## The Apollonian parabola

Some say that Archimedes and Apollonius of Perga (modern Murtina in Turkey; the center of the great yavana temple of the goddess Artemis in the days of Apollonius) were the two great yavana-s who might have rivaled Karl Gauss or Leonhard Euler in their in intellectual achievements. Beyond doubt a whole lot of modern knowledge rests on the foundation of Apollonius and in the west one could perhaps say that a Kepler or a Newton may have never shone forth had that Apollonian foundation not existed. Indeed, even today the study of Apollonius is a profound experience for the man who can do so. As with yavana religion much of their knowledge including the books of Apollonius were lost due to the ravages of pretonmāda in the west. Not surprisingly even at a later age the Lord Protector of England the duke of Somerset was burning books with yavana geometrical diagrams at the Oxford university.

Ironically in the world of its younger sister, the marūnmāda, Apollonius held a great fascination among those influenced by (Neo)-Platonism. Among the marūnmatta-s, their intellectual ibn Sina from Persia was perhaps the most fascinated by Apollonius. This allowed the preservation of much more of his works than in the west. Not surprisingly later marūnmatta-s referring to his work said that while his Qoranic studies were fine, he should be accused of kaffr-hood, and “innovation” derived from yavana and Hindu knowledge. Indeed, some mullahs suggested that he should have been cut pieces and feed to raptors. Nevertheless, his studies on yavana geometry and Hindu trigonometry has resulted in some interesting preservations which would have otherwise been lost. One such we shall discuss here is an Apollonian construction of a parabola using the geometric mean theorem. It goes thus in modern terminology:

1) Let point O be the origin. Take a point A on the x-axis at a given negative coordinate $(-a,0)$.
2) Let B be a point moving on the x-axis from the origin in the positive direction.
3) Draw a circle with diameter as $\overline{AB}$.
4) Draw the tangent to this circle at point B. It will be perpendicular to the x-axis.
5) The above circle cuts the y-axis at points C and D. Draw lines through these points which are parallel to the x-axis.
6) The above pair of lines will intersect the tangent to the circle at point B at points E and F.
7) The locus of points E and F as B moves along the x-axis is the desired parabola.

From the construction we see:
$\overline{OA}=a$; The x-coordinate of the parabola is $\overline{OB}=x$; The y-coordinate of the parabola is $\overline{OC}=\overline{OD}=y$. Given the circle used for the construction we notice that $\overline{OC}=\overline{OD}$ is the geometric mean of $\overline{OA}$ and $\overline{OB}$ because $\overline{AB}$ is the diameter of the circle (geometric mean theorem). Thus we get:
$y=\sqrt{ax}$
$\therefore y^2=ax$
which is the equation of our desired parabola.

## Leaves from the scrapbook

There were extensive memoirs in the form of electronic scrapbooks of Somakhya, Lootika and some members of their circle. Those in the know read the available excerpts due to matters of considerable interest being recorded in them. Other parts were written in a cryptic language that most could not understand what was really in them. These leaves are from the record of Lootika; some of what is contained in them might have a parallel version in Somakhya’s record.

Entry 22; mārgavakra-mīna, Rākṣasa year of the first cycle: These summer days Vrishchika and I sleep on the terrace. We are often up late talking and duly woke up late too much to our parents dislike. We watch with awe the starry vault turn above us, looking out for meteors, Ulka-s, which are pratyakṣa-s of the dreadful elephant-headed Vināyaka deva-s, four in number created by the mighty Skanda. We often try to catch the faintest objects on good nights. Vrishchika fills me in on the work she is  doing on the genetics of subtler visible quantitative traits in humans and we break our heads about the many problems in our investigation before lapsing into a pleasant sleep dreaming of being in the embrace of our kāmin-s against their firm chests. Many aspects of such a study have been surprisingly neglected as it was derided as no better than disreputable things like phrenology, physiognomy, racism and the like. While with my assistance Vrishchika has a good handle of how to study and understand the actual sequence variation at various loci, we are hard-pressed with many aspects of the quantitative genetics. Somakhya has a long distance friend named Indrasena with whom he is close and often in touch though he has seen him only once or twice. He is from the distant city of Udyānapurīṣa where Somakhya’s cousins Mandara and Saumanasa live. This Indrasena is a jack of many trades and I would love to do intellectual battle with him and over all emerge his superior. But he is quite a master of quantitative genetics and Somakhya has introduced Vrishchika to him to get to help with her roadblocks. I suspect Vrishchika has taken quite a fancy for him and might even combine forces with him against me if such a battle were to be joined. In any case they are now able to detect some interesting signals in our data which we could investigate further.

My classmate Sumalla (Sumallā) is considered very beautiful and desired by many males in my class and beyond. However, Vrishchika identified certain subtle physiognomic traits that made her infer based on her current analysis that Sumalla would develop a metabolic condition starting as early as her 25th year that will ruin her good looks and even make her less desirable by the time she nears her 30th year. She volunteered to be a data point and Vrishchika has remarkably detected four informative polymorphisms in the transcription factor-encoding genes NKX2-6, SMAD6, and MEIS1 and the neuropeptide gene NMU, which make it likely indeed that Sumalla would develop the condition Vrishchika has prognosticated. I wondered how the news should be broken to her but Vrishchika seemed quite happy to handle that – after all she’s the one who is going to be a physician. This brings me back to an interesting evolutionary angle which I had discussed at some length with Somakhya: why is it that such variation has survived? Perhaps the earlier age of mating allowed such variation to pass through is one hypothesis. An alternative is that in the past the extant of food availability selected for these traits that are deleterious in today’s dietary landscape. A further alternative is they were protective against infectious disease(s) that are no longer as threatening to fitness as the metabolic condition these variations facilitate. I wonder if Indrasena will have any success at all in obtaining DNA from skeletons he has identified in a cemetery that he believes comes form a time before the coming of the ārya-s to India.

Entry 23; madhu-biḍāla, Rākṣasa year of the first cycle: Vrishchika is making clones for making recombinant Neuromedin (NMU). I have asked her to also make a clone of a CDC123-like ATP-grasp enzyme from Legionella, which I intend using in the ambitious protein “stapling” experiment. For the past several nights we have seen a new star appear in Cygnus between the stars forming base of the neck and the head of the goose. As a result Vrishchika and I had a tangential conversation on what the beings on various planets around stars along the Milky Way stretching above us might be thinking even as they look out into their skies. Whether there would be beings out there wondering, just as we are, if someone is watching their star from a planet around another star. It made me feel that one area where we have a deeply limiting lacuna in our knowledge is the range of forms life and intelligence on other worlds might assume.

Entry 24; cala-saṃkhya, Rākṣasa year of the first cycle: I wished to learn about the new star we had seen appear in Cygnus; so I went to meet Somakhya. He too was excited about it and informed me that that it was a pulsating red giant called $\chi$ Cygni having reached its maximum. It is a Mira-type variable, whose prototypical star Mira in Cetus he had shown me few years earlier. Thereafter, I had such a engrossing session with Somakhya. It all began when I asked Somakhya about the variability of the star $\eta$ Aquilae which we had noted waiting for sleep to overtake us. He informed me that it was a pulsating star called a Cepheid. From there we moved on to a discussion of oscillators other than the $\sin (x)$-like functions. Somakhya had taught me the van der Pol equation a couple of years ago when explaining how one numerically solves differential equations but I had not paid much attention to its specifics. Today he showed me how some forms of the vdP equation produced oscillations which could show a bit of chaotic behavior. Forms of the vdP are good simulators of the curves of (semi)regular pulsating variables and we played with them a little trying to recapitulate what Somakhya explained to me as being the Blazhko phenomenon in Cepheid variability i.e. the periodic modulation of the basic mode of pulsation.

Entry 25; cala-stṛ, Rākṣasa year of the first cycle:

Simple reaction kinetics

The session I had with Somakhya was so exciting that I had to try to relay all I had learned to my sisters. To give them some background I began by explaining simple chemical kinetics using simple differential equations. Even little Jhilli was able grasp much of it suggesting she is coming of age. It was a good revision for me; hence, I am recording the highlights here. After we had discussed the vdP equation Somakhya and I had segued into other types of oscillations in nature. I am quite conversant with the simple prey-predator population dynamics model: the prey is growing at an exponential rate depending on its population size at time t in a food-rich environment. When prey and predator meet the prey is killed, so its population growth rate is negatively affected by this interaction, which scales as the product of the prey and predator population sizes. In the case of the prey its population grows when it gets food so its growth rate scales as the product of the prey and predator populations while it decreases due on death from inter-predator conflict and migration which are proportional to prey population at time t. This gives us the well-known Lotka-Volterra equation.

Lotka-Volterra prey-predator population oscillations

Then Somakhya introduced me to two interesting equation systems that have been discovered by mathematicians [Footnote 1] which can be seen as different variants of the simple LV equation that describes in population oscillations of prey and predator locked in conflict.

Emergence of chaotic population dynamics in a simple system of 3 complexly interacting species

The first of these can be seen as a model describing population dynamics of 3 complexly interacting species X, Y and Z. X, like in the LV system, grows at an exponential rate proportional to its population at time t. Its growth rate is also similarly proportional to another symbiotic species Z, which might be producing metabolites useful for its growth. In contrast, its growth rate is depressed by its encounter with a parasite Y; thus the decrease in growth rate is proportional to the product of the X and Y populations as in the simple LV system. Like in the simple LV system it grows at a rate proportional to encountering the species X and directly deriving nutrition from it; thus its growth rate is proportional to product of the X and Y populations. Additionally, the growth of Y benefits from metabolites which X produces even without Y directly feeding on X. As a result the total positive dependency of the growth rate of Y is proportional $X^2Y$. Like in the simple LV equation Y again loses individuals to death and migration arising from inter-parasite conflict at a rate which is proportional to its own population at time t. Y has no interaction with Z; hence its growth rate shows no direct dependency on it. Now species Z needs to directly interact with X to grow; hence, its growth rate is positively correlated to product of the populations of X and Z. On the other hand the increase of X by itself results in loss of resources for Z’s growth, so its growth rate shows a further negative dependency on population of X by itself and Z too loses a certain part of its population to death and migration at a rate proportional to its current population size. Remarkably, this system has a notable range of parameters where the populations X, Y and Z vary chaotically. As a result their population maps produce strange attractors similar to the famous Rössler or Lorenz attractors.

Chaotic population dynamics of four competing species with a basic logistic growth

The second model which Somakhya introduced me to was one where multiple species are growing in an environment with a fixed carrying capacity, i.e. logistic growth under the Verhulst equation. Additionally, they also negatively interact with each other, i.e. compete with each other as per the standard LV model. In such a model chaotic population dynamics can emerge even with just 4 competing species. While with 4 species the chaotic picture is seen in a relatively small parameter range, with more species chaos becomes much more common. What this tells us is that chaotic dynamics are likely a regular feature of nature. One of the species in this model under certain starting conditions seems almost certainly prone to extinction before it makes a dramatic oscillatory come back which would never have been expected unless you had learned of such dynamics.

Entry 26; ugra, Rākṣasa year of the first cycle: I reflected with my sisters on the general implications of what we had learned about chaotic behavior in nature. It could produce many impressions some simultaneous and some exclusive in the same or different persons. 1) Even among those without much conceptual grasp or generative capability the strange attractors could produce an aesthetic experience. Somakhya conjectured that randomness does not produce an aesthetic experience in most people – indeed few would call a random scatter of points as aesthetic. In completely convergent attractors like an ellipse or the oblongs produced by LV solutions several more people might have an aesthetic experience but still it might not be very widespread. But with the strange attractors certainly many more would have such an experience – something which might be related to their fractal dimension. Perhaps, this is the reason why the devāyatana-s of our ancestors tended towards increasingly fractal states. We will have to explore this more but even this aesthetic experience is perhaps much more in people who can grasp the rahasya-s. 2) In the person characterized by quantitative arrogance, it might produce a belief of being able to prognosticate complex systems. Such a person might think that they could produce models to sufficiently predict high degrees of complexity. 3) In a person who has studied nature it might produce simultaneously a sense of insight and humility. It makes one aware that behind the seemingly disordered state lie laws. Chaos in itself should be seen as a given in nature, so there could be selection for systems that factor in chaos. 4) For the historian trying to prognosticate the fate of nations it produces a strange paradox. The possibility of such chaotic behavior being common place has an almost seductive hold for those who see it in action. Like in the case of one with quantitative arrogance, one might think this is the way going forward with even more complex equation systems being able to ultimately predict history. Yet for all the swirling chaos in its undergirding macro-history does seem far less chaotic – hence one might ask why?

Entry 27; harivāhana, Rākṣasa year of the first cycle: The dreadful box returns: the red ellipse and the red hyperbola. All this began years ago when we were in school. I have hitherto only recorded bits and pieces out of fear of the thing. But now that it thrust itself back into my existence I am moved to record the whole story supplemented by Somakhya’s own records on the mathematical curiosities that surround it.

It was shortly after I and my sisters had transferred to the new school, an event which was to have a momentous impact on my existence. Intentionally or unintentionally it seemed as if I had rather quickly acquired some notoriety in my class and soon I had a lot of classmates giving me company at almost every free moment in school. At lunch break my classmates Vidrum and Hemalinga came to me and said they had a method to trisect any angle with just a compass and a ruler. I had heard from Somakhya that this was not generally possible and told them so. But they showed me a construction which really seemed to trisect the angle when I checked it using my protractor. I asked them as to how they had discovered it. Vidrum said that he had this great geometry box whose instruments would almost magically guide him towards drawing various figures. Curiously, Vidrum nearly lost that box a month before that incident and Vrishchika and I had found it and restored it to him. That morning Vidrum drew one such a figure and was puzzling over it when Hemalinga studied it and realized that it was a solution for the trisection of the angle with just a compass and a straightedge [Footnote 2].

Construction to approximately trisect a given angle

Just then I caught Somakhya going to eat his lunch at a secluded spot away from everyone else. I called him over and soon Vidrum and Hemalinga described their construction with a triumphant laugh. Somakhya half-smiled and said they were certainly wrong. Over the rest of the lunch break he pored over the figure and proved using some geometry that it was not a true trisector but a saw-tooth like function of the form:
$y=\arctan \left(\dfrac{2\sin \left(\frac{x}{2}\right)}{1+2\cos \left(\dfrac{x}{2}\right)}\right)$
He called it the approximate trisector and showed that it rather closely approximated $y=\dfrac{x}{3}$ in the domain $-\dfrac{4\pi}{3} \leq x \leq \dfrac{4\pi}{3}$

The approx-trisector function

Using this function he established that at $x \approx 124.3^o$ the difference between the real and this approximate trisection was $-1^o$. Thus, for the angles we had tried it was close or below the resolution of my protractor. Hemalinga, who was known in the school for his prodigious mathematical capacity, seemed a bit red-faced and Somakhya rubbed it in a bit further after calculating the first term of the Maclaurin series of approx trisector showing it was $y=\dfrac{x}{3}$ thus proving why it seemed to work. Two years ago Hemalinga ran into us, and as though to make a point to Somakhya, showed us that he had integrated by hand the approx trisector function – something which filled a whole page. He felt very pleased when Somakhya remarked that he could never done that without a computer.

That incident aside, coming back to the the matter at hand, by the end of that school day both Somakhya and I were feeling strangely unwell. I remarked that it might because we had not eaten since morning having skipped lunch due to the pursuit of the approx trisector. However, it did not go away and we were mysteriously unwell for 3 days without any apparent cause and during that illness had a disturbed sleep from the repeated apparition of a saw shaped like the approx-trisector function cutting through us. In course of the summer vacations that year Somakhya smashed his own finger due to my closing his eyes shortly after establishing the properties of a geometric figure which Vidrum had drawn. Early the next year a girl in our class to whom Vidrum was close had died mysteriously right in cemetery near his house. I distinctly remember Vidrum picking up his geometry box from her desk the day after the two great kṣatriya-s who respectively delight in svāhā and svadhā had visited her.

Sometime around the middle of that semester with the boring exams temporarily past us, Somakhya pointed to a purple velvet bag lying on the parapet below the window beside which his desk was stationed at school. He used to be curious about its contents. Hence, one day after school he climbed on to the compound wall and jumped forward to hold on to the parapet and clamber on to it. Thus, he reached the bag that intrigued him so much. But even as he picked it up the whole parapet came crashing down. Luckily, other than being dusted up he was not hurt and our land unlike that of the mleccha-s had no surveillance device for anyone other than me and Vrishchika to know what had really happened. From the bag he retrieved a sturdy geometry box which had elegant instruments, a slide rule with fine calibrations and stencils of ellipses, circles and hexagons and a remarkably smooth curve-fitting tape. I immediately recognized this distinctive box as being that of Vidrum and informed Somakhya that it was so. He decided to repatriate it to Vidrum the next day. To my utter surprise Vidrum refused to take it and even denied that the distinctive box was his. I was utterly puzzled by his denial but Somakhya was more than happy to keep it as its finder. There was a śūlapuruṣīya inscription indicating that the box was made in the Śarmaṇyadeśa – indeed modern Hindus never produced anything of that quality and robustness.

The next day I was still fighting with the construction when Somakhya gave me a hint: “The eccentricity of a conic is the tangent of an angle between 0 to $\dfrac{\pi}{2}$.” This immediately fired the light bulb in my head and I took that geometry box and drew out the construction. As I placed the curve-fitting tape to draw out the ellipse, it neatly folded into an elliptical path almost magically. I was amazed and drew a nice ellipse with the red pen which looked like an egg of the goddess Vinatā bearing the aquiline god. I next drew a parabola with the angle set at $\dfrac{\pi}{4}$ I wanted to show this construction to my family, so I carefully placed it in my file so that it might not get folded. That night I explained the construction to my father and drew out the sheet to show it him when to my anguish and embarrassment the parabola and the ellipse had vanished even though the rest of the construction was intact. The next day I showed it to Somakhya who was also surprised by how cleanly the conics had vanished. He had seen them himself and wondered how that could be. He brought out the red pen and drew them again himself. We ensured they were there and I again put the sheet into my file. That afternoon just before leaving home I checked the sheet again and the conics had vanished again. Startled I just threw sheet away and went home.

Some days later Somakhya had shown me how to genuinely trisect any angle using a hyperbola. He had again constructed a red hyperbola and demonstrated the construction to me. He was then talking of some peculiar properties of a related parabola and ellipse which were approximate dividers of the angle in certain ratios. He then gave the sheet with the construction for me to repeat it with my own substandard instruments. That evening after school Somakhya was approached by a classmate whose name I do not recall clearly. He belonged to a community previously classified as a depressed class and his dark grayish yellow complexion and facial features suggested that most of his ancestry derived from the mysterious tribal peoples who inhabited India before the coming of the later waves of humans. That guy hardly had any intellectual proclivities to be able hold even a limited conversation with Somakhya but he was perhaps the only guy in the class who had a deep fancy for the paper objects that Somakhya made following the Japanese way (Some of our teachers verily hated Somakhya for his love for this art which was compounded by his utter disdain towards some of them). Indeed, Somakhya used to remark that this paper folding technique of the easterner islanders marked them as a people of great creativity. Hence, whenever that guy would come with sheets of paper Somakhya would indulge him and fold those objects, like an eagle, a bear, a starfish, a walrus and boxes of different types. That guy for his part would collect and neatly preserve Somakhya’s creations in a large box. Thus, that day as Somakhya was folding paper, a lout who was also in our class, whom we knew as Sphichmukh, surreptitiously stole his bag with the geometry box and his drawing board and swiftly made away.

Somakhya was utterly disappointed and the next day he asked me for he sheet of paper on which he had drawn the hyperbolic trisection of an angle, perhaps with a longing for the stolen box. I took it out and we looked at each other in horror: the hyperbola had utterly vanished! I could read Somakhya’s mind that he was beginning to have conflicting feelings about the box. However, he could not take the theft lying down. Via several inquiries he obtained evidence that the thief was Sphichmukh. He decided to recover the stuff and said that he was setting out with Sharvamanyu and Vidrum to punish Sphichmukh and seize the stolen stuff. I saw them mount their bikes and tail Sphichmukh after school. Knowing that Sphichmukh was a lout with other hoodlums as friends they had armed themselves for the exploit. Somakhya had a bicycle chain, Vidrum a nunchaku, and Sharvamanyu a knife. Seeing all this I felt a mixture of fear, curiosity, and excitement and followed them on my bike at a safe distance to see what would happen. As they closed in on Sphichmukh he realized that his game was up, but given his usual temperament he furiously rode to the edge of a pond and threw the bag with the geometry box and Somakhya’s drawing board into the murky waters where buffaloes bathed. The three were livid with fury and it seemed they would have give Sphichmukh himself a jalasamādhi; I feared they could land in prison themselves from that. So I pedaled hard to quickly reach the three of them and suggested that rather than smiting Sphichmukh or shoving him into the pond they should complain to the school authorities and I volunteered as an additional witness for Sphichmukh’s crime. For some reason Vidrum then rather passionately asked Somakhya to let go off the box and forget about it.

Perhaps, a bit mollified by my sight they desisted from a direct attack and Somakhya complained as suggested. However, poor Somakhya had little traction with the school authorities; much to his chagrin, they informed him that since the constructions he was drawing and the board were not part of the curriculum, which was being taught at school they would not take any remedial action. They let off Sphichmukh after mildly lecturing him about the impoliteness of taking others things. Somakhya revealed to me then that more than the loss of the box which he had himself obtained for free he had lost something more precious with it. In the same bag he had kept a khārkhoḍā with the yantra of the pūrvāṃnāya with the trident and the three bhairavī-s and the 3 supine bhairava-s. With the loss of the bag he also lost his siddhi of the mantra-s of the pūrvāṃnāya.

Now we come back to the present. Starting this week we have changed our schedule due to much haranguing by our parents. All four of us get up earlier and bike to the pond and run thrice around it. Thereafter Vrishchika and Varoli return home because they spend a couple of hours preparing for their respective entrance exams before continuing with all the fun science we are doing. They seem more sincere in this business than I ever was. However, I and little Jhilli, being more carefree, go to the adjacent hall to play table-tennis for some time along with Abhirosha. Abhirosha is attempting a difficult exam for whatever she wished to do, which was quite removed from my path of life. Nevertheless, that exam had several stiff mathematical tests. I had trained her the previous year for her university entrance exam and knew that she was quite capable of surviving the impending tests in algebra, calculus and numbers. But she does not have much of flourish in geometry so she is back to consult me. Thus, I and Jhilli decided to look at her problems. There was a problem of polygons whose areas and circumferences approximate $\pi$. I remembered that Somakhya’s father had once posed that to me and Vrishchika – he wanted to see for himself if we were really what what people said about us. I let Jhilleeka solve that one. Then she took out a sheet of math-paper and showed a failed attempt of a construction of what should have been a Cartesian oval on it. As she showed it she pointed to some unnecessary lines and remarked: “Hell, where did these vertical lines come from out of the blue! This is is spooky.” Just then I caught sight of her geometry box and my jaw literally dropped: “How on earth did you get that box?” Abhirosha: “Actually, that’s bit strange. I found it by chance on the bank of the pond sometime ago.” I looked at it more closely and asked Abhirosha: “Did it come with a bag?” Abhirosha: “Not it was all by itself dented and worn but the instruments inside were intact. Seems like the famous German engineering.” I noticed that the dent corresponded to it being trampled by the hoof of a buffalo. Now I was not surprised by the strange lines that seemed to have appeared by themselves. More Abhirosha told me of it, it was apparent that our old box had come back into our lives again. I have sent an email to Somakhya detailing these strange events. I also sent one to Vidrum inquiring about the box. He responded rather quickly saying : “You guys dabble with such things anyhow. You can take care of him and please don’t get me into this matter again. I believe he was perhaps a civil engineer from a town among the Karṇāṭa-s known as Hiriyuru.

:::::::::::::::::::::::
Footnote 1: e.g. works of Sprott and the like

Footnote 2: The construction goes thus:
1) To trisect angle A in this case $57^o$ draw circular arc BC cutting the two rays of the angle at B and C.
2) Then bisect angle A to intersect $\widehat{BC}$ at point D. Connect Point D to point B and C to get segments $\overline{DB}$ and $\overline{DC}$.
3) Trisect $\overline{DB}$ to get point H as the beginning of the 3rd segment of trisected $\overline{DB}$.
4) Join point A to H to get $\overrightarrow{AH}$. $\angle{BAH} \approx \dfrac{\angle BAC}{3}$.
Let the approximate trisection of $\angle BAC=x$ be $\angle BAH= \beta$. From construction $\angle BAD =\dfrac {x}{2}$ and $\angle BAD =\dfrac {x}{2}-\beta$. In $\triangle{ABD}$ we get from construction $\angle ABD=\angle BDA$. Using sine rule we get:
$\dfrac{\sin(\beta)}{\overline{BH}}=\dfrac {\sin (ABH)}{\overline{AH}}=\dfrac {\sin(x/2-\beta)}{\overline{DH}}$

From the construction we get $\overline{BH}=2 \overline{DH}$
Thus we have: $\sin (\beta)=2\sin (x/2-\beta)$

$\sin (\beta)=2\left(\sin (x/2)\cos (\beta)-\cos(x/2)\sin(\beta)\right)$

$\tan(\beta)=2\sin(x/2)-2\tan(\beta)\cos(x/2)$

$\tan(\beta)=\dfrac{2\sin(x/2)}{1+2\cos(x/2)}$

Thus we get: $\beta= \arctan \left(\dfrac{2\sin \left(\dfrac{x}{2}\right)}{1+2\cos \left(\dfrac{x}{2}\right)}\right)$

## śūlapuruṣasya vicāraḥ

Somakhya’s cousin Babhru and his family was visiting him during the vacation before they were to join university. After lunch he convinced Vidrum and Sharvamanyu to come over to his house when they got the chance for he thought more might be merrier when it came to casual matters. Upon their arrival they spent some time playing mock cricket in their backyard and when the sun rose high they went into Somakhya’s room/home-lab. Sharvamanyu: “What have you guys been doing otherwise.
Somakhya: “I have been trying to get Babhru to learn the art of analysis of old texts with an elementary Rāmāyaṇa which our old teacher Shilpika’s husband had published.”
Vidrum: “We have been taking your words seriously. Sharva and I have started brushing up the old language and attending daily classes till our fates regarding university entrance are decided. But I must say it is rather depressing.”
Somakhya: “Why?”
Vidrum: “The teacher is doing the vairāgya-śatakam of prince Bhartṛhari. It makes one see life in a rather somber light.”
Babhru: “Our teacher too would go into raptures in course of reading from that text.”
Somakhya: “Raptures? Perchance did you mean the śṛṅgāra-śatakam?”
Sharvamanyu: “Whichever, who would want the fate of prince Bhartṛhari?”
Somakhya: “Why don’t you try this one instead, the Kucumāra-pañcāśikā with its bhāṣya by Anaṅgadāsa. It might provide a palliative for all the vairāgya you have been receiving.”
Vidrum picked up the book from Somakhya’s shelf and read out the opening vākya: “makaradhvaja-ratibhyāṃ namaḥ । kandarpamusalo vā kāmāyudho vā madanāṅkuśo vā +ānandasya daryā dvāra-bhettā ।” Babhru and Sharvamanyu: “Man! This guy Kucumāra wastes no time getting on with business!”

Sharvamanyu: “Guys may be we should get started to see the historical museum at the university. If we leave right away we might get an hour before it closes.” Somakhya: “That sounds like a great idea. Babhru has been long wanting to see it. Let’s go.” Thus, they left for the museum where they saw arms from from the 1600s down to the end of the 1800s. There were swords, knives, bows, muskets and more exotic stuff like the metal claws of the type the founder of the Marāṭhā nation had used to disembowel a giant marūnmatta, some torture instruments used by the Mohammedans and Christians, a drug bong used by a noted Hindu leader and many other such things. There were also letters of some notable historical figures and models of forts and other defensive constructions. Babhru who had never seen anything like that before was greatly excited by the visit and sneaked in some photos when the guards were not looking. But soon the hour was over and they had to leave.

Back at Somakhya’s house they stationed themselves in the garden and were swiping their way through the photos Babhru had surreptitiously taken. As they were checking out the pictures of the bows Somakhya remarked: “With Babhru around to help me we have managed to make three new bows. Let’s check them out. Somakhya brought out four bows, the three new ones and his old one and also a bullet-thrower that they had made. For sometime they tried them out at targets placed in the backyard. In course of that Vidrum remarked: “Do you think with your long bow here you could have pierced that armor labeled sūfī ghāzī Nizam al Din we saw in the museum. I heard a story that a Telugu warlord Peḍḍa Nāyaḍu killed a huge sūfī who was encased in such an armor by shooting his heart through it. His descendants still keep that armor he took as a trophy.”
Somakhya: “I think yes but it would need some training to get to Nāyaḍu’s level of strength.”
Sharvamanyu: “I am pretty sure we can get there but we need to be training more regularly. Somakhya, I guess you have just resumed this summer?”
Babhru: “This sounds more like a tall tale like they make up for Pṛthivirāja Chāhamāna. I have been trying these bows and even my best shots only go in up to the full arrow head. Somakhya calculated that we need the force to pierce at least half the length to pierce medieval Moslem armor. These weapons were probably only for Hindus who fought bare-bodied as I read in the work of eminent historian Jadunath Sarkar.”

Sharvamanyu: “Babhru, that’s pretty baseless to me. I don’t think you have really seen what bows can do. But it will take some practice.”
Vidrum: “I would rather believe Nāyaḍu’s descendants rather than some Jadunath Sarkar.”
Babhru: “Whether you like it or not Nāyaḍu’s farrago is no substitute for the meticulous records of the Mohammedan historians cataloged by the likes of Sarkar.”
Somakhya: “Well I can confirm from this account from the mouth of the Mohammedan himself. It is said that the Shaikh Ali Pehlavan was a giant sūfī who was the  head of a forward volunteer force of the Khalji’s jihad into south India well before the main army. Even as the Turks were still consolidating in the north he led one of the first jihads into south India, which I am sure none of your eminent historians in Delhi told you about. Disguising as innocuous fuckirs they penetrated through the opening created during the Jihad in Mālava and filtered through the Seuna Yādava kingdom into the southern land. There in the vicinity of where the Tungabhadra joins the Krishna they were uncovered by Nāyaḍu who launched a swift attack killing them. Indeed, the two further sūfī vermin Shaikh Shahid and Pir Jumna continued from where Pehlavan had failed. But they too were promptly killed along with all their followers by the men sent by another nāyaka.”
Sharvamanyu: “Somakhya pointed to how that account mentioned one of the sūfī-s of that camp clobbered many naked jaina-s to death before himself being shahidized by a blow from a mace. It would be great to have a mace like that. These sūfī jāragarbha-s were like today’s Mohammedan terrorists in every sense running amok among the hapless citizenry seeking śahadat in course of such jihad-s. They were truly the vanguard of the army of Islam that was to follow.”
Babhru: “Thank you for the interesting account. But in a sense this proves my point. You have been able narrate this history because of the meticulous record of the Islamic sources.”

Vidrum: “Why is it for everything we have to cite one marūnmatta historian or another to convince ourselves? I am sure you would not have taken what I said from the Hindu perspective without this confirmation?”
Babhru: “My history teacher was none other than the eminent historian Dr. Dhurtaprasad Sharma. He presented the thesis that Hindus lacked the concept of history as a science. They only had epic mythology. He said that this rigorous approach to history began with Herodotus and was picked up by the great Moslem chroniclers and finally by Karl Marx and his successors. I know my cousin Somakhya disagrees with this thesis but it does seem to me that we have very little to show by way of a rigorous history.”
Vidrum: “I don’t quite understand why we need to bring in rigor into everything. What matters to me in these narratives and memories is the raw sentiment it arouses in us. The clash our ancestors engaged in: can we ensconced in our comfortable homes ever use rigor to capture what it means to be in the middle of a life and death struggle with a blood-curdling, blood-spilling Pehlavan of a sūfī and his murderous fellow beards who have just burst into your village to seize your women?”
Sharvamanyu: “Well said indeed! I’m reminded of Lootika’s sister Vrishchika’s remark about our mathematically gifted classmate Hemalinga: ‘aiming for rigor he reduces the otherwise lively subject to a corpse with rigor mortis’.”
Babhru: “While I certainly do not want to trivialize our brave freedom fighters, I am afraid your sentiments about objective history only reinforce the image of the Hindu’s lack of genuine history. It runs deep. Think about it: why did we not ever produce a Thucydides? Instead we had Kaḻhaṇa-s who tried to pass poetry and epic myth as history – evidently they prized these sentiments much like you all.”

◊◊◊◊

As they were finishing up lunch Somakhya mentioned the discussion they had on history and the conversation they had. Lootika: “This reminds me of the śūlapuruṣa Spengler. He had something very interesting on that. I have been wanting to talk with you about that for sometime.” Somakhya: “Good point but would you not want to look at your data: remember a scientist has no vacation.” Lootika smiled and said: “I thought you were going to stay for sometime; we have the whole day before us. Can you help me find it in his prolix tome?” So saying she passed him the text on her computer. Having located the relevant parts Somakhya asked her to read it out aloud.

Lootika remarked that it was very prolix and heavy prose and read on: “Lastly, the words History and Nature are here employed, as the reader will have observed already, in a quite definite and hitherto unusual sense. These words comprise possible modes of understanding, of comprehending the totality of knowledge — becoming as well as things-become, life as well as things-lived — as a homogeneous, spiritualized, well-ordered world-picture fashioned out of an indivisible mass-impression in this way or in that according as the becoming or the become, direction (“time”) or extension (“space”) is the dominant factor. And it is not a question of one factor being alternative to the other. The possibilities that we have of possessing an “outer world” that reflects and attests our proper existence are infinitely numerous and exceedingly heterogeneous, and the purely organic and the purely mechanical world-view (in the precise literal sense of that familiar term) only extreme members of the series. Primitive man (so far as we can imagine his waking-consciousness) and the child (as we can remember) cannot fully see or grasp these possibilities. One condition of this higher world-consciousness is the possession of language meaning thereby not mere human utterance but a culture-language, and such is non-existent for primitive man and existent but not accessible in the case of the child. In other words, neither possesses any clear and distinct notion of the world. They have an inkling but no real knowledge of history and nature, being too intimately incorporated with the ensemble of these. They have no Culture.

And therewith that important word is given a positive meaning of the highest significance which henceforward will be assumed in using it. In the same way as we have elected to distinguish the Soul as the possible and the World as the actual, we can now differentiate between possible and actual culture, i.e., culture as an idea in the (general or individual) existence and culture as the body of that idea, as the total of its visible, tangible and comprehensible expressions — acts and opinions, religion and state, arts and sciences, peoples and cities, economic and social forms, speech, laws, customs, characters, facial lines and costumes. Higher history, intimately related to life and to becoming, is the actualizing of possible Culture? We must not omit to add that these basic determinations of meaning are largely incommunicable by specification, definition or proof, and in their deeper import must be reached by feeling, experience and intuition. There is a distinction, rarely appreciated as it should be, between experience as lived and experience as learned (zwischen Erleben und Erkennen), between the immediate certainty given by the various kinds of intuition — such as illumination, inspiration, artistic flair, experience of life, the power of “sizing men up” (Goethe’s “exact percipient fancy”) — and the product of rational procedure and technical experiment.

Lootika paused and remarked: “Hier der Unterschied zwischen Erleben und Erkenne ist von zentraler Bedeutung. In course of our past discussions I have thought about this point and have come to believe that there are two distinct histories, one which indeed lies in the realm of feeling, experience and intuition and the other which as he says belongs to the realm of rational procedure and technical experiment. We would say: aham īkṣe ahaṃ vicetāmi tayor madhye bhedaḥ । This has come out in the discussion you narrated involving your cousin and friends. Now it would seem to many minds these two are not bridgeable and they reside in one or other realm…”

Somakhya: “Wait Gautamī! While I am often amused or should I say saṃhṛṣṭa by the points where we seem to so naturally converge, I must ask you to pay more careful attention to the following: ‘…A homogeneous, spiritualized, well-ordered world-picture fashioned out of an indivisible mass-impression in this way or in that according as the becoming or the become… And it is not a question of one factor being alternative to the other. The possibilities that we have of possessing an outer world that reflects and attests our proper existence are infinitely numerous and exceedingly heterogeneous, and the purely organic and the purely mechanical world-view only the extreme members of the series.‘ Thus, the two histories you point out are not two unbridged worlds in themselves. They are so as you rightly apprehended in the minds of many but by themselves should be seen as only poles of a spectrum between which lie many distinct possibilities like the tints and shades of a color.”

Lootika: “But then O Jāmadagnya, the śūlapuruṣa does clearly distinguish two entirely distinct modes of apprehension rather than a whole spectrum. Regarding the various types of intuitions he talks about he says: ‘The first [i.e. the intuitions] are imparted by means of analogy, picture, symbol…‘ Regarding the rational procedure and experiment he says: ‘the second by formula, law, scheme.‘ He then elaborates: ‘The become is experienced by learning — indeed, as we shall see, the having-become is for the human mind identical with the completed act of cognition. A becoming, on the other hand, can only be experienced by living, felt with a deep wordless understanding.‘”

Somakhya: “It is true he distinguishes two types of apprehension but then pay attention to what he say down stream; Bārhadukthī, please read on and we shall consider it further.”

Lootika adjusted her spectacles read on: “It is on this that what we call “knowledge of men” is based; in fact the understanding of history implies a superlative knowledge of men. The eye which can see into the depths of an alien soul — owes nothing to the cognition-methods investigated in the “Critique of Pure Reason,” yet the purer the historical picture is, the less accessible it becomes to any other eye. The mechanism of a pure nature-picture, such as the world of Newton and Kant, is cognized, grasped, dissected in laws and equations and finally reduced to system: the organism of a pure history-picture, like the world of Plotinus, Dante and Giordano Bruno, is intuitively seen, inwardly experienced, grasped as a form or symbol and finally rendered in poetical and artistic conceptions.
She continued: “Aurva, the history that your cousin took to be real history would be the one of the śūla-puruṣa’s pure nature-picture while that which declared as non-history by him and his professor would be the śarmaṇya’s history-picture! Rather paradoxical indeed. Thus, we see two starkly different apprehensions of history.”

Somakhya: “What I posit are that these two are parts of a spectrum that the śūlapuruṣa in a sense alludes to. While we both know that we need not take everything he claims at face value, paying some attention to the following, though abstrusely worded might be useful to get to the point. O jālikā read this part.”

Lootika read it out: “The laws of nature are forms of rigorous… It becomes easy to see why mathematics, as the ordering of ‘things-become’ by number, is always and exclusively associated with laws and causality. Becoming has no number…Pure becoming … is in this sense incapable of being bounded. It lies beyond the domain of cause and effect, law and measure. No deep and pure historical research seeks for conformities with causal laws — or, if it does so, it does not understand its own essence. At the same time, history as positively treated is not pure becoming: it is an image, a world-form radiated from the waking consciousness of the historian, in which the becoming dominates the become. The possibility of extracting results of any sort by scientific methods depends upon the proportion of ‘things-become’ present in the subject treated…the higher the proportion is, the more mechanical, reasonable, causal, history is made to appear. But when this content of ‘things-become’ dwindles to very little, then history becomes approximately pure becoming, and contemplation and vision become an experience which can only be rendered in forms of art …

This contrast lies at the root of all dispute regarding the inner form of history. In the presence of the same object or corpus of facts, every observer according to his own disposition has a different impression of the whole, and this impression, intangible and incommunicable underlies his judgment and gives it its personal color. The degree in which ‘things-become’ are taken in differs from man to man, which is quite enough in itself to show that they can never agree as to task or method. Each accuses the other of a deficiency of “clear-thinking”… Nevertheless, we must not lose sight of the fact that at bottom the wish to write history scientifically involves a contradiction. True science reaches just as far as the notions of truth and falsity have validity: this applies to mathematics and it applies also to the science of historical spade-work, viz., the collection, ordering and sifting of material. But real historical vision (which only begins at this point) belongs to the domain of significances, in which the crucial words are not ‘correct’ and ‘erroneous’ but ‘deep’ and ‘shallow’… Nature is to be handled scientifically, History poetically.

Somakhya: “Note that despite the concluding sentence he is not denying the process of what he calls ‘historical spade-work‘. The notable point he makes is that in each person’s apprehension of history, given the same corpus of spade-work, there will be difference in the amounts of what he calls the ‘become’ and the ‘becoming’, which constitute the individual’s whole picture. Thus, we have the spectrum of states between the two extremes which he characterized by those peculiar terms. Indeed, I would say in Babhru’s vision, undoubtedly installed by his instructors at Indraprastha, history practically ends with the spade-work of fact-gathering. He genuinely thinks that this *is history*. Where I would agree with the śūlapuruṣa is that it is just the beginning and by no means can be considered a complete vision of history. Vidrum’s vision of history in contrast is a collection of fragments of the experiential realm from which kāvya is born in bardic minds. So while Babhru inculpated us of utterly lacking history I must say that we have what the śarmaṇya has called the purer expression of history, whose seeds are clearly experienced by Vidrum.”

Lootika: “I guess Babhru fears the dangerous door of complete subjectivity. Indeed at the center-right debate club which Vrishchika and I attended there was the talk of convincing white indologists by writing objective rebuttals. It struck me that it was not at all an issue of facts per say.”
Somakhya: “hanta! I was not spared of that either. It was none other than our friends Sharvamanyu and Vidrum who tried to seek my assistance in writing such material to post on the internet. You are right; now this is what the śūlapuruṣa alluded to when he talks of the same ‘object or corpus of facts‘ being differently apprehended by different people. The white indologist and their native or Japanese imitators are never going to ever see it like you or me do notwithstanding the number of additional facts we bring to the table. The same things will always produce the same divergent images in them and in us. And if you were to think that is purely subjective, I would say that it is not necessarily the case: In fact it is rather notable that the same body of facts will produces tolerably predictably divergent images in the two groups. In the past there were mleccha historians who could see the facts in large part like ourselves – e.g. Kincaid or Tod, which would also argue against pure subjectivity. When a modern mleccha claims to be ‘correcting’ these earlier mleccha-s through his revised understanding he is indeed falling into the misapprehension the śūlapuruṣa had raised. That is why I admit that he has point when he says history is not just the spade-work of the facts.”

Lootika: “This brings me to a peculiar situation. The center-right fellows had called a banker patronized by the current government of Pratap Simha who delivered a windy speech on why our ārya ancestors did not come to India from the steppes of Eurasia but were rather autochthons of the subcontinent. Most of the fellows there excepting perhaps Sharvamanyu and Vidrum whom we have educated, seemed to fall in line with the nonsense he spouted. As we have seen no amount of facts seem to be able to correct this awful misapprehension of these compatriots of ours. So it would seem that there is a kind disconnect between the factual and experiential when we come to the domain of apprehended history.”
Somakhya: “Indeed, the fascination for the autochthonous ārya-s brings home that issue of that second entity that constitutes history beyond the the śūlapuruṣa-s ‘become’ or the array of facts. This case is rather extreme in that despite the facts being laid out plainly on the table for the two sides to see the proponents of the autochthonous ārya-s seem as if mind-blind despite seeing the facts. This truly illustrates the force of that second constituent the ‘becoming’ of the śūlapuruṣa, which dominates their apprehension of history.”

Lootika: “Moreover it indicates that the misapprehension of history can arise from either constituent: From the force of the an unprincipled ‘becoming’, even as the awakening of a yogin without grounding in the āgama, or from the absence of a sufficient corpus of facts. Verily, we see the latter in the śarmaṇya despite the insights he might have in the intuitive realm. For instance he says: ‘It is the Western world-feeling that has produced the idea of a limitless universe of space — a space of infinite star-systems and distances that far transcends all optical possibilities — and this was a creation of the inner vision, incapable of all actualization through the eye, and, even as an idea, alien to and unachievable by the men of a differently-disposed Culture.‘ Now, the idea of a huge number of unseen star-systems distinct from the one in which we reside was very much a part our own tradition long before the occidental world of Spengler ever came into being. Indeed, it was taught by none other than your ancestors the Bhṛgu-s to the Bharadvāja-s. And even within our tradition the nuance of the very vast universe as opposed to an infinite one was debated between us and the veda-virodhaka-s following the cults of the one who had arrived and the naked one. This ignorance of the śūlapuruṣa arises from the lack of facts, which then made him see such conceptions of space as a unique realization of his world as opposed something which had already arisen in different intellectual milieu among the ancients. ”

Somakhya: “No doubt deep lacunae or misapplication of one or other constituent can prove greatly damaging to the perceived historical vision. I would say in the larger sense this is so even in science, which our śūlapuruṣa tends to place purely in the domain of the laws, i.e. the numerical relationships between the facts. But this alone is hardly sufficient in science. One can have a very accurate purely numerical astronomy which does perfectly well as both a descriptive and a predictive device, yet the deeper insight behind it can be completely absent. The great kṛśapuruṣa, Newton performed many brilliant feats of mathematics and physical experimentation but was able to arrive at the profound penetration like the inverse square law or for that matter even that thing in mathematics we call calculus only due to application of that intuitive component of knowledge-production. Likewise in the realm of history.”

Lootika: “In science both you and me know that well from our actions. We also know that the realm of intuition has its own contours leading to distinct apprehensions that the śūlapuruṣa would call ‘deep and shallow’. Though, I would add that even here there is, at a more basic level, right and wrong because the very superficial could be wrong in a general sense. However, we do find it hard to describe the method of that intuitive insight to others. I noticed that my anujā-s simply would not ‘get it’ when I would try to verbally convey the guhya rahasya-s to them; they would eventually get there only upon performing their own karman following my instructions. Perhaps, that is why Spengler places much weight on JW von Goethe’s vākya: ‘No man can judge history but one who has himself experienced history.‘ But when it come to describing his own intuitive insights in the realm of the historical Spengler lapses into what in his otherwise precise but tedious prose can be called poetic. Look, he says: ‘Countless shapes that emerge and vanish, pile up and melt again, a thousand-hued glittering tumult, it seems, of perfectly willful chance — such is the picture of world-history when first it deploys before our inner eye. But through this seeming anarchy, the keener glance can detect those pure forms which underlie all human becoming, penetrate their cloud-mantle, and bring them unwillingly to unveil.‘ This would almost sound like Yājñavalkya-s conclusion to the śruti of the Vājasaneyin-s where he calls upon the deva Puṣaṇ: ‘hiraṇmayena pātreṇa satyasyāpihitaṃ mukham । tat tvaṃ puṣann apāvṛṇu satya-dharmāya dṛṣṭaye ।‘ (The face of the true is concealed by a golden vessel O Puṣaṇ uncover that so that we may see the nature of the true.).”

Somakhya: “Ah there āṅgirasī you seem to be at doorstead of the method: This is what was expounded by JW von Goethe in his difficult to understand work Farbenlehre (Color theory). There he offers an alternative way of glimpsing the underlying foundations of the observed phenomena. While most kṛśapuruṣa-s held the view that the laws of nature cannot be self-evident and can only be captured by rational analysis, Goethe offers a distinct process of accessing what he called the Urphänomen. He says that any field of experience can be reduced to the fundamental perception of the Urphänomen or the archetypal phenomenon and this would be the self-evident manifestation of the natural law. This has been related to the very sense in which the intellectuals of the yavana-s of yore used the word theorio – to behold – what the śūlapuruṣa Werner Heisenberg pointed as mapping to the episteme and the dianoia of the Platonists. Heisenberg says: ‘Episteme is precisely that immediate awareness at which one can halt and behind which there is no need to seek anything further. Dianoia is the ability to analyze in detail, the result of logical deduction [Footnote 1].‘ Of the former we would say: tasya dṛṣṭyāṃ parokṣaṃ pratyakṣaṃ bhavati ।. It was such a process by which Goethe apprehended the basis of homeotic transformation of floral whorls to arrive at the theory of the origin of flowers (his Versuch die Metamorphose der Pflanzen zu erklaren) and more generally biological evolution. At the heart of this process lies the intuition of finding the right homologies in the sphere of understanding. The successful distinction between homology and analogy is of central importance. In our old world was termed the upa-ni-ṣat, that which one glimpses upon performing karman and is hard to explain in words – a pratyakṣa has to be obtained by action. I would wager that many, if they were to listen in on this conversation of ours, may still not get the method we are talking about.”

Lootika: “Perhaps, hence Goethe too chose to express some of his realization in a poetic form as he said of nature, almost though an early sāṃkhya sage would say of prakṛti: ‘There is everlasting life, growth, and movement in her and yet she does not stir from her place. She transforms herself constantly and there is never a moment’s pause in her. She has no name for respite, and she has set her curse upon inactivity. She is firm. Her tread is measured, her exceptions rare, her laws immutable… Life is her most beautiful invention and death her scheme for having much life [Footnote 2]…’ That last sentence is the expression of insight; probably that’s why Spengler thought Goethe knew of all old Charles had to offer.”

◊◊◊◊

Later that evening as Somakhya was returning home Lootika accompanied him till the point where the road lead to Vidrum’s house. There they met Vidrum who described how Sharvamanyu had had an encounter with some marūnmatta-s from the dargah of Mohammadwadi who were about to engage in arson of vehicles – a tumultuous tale for another occasion.

Vidrum: “The debate with Babhru the other day resulted in us wandering away from the matter of ghāzī sūfī-s. I was wanting to ask you to give me a run down of how they fit into the larger picture of the advance of marūnmāda?”

Somakhya: “Many glimpse local historical incidents individually but identifying the right connections might appropriately place them in the larger canvas. Even as the advance of the marūnmāda in Jambudvīpa was spear-headed by the turuṣka-s, like an advance guard fanning out in front of them, even before the regular army of Islam had been deployed, the ghāzī-sūfī-s marched ahead penetrating deep where marūnmāda had never been seen before. While this advance was stopped, it was clearly able to provide intelligence for the formal army of Islam under the Amir al Momīn. Now this was not an isolated pattern. The same pattern was followed in the west against the sister Abrahamism of pretonmāda. The ghāzī-sūfī-s who provided the religious foundations and the sharia for the Osmans and other Turkic spear-head groups in the west were also the ones who led the forward incursions into the lands of the mleccha-s. Indeed, much of the Osman akinci-s were comprised of such ghāzī-sūfī-s to start with. Their romantic ghāzī literature is quite a mirror image of the same found among the ghāzī-sūfī-s operating against the Hindus. For instance, the ghāzī-sūfī-s of the west also used heavy metal clubs to batter the śavasādhaka-s, much like those in Bhārata bludgeoning the jaina-s and Hindus.

Then there were the Turkic bābā-s who played a role parallel to the ghāzī-sūfī-s in the west like the mariner ghāzī bābā Umur Pasha or the miracle-monger Saltuk who switched between ghāzī and bābā roles in converting Turkic, Iranic and Mongolic peoples around the Black Sea. Likewise, the Mohammedan bābā-s of India fought yogins and human yoginī-s, subverted Hindus with miracle displays, and demolished temples. Such bābā lineages could quickly switch gears from such a state to that of the ghāzī, even as a fungus switches its mating type – such indeed was the genesis of the ghāzī Tipu Sultan in south India. Our people have reached such state of depravity that the word bābā, which applied to such knaves has now become the appellation of choice of pāṣaṇḍa-s among our peoples. Indeed the original bābā-s contributed much to the emergence of a culture of intellectual non-achievement both in the Turkic-Mongol zone as well as India with these tendencies rubbing onto the Hindus. Thus, even if an Ulugh Beg were to arise he would be quickly replaced by mad fuckeers who are worth nothing.

Now, this is not limited to the third Abrahamism; cognates of these exist within the second in the form of the Templar knights and other such orders. They were the front line of the advance both against he marūnmatta-s and our distant heathen cousins the Lithuanians. You know well that they too have their own knaves of the bābā type. Thus, when one observes the larger system of Abrahamism it is rather notable its sub-branches produce similar types. Moreover, these varieties like the sūfī and the bābā are rather parallel in their role in spreading both flavors of the Abrahamistic meme even if at the cost of the their own fitness. One who sees the large picture realizes that just as certain genes which originated in bacteria, such as what we termed the ‘genes of the apoptotic complex’, repeatedly resulted in similar cellular signaling behaviors in different eukaryotes into which they were transferred, Abrahamism too elicited the repeated emergence of similar tendencies in the various groups which it attacked or infected. Sadly, few Hindus are able to see these well enough in order to be able to participate effectively in that conflict which is intrinsic to human existence.”

::::::::::::::::::::::::::::::::
Footnotes:
[1] This treatment of Goethe and Heisenberg is based on: “Goethe’s theory of color and scientific intuition” by AG Zajonc in American Journal of Physics, 44, 4: 327-333
[2] “The Scientific Studies” by Goethe, Johann Wolfgang von edited and translated by D. Miller

## Some meanderings among golden stuff

There are some angles that we often encounter in the construction of the golden ratio and its use in religious art. The first is the most obvious is the angle $\dfrac{2\pi}{5}=72^o$ which is the angle made by the diagonals connecting a side of the regular pentagon to a vertex. Thus, for a unit pentagon $\phi$ is the length of its diagonals. This captures the quintessence, literally the five-ness of $\phi$. This angle was used in Hindu tradition as can be seen in the figure below. Here the angle and $\phi$ are deployed in one of the most remarkable religious works from the Vaṅga country (region of modern Balurghat) — a liṅga on the four sides of which are the four kula-yoginī-s of the directional āmnāya-s. This is keeping with the deployment of the golden ratio in other figures of the kaula tradition like the Kubjikā and śrī yantra-s.

In the previous note on finding the translation vectors for the golden construction we encountered the famous angle $\theta=\sin^{-1}\left(\dfrac{2}{2\phi-1}\right)=\cos^{-1}\left(\dfrac{1}{2\phi-1}\right)=\tan^{-1}(2) \approx63.43^o$. This angle is the supplementary angle of the dihedral angle of a dodecahedron that quintessentially quintessent object. This angle comes up often in the construction of golden ratio with the bhujā-koṭi-karṇa-nyāya, which we place here along with other miscellany for the record. Indeed, Posamentier and Lehmann point out in their wonderful monograph on the golden ratio that Johannes Kepler had said: “Geometry harbors two great treasures: One is the Pythagorean theorem, and the other is the golden ratio. The first we can compare with a heap of gold, and the second we simply call a priceless jewel.” While these are rather elementary constructions, we are just putting down a few of them for own record and the instruction of those in need.

Construction 1:

1) Draw two unit squares sharing a common side $\overline{DB}$.
2) Bisect $\overline{DB}$ to get midpoint H. With H as center draw a circle of radius 0.5
3) Draw $\overline{AF}$. $\overline{AF}$ cuts the circle at points I and J. Now $\overline{AJ}=\phi$. Also $\angle BHA=\tan^{-1}(2)$
From the construction we have:
$\overline{AJ}=\overline{HJ}+\sqrt{\overline{AB}^2+\overline{BH}^2}=\dfrac{1}{2}+\sqrt{1+\dfrac{1}{4}}=\phi$

Construction 2:

1) Draw three unit squares with each adjacent one sharing a common side.
2) Draw $\overline{ED}$. Note $\angle BED=\tan^{-1}(2)$. Bisect $\angle BED$ to get point K. $\overline{AK}=\phi$
From construction we have:
$\overline{ED}=\sqrt{\overline{BE}^2+\overline{BD}^2}=\sqrt{1+4}=\sqrt{5}$
Using the angle bisector theorem we have:
$\dfrac{\overline{BK}}{\overline{DK}}=\dfrac{\overline{BE}}{\overline{DE}}$. Let $\overline{BK}=x\; \therefore \dfrac{x}{2-x}=\dfrac{1}{\sqrt{5}}$

$\therefore x=\dfrac{2}{1+\sqrt{5}}=\dfrac{1}{\phi}\therefore \overline{AK}=1+\dfrac{1}{\phi}=\phi$

Construction 3

1) Draw a square ABCD with sides of length 2 units.
2) Obtain midpoint E of $\overline{CD}$ and join it point A and B to obtain $\triangle ABE$. Note $\angle ABE=\tan^{-1}(2)$
3) Draw incircle of $\triangle ABE$ having determined incenter I. It touches $\overline{AB}$ at F.
4) Using $\overline{FI}$ as radius and F as center draw another circle to obtain point J. $\overline{AJ}=\phi$.
From construction we have:
$\overline{AE}=\overline{BE}=\sqrt{5}\; \therefore perimeter(\triangle ABE)=2+2\sqrt{5}$

Radius of incircle $r_i=\sqrt{\dfrac{(1+\sqrt{5}-\sqrt{5})(1+\sqrt{5}-\sqrt{5})(\sqrt{5}-1)}{1+\sqrt{5}}}=\dfrac{1}{\phi}$
$\therefore \; \overline{AJ}=\phi$

A construction such as these can be also used to easily construct a model of the great pyramid of king Khufu of Egypt. We illustrate this below using construction 2 with 3 unit squares:

1) The first of these is used to construct a square of side 2 units BEDC, which forms the base of the pyramid.
2) We then use the above construction to obtain $\overline{OF}=\phi$.
3) We then deploy the geometric mean theorem in the vertical plane on $\overline{OG}$ and $\overline{OF}$ to obtain $\overline{OA}=\sqrt{\phi}$, which will be the height of the pyramid.
4) We then join the vertices of the base square BEDC to form the model pyramid of Khufu.

One will notice that the cross-section of the great pyramid is the $\triangle GAL$ with base 2 and isoceles sides $\phi$. This triangle is the same as the great śrīkaṇṭha triangle of the śrī-cakra. Possibly the Hemiunu used such a construction to model his pyramid. There has been some debate since the modern rediscovery of the golden ratio in the great pyramid as to whether the Egyptians really knew of it or not. When we account for the the remarkable casing stones and the damage over the centuries the dimensions of the great pyramids does come strikingly close to the above constructed model. The clinching factor is the statement of the yavana Herodotus who evidently recording an Egyptian tradition noted that the square of the height of the great pyramid is equal to the area of triangular faces. From the above construction one can see that this implies that the pyramid had to have the height $\sqrt{\phi}$ and the isoceles sides of the cross-section of through the middle of the square base being $\phi$. Thus, some form of expressing the golden ratio, perhaps in the form of a construction like the above was known to the Egyptians.

This makes the great pyramid at ~2560 BCE perhaps the first monument in world history to encode this famed ratio. It is simultaneously a stark reminder of a great ancient civilization and how it can vanish despite it achievements. But due to the encoding of this information on a monumental scale a glimpse of ancient Egypt’s knowledge has come down to us. Indeed the scale and the brilliant engineering of the pyramids has helped them survive the attempts to erase them: Al-Malik al-Aziz Uthman the son of the counter-crusader Salah ad-Din who wanted clean Giza of its jāhilīyah devoted a large amount of labor to demolish the pyramids but only having succeeded in damaging the smallest of the big three eventually gave up. Then the Mamluqs of Nasir-ad-Din al-Hasan carved out the casing stones of the great pyramid to build masjids but could not get down the colossus. However, there is no guarantee that in the coming years a Mohammedan upheaval in Egypt does what their predecessors wished but failed to do.

This brings us to whether the Hindus and yavana-s obtained the idea of using $\phi$ from the Egyptians. This is plausible since they are seen using a construction that embodies this ratio much before the other two. The yavana-s show a clear record of the ratio and specific constructions for the first time in the work of Euclid. Given Euclid was recording earlier geometric knowledge it might have been known sometime before him as suggested by $\phi$ used by Phidias in several architectural features of the Parthenon. This was around the same time Herodotus records relationship concerning the great pyramid. Hence, we at least have evidence for Greek-Egyptian contact where a construction involving this ratio is recorded supporting the possibility that the yavana-s obtained the concept from the Egyptians before it was mathematically formalized in Euclid.

The Indian situation is less clear but offers certain interesting clues. The as yet mysterious Harappan civilization was undoubtedly in contact with Egypt at a time reasonably close to when the great pyramid was constructed. Yet, I have so far not found any evidence for the use of golden ratio-related constructions in the Harappan artefacts that I have seen. The Indo-Aryan tradition abounds in extensive geometric constructions related the śrauta ritual. There is no evidence for the use of golden ratio-related constructions in the early Indo-Aryan śrauta tradition among the bricks, altars or ritual halls. Since this tradition was something derived from the common ancestral tradition that also seeded Greek geometry, it is likely that in old Indo-European tradition the golden ratio did not play any major role. However, in the late Atharvavedic tradition we find the construction of a regular pentagonal altar for Vāyu where the ratio could have been involved (“vartulaṃ pañcakoṇaṃ…vāyavyāṃ pañcakoṇaṃ tu vāyavyeṣv api karman ||“).

On the other hand we see a completely distinct mathematical approach which yields a series whose limiting ratio is $\phi$: the Meru of Piṅgala. This at its latest would have been close to Phidias in antiquity but there is no evidence that the numbers of the Meru were used in a geometric construction. With the rise of recognizable Indian iconography, i.e. the iconographic tradition that has remained rather conservative since then, suddenly the golden ratio appears in certain examples. Below is one of its early appearances in a tāthāgata context from Sarnath.

While iconographic examples such as these might be disputed, the śrīcakra is a rather clear illustration. Any direct connection to yavana tradition for these Hindu examples can hardly be established. Moreover, the ratio does not play any role to our knowledge in the classic Hindu successor mathematical tradition of the śrauta constructions. Thus, it appears to have been a para-mathematical tradition that primarily survived in a religious context illustrating that classical Hindu mathematical tradition does not encompass all of Hindu mathematical knowledge. Other examples of such include the Platonic solids in bead manufacture in ancient India or the ellipse in temple architecture.

## A golden construction

Anyone with even a small fancy for geometrical matters would have at some point in their lives played with the golden ratio ($\phi=\dfrac{1+\sqrt{5}}{2}\approx 1.61803398875$). Indeed, we too have had our share of fun and games with the golden ratio. In course of this we stumbled upon what seemed to us an interesting geometrical problem although it is likely to be seen as a trivial issue by mathematicians. $\phi$ was apparently rather important to the yavana-s. Among the Hindus as we have noted before, it was used in the construction of the śrīcakra, the primary yantra of the śrīkula tradition.

Problem and the rules of the game
Starting material: A unit square. For convenience we place one vertex at origin and one side on the x-axis.
Objective: To construct a recursive golden rectangle and a golden spiral of any given resolution using this unit square. At the risk of sounding slow-witted I must emphasize this is not the same as drawing a golden rectangle and sectioning it.
It is well-known that the recursive golden rectangle is constructed by starting with a unit square and repeatedly drawing squares which are scaled by a factor of $\phi-1=\dfrac{1}{\phi} \approx 0.61803398875$ and arranging them in an inwardly spiralling fashion. The golden spiral is obtained by drawing a quadrant arc using one vertex of each of these squares as the center and the side of the square as the radius such that we get a smooth curve. Figure 1 shows a few iterations of this being done manually.

Figure 1

Now doing the above is a tedious manual procedure with its limitations. So the question was can we do it relatively automatically taking advantage of a modern construction software like GeoGebra. One could technically compute the coordinates of each new square and draw them out with such a program but this is tedious too and needs some rather unappetizing programming sleights. Instead, we wish to achieve this construction using only the following three allowed operations, which can be easily automatically repeated in construction software (e.g. GeoGebra) to obtain a recursive golden rectangle of spiral of any desired resolution:
1) A single vector which can translate (rectilinear displacement) the starting unit square by a given distance in a given direction. Figure 2 shows the displacement of our unit square ($square_1$ by the vector $u$ to get $square_2$.
$u=\begin{bmatrix} -1.66\\ 0.73 \end{bmatrix}$

Figure 2

2) Dilation (scaling) with the origin as the center of dilation. This operation allows one to scale an object (in our case the unit square) by any factor such that ratio of the distance of any point on scaled square from origin to distance of its equivalent point on the the unit square from origin is equal to the scaling factor. Figure 3 shows such a dilation operation performed on $square_2$ by $scaling\;factor=\dfrac{1}{\phi}$ to get $square_3$.

Figure 3

3) Rotation of an object about origin by a constant angle in one direction. Figure 4 shows the rotation of $square_3$ by $90^o$ to get $square_4$.

Figure 4

To get a flavor of how this works when done recursively we start with our unit square $square_1$, which is what we need to construct the golden rectangle, and inscribe a circular quadrant inside it. We then apply following series of operations recursively on these two object: Translate by vector $v$, then dilate by a certain scaling factor and then rotate by a given angle. For obtaining the golden rectangle, one can find the rotation angle easily. As one can see from the manual construction in Figure 1, for a recursive rectangle each successive square must be rotated by $90^o$. The scale-factor is also obvious because we are aiming for a golden rectangle; the square must be successively scaled by the factor of $\left ( \dfrac{1}{\phi}\right )^n$, where $n=0,1,2,3\;...$ for each iteration. But these two are not enough as can be seen in Figure 5. There we apply the above rotation and dilation transformations on $square_1$ with an arbitrary translation vector $v$ thus:
$Rotate[Dilate[Translate[square_1, v],\;b^n], \dfrac{n\pi}{2}], n=0\;to\;10$
where the vector is:
$v=\begin{bmatrix} -1.23\\ 0.42 \end{bmatrix}$

Figure 5

We get a spiral arrangement of the squares and the quadrant arcs but clearly this is not the golden rectangle or spiral. Hence, the big question is how do we find the right translation vector and how many such vectors exist which can produce a recursive golden rectangle with the unit square.

Determination of translation vector and construction of the desired golden entities
The determination of the translation vectors to produce the golden rectangles and spirals involves an interesting construction. We are not going to repeat the well-known constructions of the yavana-s by which they obtained $\phi$ and $\phi-1=\dfrac{1}{\phi}$. We start with segments of these values already pre-constructed:
1) Let the unit square be ABCD with point A at origin.
2) From point A in the direction opposite to side $\overline{AB}$ along the same straight line mark the point G at distance $1-\dfrac{1}{\phi}$ and point H at distance $\dfrac{\phi}{2}$.
3) From point A in the direction opposite to side $\overline{AD}$ along the same straight line mark the point F at distance $\dfrac{2-\phi}{2}$ and point E at distance $\dfrac{1}{\phi}$.
4) Draw $\overleftrightarrow{GF}$. Drop perpendiculars to $\overleftrightarrow{GF}$ from points H and E to meet it at points I and J.
5) One will notice that $\overline{IJ}\cong \overline{AB}$ i.e. it is congruent to the sides of the unit square.
6) Now use $\overline{IJ}$ to complete the construction of square IJKL which is congruent to our unit square ABCD.
7) Draw vectors $u_1, u_2, u_3, u_4$ that connect point A to points I, J, K and L. Any of these 4 vectors can serve as translation vectors with the above dilation and rotation factors to give us 4 golden rectangles from the unit square ABCD and 4 golden spirals from the 4 quadrant arcs inscribed in square ABCD (Figure 6 and 7).

Figure 6: single example of golden rectangle/spiral

Figure 7: All four golden rectangles and spirals

This construction reveals some interesting features:
1) The angle between the vectors $u_1$ and $u_2$ is $\dfrac{3\pi}{4}=135^o$ and each vector is separated from the adjacent one by $\dfrac{\pi}{4}=45^o$. Thus the 4 vectors together form a trisection of the angle $\dfrac{3\pi}{4}$.
2)As noted above the square IJKL which determines the end points of the four translation vectors is congruent to the starting unit square but from the construction we can see that it is rotated with respect to it by an angle of $\theta$, where $\theta=\sin^{-1}\left(\dfrac{2}{2\phi-1}\right)=\cos^{-1}\left(\dfrac{1}{2\phi-1}\right)=\tan^{-1}(2) \approx63.43^o$. This angle is the supplementary angle of the dihedral angle of a dodecahedron.
3) From the construction we can show the shortest of the vectors is:
$u_2=\begin{bmatrix} \dfrac{2-\phi}{2\phi-1}\\[10 pt]-\dfrac{2-\phi}{2\phi-1}\phi \end{bmatrix}$

$|u_2|=\dfrac{(2-\phi)\sqrt{2+\phi}}{2\phi-1}$
3) The six ratios of the magnitudes of the 4 translation vectors can be expressed as relationships featuring $\phi$ thus:
$\dfrac{|u_3|}{|u_1|}=\phi$

$\dfrac{|u_4|}{|u_2|}=\phi^3=2\phi+1$

$\dfrac{|u_4|}{|u_1|}=1+\dfrac{1}{\sqrt[3]{\phi}}$

$\dfrac{|u_3|}{|u_2|}=2+\dfrac{2}{\sqrt[3]{\phi}}$

$\dfrac{|u_1|}{|u_2|}=\sqrt{2}\phi$

$\dfrac{|u_4|}{|u_3|}=\dfrac{\phi}{\sqrt{2}}$