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.”

Lootika: “We asked our mother to buy the śṛṅgāṭaka-sūpa for us. She refused as ever barking at us and conjuring up images of various helminthic infections of the brain, Entamoeba, and Balantidium. But all that fell on our deaf ears and we were throwing a tantrum. Our mother stood there and watched to see if the cartman’s procedure was hygienic enough for her standards. Then she suddenly remarked: ‘it is not a bad idea if you get some immunity. The śṛṅgāṭaka-s are bhojya and moderately bhakṣya, so I’ll get you all one each. But not that chickpea side-dish. Instead, I’d substitute it with something at home.’ So she told that vendor that she would take six freshly made śṛṅgāṭaka. He was about to dispense them in pieces of paper which he tore from a book and handed to me and my sister. She forbade him and instructed him to transfer the śṛṅgāṭaka-s directly from his kaṭaha to an empty box in one her bags. Holding that page from the book felt like coming in possession of a Japanese yokāi. In the light of the lantern on the food cart, I caught the printing on the page, which was made of good American paper – it displayed a potash apparatus. Puzzled, I asked I could see the whole book. The cart-man gruffly handed it to me. Leafing through it even as the śṛṅgāṭaka-s were being transferred I realized it was something to possess. I asked my mother if she could buy it from him. My mother gave him a few ₹s extra and got the book for me. That evening eating that śṛṅgāṭaka with my mother’s pickle and reading the very Gattermann, which you just picked off the shelf, I felt I was in the abode of Indra. I wished my mother had bought us at least one more of those śṛṅgāṭaka-s, but she instead also meant to give us another form of immunity. After we were done with dinner, she brought out images of the slices of the brain of a man who died recently. ‘Cysticercosis’ she remarked even as I was shocked to see the ghastly pitted cerebrum. She explained the locations in the brain and where all those lesions were. Then she showed us sections of a liver infected by Entamoeba along with rupture where the amoebae had entered the patient’s lung. Vrishchika was excited beyond words seeing those and made copies which she stored on her computer. I retired to the lab that I had just initiated to do some experiments inspired by Gattermann. When I returned to sleep that night, Vrishchika who lay on the mat beside me was excitedly talking about her readings in parasitology. ”

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.

It is a bright morning with the sun shining amidst the coconut trees. The train is headed to the town of Kumbhaghoṇa in the Dravidian country. Among the throng of travelers is a young man who appears to be in his 20s but bears the mien of one who has seen a lot of life. By the ūrdhvapuṇḍra he wears it is clear that he was a member of the northern branch of the dominant vaiṣṇava sect of that part of the country. It seems to be clearly a different era for among the travelers are soldiers with rifles from a bygone time. They are talking about a great rebellion of god-fearing Mohammedans that they have just put down. Keeping with this we also see a couple of English panjandrums with their hats and revolvers – clearly upholders of the English tyranny in the subcontinent. The said vaiṣṇava is seated beside the window and intensely looks out once in a while but for the most part, is immersed in reading a tome titled the ‘On the origin of species…’ Sometimes he raises his head and makes some notes in a notebook. On reaching Kumbhaghoṇa he spills out of the train with his tin box along with a mass of other travelers and heads towards a stand of bullock carts. After choosing a special one which he evidently seemed to have prearranged he heads towards a village some distance from the town. Upon reaching the village he is seen directing the cart-man to one side of it where stands an exceedingly old temple that does not appear to be in particularly active service. Having bought a ghee lamp, some flowers and basil leaves he goes to the temple. There is no priest there nor is are there any other visitors besides him. Having lit the lamp he does a pradakṣiṇa to the deities. As he completes his pradakṣiṇa he is approached by another temple visitor. He respectfully asks the vaiṣṇava: ‘Oh brāhmaṇa, are you the arcaka of this shrine?” He answers: ‘no, I am from the Karṇāṭa country; I’m visiting here.’ The other man: ‘But you seem to be a learned brāhmaṇa, maybe you can answer my question.’ The vaiṣṇava: ‘Maybe, go ahead.’ The other man: ‘I am aware that the deities of the temple are Kṛṣṇa, Rukmiṇi, Balrāma, Pradyumna, and Aniruddha. But who is the fifth male deity? Some say he is Rāma but others say that is not so.’ Our vaiṣṇava says: ‘ You may say so and take it to be this way; but that is not Rukmiṇi, she is devī Ekānaṃśā. That other deity you can take to be Sātyaki. The man:’OK. Who is this Sātyaki?’. The vaiṣṇava: ‘He is an incarnation of the chief Sādhya, a class of gods who formerly carried out the orders of the śrīman Nārāyaṇa. He was an incarnate hero in the days of the Bhārata to aid the līlā-s of incarnate Nārāyaṇa and Nara. You must hear the Bhārata when it is recited by the Bhāratiyār who visit the village. Having said this we see the vaiṣṇava seek a corner of the temple where he is seen doing japa of the mantra:’vavande vāsudevaṃ ca saṃkarṣaṇaṃ ca । namāmi pradyuṃnaṃ sadyojātam aniruddhaṃ ca । ekānaṃśāṃ prapadye sādhyaṃ prapadye । oṃ ॥‘.

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.

That Ayyangār, whose story I gave you a glimpse of, eventually reached the end of his allotted span of time. As Yama’s dogs with their broad muzzles were about to shred him to pieces, he cried out: “Vāsudeva! Balabhadra! Pradyuṃna! Aniruddha! Ekānaṃśā! and he was borne away to join the vast retinue of Viṣvaksena in the loka known as Vaikuṇṭha. Nobody around him really understood what he had researched and discovered in his life. His best student had only 1/24th of his genius. Before his death, the vaiṣṇava asked him to carefully study that fungus he had discovered on the pods. He never did so but to his credit, he continued to culture it on a bark culture devised by the old Ayyangār. Then he passed it on to his student to do his Ph.D. on that but he made no serious headway beyond continuing to culture it. That student became a lecturer at my university. There was a curious brāhmaṇa student with origins in the Karṇāṭa or Drāviḍā country who was in the bachelors program at that time in his department. He found out that the fungus made a potent cytotoxic compound. This piqued my attention and I decided to determine its structure and try to synthesize it as part of my Ph.D. project. After some effort, I showed that it was a protocatechuate ester derivative of a sesterterpenoid with four sulfurs in a tandem linear linkage.”

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

Vidrum continued the relay undistracted by his friends’ excitement: “Puffed up with my success I decided to use the rest of the time I had on my stipend to attempt a synthesis. It was a tall order and I could not get the epitetrathio linkage and struggled with the heptagonal ring with the oxygen in it. But my successful purification, structure determination and good progress towards the synthesis gave me a respected paper and a fellowship to work upon graduating in Japan. My adviser was jealous of me because he had failed when he applied to the same Japanese fellowship. Also, this cytotoxic compound with a therapeutic potential could bring me some recognition and money. So he decided to thwart me. A couple of years before that point a girl of great beauty but no ability had joined the lab. She was clumsy from day one. She broke an expensive cuvette of the spectrophotometer the first week she was in the lab. She then caused a fire with tert-Butyllithium. But my adviser kept her in the lab for some reason. Perhaps, it was because just like me he too was greatly infatuated by her. Accordingly, I was excited even more than about receiving my Japanese fellowship when she suggested that we spend the weekend at the hotel with some drinks. When I went to use the rest-room she seemed to have slipped a mixture of ganja and opium into my drink. She was doing so under the instructions of our adviser who wished to bring a drug charge on me and thwart my going to Japan. But clumsy as she was, she overdosed the opium and I expired as a consequence. The adviser handed my work to her and she graduated with a Ph.D. for doing nothing. Having passed into the state of a phantom I wanted revenge on them but my vīrya was entirely drained due to the excesses I had engaged in with my killer on the day of my death. Thus, I was consigned to being a benign phantom tied to my favorite book. My brothers had no value for my books sold them to the paper-recycler from whom the śṛṅgāṭaka man obtained it to make his paper cups. It was then that I was recovered by this brāhmaṇa lady here. She and sisters are armed with various mantra-s; hence, I remained incapable of movement as they generally performed powerful digbandha-s to protect themselves. But now good man you have set me free; hence, I’ll take my revenge and come back to do you a good turn.”

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 | Tagged , , , , , ,

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.

ellipse_excentric_circle_Fig1Figure 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?

ellipse_conics_Fig2Figure 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).

ellipse_conics_Fig3Figure 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.

ellipse_excentric_circles_Fig4Figure 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.

ellipse_from_c2_Fig5Figure 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.

ellipse_from_c1c2_Fig6Figure 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.

ellipse_hearts_Fig7Figure 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:

ellipse_ovals_Fig8Figure 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

ellipse_lemniscate_Fig9Figure 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.

ellipse_animation_fig10Figure 10

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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.

Square_interior_problem

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 | Tagged , , ,

Residues of squares, sequence curiosities and parabolas galore

Squares and their residues
This is an exploration of number triangles in the same vein as some other such we have previously described . It resulted in some observations that seemed interesting to us. Some are perhaps trivial but some seem puzzling to us probably on account of our very meager mathematics.

Consider the triangle of squares T_s. The j^{th} row of T_s is comprised of squares of all integers 1:j; illustrated below is the tip of this triangle.

\begin{tabular}{|*{15}{r|}} \cline{1-1} 1 \\ \cline{1-2} 1 & 4 \\ \cline{1-3} 1 & 4 & 9 \\ \cline{1-4} 1 & 4 & 9 & 16 \\ \cline{1-5} 1 & 4 & 9 & 16 & 25 \\ \cline{1-6} 1 & 4 & 9 & 16 & 25 & 36 \\ \cline{1-7} 1 & 4 & 9 & 16 & 25 & 36 & 49 \\ \cline{1-8} 1 & 4 & 9 & 16 & 25 & 36 & 49 & 64\\ \cline{1-9} 1 & 4 & 9 & 16 & 25 & 36 & 49 & 64 & 81 \\ \cline{1-10} 1 & 4 & 9 & 16 & 25 & 36 & 49 & 64 & 81 & 100 \\ \cline{1-11} 1 & 4 & 9 & 16 & 25 & 36 & 49 & 64 & 81 & 100 & 121 \\ \cline{1-12} 1 & 4 & 9 & 16 & 25 & 36 & 49 & 64 & 81 & 100 & 121 & 144 \\ \cline{1-13} 1 & 4 & 9 & 16 & 25 & 36 & 49 & 64 & 81 & 100 & 121 & 144 & 169 \\ \cline{1-14} 1 & 4 & 9 & 16 & 25 & 36 & 49 & 64 & 81 & 100 & 121 & 144 & 169 & 196 \\ \cline{1-15} 1 & 4 & 9 & 16 & 25 & 36 & 49 & 64 & 81 & 100 & 121 & 144 & 169 & 196 & 225 \\ \cline{1-15} \end{tabular}

We then obtain the square residue triangle T_{sr} from it by following operation, T_{sr}[j,k]= T_{s}[j,k] \mod j. The first 15 rows of this triangle are shown below.

\begin{tabular}{|*{15}{r|}} \cline{1-1} 0 \\ \cline{1-2} 1 & 0 \\ \cline{1-3} 1 & 1 & 0 \\ \cline{1-4} 1 & 0 & 1 & 0 \\ \cline{1-5} 1 & 4 & 4 & 1 & 0 \\ \cline{1-6} 1 & 4 & 3 & 4 & 1 & 0 \\ \cline{1-7} 1 & 4 & 2 & 2 & 4 & 1 & 0 \\ \cline{1-8} 1 & 4 & 1 & 0 & 1 & 4 & 1 & 0 \\ \cline{1-9} 1 & 4 & 0 & 7 & 7 & 0 & 4 & 1 & 0 \\ \cline{1-10} 1 & 4 & 9 & 6 & 5 & 6 & 9 & 4 & 1 & 0 \\ \cline{1-11} 1 & 4 & 9 & 5 & 3 & 3 & 5 & 9 & 4 & 1 & 0 \\ \cline{1-12} 1 & 4 & 9 & 4 & 1 & 0 & 1 & 4 & 9 & 4 & 1 & 0 \\ \cline{1-13} 1 & 4 & 9 & 3 & 12 & 10 & 10 & 12 & 3 & 9 & 4 & 1 & 0 \\ \cline{1-14} 1 & 4 & 9 & 2 & 11 & 8 & 7 & 8 & 11 & 2 & 9 & 4 & 1 & 0 \\ \cline{1-15} 1 & 4 & 9 & 1 & 10 & 6 & 4 & 4 & 6 & 10 & 1 & 9 & 4 & 1 & 0 \\ \cline{1-15} \end{tabular}

The rows of T_{sr}
If we look at each row of T_{sr}, we see a mirror symmetry upon excluding the last term which is always 0. Leaving out the final 0, the first and the last n elements of the j^{th} row are the squares 1^2:n^2<j. Thus, the same squares 1^2:n^2 are found at the beginning of each row from j=n^2+1 to j=(n+1)^2. After that a new square (n+1)^2 gets added to this run of squares at the beginning and end of the row. The reason for this is rather obvious: All 1^2:n^2<j will leave themselves behind as residues of the modulo operation with j. j^2 will result in a residue in the form of the terminal 0 of that row. Then (j-1)^2, (j-2)^2 ... (j-n)^2 for n^2<j will leave residues n^2:1^2, i.e., in the reverse order as the starting run. The middle part of each row between these square terms may or may not be square terms but will also symmetrically expand. The pattern of residues is unique for each row of T_{sr} but the basic symmetry of the parabolic ascent and descent is common to all. This is illustrated in Figure 1

square_mod_Fig1Figure 1. Figure 1 shows 9 different rows of T_{sr}. The first row — j divisible by 4; second row — other numbers; third row — primes. The successive values are alternately colored blue and red.

We observe that for j divisible by 4 we get a supersymmetry, wherein each symmetric half of the row is further symmetric (Figure 1, row 1). All other j show a pseudo-supersymmetry (Figure 1, rows 2 and 3). All j also display a central parabolic region but its particular form varies from number to number.

The columns of T_{sr}
Now, if we look at T_{sr} vertically, every column k starts with a 0. It eventually terminates in a continuous run of square terms with the value k^2. Thus, the first column terminates in a continuous run of 1s, the second column in a run of 4s, the third in a run of 9s, so on. In a given column, between the starting 0 and the first k^2 term we observe several notable patterns:
1) Immediately after the starting 0 there is an ascending sequence of square terms. This becomes apparent from the second column, i.e. k=2 onwards, where after 0 we get a 1. For k=3, the sequence is 1, 4 — a new square term 4 is seen. 1,4 then continues as the initial ascending sequence of square terms till k=6. At k=7, we get a new square term 9; thus yielding the ascending sequence of square terms 1, 4, 9, which continues till k=12. At k=13, we get next square term 16, yielding the ascending sequence of square terms 1, 4, 9, 16. Thus, we derived the general formula for the number of terms n in this ascending run of square terms as:
n=\textrm{round}(\sqrt{k}); with the sequence of square terms being 1, 4, 9… \left(\textrm{round}(\sqrt{k})\right)^2
Thus, the new square term gets added when k=n^2-n+1, i.e. at 1, 3, 7, 13, 21, 31, 43, 57, 73, 91… (see below for more on this sequence)

2) The next pattern is more complicated. Starting from the largest and last square term of the initial ascending run of square terms of the column k (described above), we start seeing a series of descending runs, until we hit a terminal 0, which precedes the concluding continuous sequence of square terms k^2. To understand this let us examine the column k=7 of T_{sr}. Below are its first 46 terms:
0, 1, 4, 9, 5, 1, 10, 7, 4, 1, 15, 13, 11, 9, 7, 5, 3, 1, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 49, 49, 49
The first term is the starting 0. Terms 2, 3 and 4 are the 3 ascending square terms 1, 4, 9. Terms 44:46 are the first 3 of the terminal continuous run of square terms k^2=49. In between are the descending runs that are under consideration. From term 4 which is 9 we descend to 5 and then to 1 (terms 5 and 6 of above sequence); this is a descent by 4. Then we have 10, 7, 4, 1, which is a descent by 3. Then we have 11, 9, 7, 5, 3, 1, which is a descent by 2. Then we have 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, which is a descent by 1. For all columns we have a descent by 1. For higher k we see the addition of new runs of descent by 2, 3, 4 ,so on. Thus, for column k=7, we have descents by 4, 3, 2, 1.

3) We notice that in T_{sr}, for a column k, these descents by 1, 2, 3… begin with a certain number that keeps increasing. The descents by each number continue till they hit 0 or the positive integer nearest to 0, which can be attained via such a descent. Thus, the descent by 1 sequence will always terminate in a 0. However, as we saw above, for k=7, the descent by 2 starts with 11; so the integer nearest to 0 that the run can reach is 1. Now if we look at the descent by 1 sequence, the first column, k=1, has just 0 before the continuous run of 1s; so we can take the start of the descent by 1 as 0. For k=2 the descent by 1 begins with 1; for k=3 it begins with 4; for k=4 it begins with 7; for k=5 it begins with 12. Similarly, we can identify the starting integers with which the descent by 2, 3, 4… begin for successive columns of T_{sr}. This is shown in table 1.

\begin{tabular}{|l|r|r|r|r|r|r|r|r|r|} \hline k \textbackslash d & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline 1 & 0 & & & & & & & & \\ 2 & 1 & 0 & & & & & & & \\ 3 & 4 & 1 & 0 & & & & & & \\ 4 & 7 & 4 & 1 & & & & & & \\ 5 & 12 & 7 & 4 & & & & & & \\ 6 & 17 & 10 & 6 & 4 & & & & & \\ 7 & 24 & 15 & 10 & 9 & & & & & \\ 8 & 31 & 20 & 13 & 12 & 9 & & & & \\ 9 & 40 & 25 & 18 & 13 & 11 & 9 & & & \\ 10 & 49 & 32 & 22 & 16 & 15 & 10 & 9 & & \\ 11 & 60 & 39 & 28 & 21 & 16 & 13 & 9 & 9 & \\ 12 & 71 & 46 & 33 & 28 & 19 & 18 & 11 & 8 & 9 \\ \hline \end{tabular}
Table 1. The starting integer of each descent run by d=1, 2, 3...9 (horizontal) for the first 12 columns of T_{sr} (vertical) are shown.

Thus, we get sequences of the starting integers of the descent by a given d in the successive columns of T_{sr}:
d=1; 0, 1, 4, 7, 12, 17… After the starting 0 here successive terms differ by 3, 3, 5, 5, 7, 7…
d=2; 0, 1, 4, 7, 10, 15… After the starting 0 here successive terms differ by 3, 3, 3, 5, 5, 5,7,7,7…
d=3; 0, 1, 4, 6, 10, 13…
So on, as in Table 1. For the first three of these sequences, i.e. the starting integers of the descent by 1, 2 and 3 in the successive columns of T_{sr}, we were able to derive general formulae for the n^{th} term of these sequences:
d=1; \; \left \lfloor \dfrac{n^2}{ 2} \right \rfloor+n

d=2; \; \left \lfloor \dfrac{n(n+2)}{3}\right \rfloor-1

d=3; \; \left\lfloor\dfrac{n(n+6)}{4}\right \rfloor

square_mod_Fig2Figure 2. The number of edges in the graph are shown above it in red . The number of unit square cells in the graph are shown in blue below it.

Remarkably, for the first of these sequences, d=1, we were able to find a clear geometric interpretation. It corresponds to the number of edges found in the alternating square-rectangle graph, which is formed by extending the previous element by a unit (Figure 2, panel 1). This sequence has been described in OEIS as appearing in several surprising contexts relating to packing of shapes and beyond. However, to our knowledge this is the first time it has been found emerging in the triangle T_{sr}. The second of these sequences corresponding to d=2 may also be interpreted as the number of edges in a growing graph of unit squares (Figure 2, panel 2), but describing the growth pattern is less simple than what we see for the case of d=1. One can imagine that these sequences provide analogies for the growth of cell-layers under certain constraints. We are yet to find a clear geometric interpretation for the case of d=3. We have not yet been able to find general formulae for d \ge 4. It is something that mathematicians might be able to find.

4) Then we ask the question as to how many terms the total set of runs of descent in given column k of T_{sr} occupy. For k=1, it is obviously 0. For k=2 the terms are 1,0; thus, the number of terms is 2. For k=3, the terms are 4, 3, 2, 1, 0; thus the number of terms are 5. For k=7, which we saw above, the total number of terms occupied by the runs of descent is 40. We derived the general formula for the total number of terms in the runs of descent for a column k:

k^2-k-\textrm{round}(\sqrt{k})+1

This to our knowledge is a novel sequence whose significance in the context of T_{sr} has not been previously described.

5) From the above results it also becomes apparent that the total number of terms in a column k of T_{sr} before the terminal continuous run of square terms k^2 is given by formula k^2-k+1. This sequence 1, 3, 7, 13, 21, 31, 43, 57, 73, 91, 111 … is another famous sequence with a clear geometric significance. It is the number of cycles that exist in a pyramid graph (also known as a wheel graph). It has many other interesting properties (see OEIS), but to our knowledge, this is the first report of its occurrence in the context of T_{sr}.

Thus, T_{sr} is the mother of many interesting sequences that show up in other contexts with various geometric/topological implications.

The overall structure of T_{sr}

Square_mod_Fig3Figure 3. In the left panel T_{sr}[300,300] is plotted using a different color for each value. In the right panel the values of T_{sr}[300,300] are modulated using \textrm{arcsinh}(x) to reveal the internal structure more clearly.

The above-described row-wise and column-wise patterns in T_{sr} combine to give rise to a further intricate structure that becomes apparent if we visualize T_{sr} graphically by plotting each cell of the triangle in a different color (Figure 3). We see a central family of nested conics along the median of the triangle. This is flanked by further such curves along the medians of the two triangles formed by the median of T_{sr} and also their flanks. A mathematician might be able to provide the equations to describe these curves in T_{sr}.

Finally, if we take T_{sr}[j,k] up to a certain value of j, k, we can ask what are the frequencies of the various integers in it (Figure 4) ?

Square_mod_Fig4Figure 4. The frequencies of the all integers occurring in T_{sr}[1000,1000]. The brown line is the best linear fit for the frequency of the successive perfect squares. The unusually low (red) and high (violet) values among the non-squares are marked.

It is immediately apparent that the perfect squares occur at much higher frequencies than the non-squares. Why this is the case can be generally understood from the above discussion on the rows and columns of T_{sr}, which over-represent perfect squares. The frequencies of successive perfect squares decrease almost linearly (Figure 4, fit line). Among the non-squares, we observe that unusually low and unusually high frequencies are often seen for certain numbers just before or just after the perfect squares (Figure 4). The exact basis of this pattern is in need of an explanation.

The linearization of T_{sr}
Another way to examine the structure of T_{sr} is to linearize it. This is done by concatenating successive rows to generate the sequence f. The first few terms of f are shown below:
0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 4, 4, 1, 0, 1, 4, 3, 4, 1…
We then plot f such that the first term is placed at 0 with f[n] being y-coordinate and n being the x-coordinate (Figure 5).

Square_mod_Fig5Figure 5. A plot of f with successive terms alternately colored red and blue. The bounding parabola (green) and the primary parabolic attractors (pink) are plotted over f.

Given that f is created by linearizing a triangle it is bounded by a parabola (Figure 5) of the form:

y=\dfrac{-1+\sqrt{1+8x}}{2}

Under this parabola the plot of f reveals a complex pattern, which is clearly non-random. There are curves, which might be termed attractors, on which the points tend to occur more frequently than elsewhere:
1) The most obvious attractors are straight lines of the form y=k^2; where k \in \mathbb{N} (Figure 5).
This relates to the preponderance of square terms in T_{sr} (Figure 4).

2) The next most visible attractors are the parabolic arcs that start from the bounding parabola and run towards the x-axis (Figure 5). We term these the primary parabolic attractors. The coordinates of the starting points of n^{th} parabolic attractor, ( x[n], y[n]) are specified by the following algorithm:

Let r_1[j]= j^2;\; j=1, 2, 3...

r_2[k]=\frac{r_1[j]+r_1[j+1]}{2}

r=\textrm{sort}(r_1,r_2)

y[n]=2r[n]

x[n]=y[n] \cdot r[n]

The n^{th} primary parabolic attractors themselves are defined by the equations of the form:

y=a[n]-\sqrt{bx-c[n]}

We empirically determined that b=1.89.

a[n]=\left \lceil \left( n+\frac{2}{3} \right)^2 \right \rceil

Thus, a is a sequence whose first few terms are: 3, 8, 14, 22, 33, 45, 59, 76, 94, 114, 137, 161, 187, 216, 246, 278, 313, 349, 387, 428, 470, 514, 561.
We were unable to derive a general formula for c[n]. However, we empirically determined the first few terms of c to be: 1, 25, 50, 100, 100, 200, 200, 400, 480, 600, 700, 1000, 1100, 1400, 1550, 2100, 2000, 2700, 2800, 3250, 3250, 4200, 3900.
Perhaps a mathematician could derive the a general formula for sequence c and also provide a means of deriving the value of b from first principles.

3) Finally, a closer look reveals secondary attractors in the form of fainter parabolic arcs and lines. We have not attempted to derive their equations. A general scheme for identifying them would be of interest.

In conclusion the humble T_{sr}, which can be derived using elementary school mathematics, spawns some interesting patterns and sequences that might uncannily appear in other places.

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A poll on peoples’ beliefs on reincarnation

Using the Twitter’s poll tool we conducted a poll which gave the below results:
Reincarnation happens:
Believe so but there is no evidence for it: 52%
Sure, have the evidence for it: 20%
No, just mumbo-jumbo: 11%
Do not know: 17%
388 respondents
Most of the people who are in my sphere on Twitter are Hindus from India and North America, with smaller numbers from East Asia and Europe.

Hence, the above result is quite in line with reincarnation being a widespread belief among Hindus. With all the caveats of sample size and this poll not being “scientific” and the like, it is still notable to us that the majority believe in it. 72% in one way or another. Some complained that the negative alternative was too strong. Others said that it was self-evident to them hence the alternatives did not capture that special experience of their self-evident knowledge of reincarnation. All those are valid but we think these alternatives mostly capture the “big picture”.

In Hindu tradition, reincarnation makes its appearance in the late Vedic tradition in the āraṇyaka-s and upaniṣat-s while being conspicuously absent in the early layers of their tradition. Yet parallel beliefs are encountered among their cousins the yavana-s and in a muted form (perhaps due to Zarathushtra) among the Iranians (at least as far as the Zoroastrians are concerned). Thus, it could have been an older belief that came back to the mainstream in the later Vedic layers. It is central to the philosophies of the nagna and the tathāgata suggesting that the belief in reincarnation had taken deep root by around 2500 YBP. Thus, Hindus holding such a belief today is unsurprising. Medieval Abrahamistic thinkers were terrified by this idea taking root in their peoples under Hindu and tāthāgata influences as they thought it to be damaging to their cults. However, this belief exists widely outside the Hindu world, even among peoples smothered by the hand of Abrahamistic delusions. The Turks in Turkey for example widely hold such beliefs; so also do Lithuanians and First Americans to name a few, who have not been in the Hindu sphere of influence.

Our poll and anecdotal conversations we have had over the years suggests that even though the majority of Hindus might not have seen evidence directly, i.e. either through their own experience or seeing cases first hand, they seem to believe in it. Hence, it is one of the few aspects of the Hindu religion that seems to rest in large part on belief. Yet, Hindus were always somewhat uncomfortable with this facet of their religion resting purely on belief. Thus, even before the days of the tathāgata, we hear of an ikṣvāku monarch performing experiments with criminals condemned to death in order the determine the “substance” that left the dead body to reincarnate in another. These experiments, however poorly conducted, were seen as providing evidence for a reincarnating substance and thus providing an empirical basis for the claim. Remarkably, even in our age, one of my own śruti teachers believed that such an experiment had been more recently repeated and the results of yore had been reproduced. Others in the past were less sanguine about such experiments and declared that they found evidence from more indirect sources. The most commonly stated evidence is that of people recalling texts or special knowledge at a very young age, apparently without any exposure. Also included in this class of evidence, is the recollection of events of past births that can then be verified independently. At least 77-78 people in our poll responded that they had evidence – so we assume that quite a number of people feel that they have seen such signs even today.

Finally, in all our inquiries it has been apparent that to most people it is not clear as to what really reincarnates. This was apparent in the responses to the poll. One respondent said that he took reincarnation as an axiom of his worldview. However, it is not clear if a completely consistent worldview has emerged for him by taking reincarnation as an axiom. Inquiries with Hindu teachers who believe in reincarnation usually results in the following answer: There is something called a sūkṣma-śarīra. This is said to “envelop” the ātman and keeps a record of the karman balance of an individual. As long as it is non-zero it results in the ātman experiencing reincarnation. However, when that balance is zero the ātman no longer has the experience of identification with a śarīra attains the state of mokṣa. Of course, there are subtle variations on this theme depending on the teacher and also some more drastic differences with the bauddha-s and jaina-s. Yet they all seem to agree that there is something like the sūkṣma-śarīra that reincarnates. Some of the believers might even give a more precise account of that sūkṣma-śarīra and associate it with the manas or buddhi. Several modern believers (including some Hindu teachers) and investigators of reincarnation state that it is more physical. They say that birthmarks and developmental malformations correspond to lesions from the past birth.

Today it is thought that only cārvāka-s hold that reincarnation does not happen. Many Hindus fear that if this belief was proven wrong then the Hindu religion might crumble. Whether this belief has a leg to stand or whether the Hindu religion will really crumble if it is shown to be false are different questions. There is also an even more fundamental question of whether the notion can be tested at all. Those will be discussed separately in a very different format if the gods are favorable to us. This is just a brief record of the beliefs on the matter of reincarnation that might provide a preamble to that discussion, whenever it happens.

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Making of a modern-day mantra-śāstra pamphlet

Originally, all mantra instructions were oral, keeping with the spirit of the śruti. But over time, starting probably around the Mauryan age, written manuals began to supplement the oral teachings. Thus, through the ages practitioners of the mantra-lore have made pamphlets for self-use, use by students or more general distribution. Until the English conquest, the Hindu, despite having familiarity with the printed mode of text dissemination, preferred the written mode. However, one of the signs of Hindu modernity was the production of printed and subsequently electronic texts. These mantra-manuals were originally written entirely in Sanskrit but with the decline of the use of ārya-speech outside the ritual, bilingual manuals started making their appearance. What we present here is one such bilingual manual. It is made taking advantage of TeX, the famous and innovative typesetting system of Donald Knuth, which allows you to make nice-looking documents. The TeX distribution used is MiKTeX2.9 and the typesetting engine is XeLaTeX. The text is input using the TeXworks environment, which comes with MiKTeX for editing and typesetting. The Devanagari and IAST conversion from an ITRANS input was achieved using the Sanscript converter (http://www.learnsanskrit.org/tools/sanscript). The font for the Devanagari script is the Siddhanta font of Mihail Bayaryn. The manual presented here is a reasonably comprehensive discourse on “The Sāvitrī and the upāsanā of the Deva Savitṛ” (pdf format). It is primarily aimed at a serious student with some familiarity with the śruti.

The source .tex file may be obtained here.

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Some words on mathematical truth, scientific conviction and the sociology of science

Sometime in the bronze age more than one group of humans, including our own Aryan ancestors, discovered that the squares of the two legs of a right triangle sum up to the square of the hypotenuse. This is the famed bhujā-koṭi-karṇa-nyāya, which remains true to this date in Euclidean space. In contrast, only a few of the scientific theories of the bronze age have survived in any form close to how they were originally proposed. Coeval with this momentous mathematical discovery, in the bronze age, most civilizations thought that the sun and the planets go round the Earth. Then a few millennia later the counter-hypothesis that the earth and the planets go around the sun took birth. But it took a long time for the older hypothesis to be falsified and the new one to take root. The new one stood the test of all subsequent falsifications but its actual form underwent many further modifications. This flow of the scientific process has been presented in its idealized formed by the Jewish intellectuals Popper and Kuhn. However, it should be kept in mind that the actual process of science rarely follows the post-facto idealized presentation. In any case, the primary lesson from this abstraction of the scientific process is that science is rather different from the mathematics in one matter.

A mathematical truth once discovered remains pretty much the same. This truth is established by what is termed as a proof in mathematics, which itself is based on an underlying set of axioms (for now we shall set aside the big issue of Gödel’s theorems). The form of the statement of such a mathematical truth, a theorem, might change over time due to the concept of “mathematical rigor” affecting the nature of the proof which is supplied for it; nevertheless, its essence remains pretty much the same. However, unless a scientific matter can be trivially reduced to an underlying mathematical theorem, there is no such truth in science as there is in mathematics. Instead, there are only falsifications and attempted falsifications. A scientific statement which survives all subsequent falsification attempts may be considered a scientific “truth”. More correctly, it may be considered a scientific conviction because, for the most part, it is established in a way quite different from the mathematical truth arrived at by the device of a proof.

Donkey
Figure 1

Yet, there is a basic similarity of a key process used in both mathematical and scientific discovery. The investigation begins with a body of observations. For example, one observes that whatever triangle one draws or conceives the sum of two of its sides is always greater than the third. This can be easily proved under the axioms of Euclidean geometry as in Figure 1 thereby becoming the mathematical truth, the Donkey’s theorem. In science too we begin in the same way by gathering a mass of observations. Then one makes a proposal to explain that mass of observations, which may be termed the scientific hypothesis. Here is where things get different between mathematics and science. The proposal is considered truly scientific only if it offers a specific “prediction”, which can then be tested usually by another set of observations. If these new observations falsify the original proposal, then the hypothesis is no longer considered as a valid one and a new proposal has to be sought to explain the observations. Now, scientific conviction regarding a hypothesis gets established by a large body of supporting empirical observations. This is quite contrary to mathematical proof. A large body of empirical observations supported Fermat’s last theorem, which was then finally proved. All observations within our current reach support the hunch that the logarithmic integral \textrm{Li}(x)> \pi(x) but Littlewood proved it to be false. Similarly, the Mertens conjecture regarding the value assumed by the Mertens function has been proven to be false but no current empirical observation has reached the point where it is really false. Thus, mathematical truth is very different from scientific conviction – a corresponding body of observations as those ‘testing’ the \textrm{Li}(x)> \pi(x) or Mertens conjectures would have made for a strong scientific hypothesis yet that body contributed nothing to the truth of the respective mathematical statements. In this regard it might be pointed out that the mathematicians tend to term their hunches or even well-tested but unproven convictions as conjectures. Some of these which are supported by a large body of downstream evidence but still remain unproven are dignified by the term ‘hypothesis’, e.g. the Riemann hypothesis regarding the connection between the Zeta function and the prime numbers. Finally, it should be stated that even when scientific conviction is established upon successful hypothesis-testing, underlying it is a probabilistic statement. This usually takes the form that given the body of testing observations, the chance of an alternative hypothesis as opposed to the chosen one explaining the observations is some low value.

Often, getting a valid body of observations is itself a limiting factor in science because one may or may not have had the technology in the first place to generate such observations. Further, even with the technology in place, the observation collection might have other practical roadblocks like the capacity of the human or machine observers. Thus, a big part of science is the collection of a clean body of observations – this is often overlooked in narratives privileging the hypothesis-creation step. The availability of technology again plays a central role in the testing of the hypothesis. The observation of gravitational waves or the Higgs boson are classic examples of this. The specific predictions were made a long time ago by the respective hypotheses in these examples. However, we needed all this time for technology to catch up to make the test of the hypothesis.

The role of the idea of proof in establishing mathematical truth, pioneered by the yavana thinkers, played a huge role in their thought process and also that of the traditions which borrowed from them like the Mohammedans and the later Europeans. Among the Hindus, a parallel concept of proof from a set of axioms developed from the linguistic tradition culminating in the work of the sages Pāṇini, Kātyāyana, and Patañjali. The great Pāṇini, after an expansive data-collection foray, created the clean data set of the gaṇapāṭha. This formed the basis of developing a system of proof for a linguistic observation based on certain axioms. As an example, let us take the word mahoraskaḥ meaning ‘he who has a broad chest’, which is a bahuvrīhi compound. How do you “prove” the formation of this compound word from the constitutive root words mahat and uras. Following Pāṇini you get the below proof.

mahat~su+uras~su-> mahat~su+uras~su+ka~p-> mahat+uras+ka-(ānmahataḥ…)-> mahā+uras+ka-> mahoraskaḥ |

Here,’~su’ is a Pāṇinian meta-element, much like the construction of the circle in the above proof of the Donkey’s theorem. It is indicated by Pāṇini’s sūtra: anekam-anya-padārthe | (2.2.24). Likewise, the ending is specified by a samāsānta-sūtra. In this case the uras~su triggers the samāsānta-sūtra: uraḥ prabhṛtibhyaḥ kap | (5.4.151), which brings in the ending and the meta-element ‘+ka~p’ for the ending. Once that has been docked to the terminal one applies the sūtra concerning the meta-elements: supo dhātu-prātipadikayoḥ | (2.4.71), which directs the deletion of the meta-elements. This then triggers a transformation of one or both of the combined elements by a samāsāśrayavidhiḥ. In this case, it is: ān-mahataḥ samānādhikaraṇa-jātīyayoḥ | (6.3.46) which causes a transformation of the mahat to mahā. Then it triggers the sandhi-sūtra-s, which in this case are akaḥ savarṇe dīrghaḥ | (6.1.101) and ādguṇaḥ | (6.1.87) which finally result in mahoraskaḥ (Footnote 1)

Thus, this system provides a means of “proving” the formation of a compound as per the Pāṇinian axioms.

While, as we saw above,  there is a distinction between scientific conviction and mathematical proof, the “hidden hand” of geometry underlies the establishment of a scientific conviction. In physics this is actually not so hidden – it might be directly operating via the reduction of the physics to an underlying mathematical expression. Alternatively, the types of hypotheses that can be created are seriously constrained by underlying geometric truths. This latter expression is also seen in chemistry to a great extent. In biology too we find that the geometric constraints of hypotheses to be a serious player, often but not always relating to the underlying chemistry. In fact, we go as far as to say that the geometric constraints layout even part of the basic axioms from which biology should be built. However, we posit that in biology a second underlying element is critical in constraining the hypothesis that can be formed. This takes the form of the grammatical structures similar to those analyzed by the school of Pāṇini in the analysis of the Sanskrit language. One may see this earlier note for some details (section: An ideal realm with a syllabary?). In conclusion, having an eye for these underlying geometric constraints and the parallel “linguistic” constraints allows one to formulate hypotheses that can produce genuine scientific convictions, especially in biology.

In practice, such an understanding regarding hypothesis-formation, while widespread among physicists and in large part among chemists, is not common among biologists. They have neither a clear idea of the foundational axioms nor the foundational theories of their science. They can still be effective at gathering data, but the pressure from the funding agencies for “hypothesis-driven science” has resulted in a fetish for poorly framed hypotheses or pseudo-hypotheses that are not really capable of producing genuine scientific convictions. However, biology, particularly its study at a molecular level, has drawn a lot of money due to its direct relationship to the human condition via the promise of medical advances. This money, like most other monetary incentives, is available in a competitive manner to biologists. With the competition for money comes the opportunity for winners to lead a life of mores, or even a larger than life existence with wide-ranging world travel at public expense. There are other non-monetary benefits – fame, and adulation via vanity articles in the popular press (e.g. note the vanity article on Voinnet, a French fake researcher in RNA biology in the Science magazine prior to his suspension for faking. He was also conferred some big award and one of his commenders even felt he should have been given the Nobel prize). The display of success in order to win the next round of funding is typically achieved through publications in certain prestige venues, like what the Chinese and the Koreans call CNS (the Cell journal and the magazine Nature and Science). Sometimes just raking up a large number of publications in other respected venues might also do the trick. The availability of big money also allows investigators in this field to run labs like sweatshops and lowers the bar for the employment, thereby letting in a body of less-discerning and/or less-intelligent people into the field. In fact, the widespread lack of foundational knowledge has allowed such individuals to even prosper widely – almost the equivalent of having physicists or engineers with a poor understanding of Newtonian mechanics. Moreover, the widespread lack of foundational knowledge leads to a tendency of it being better to be “vague rather than wrong” – an inverse of the correct scientific attitude (voiced by mathematical thinker Freeman Dyson): “it is better to be wrong than vague.” This manifests in molecular biology and allied fields like immunology in the form of an emphasis on phenomenology and vague models rather crisp biochemical predictions (of course on the other side there is also physics-envy manifesting in the form of worthless mathematicization that yields little biological insight). With such a system in place, we are left with an explosive situation – an unsurprising call to the only too human urge to cheat.

This cheating has taken two major forms: 1) rampant plagiarism; 2) production of fake results. The first is primarily a sociological problem arising from the urge to sequester all the spoils for oneself. However, it also feeds the extensive misrepresentation of scientific results and inflation of particular findings in order to gain an edge against competitors. Not surprisingly, it creates a rather unhealthy social system within science. The second is fundamentally damaging to the science itself for it fills the field with noise. This is compounded by both the drive to publish a large number of worthless papers and the fetish of peer review orchestrated by cartels which work as echo-chambers. As a result, it becomes difficult for the inbuilt corrective mechanisms of science to clean up the mess in piling mass of literature. While I have taken molecular biology as the centerpiece here, it appears that this is a more general problem. It might actually be even more rampant in fields like psychology and also the area of applied medical and nutritional research. This should not be just a cause of concern for the scientists in the field because 1) a lot of research is done on public money; 2) a lot of this research informs medical practice which directly impinges on the health of people; 3) unscrupulous practice in publicly funded science will seep through (via cartel formation) to commercial medical research and practice leading to more suffering for the patients – a striking example in recent times is that of the Italian ‘celebrity’ doctor who claimed to perform tracheal transplants only to end up consigning several of his patients to gruesome deaths; he was prone to faking his scientific results and credentials.

Is there a way out of this? At this moment that does not look easy to me. Very powerful people in Euro-American science are part and parcel of the problem. Those who have read this story of ours before will get a hint. The whole attitude within Euro-American science need to change and some of that has deep connections to the Abrahamistic undergirding of their culture. Sadly, the negatives are worsened by either the ‘gaming’ of or the imitation of the Euro-American system to different degrees by all the eastern nations (China, Korea, Japan, and India being the chief among them). In all this, we see the wisdom of father Manu that the brāhmaṇa’s ethic is needed for such pursuits and that the brāhmaṇa should keep a low-profile staying away from this business of feasting on adulation.


Footnote 1: This example was taken from a learned paṇḍitā Sowmya Krishnapur’s lecture on the bahuvṛīhi compound.

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