of the ellipse. Figure 7 shows the relationships between a system of ellipses and their corresponding ellipse-hearts. They might define the outlines of certain leaves or the dehisced pod of the bastard poon tree.

Figure 7

The Cassini curve of an ellipse

Every ellipse is associated with a confocal Cassini curve sharing parameters with the ellipse.

Even though the Cassini curves are well-known, that they are intrinsically associated with every ellipse does not seem to be common knowledge (at least as far as we know). This is despite the historical associations of a version of the Cassini curve, the Cassini oval. Hence, it excited us considerably when, in our youth, we discovered the two to be intimately linked. The curve itself was discovered by the astrologer and mathematician Giovanni Cassini in an interesting context: In the west, as in India, the transition from geocentricity to heliocentricity was neither immediate nor uncontested. Cassini, despite being a prodigious observational astronomer, believed that the sun went around the earth in an oval orbit, which was defined by one lobe of the curve now known as the Cassini ovals. However, later in his life he realized that he was totally wrong and that Kepler was right in describing the orbit of the earth around the sun as an ellipse rather than an oval.

Now, how is the Cassini curve associated with the ellipse? It arises from the following question: Given an ellipse with a point $P$ on it, let points $A$ and $B$ be the feet of the perpendiculars dropped from the two foci of the ellipse $F_1, F_2$ to the tangent at $P$. What would be the locus of the points of intersection $F$ and $E$ of the circles with radii $r_1=F_1A$ and $r_2=F_2B$ as $P$ moves along the ellipse. We solved this thus:

Figure 8

From the above discussion it becomes clear that both $A$ and $B$ will lie on the circumcircle of the ellipse $c_2$ (Figure 8). As both are pedal points they would define parallel lines $\overleftrightarrow{AD}$ and $\overleftrightarrow{BC}$ which form the rectangle $ABCD$ inscribed in the circle $c_2$. From this rectangle (Figure 8) it becomes clear that the $\overline{AF_1}=\overline{CF_2}=r_1$ and $\overline{BF_1}=\overline{DF_1} =r_2$. We then apply the well-known Euclidean intersecting chords theorem on $AD$ and the major axis of the ellipse $V_1V_2$. Thus we get:

$r_1\cdot r_2=\overline{V1F_1}\cdot \overline{V2F_2}=(a-c)(a+c)=a^2-c^2=b^2$

Thus, the product of the two pedal segments of an ellipse is a constant equal to the square of the semiminor axis: $r_1\cdot r_2=b^2$. Now, the bipolar equation of the form $r_1\cdot r_2=b^2$ defines the Cassini curve. From this bipolar equation we can derive its Cartesian equation:

$\left((x - c)^2 + y^2 \right) \left((x + c)^2 + y^2\right) = (a^2 - c^2)^2=b^4$

Figure 9

The form taken by the Cassini curve depends on the eccentricity of the ellipse. When $e_c=0$, the ellipse, circumcircle $c_2$ and the Cassini curve all become a coincident circle. When $e_c=\tfrac{1}{\sqrt{2}}$, the curve crosses over to become the lemniscate of Bernoulli (Figure 9). When $1>e_c> \tfrac{1}{\sqrt{2}}$ it becomes two separate oval loops and is the classic Cassinian oval. When $\tfrac{1}{\sqrt{2}}>e_c>\tfrac{1}{\sqrt{3}}$ the curve is bilobed with central dimples but a single loop. When $e_c \le \tfrac{1}{\sqrt{3}}$, the curve takes the form of a single biaxially symmetric convex oval. For values close to the minimum of the range of $e_c$ the Cassini curve approximates a single circle while close to the maximum it approximates two small circles. In conclusion, the eccentricity of the ellipse $e_c$ is entirely sufficient describe the range of shapes adopted by the Cassini curve. Indeed, we can conceive, the three curves, namely the ellipse circumcircle $c_2$, the ellipse and the corresponding Cassini curve as three degrees of response to the eccentricity parameter. The circle $c_2$ represents the $0{th}$ degree in that it does not change at all with $e_c$. The ellipse represents the first degree response in that in flattens uniformly along the minor axis with increasing $e_c$. The Cassini curve represents the even more exaggerated second degree response in that it starts of by flattening along the minor axis even more rapidly than the ellipse. Thus it first dimples, then crosses over as the lemniscate and finally breaks apart into the two loops of the oval. Thus the first degree response is a conic while the second degree response is a toric section (i.e. a section through the torus as the Cassini curves). Figure 10 shows an animation of the evolving curve with changing $e_c$.

Figure 10

Posted in Scientific ramblings |

## The mathematics class

It was a dreary autumn day, the same year Lootika had joined their school. The apabhraṃśa class had just gotten over. Somakhya’s head was spinning with all the confusing genders of the vulgar apabhraṃśa that was dealt with in the class by a positively sadistic teacher. The genders of the nouns in that apabhraṃśa were mixing with the genders in another northern apabhraṣṭa tongue they were supposed to learn, and further, they began seeping into the genders of his own brahminical speech threatening to corrupt it. “What a mess he thought to himself” and remarked to his partner Vidrum: “How I long for the day we will be rid of these apabhraṃśa-s upon graduating from school.” Vidrum: “Don’t say that too loudly on the streets or our apabhraṃśa enthusiasts might lynch you then and there. At least, unlike my parents, we don’t have to study yet another deśa-bhāṣā in some other zilebia-like script.” Their conversation was cut short by the commencement of the next class with the arrival of the ever-irascible mathematics teacher, who marched in gaily swinging his cane. The class arose and wished him in unison: “namaste master-jī”.

He stroked his beard, then twirled his mustache, looked around the class with an air of disdain, and remarked in a gruff voice: “You idiots have been making much noise. Let me bring you down to earth. Today I am going to give you all the surprise supplemental problem. It is exceedingly elementary Euclid and you will have a maximum of 7 minutes to solve it. The first five solve it should come with their notebooks and place it on my desk. If you get it right you will get 5 extra marks in the mid-semester exam. Those who solve it within correctly 7 minutes but are not the first 5 to place your notebooks on my desk will get 1 extra mark. If you get it wrong but still have the audacity of coming up and showing the answer you will get minus 5 marks. Those who don’ t solve it in 7 minutes will take it home and complete it and show it to me tomorrow. If you have gotten it wrong then I’ll issue a punishment: the boys will get three whacks from a full swing of my cane, while the girls will have to kneel-down outside the class. Now take down the problem:

Given a unit square, if a point lies on the same plane as the square at not more than a unit distance simultaneously from each of the four vertices of the square then what will be: 1) the minimum distance it can reach from any side of the square; 2) what fraction of the area of the square can the point be located in.” He then yelled: “The clock starts now!”

A silence of terror rippled through the class and they hit their notebooks to solve the problem. Soon, Hemaling dropped his chair down and made a loud clanging noise to distract the rest of the class and barreled between the rows of benches, like a bandicoot down a drain, to reach the teacher’s desk and place his notebook. A little later Somakhya walked up to the teacher’s desk as though taking a stroll in the Manorañjanodyānam to place his notebook. Sometime later Gomay rushed like a dung-beetle rolling its ball to take his answer to the teacher, who was pacing around with an eagle-eye to catch any episodes of cheating. Thereafter Lootika took her book and quietly placed it on his desk and returned to her own. Then Sharvamanyu, Tumul, Vidrum, and Nikhila ran to the teacher’s desk, with each of the three boys trying to edge out the other. The teacher asked them to behave themselves and said he would consider all four of them as valid submissions.

Finally, the time of reckoning arrived. The teacher looked through their solutions and made his remarks: “Hemaling your workout is so neat that it looks as though typeset with LaTeX but you have drawn no figure.” Hemaling: “Sir, I saw no need for any figure when the algebra was so obvious.” Somakhya, you have shown no algebra but your construction is aesthetic with proper annotations of the solution on it. So full marks for both of you Hemaling and Somakhya! Your handwriting sucks Gomay! But your solution is right; so you too will get 5 marks too.” Then he looked seriously at Lootika and stroking his beard thunderously remarked: “Lootika, stand up! Your answer is correct and neatly presented.” Then raising his voice he continued: “But remember if you think you are a smart alec, we punish those types severely at our school. You have been a bad girl today. You thought I did not see you, but I caught you helping your partner Nikhila. There is no way she could have solved it in the time she did if you had not helped her. As you are a new student and might bring a great name to our school in the future, I am sparing you the punishment… But you will get 0 extra marks despite having solved the problem correctly.” Lootika quietly sat down. He roared: “Lootika, stand up again! Apologize for your misdemeanor! You are not supposed to just sit down like that.” After Lootika had apologized, he turned to Vidrum: “You have tremendously improved this year and have been showing great diligence in class. Hence, you will get not just 5 but 7 extra marks for the correct solution.” He continued: “ Tumul, you have not calculated the exact form of $\sin(\tfrac{\pi}{12})$, so you will get only 1 extra mark.

Then the teacher twirled his mustache and with cane pointed to Sharvamanyu: “You buddhū. You have made a mistake. Still, you had the temerity to try to push your classmates to reach my desk! You will get -7 marks. Sh: “Sir, I see I made a mistake but it was inadvertent. I know it should be a negative sign instead of a + sign for $\sqrt{3}$.” The teacher barked at him: “You, moron. Having made the mistake you still want to talk back. Do you want my boot to kiss your ass? You are punished. Go and stand outside the class for the rest of the period. Then you shall write out the solution to this problem over and over again 196 times by the time of tomorrow’s class.” He then turned to Nikhila: “You stupid girl. Do you think you can learn mathematics by copying from that bespectacled girl? You are punished. -10 marks for you and you will go and stand outside the class for the rest of the period. If are dishonest like this in the exam I will see that, that you fail the year.” Nikhila burst into tears and left the class.

That evening after school was over, a throng of girls went rushing to Lootika to get her help in solving the problem. Somakhya went home to get something to eat and then mounted his bike to go to Vidrum’s house. Unfortunately, Sharvamanyu could not join him on the way because he had the repetition to do as part of his punishment. Having parked his bike inside Vidrum’s house, Somakhya went with him to buy some kite-string from the nearby market-stall. As they were returning they caught sight of Lootika at the corner of the road with Nikhila who was still crying. Lootika waved out to them and asked what are you doing this evening? Somakhya: “We are headed to fly a kite.” Lootika: “Fly a kite? Let me join you all. I have long wanted to get one up but have repeatedly failed to do so.” Lootika then turned to Nikhila: “Please, go home now and rest. It is just another day at school. You don’t have to tell your parents what happened today. I did not. As I told you so many times, you should not be taking such incidents at school too seriously. One of the main reasons, perhaps for people like I and Somakhya, the only one, for going to school is to learn to acquire a tough skin. Life is rough and full of conflict – hence, one should learn to keep fighting and not be afraid of authority. Yes, today we were caught and castigated but tomorrow, by Indra, we will come up with even cleverer devices by which we might evade those standing in our path.” Nikhila half-sobbing: “Lootika thank you for all the support. I hope I am able to pull this off when I get home.” Lootika: “Remember, I am there to help you with any curricular troubles that might arise. But ‘vāsāṃsi jīrṇāni’. Someday all friends must part ways from dispersion or death, so remember that to face life alone one needs to be tough and even then one has a finite probability of breaking.”
Somakhya: “That’s true. It is also very human to ask your partner for the solution. While it might be a gray issue in the context of an examination, don’t get too worked up by the day’s incidents and just forget about it.”

When Nikhila rode away on her bike, Vidrum turned to Lootika and said: “Are you not poisoning her with sort of radical ideas. Were you not abetting dishonesty in class today? You seem totally unbothered by what happened in class today? What would you say of the lesson we had in the apabhraṃśa class, wherein tyrant Akbar is said to have punished three robbers differently. The first one he just verbally reprimanded and sent home. The second he gave a whacking with his rod and sent home. The third’s face he blackened and paraded him on a donkey. The first committed suicide, the second left Delhi and the third just went home and had a nice bath. Nikhila is being like the first one while you, it would seem are calling on her to be like the third?”
Lootika: “See, Vidrum. By my secretive teaching, even though under difficult circumstances, she at least learned how to solve such a problem. Do you think śmaśru-dāḍhī-masterjī’s yelling and dramatics would have taught her any better? So what I did was actually beneficial for my friend and class-partner. It is not that we had committed a sin like theft, like in that story. It is just unfortunate we got caught. Plus, if she is so sensitive, how would she navigate life where we would face even rougher incidents?”
Vidrum: “You have a point. Let us get moving with the kite.”

As Vidrum and Somakhya had their kite up and seeing no one else in the vicinity who might cut their string, they gave it to Lootika to fly. Vidrum still happy over the day’s events asked Somakhya: “Did you tell your parents about your performance in today’s surprise math test.” Somakhya: “What’s there to tell about such minor things. My father whose thoughts are often embedded in higher mathematical realms will think something is wrong with me if boast to him about solving some exceedingly elementary Euclid as the masterjī put it. My mother would hardly be excited by something so minor. It is your day today. You won squarely with a sevener even though we beat you to the solution in speed.”
Vidrum: “That was unexpected indeed. Of course, I accept the fact you and Hemaling beat me to the post. But the math-tyrant is a real sadist what a punishment he imposed on poor Sharvamanyu. His mindset makes me think he was born a marūnmatta in is last life.”
Somakhya: “As Lootika said it is just another day in school. Indeed he imposed a harsh punishment on Sharva. I too made a sign error and luckily corrected it before I drew the final figure. It is hard to guess the causes for the psychology of such types. Maybe his wife beats him with a broomstick at his home and he takes it out on us. Or maybe he thinks he is Akbar himself. Who verily knows?”

As Lootika was utterly lost in the excitement of flying the kite she asked the other two: “How did you figure this out. I did all the things we did today with my sisters only to have my kite repeatedly shredded before it even got up to a few feet. But today it seems like magic – I seem to finally have a hang of getting it up.” Somakhya: “Long before we knew you, Vidrum and I spent many a day suffering the same frustration as you. We looked like idiots when everyone else would get their kites flying. Then my father told me that I was indeed an idiot and had me pay closer attention to design and aerodynamics. He instructed me how to make a kite like with polyester cloth, shaped like a delta and also how to design the right attachment for it in the form of a central keel with a tether hook. Then he told us how to position ourselves with respect to the wind. Such a wind does not blow on top of my house but it is ideal at Vidrum’s place – it is indeed all about Vāyu as the Kākṣaseni had learned in the days of yore. Thus, we too reached the skies. Now that you know the rahasya you too can impart it to Vrishchika and others.”

Appendix: Solution the question posed in the class.

The blue curvilinear quadrilateral is area the point in the question can occupy:
$\dfrac{\pi}{3}+1-\sqrt{3} \approx 0.3151467$

Posted in Life |