The Platonic culmination of Euclid and the pentagon-hexagon-decagon identity

Why did great sage Pāṇini compose the Aṣṭādhyāyī? There were probably multiple reasons but often you hear people say that he wanted to give a complete description of the Sanskrit language. That was probably one of his reasons but was it the central reason or the most important driver of his endeavor. From his own language and the evolution of the language to his times we can see that the Vedic register of Sanskrit was well known but already largely a thing of history in terms of being a spoke language. Yet, Pāṇini devotes an enormous amount of effort in describing it accurately and it still forms the foundation of our effort to understand the earliest words of our ārya ancestors. Thus, understanding the Vedic language so that we can properly understand the religion of our ārya ancestors and perform rituals as enjoined in the śruti was the major reason for Pāṇini-s effort. This indeed is acknowledged by his great commentator Patañjali. One who is well-acquainted with the śruti also realizes that the ārya-s conceived this linguistic framework for understanding the śruti and performing the rituals enjoined by it as having a deeper significance — it offered an means of apprehending the nature of the deva-s and understanding the universe. Hence, the great Aṅgiras Dirghatamas auccāthya states in the śruti:

ṛco akṣare parame vyoman
yasmin devā adhi viśve niṣeduḥ |
yas tan na veda kim ṛcā kariṣyati
ya it tad vidus ta ime sam āsate || RV1.164.39

In the syllable of the ṛc set in the highest world,
therein all the gods have taken residence,
he who doesn’t know that what can he do with the ṛc?
verily only they who know can sit together [in this ritual session].

It was with such considerations in mind that the sages Pāṇini and Patañjali composed their works. Thus, their works, which aimed at developing a certain comprehensiveness of the framework from as small as set of foundational axioms and with as much internal consistency as possible, never lost focus on its relationship with the “root”, i.e. the śruti.

Just as Pāṇini is for the ārya-s, Euclid is to their yavana cousins. His geometric system was in many ways parallel to the vyākaraṇa of Pāṇini. Like the work of Pāṇini it served as the framework of knowledge at large among the yavana-s, but what was the central reason for its composition? In the modern Occident which considers Euclid as one of the elements of their intellectual undergirding this is either completely ignored or wrongly stated as being the need to compose a comprehensive secular geometric text. However, we hear from Euclid’s great commentator, a sage in his own right, Proclus, that the reason was to ultimately develop a geometric framework to understand the “roots” (rhizomata) of existence. These were first expounded by Empedocles in his verse:

And first the fourfold root of all things hear!–
White gleaming Zeus, life-bringing Hera, Aidoneus,
And Nestis whose tears bedew mortality.

Empedocles identified the “roots” with the four deities Zeus (the fiery element= Skt: agni), Hera (the gaseous element= Skt:vāyu), Aidoneus (the solid element= Skt: pṛthivī) and Nestis (the liquid element= Skt:ap). One may compare the initial part of the Empedoclean system with the proto-Sāṃkhya system described in the upaniṣat of the Taittirīya-s.

Plato, who came after Empedocles, incorporated these “roots” into his theory of knowledge as “stoicheia”, which literally mean letters of the alphabet. This hints a parallel to the ārya usage that might lurk behind Plato’s stoicheia because indeed the akṣara-s, the Sanskrit equivalent of stoicheia, are at the foundation of the Hindu system as encapsulated by Pānini in his Māheśvara-sūtrāṇi. However, the sense in which Plato constructed his stoicheia was not by using linguistic analogs but by using geometric ones. He chose for this purpose the only regular solids that can be inscribed in a sphere — i.e the 5 Platonic solids. Plato is said to have learned of these from his predecessors like Pythagoras and his mathematical interlocutor Theaetetus. He incorporated them into an elaborate geometric “atomism” wherein the the fiery element was the tetrahedron, the solid element was the cube, the gaseous element was the octahedron, the stuff of the universe as a whole was the dodecahedron and the liquid element was the icosahedron. However, as his polyhedral system had 5 solids his system ultimately had to break away from the 4 element system of the old yavana-s and converge towards the 5 element system of the ārya-s.

Plato’s polyhedra may be seen as “molecules” because beneath them there lay his true atomic entities namely the 30-60-90 and the 45-45-90 right triangles (the 3 angles of this triangle are given in degrees). Two of these combined to give the equilateral triangle and the square which in turn constituted the four elemental polyhedra (sort of like the dvayāṇu theory of Akṣapāda Gautama). So in a process like the boiling of water into the gaseous state under the influence of the fire tetrahedron Plato saw the water icosahedron break down into equilateral triangles and reconstitute gaseous octahedra. In his system the solid cube had to keep out of these of inter-conversions. Aristotle was uncomfortable with this but the later Platonists including Proclus saw this a genuine feature of the solid element. Proclus explained it by stating that the solid element might be subdivided in interaction with the other two but does not ultimately transform.

In any case, for Proclus the foundation of understanding was based in mathematics. Indeed, he says that it “purifies and elevates the soul even as the goddess Athena dispersed mist obscuring the intellectual light of understanding.” Hence, it was critical to have a complete theory from few starting axioms to construct the 5 Platonic solids that form the basis of existence. This, he says, is the ultimate objective of Euclid — a complete theory from axioms to construct the Platonic solids. In support of this understanding one may note that the key constructions of the Platonic solids come in the last book of the Elements of Euclid. As per Proclus, Euclid was a Platonist who had studied with associates and students of Plato such as Philippus and Theaetetus before composing his treatise. Now the tetrahedron and cube are easy to construct from equilateral triangles and squares. The octahedron is slightly more involved by is also quite easily achieved as a equilateral bipyramid on a square base. However, the dodecahedron and the icosahedron are much more of a challenge. It is this context we encounter the “culmination of Euclid” in the form of the famous pentagon-hexagon-decagon identity of Euclidean geometry. Below we shall journey through this identity, and note its proof and how it specifies the icosahedron. Once, we have an icosahedron we can construct the dodecahedron as its dual.

penta_hexa_decaFigure 1

Consider a regular pentagon, hexagon and decagon inscribed within the same circle (Figure 1). Let their sides be P, H, D. Then, what is the relationship between their sides? The answer to this is provide by Euclid in his Elements, Book 13, Proposition 10:

The pentagon-hexagon-decagon identity( P.H.D): If an equilateral pentagon is inscribed in a circle, then the square on the side of the pentagon equals the sum of the squares on the sides of the hexagon and the decagon inscribed in the same circle.

penta_hexa_deca_BKHFigure 2

Alternatively, if the sides of a regular pentagon, hexagon and decagon respectively form the hypotenuse, greater and smaller legs of a right triangle then their circumcircles are congruent (Figure 2).

Now, the the proof for this as given by Euclid is not entirely obvious and, at least to us, it is unclear if he even gave a complete proof for it. Nevertheless, we can prove it using all the material he has described up to that point and the commentary of Proclus. For the proof one needs to first define that special ratio known as the Golden ratio \phi:
\phi^2=\phi+1
Since this is a quadratic equation it has two roots. We call the larger one of these \phi \approx 1.618... and by definition the smaller one is \tfrac{1}{\phi}=\phi-1. Proclus tells us that it was revealed to the Pythagoreans by the Muse goddess who came to them. He then says that Plato recognized its importance and made conjectures on them which his mathematician student Eudoxus then proved using geometry. It is evidently this material of Eudoxus that is presented by Euclid — for example, he provides a means to construct \phi without calling it that in Book 2 of his Elements. Now this ratio has a special relationship to the isosceles triangle with angles \tfrac{2\pi}{5}, \tfrac{2\pi}{5}, \tfrac{\pi}{5}: in such a triangle the ratio of either of the congruent sides to its base is \phi. This Golden ratio triangle is central to the proof of the P.H.D identity.

penta_hexa_deca.proofFigure 3

Figure 3 depicts the construction required for our proof. We observe that the said pentagon of our identity is in green, the hexagon in blue dotted segments and the decagon in red. The radius of their circumcircle is the same as the side of the hexagon H. From the construction we see two instances of the above-mentioned Golden ratio triangles: One with sides H, H, D and another with sides D, D, 2x. From those triangles we can write the below:

\dfrac{P^2}{4}=D^2-x^2\\[10 pt] \dfrac{P^2}{4}=H^2-(H-x)^2=2Hx-x^2\\[10 pt] \dfrac{P^2}{2}=D^2+2Hx-2x^2\\[10 pt] \dfrac{H}{D}=\dfrac{D}{2x}=\phi \\[10 pt] x=\dfrac{D^2}{2H}\\[10 pt] P^2=2D^2+4Hx-4x^2 = 4D^2-\dfrac{D^4}{H^2}\\[10 pt] P^2=D^2(4-\dfrac{1}{\phi^2})=D^2\left(4-(\phi-1)^2\right)\\[10 pt] P^2=D^2\left(4-\phi^2+2\phi-1 \right)=D^2\left(3-1-\phi+2\phi \right)\\[10 pt] P^2=D^2\left(2+\phi \right)=D^2\left(1+1+\phi \right)=D^2\left(1+\phi^2 \right)\\[10 pt] P^2=D^2+\phi^2D^2=D^2+H^2 \quad _{\blacksquare}

If one were to assume the theorem of Brahmagupta or Ptolemaios regarding the diagonals and sides of a cyclic quadrilaterals one can prove it by an alternative path. We leave this for the geometrically inclined reader to work out.

icosahedron_P_H_DFigure 4

The remarkable discovery of the yavana-s was the relationship of this P.H.D identity to the icosahedron. Consider the icosahedron in Figure 4. By definition all edges of the icosahedron are equal and all faces are equilateral triangles. Now, the edges of the icosahedron are congruent to the sides of the pentagon P in the P.H.D identity. Thus, if one constructs the circle as in Figure 4 we get the circumcircle as in Figure 3. With this circle we can construct the sides of the decagon D as above, e.g. \overline{PQ}=D in Figure 4. Then, the relationship it has to the icosahedron is that \overline{AB}=\overline{PQ}=D. The 5 faces of an icosahedron each of which share a common edge constitute a pentagonal pyramid. The height of this pyramid is D (Figure 4). Now, by construction the radius of the circumcircle of the pentagonal bases of such pyramids in the icosahedron is of length H, i.e. the side of the hexagon in the P.H.D identity (Figure 4; \overline{BC}=H). Interestingly, if we visualize the icosahedron as being made up of two pentagonal pyramids stuck to a central band of 10 facets then the distance between the pentagonal bases of the two pyramids is H (Figure 4; \overline{QR}=H). Thus, the planar P.H.D identity is able to describe the icosahedron in 3D space and we can use it to construct this non-obvious Platonic solid. In order to prove this relationship between the P.H.D identity and the above landmarks of the icosahedron we have to effectively prove that \triangle{ABC} \cong \triangle{PQR} (Figure 4). They are right triangles and their hypotenuse and legs are respectively P, H, D.

penta_hexa_deca.GRD_icosaFigure 5

In order to prove that we have first construct rectangles inside the icosahedron like the one shown in Figure 5. From the above Golden triangle (Figure 3) and its relationship to the pentagon (used in the compass and straight-edge construction of the pentagon; see this note) we can see that such an internal rectangle of the icosahedron will be a Golden rectangle, i.e. ratio of its non-equal sides would be \phi.

penta_hexa_deca_icosahedroFigure 6

We then use this rectangle to make an orthographic projection of the icosahedral construction in Figure 4 on to a plane (Figure 6). In the right panel of Figure 6 we see how this Golden rectangle is placed with respect to key landmarks of the icosahedron. In the left panel we see the actual projection of icosahedral construction in Figure 4. This preserves the mutual relationships of \triangle{ABC}, \triangle{PQR} between 3D construction and the planar projection. We construct \overline{XY}. By examining this figure we observe \overline{XY} || \overline{AC}. Thus, \angle{ACB} \cong \angle{XYP}=\theta. From this, one can show that \angle{PRQ}=\theta. By the nature of the projection we preserve the equivalence of the edges of the icosahedron such that \overline{AC} \cong \overline{PR}. Thus, by Side-Angle-Angle test we can show \triangle{ABC} \cong \triangle{PQR} \quad _\blacksquare

Thus, we complete our journey through one of the profound aspects of Euclidean space — a thread passing through its defining feature the bhujā-koṭi-karṇa-nyāya, the three regular polygons and the regular polyhedron, the icosahedron. When we were young, on account of our fascination for viral capsid structures, we closely studied the geometry of polyhedra by practical means, i.e. making them out of paper. It was in course of this we learnt of the P.H.D identity. We were able to practically confirm that for ourselves based on our models. However, it took us some time before we actually apprehended the proof.

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Leaves from the scrapbook-4

As described here these entries are from the scrapbook of the second of the caturbhaginī, Vrishchika.

Entry 1; kāmāturā, year Siddhārthin of the first cycle: Indrasena departed for grad school last week and I am feeling a bit of loneliness of a kind I did not even feel when Lootika left. In the midst of all this I did not even register the fact that I have completed my basic medical degree in good time. But dear my agrajā’s words come to mind that the true prowess of a person becomes apparent only when they prove themselves all on their own against hostile forces. I was also reminded of her admonishing me that I was way too swayed by sentiments of attachment and eros. But sometimes I feel concerned that she might have taken that too far — she has totally lost all contact with Somakhya for an year now. At least now Somakhya has Indrasena beside him but my pretty agrajā is all by herself battling hostile mleccha-s in the quest for glory.

Thankfully, due to the favors of the gods I am likely to get any internship I would like, but that is just practice. It goes without saying that I would be pursuing experimental medicine for that was how I always supposed to be doing. In any case, I have got a waiver to already start accompanying and assisting my senior Vidrum on the rounds till my program formally opens. I guess he harbors some edginess towards me right from school days but still tolerates me because of all the curricular help I have rendered him over the years and his friendship with Lootika and Somakhya. One of my reasons for writing this entry was an interesting case that confronted us last week — it brought a closure to me in an unexpected way. A young mleccha man arrived with acute cardiac symptoms. He had a generally culprit-free angiogram but rapidly deteriorated and died a couple of hours after admission. Since it was an unusual case we pursued it further for it might teach us something we did not know. One thing which became apparent was that he did not show any elevation of cardiac-specific troponins in the blood. However, his cardiac muscles showed a dramatic fall of calcium ions in the sarcoplasmic lumen a sign that his SERCA calcium pump was compromised. Yet, it was totally normal in sequence.

An interview with his parents who was accompanied by an official of the embassy of the said mleccha country revealed something interesting. They were working for a NGO for the “inculcation of scientific temper among Dalits and tribals”. I wondered what that could mean. They were full of pride on their son and declared that he was a great genius from childhood and a “Do it yourself” scientist. However, they said he had a strong suicidal tendency; hence, due to certain episodes they had brought him over from their country to stay with them for sometime and cool off. They showed me some of his notebooks, which indicated that he was indeed some kind of a polymath with ranging interests and insights. Moreover, he was a DIY molecular biologist. That stirred something in me for it reminded me of none other than myself and my sisters. The scan of his notebook revealed that he had engineered a variant of the SERCA inhibitory peptide phospholamban, which was effective in shutting down SERCA in the heart. A closely analysis of his tissues revealed that it was by means of that peptide he had sent himself to king Vaivasvata — he thought no one would ever figure out that it was a suicide. Via Vidrum I provided a report to his parents and the official and closed the case.

I was struck by the strangeness of all this. His death reminded me that often a great tree grows in a jungle and then one fine day is struck down by the indra-senā or the wrath of the pitiless Vāyu as our Iranian cousins would have said. Thus, indeed this young man had seen many realms of knowledge but no one knew that he had reached so deep for he had fallen like that tree in a lonely boreal forest with no one to hear the crash of its fall and thereby judge its great mass. But the case brought to something more sinister to my mind: once when we were young and had expressed our svābhāvika-siddhi-s for the first time Lootika and I had made the ghost of a brāhmaṇa speak. He prophesied my vīra but at the same time mentioned that a prativīra would also emerge. But the ghost added that we would not be affected by the prativīra for he would attain Vivasvān’s mace-wielding son even before he could engage in a contest with us. I realized that the individual was the prativīra and that he had passed away just in front of my eyes. That also gave me a signal because Indrasena had told me via the kauberī-vidyā that after the prativīra has departed we will face the true contests of our life even as I proceed towards the bus stop at Mlecchadigdvāravṛti.

Entry 2; rakta, year Siddhārthin of the first cycle: Things seemed to precipitate by the day as though drawn by fate inexorably towards a certain preferred unfolding of events things. Or at least that is how the human mind tends to rightly or falsely pick patterns from the noise in the background. I was part of the team attending to a trauma case involving activists of the mandira vahīm̐ banega group who were holding a meeting for mobilization for the reconstruction of the temple for the Ikṣvāku heroes at Ayodhyāpuri. As they were proceeding to their office beside the Vināyaka shrine on the Somamārga for a meeting they were attacked by assailants who had vanished almost without a trace. The eyewitness declared that the shots came from the snaking queue, which was awaiting a darśana at the shrine and that the assailants created some chaos in the line by shooting in the air and took advantage of that to merge into the crowd and escape. Some even insisted that the assailants wore saffron robes and were actually worshipers at the shrine. In any case the mayhem left one of the activists with a bullet lodged in his left clavicle. We were able to quickly stabilize his wound and save his life. However, the secretary of the mandira vahīm̐ banega committee had succumbed to his injuries by the time he was brought to us and the only task we were left with was the autopsy. He had two bullets which had entered his skull through the occipital region — one had exited through the right frontal while the other was lodged in the brain.

There was some discussion after the autopsy which involved one of our seniors the resident Samikaran and Vidrum’s new friend, another resident Marxenga Sen. They both remarked that his death was probably fully deserved because he was there to cause trouble. They added: “After all had the Chief Justice Shashiyabh not declared that such people do not want the country to ever be in peace.” Duly, I did not pass the opportunity to rile them by stating that it was not the mandira vahīm̐ banega group but their mustache-less bearded friends who were the biggest cause of not just the country not but the world not being at peace. They went on like a person on bhaṅga or a song playing in a loop on how the assailants were certainly Hindu terrorists as they were saffron-clad and visiting the temple and that it was an internal gang-fight among the Hindu terrorists. I quickly made my exit wondering when these might join the beards or the well-known socialist terror movement.

Just then my mother had called me and asked me to catch sleep in the residents’ lounge. She did not want me coming back home that night because apparently there was an operation to apprehend Saif ad-din as-Sullami on way between the hospital and home. She said that she would pick me up in the morning. That name sounded vaguely familiar; so, I looked him up and found that he was known to be a peddler of a flavor of music called sufi-pop, which accompanied gyrating dances that were popular with a section of the educated youth with a proclivity towards debauchery. There was a rumor in certain circles that he was also involved more serious stuff like using his musical outlet for love-jihad and running a hitman service for a national and international clientele of criminals. It was seemed that he was behind the shooting that took place earlier in the day and had also played a big role in the UCC riots. I wondered his men were among those who had tried to kidnap me last year during the UCC riots.

As that scary incident flashed in my mind I took out my knife and garala-śaṅkula and placed them on my desk. Jhilleeka had improved the design of the garala-śaṅkula such that it was less-likely to jam as it had done during my encounter with the marūnmatta-s. I even have a third weapon, the śilī which Indrasena had gifted me along with the instructions of its use. It is a small peg which can be used to impinge on particular points that were termed marman-s in the old Hindu medical tradition. Our ancestors held that: “marāyantīti marmāṇyucate ।”; hence, pressing at those points with the śilī could in the least incapacitate the attacker. These points of the old tradition are mostly a secret and mentioned in the old medical literature. In Cerapada there is also a legend that the twin Aśvin-s appeared on earth to teach the secrets to their kṣatriya-s to counter their drāviḍa cousins to the east, while the latter were favored by Bhava. I realized that one could fairly precisely strike these marman-s by locating certain superficial blood vessels, fascia, tendons and aponeuroses, which tradition called sīrā- māṃsa- snāyu- and saṃdhi-marmāṇi. I had a good sense of their location from actual anatomical examination. I also felt I was much better than my dear agrajā in wielding these implements and could actually cause some serious damage even if I were to be completely overwhelmed by a male assailant. But perhaps this is only feeling arising from the human tendency to overestimate oneself.

Entry 3; rasa, year Siddhārthin of the first cycle: On my way to college I pass by a little park, which originally had a small shrine in the middle of it. Several years ago out of curiosity Varoli and I went to see what was there and found it to house the statue of a man called Rasgol-bābājī. There, a woman told me that miracles could be witnessed by praying to the said bābājī. She explained that Rasgol-bābājī had the power of a 1000 Sai-fakīr-bābājī-s, a 100 śrī-śrī-108-bābājī-s and 10 das-ser-jigar-bābājī-s. Even as I was taking in the magnitude of the hepatomegaly of the last of them, she exuberantly mentioned how bābājī used to appear in her dreams to warn her of all manner of impending disasters. She then went on to say that his student Khambhā-vālā-bābājī was now in charge and that he would occasionally visit the park to dispense miraculous medications that could “cure any disease”. Sometime later I noticed that a new shrine had emerged next to that of Rasgol-bābājī. Apparently, his successor Khambhā-vālā-bābājī had joined his master in the abode of the buffalo-rider and was now consigned to a samādhi beside him.

Vidrum had been informing me of the multiple cases of lead and mercury poisoning he has been encountering in the past couple of years. We checked the data an realized that there was indeed a significant spike. We wondered what might be cause for those cases as our state’s environmental agency had given a clean-chit for our water and food sources. The answer dawned on us with a crash yesterday. We had a “god-man” rushed to our care by his throng of devotees. They claimed he was under an abhicāra from his rival śūl-vālī-mātājī. They added that he could cure himself but his symptoms were so severe that they need us to give him some medication so that he could become stable enough to counter the abhicāra or prepare medications for himself. He was not cooperating constantly stating he could cure himself but was clearly in deep agony. His cheeks were red and he was complaining that scorpions were constantly running under his skin. Reading my name on my badge he became delirious and started yelling that the scorpions running under his skin had suddenly manifested a dangerous woman. Before I could even process what happened next, one of his votaries rushed at me and started pulling my hair while another grabbed my hand. Thankfully, Vidrum who was beside me acted quickly and knocked out the guy who was grabbing my hand. That gave me a breather by which time I had drawn my śilī and bore it down on the exposed sīrāmarman on the attacker’s hand. His hand went limp and I freed my hair and struck him again on his skull with śilī thus making him fall. Vidrum and others signaled me to retreat as they handled the case. The scary memory of the murder of a physician near my school by the irate relatives of a patient flashed in my mind but in the adrenaline rush I was feeling elated as I had never felt recently — I had knocked out an attacker successfully with the śilī! In my excitement, I sent off mails to Indrasena and my sisters describing the incident and asked them not to tell my parents.

Later that day, I heard that Vidrum had referred the god-man to the poison-control department. There the god-man’s anuyāyin-s had caused quite a bit of trouble and took their man away after the department told them that they had to stop him from using mercury and lead and that he had to remain in the hospital for an attempt at chelation therapy. Vidrum said that he was considering reporting the man to the poison-control authorities because he believed he might be behind the metal-poisoning cases we had seen. Out of the blue, Marxenga came to me and started haranguing me that “right wing” types like me were the cause of emergence of likes of that god-man. Vidrum for a change shooed her away. He told me that the supposed holy man was a certain Gajpuṭ-bābājī, the student and successor of the late Khambhā-vālā-bābājī. I looked him up and realized that he ran a popular channel that was frequented by many believing in his miraculous “medicines” for whitening the skin, curing malignancies, diabetes and the like. Vidrum asked me regarding his peculiar name. I explained to him that evidently he had taken it up from a traditional kiln known as the gajapuṭa, which among other things was used in baking mercurials. This led me to investigating his channel and I saw that he had a show in which he described the preparation of a leaden, golden and silver rasagola-s for people to wear. There he took elemental mercury heated it successively with the juices of garlic, aloe, wood-sorrel, betel leaf and the triphala-s, each for the duration of a week in his famed gajapuṭa that had lent him his name. At the end of it he claimed that he had pacified the mercury and started handling it with his bare hands mixing it with powdered silver. Then he squeezed out the mercury remaining from the amalgamation and continued repeatedly heating it with the pastes of various herbs prepared with gomūtra, ajamūtra, nārīmūtra of a menstruating woman and milk. At the end he arrived at an amalgam globe that he advised his devotees to wear or keep in their mouth as a panacea for all kinds of diseases. With that it was clear how our failed rasa-siddha had landed up in our hospital and had probably also sent many of his votaries over too with his mercurials, irrespective of the powers of his rival śūl-vālī-mātājī. I wait to learn more of his fate and that of his enterprise.

Entry 4; vi, year Siddhārthin of the first cycle: A lecturer in the geology department at the university called on me. He was known to Somakhya and had been friendly with him. It was through him he had a passing acquaintance of Lootika and me. He had with him a visitor from Argentina. He had come to ask me if I could somehow arrange to use the CT scanner in the hospital for scanning a rock with a fossil that he had found at Utnoor in Telangana. I asked him for more details. He told me that the Argentinean was an expert on such extinct reptiles and believed that the skull he had found was notable. The Argentinean told me that he believed it was fossil of utmost importance and a relative of the dinosaurs. I took a closer look and could make out the partial contours of a skull. It looked too primitive to be a dinosauromorph or a pterosaur. I realized it might be the skull of an aphanosaur and remarked so. The Argentinean was surprised and he said that nobody he had encountered in India till now had known about aphanosaurs. I told him that I immediately realized its significance because it promises to be the first complete skull of an aphanosaur and we could test the hypothesis of their being close to the ancestry of the greater dinosaurian lineage. I agreed to help them with arranging the scans on condition that they would give me a chance to study the skull too. They agreed to include me as an author on the study. I am sure Somakhya would be as jealous as he can ever be of me when he hears of my luck in this regard.

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From Plato to Euler and back

This is primarily meant as an educational handout on some very basic theorems of geometry that one might have studied in school. Some educated adults whom we asked about these had either forgotten them or claimed to have never studied them. Hence, we provide these here for those who might be interested. In the below discussion all angles are in radians unless explicitly specified otherwise.

There are only 5 Platonic solids
The first question is at least as old as the times of Plato: Why are they only 5 Platonic solids? Platonic solids are regular polyhedra, i.e. they have all their faces as the same regular polygon. The simplest regular polygon is an equilateral triangle. Its angle is \tfrac{\pi}{3}. In order to fold a solid the sum of the angles of the polygonal faces must be less than 2 \pi radians because 2 \pi is the sum of the angles about a point on a plane in Euclidean space. Further, to make a solid we need at least 3 polygonal faces meet at a vertex. Hence, with an equilateral triangle we can make solids with 3 or 4 or 5 faces meeting at a vertex, 3 \times \tfrac{\pi}{3}; 4 \times \tfrac{\pi}{3}; 5\times \tfrac{\pi}{3}<2\pi. These respectively yield the tetrahedron, the octahedron and the icosahedron (Figure 1). With the next regular polygon, a square, we can get 3 squares to meet at a vertex, i.e. 3 \times \tfrac{\pi}{2}. This yields us a cube. Next, with a regular pentagon we can get three pentagonal faces at vertex, i.e. 3 \times \tfrac{3\pi}{5}. This gives us the dodecahedron. Thus, we can have only 5 Platonic solids in 3-dimensional space _\blacksquare.

In principle, one can join the centers of adjacent faces of regular polyhedron to get another regular polyhedron due to their inherent symmetry (Figure 1). Performing this operation one can also see that the Platonic solids have a duality with another Platonic solid: (cube-octahedron) – (tetrahedron-tetrahedron) – (dodecahedron-icosahedron). In each dual pair there is one of the three formed by equilateral triangle faces. Given that the duality operation on the tetrahedron yields a congruent tetrahedron, you again will have only 5 Platonic solids. It made a such a profound impression on me when I first discovered this for myself as a kid that I could entirely appreciate the profound impact the discovery of these solids (apocryphally by Pythagoras and Theaetetus) had on Plato of yore. It tells you some very basic thing about Euclidean space.

Platonic_solidsFigure 1. The 5 Platonic solids and their duals.

The sum of the angles of a convex polygon
A polygon is a closed path with B vertices where adjacent vertices are connected by a single edge. Hence, it follows that the polygon has B edges or sides as well (Figure 2). This gives our next question: what is the sum of the angles of a polygon? The simplest polygon, the triangle, has angles summing to \pi radians in Euclidean space. Every other planar convex polygon of B vertices can be constructed from B-2 triangles all of which share a common vertex (Figure 2). From Figure 2 it is apparent that the sum of angles of the polygon is equal to the sum of the angles of the constituent triangles. Thus, the sum of the angles of a convex polygon is (B-2)\pi radians. This result was first derived by Proclus.

polygon_triangulationFigure 2. The sum of the angles of a convex polygon.

The triangulation theorem
Let a convex polygon have B vertices and I internal points (Figure 3). We triangulate it by connecting the internal points to the vertices of the bounding polygon or each other without any edge crossing over such that the whole polygon is dissected into F triangles (Figure 3). We then ask: Is there are relationship between B, I, F. The answer is the triangulation theorem, which states that F=2I+B-2.

triangulation_theoremFigure 3. Triangulation of a polygon.

Proof:
1) The sum of the angles of the bounding polygon (as shown above) is (B-2)\pi.
2) The sum of all the angles on a plane sharing common vertex is 2\pi radians. Hence, the sum of the angles at all internal points is 2I\pi.
3) Since the sum of the angles of a triangle is \pi, the sum of the angles of all the F constituent triangles of the triangulation of the polygon is F\pi. From Figure 3 it can also be seen that it is (B-2)\pi+2I\pi.
4) F\pi=(B-2)\pi+2I\pi; \; \therefore F=2I+B-2\;\; _\blacksquare

Euler’s theorem

polygon_EulerFigure 4. Two polyhedra and the polygonal tilings derived from them.

The above result regarding the triangulation of polygons in 2D space Euclidean space and Plato’s discovery that only 5 regular polyhedra exist in 3D space are both related to the sum of all the angles on plane about a vertex being 2\pi. This in turn leads to the celebrated theorem of Euler regarding polyhedra, which while deceptively simple had to wait for 2 millennia since Plato’s days. Consider the two polyhedra in Figure 4. The first is a pentagonal pyramid while the second is a hexagonal prism. Let F be the number of faces, V the number of vertices and E the number of edges of the polyhedron. For the pyramid in question we have F=6; V=6; E=10. Thus, 6+6=10+2. For the prism in question we have F=8; V=12; E=18\; \therefore 8+12=18+2. You can try this out for other polyhedra like those in Figure 1. We can as the question: Is there a general relationship here? We observe that always: F+V=E+2. This is Euler’s theorem and and we prove it below.

Proof:
1) We first reduce the 3D polyhedron to a 2D graph (Figure 4). The way we do this is by choosing one face of the polyhedron as the bounding polygon. We then flatten all other faces inside that bounding polygon while still maintaining the topology of the face. This means that a triangular face remains a triangle , a quadrilateral remains a quadrilateral, and a n-gon remains a n-gon in the flattened 2D graph. Thus, the 2D graph is a polygonal tiling of the selected face polygon of the starting polyhedron. This is easy to conceive for pyramids and prisms (Figure 4). It is more complicated for some of the Platonic polyhedra (Figure 5).

polygon_Euler_reductionFigure 5. The polygonal tilings derived from the Platonic polyhedra.

Thus, we observe that the 2D graph retains all edges and vertices of the starting polyhedron. However, since it is constructed by flattening all other faces onto one face we lose that face and the resulting number of tiling polygons is 1 less than the number of polyhedral faces. Thus, the 3D equation F+V=E+2 becomes F+V=E+1 for a 2D polygonal tilings. Proving the latter in 2D effectively results in proving the former.

2) We define the number P=F+V-E for a given polygon tiling. Our objective is to prove that always P=1. We next triangulate all non-triangular polygons in a given polygon tiling into triangles. Let us define T=F_t+V_t-E_t as the number we get from the faces, vertices and edges after triangulation. From Figure 6 it is apparent that for each new edge we add during triangulation we get a new face. Thus, P=T. Hence, proving T=1 will prove P=1.

triangulation_conservationFigure 6. Conservation of F+V-E upon polygon triangulation.

3) Now that we have triangulated the polygonal tiling, we can apply the triangulation theorem on the resulting graph to show T=1. The faces F in the triangulation theorem is the same as the F_t in our triangulated graph; hence we can write F_t=2I+B-2. Further, the number of vertices V_t of the triangulated graph is the sum of the number of boundary vertices of the bounding polygon and the internal points. Thus, we get V_t=I+B. Let s be the sum of the number of sides of all triangles in the triangulation redundantly counting all shared edges for each triangle. Since there are F_t triangles we have s=3F_t. Now s can be expressed in another way. B will be the number of all the sides of triangles making up the boundary polygon. The E_t-B will be the remaining edges. Now they will fall on the side of 2 triangles each (e.g. see Figure 6) so to make up s we have count them twice. Thus, we can write s=B+2(E_t-B).

Thus, we get:
B+2(E_t-B)=3F_t; \; \therefore B=2E_t-3F_t
From the triangulation theorem and I=V_t-B we can write: F_t=2V_t-2B+B-2; \therefore F_t=2V_t-B-2
Plugging the above value of B into this we get: F_t=2V_t-2E_t+3F_t-2; \; \therefore F_t+V_t-E_t=T=1
Since P=T, it follows that F+V=E+1 for a polygonal tiling. From this it follows that:
F+V=E+2 \;\; _\blacksquare

Back to the Platonic solids
We may next ask: Given Euler’s theorem can we prove that there are only 5 Platonic solids? To answer this we need to prove two further relationships first. For this we take into account that a Platonic polyhedron has regular polygons as its faces and that each vertex must necessarily belong to the same number of polygonal faces. Let n be the number of edges of the regular polygonal face of a Platonic solid. Let m be the number of edges of meeting at any given vertex of a Platonic solid. Let s be the sum of the number of sides of all polygonal faces of a Platonic solid counting redundantly. Given that we have F faces in the solid, we get s=nF. Given that we have V vertices with m edges meeting at each of them we get s=mV. Finally, we note that each edge of the polyhedron simultaneously belongs to 2 of its polygonal faces; hence, we get s=2E. From this we get the relationship for any Platonic solid: 2E=nF=mV. For example, in an octahedron we have E=12, n=3, m=4; thus, 2\times 12=3 \times 8 = 4 \times 6=24.

From the above we have for a Platonic solid: F=\tfrac{2E}{n}, V=\tfrac{2E}{m}

Next, we plug the above into Euler’s formula F+V=E+2 to get:
\tfrac{2E}{n}+\tfrac{2E}{m}-E=2\\[7pt] \tfrac{2}{n}+\tfrac{2}{m}-1=\tfrac{2}{E}\\[7pt] \therefore \tfrac{2}{n}+\tfrac{2}{m}-1>0\\[7pt] \therefore \tfrac{2}{n}+\tfrac{2}{m}>1

By multiplying both sides by nm we get the Eulerian inequality for a Platonic solid: 2m+2n>nm

We can now use this to prove that there can be only 5 Platonic solids. For this we should first keep in mind that to have a solid at least 3 faces should meet at a vertex; hence, minimally m=3. We then begin with a triangular face n=3 and plug it into the Eulerian inequality:

2m+6>3m;\; \therefore m<6
Thus, for a triangle, m=3, 4, 5: these yield the tetrahedron, octahedron and icosahedron.

Next we take a square face n=4
2m+8>4m;\; \therefore m<4
Thus, we can only have m=3 for a square face and we get the cube.

Next we take a pentagonal face n=5
2m+10>5m;\; \therefore m<\tfrac{10}{3}
Thus, we can only have m=3 for a pentagonal face and we get the dodecahedron.

Next we take a hexagonal face n=6
2m+12>6m;\; \therefore m<3
Thus, we cannot get any polyhedron because m<3. The same applies for any n>6. Thus there can only be 5 Platonic solids _\blacksquare

Tailpiece
While we can have infinite regular polygons, only the first 3 of them yield a total of just 5 Platonic solids. This is a strong constraint in 3D space; thus, one would expect these Platonic ideals to occur as reflections throughout nature. Indeed, that is the case and we may note the following:
1) The 4 bonds formed by the sp^3 hybridized orbitals of carbon define a tetrahedron. Similarly, phosphates PO_4^{3-}, such as those forming the backbone of DNA and RNA also assume a tetrahedral geometry. Fe_3O_4 assumes tetrahedral crystals. Thus, the tetrahedron is a fundamental structure of nature and life.
2) The cube is a ubiquitous habit of crystals: NaCl is a good example of such.
3) The octahedron is the habit of diamond and alum KAl(SO_4)_2 \cdot 12H_2O.
4) The dodecahedron is the habit of the quasi-crystal of the holmium–magnesium–zinc alloy. The RNA of certain nodaviruses is packed in a dodecahedral form within their capsids.
5) The icosahedron is famously the form of the capsids of numerous viruses. It is also seen in some non-viral protein assemblies like those which form the propanediol utilizing and ethanolamine utilizing bacterial microcompartments.

In the human world, a deliberately made cube used as a gaming die and a fixed weight was seen among the Harappan people of bronze age India. Harappans also made regular tetrahedral weights, such as those found in Mohenjo daro. Some believe that the occurrence of pyrites which are found in certain places in Italy and naturally assume the forms of cubes, dodecahedra and icosahedra might have inspired the yavana-s to think about these solids. Among the yavana-s, we are informed by Proclus that Pythagoras was the first to discover the Platonic solids. However, there is no evidence that he actually knew all 5 of them and that there were only 5. Plato mentions them in his Timaeus and by then it was known that there are 5 of them. It is believed that Plato obtained that knowledge from Theaetetus, his mathematical interlocutor, who is believed to have first recognized all 5 of them. However, the evidence for this comes from an apocryphal commentary on Euclid. Plato thought that the tetrahedron, octahedron, cube and icosahedron corresponded to the 4 “elements” of Greek tradition: fire, air, earth and water. He then added that a god used the dodecahedron “for embroidering the constellations on the whole heaven.” This is perhaps an allusion to the 12 Zodiacal constellations of the Greeks. It is also held that Theaetetus’ explorations inspired the Elements, where Euclid presumably following his successors gives these 5 Platonic polyhedra and indicates that there are only 5 of them. Material evidence for their recognition in the Classical world comes from the discovery of the icosahedral dice from Ptolemaic Egypt. Further, the Romans made metal dodecahedra and icosahedra (mostly the former), whose function remains unknown to date as far as we know.

In Indian mathematics we are unaware of a specific mention of Platonic solids as a group. However, we have evidence for a “ghost-lineage” of Hindu mathematical knowledge regarding Platonic solids in the form of beads shaped as those solids from at least the Gupta age. There is a persistent belief among white indologists and their imitators that Hindu mathematical tradition is a poor derivative of the Mesopotamian and Greek traditions. However, there is absolutely no evidence for transmission of Greek texts with knowledge of Platonic solids to Hindus prior to the 1700s of the common era. Thus, there is no evidence that the Indian production of Platonic solids was inspired by the Greek textual tradition. One may point out that their emergence in Indian art is approximately contemporaneous with their emergence in material depictions in the Classical world. However, the Indian versions are not used as dice as in Ptolemaic Egypt or as made like or of the large size of their Roman counterparts. They are part of a distinctly Indian use in jewelry. Indeed, in this regard one may point out that already in the Sāmaññaphala-sutta-84 the tāthāgata mentions such a well-polished octahedral bead. One may also note that some tāthāgata atomic theories in India saw substances as being formed from their constituent molecules by an octahedral packing. It is also notable that this style of polyhedral beads was transmitted to Myanmar, Thailand and Vietnam along with early gold-working traditions and at least dodecahedral and octahedral beads have been found in those regions.

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Śiva-gaṇa-s and Andhakāsura-vadha in the Vāmana-purāṇa

On Twitter, one of our acquaintances going by the name @GhorAngirasa had a discussion on the significance of the number 66 in śaiva tradition. That reminded us of an unfinished article where we had noted this number in the context of a śaiva paurāṇika narrative. This lengthy discursive article looks at some additional features of the ancient tale of the killing of Andhaka by Rudra beyond what we had discussed earlier in a note on the evolution of the “fertilizing sweat” motif. Accordingly, we took this piece out of the shelf and worked it up so that it could be minimally presentable for publication. It provided as a pdf file because of the length and the number of illustrations in it.

Śiva-gaṇa-s and Andhakāsura-vadha in the Vāmana-purāṇa 

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Packing constants for polygonal fractal maps

Among the very first programs which we wrote in our childhood was one to generate the famous Sierpinski triangle as an attractor using the “Chaos Game” algorithm of Barnsley. A couple of years later we returned to it generalize it as a map in the complex plane. Consider the polynomial equation,

z^m+1=0, where integer m=3,4,5...

The roots of this equation, z_j: z_1, z_2... z_m, define the vertices of a m-sided polygon in the complex plan. For example, if m=3, we get an equilateral triangle defined by the roots z_1= \tfrac{1}{2}+\tfrac{\sqrt{3}i}{2}, z_2= -1, z_3= \tfrac{1}{2}-\tfrac{\sqrt{3}i}{2}.

With this in place the Chaos Game map is defined for a given m as:

z_{n+1}=r(z_n+z_j),

where z_j is one of the m roots chosen randomly with equal probability as the others in each iteration of the map and 0<r\le 1. If r=1 for any m we get a random-walk structure (Figure 1).

chg06_1Figure 1

For other m, r we get chaotic maps and for particular values of m, r we get attractors with a fractal structure. Thus, the Sierpinski triangle is obtained with m=3, r=\tfrac{1}{2}. This fills the triangle defined by z_1, z_2, z_3

chg03Figure 2

For m=4, r=\tfrac{1}{1+\sqrt{2}} we get the fractal street block (Figure 3).

chg04Figure 3

We currently revisited this map because of a curious problem that emerges when we continue this operation as below for further polygons. For m=5, r=1-\tfrac{1}{\phi} (where \phi=\tfrac{1+\sqrt{5}}{2}, the Golden Ratio) we get the fractal pentagons surrounding the interior penta-flake (Figure 4).

chg05Figure 4

For m=6, r=\tfrac{1}{3} we get the fractal hexagons surrounding the interior Koch’s snowflake (Figure 5).

chg06Figure 5

For m=7, r=\tfrac{1}{S} (where S=2+2\cos\left(\tfrac{2\pi}{7}\right), the Silver constant. The Silver constant is an algebraic number which is the real root of the cubic x^3-5x^2+6x-1) we get the fractal heptagon necklace surrounding the interior hepta-flake (Figure 6).

chg07Figure 6

For m=8, r=\tfrac{1}{2+\sqrt{2}} we get the fractal octagon necklace (Figure 7).

chg08Figure 7

For m=9, r=\tfrac{1}{2+2\cos\left(\pi/9\right)} we get the fractal nonagon necklace (Figure 8).

chg09Figure 8

For m=10, r=\tfrac{1}{1+2\phi} we get the fractal decagon necklace (Figure 9).

chg10Figure 9

For m=11, r\approx 0.2209 we get the fractal hendecagon necklace (Figure 10).

chg11Figure 10

For m=12, r=\tfrac{1}{3+\sqrt{3}} we get the fractal dodecagon necklace (Figure 11).

chg12Figure 11

Given that these attractors are fractal, they have an infinite perimeter but occupy a finite area. Thus, one can define a common feature for the above attractors namely “tangency” of the fractal elements, i.e., each polygonal unit is distinct from its neighbor with the same scale-factor but at same time makes a contact with it like a tangent. While for the triangle and the square this definition is a bit murky, it is clearly visible from the pentagon onward. It can be contrasted with other fractal attractors obtained by this method where the elements overlap or are shared. For instance for m=12, r=\tfrac{1}{2+2\cos\left(\pi/6\right)}, we have the expected dodecad structure but two dodecad sub-elements are shared by each of the adjacent elements (Figure 12).

chg_12_wheel_timeFigure 12

In the above examples (Figure 2-11) we have determined for each polygon the value of r which results in a tangent attractor:

m=3, r=\tfrac{1}{2}; m=4, r=\tfrac{1}{1+\sqrt{2}}; m=5, r=1-\tfrac{1}{\phi}; m=6, r=\tfrac{1}{3};\\   m=7, r=\tfrac{1}{S}; m=8, r=\tfrac{1}{2+\sqrt{2}}; m=9, r=\tfrac{1}{2+2\cos\left(\pi/9\right)}; \\  m=10, r=\tfrac{1}{1+2\phi}, ; m=11,  r\approx 0.2209; m=12, r=\tfrac{1}{3+\sqrt{3}}

For m=5..8 it is the reciprocal of 2+2\cos\left(\tfrac{2\pi}{m}\right)=4\cos^2\left(\tfrac{\pi}{m}\right), which are known as Beraha constants ( B_n) that appear in graph-coloring theory. However, for m=9..12 this principle clearly breaks down: e.g. for m=12, we see that \tfrac{1}{B_{12}} yields the overlapping attractor (Figure 12). We have been able to obtain closed forms for the r that yield tangent attractors for all m except 11, where we report an empirically determined approximate value. What is the closed form expression for it? This leads to the question of whether there is a general formula to obtain our r that results in tangency? To our knowledge these have not been answered.

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Cows, horses, sheep and goats

It goes without saying that humans are what they are today because of cows, horses, sheep and goats. Hindu civilization in particular is in the very least the product for the first two, while remaining two contributed to it from “behind the scenes”. The major civilizational transitions in human history were built atop the bones of these four animals: the Neolithic developments in West Asia, the urban Nilotic civilization in Egypt, the rise of the Indo-Europeans on the Eurasian steppes, the rise of the Turco-Mongolic peoples on the same steppes but at their eastern end, and the expansions of the African pastoralists. We were curious if the present distribution of these animals has any notable features or relationship to their history. Accordingly, we used the UN Food and Agriculture Organization data on these animals to visualize their distribution across different countries and continents. Table 1 shows their total population (here the term “cows” stands for what is provided in the FAOSTAT database as cattle).

Table 1. The latest UN FAOSTAT population data
\begin{tabular}{lr} \hline Animal & Population \\ \hline Cows & 1491687240 \\ Sheep & 1202430935 \\ Goats & 1034406504 \\ Horses & 60566601 \\ \hline \end{tabular}

We first plot the populations of each pair of these four animals for all countries that possess both of them using the \log_{10} scale (Figure 1) and tabulate the correlations between them in Table 2.

Livestock_correlationsFigure 1

Table 2. The Pearson correlations between life stock on the \log_{10} scale
\begin{tabular}{rrrrr} \hline & Cows & Horses & Goats & Sheep \\ \hline Cows & 1.00 & & & \\ Horses & 0.85 & 1.00 & & \\ Goats & 0.81 & 0.66 & 1.00 & \\ Sheep & 0.80 & 0.72 & 0.83 & 1.00 \\ \hline \end{tabular}

The populations of the horses and cows are the most correlated pairs. The sheep and goats are the next most correlated. Thus, animals of comparable size tend to be more correlated in their populations. The horse-cow pair is also the primary pair of the old Indo-European pastoralism. It is possible there is a historical echo of the Indo-European conquests that made this pair a widely adopted unit.

Livestock_ratiosFigure 2

We then looked at the distributions of the ratios for these six pairs (Figure 2). In each plot we mark the following: the mean(red), India (blue), Mongolia (green), Kazakhstan (gray), Kyrgyzstan (cyan). We can see that the plot for the cow:horse plot is the least skewed and is approximately normal. India presents a large country that is heavily skewed in the favor of the cow over the horse. The horse evidently did not do very well over larger swaths of India unlike the cow. On one hand this had its consequences for the Indians in course of their conflict with horse-borne invaders. On the other it made India the land of cows. In contrast, Mongolia, Kazakhstan and Kyrgyzstan show the reverse trend, being horse-heavy pastoralist nations. Kazakhstan was one of the first sites of horse-domestication in horse-centric pastoralist Botai culture. This form of pastoralism persisted on the eastern steppes among the Altaic type of peoples long after the Botai people became extinct after colliding with our Indo-Iranian ancestors . Mongolia still retains this state in its most drastic form.

Mongolia is also unusual in favoring both goats and sheep over cows. Notably it also has a comparable number of sheep and goats. The sheep:goat ratio shows multimodality with some countries investing heavily in favor of sheep over goats. This is where the two steppe nations Kazakhstan, Kyrgyzstan differ from Mongolia. India in contrast weighted in favor of goats over sheep, again reflecting the environmental conditions favoring goats. Further, goat pastoralism might have come early to India along with the Iranian farmers who played a major role in the foundation of the Harappan civilization. Nevertheless, sheep pastoralism eventually spread deep into peninsular India with the use of sheep milk and a memory of it is preserved in the origin myths of the Reḍḍi-s of Andhra. The goat:horse ratio is also notably multimodal suggesting that originally goats and horses defined very different pastoralist niches. The effects of this ancient niche distinct appears to persist to this date.

Livestock_Pop_changeFigure 3. India (blue), Mongolia (green), world (black)

The above are based on the latest population statistics but one can also ask about their dynamics. In this regard the FAOSTAT database has data for many countries and the world for about 57 years. One simple way to look at population change in this period is fraction of population change with respect to the start and end of this period. This can be assessed using the formula:

f_{pc}=\dfrac{2(p_f-p_i)}{p_f+p_i}

Here f_{pc} is the fraction of population change, p_i the initial population and p_f the final population. The distribution of f_{pc} for the four animals is plotted in Figure 3. Except for horses which have shown a slight decline, the populations of the other animals have shown growth world-wide. One sees that, except for sheep, the f_{pc} for India is lower than for the world for the remaining animals. In contrast, for Mongolia it is higher than for the world define a truly pastoralist nation. The Indian horse populations have shown a notable decline in this period.

Livestock_dynamicsFigure 4. y-axis in \log_{10} scale.

This measure only gives a coarse view of the population change. Hence, we looked at the absolute population change for 3 nations (Mongolia, India, USA) in Figure 4 over a 57 year period for which data exists. The dynamics of the precipitous decline of the horse in India is apparent. As noted above, much of India is not suited for the horse and in the absence of the old military and royal-display derived pressure to keep the horse numbers high it has mostly had a free fall. USA shows almost the reverse trend with respect horses and cows. We are not sure if this trend for cows relates to the decline in beef consumption among the Americans. The India cattle situation two booms followed by busts. The turning points for these busts seem to correspond to El Niño-related droughts and it needs to be see if they were indeed the triggering factors for the declines in Indian cattle. In Mongolia there seem to be generally similar trends for both cattle and horses. The marked rise in productivity after the fall of the Soviet Empire suggests the release from Soviet collectivism allowed the Mongols to recover their traditional pastoralist lifestyle. The Mongolian situation also shows the strongest evidence for climate effects, given that after the Soviet collapse the Mongolian cattle and horse populations have shown similar busts and booms. These seem to correspond to the aftermath of the severe Mongolian winters known as the jud-s that take a heavy toll on the animals and the pastoralists. However, the warming in northern latitudes might be allowing a rapid bounce back from the juds.

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Fraudulent science by Indians: some really bad news

Around 2011 we were approached by a researcher of Indian origin for a collaboration in biochemistry regarding a family of proteins whose biochemical functions we had uncovered. After more than an year of dealing with him, it became clear that his research practices were questionable. Based on information from a whistle-blower we then reached the conclusion that he was involved in outright scientific fraud. This prompted us to investigate his work a little more and we found at least 15 published papers of his with fraudulent data. This brought to our mind one of the earliest cases of fraud in molecular biology by an Indian postdoc from Mumbai working with Jim Watson. A broader investigation revealed that the person whom we were dealing with was merely one of at least five researchers of Indian background all with professorial positions in the United States of America and India who were involved in a very similar pattern of fraud. This concerned us for it brings a bad name to researchers of Indian origin. However, simultaneously we also observed several examples of similar fraud by people of European, Jewish, Chinese and Japanese ethnicity. Indeed, very recently we had another case to deal with involving comparable, painful fraud by Chinese collaborators. Hence, we thought this is a universal problem with no special predilection for such fraud among Indians.

However, as the days went by since our original encounter with fraud, we seemed to accumulate more and more cases of Indians engaging in such fraud. Recently, we uncovered yet another case of fraud involving a family of proteins whose evolution and biochemistry we had helped characterize in the first decade of the 2000s. This again involved an Indian lead author giving us the gnawing doubt that things were not right among our people. This prompted a colleague and me to look at federal registry of scientific misconduct issued by The Office of Research Integrity (ORI) of the Department of Health and Human Services (DHHS), USA to see if some of the cases we had detected were in it. This register records cases in which fraudulent research performed using public money in form of grants given by the DHHS were reported to the ORI for examination. While the federal register released by ORI does not cover all cases of fraud performed using public funds, it has at least 60 confirmed cases along with the details of the case and the punishment recommended by ORI. These 60 cases were in period from 2011 to 2019 CE. The majority of them were biochemistry/molecular biology fraud of the kind we had encountered in our starting investigation: 1) manipulation of gel image; 2) creation of fake images of biological material; 3) Some cases of simple plagirism; 4) Out right creation of fake numerical data.

We realized that these 60 cases could provide a means of examining if Indians were particularly prone to fraud or not. Notably, the cases released in the register allowed the names to be classified quite unambiguously into 6 ethnic groups (Table 1). Among West Asians there are representatives from each of the 3 Abrahamistic religions of the region.

Table 1
\begin{tabular}{lrrrr}  \hline  Origin & Count & Percentage & Cheaters/million & Cheaters/1000 \\  \hline  European & 27 & 45.00 & 0.14 & 0.67 \\  Indian & 20 & 33.30 & 6.29 & 4.72 \\  Chinese & 7 & 11.70 & 1.85 & 0.57 \\  West Asian & 4 & 6.70 & 0.38 & 2.32 \\  Korean & 1 & 1.70 & 0.59 & 0.25 \\  Japanese & 1 & 1.70 & 0.77 & 0.33 \\  \hline  \end{tabular}

Indians constitute 33\% of the fraud cases in the federal register. This itself is quite striking because it is rather clear that they do not make up that high a fraction of the biomedical research workforce, which is being probed here. Now, the simplest normalization for the counts is by the representation of these ethnic groups in the USA. This population data can be assembled quite easily by an internet search for the period under consideration. Normalization by this population share is shown in column 4 of table 1 as the number of cheaters per million of the population. By this reckoning and Indian would be nearly 46 times as likely to be involved in fraud as a person of European descent.

One could object that the US population share of the ethnic groups is not a valid normalization for it does not accurately reflect their representation in the biomedical work force. While the latter part of the statement might be true it is amply clear to anyone in the business that Indians do not constitute a fraction greater than Chinese in this workforce, making this objection quite facile. Yet, one would want to perform a more objective normalization based on the proportions of these groups in the workforce. This data is much harder to get in a clean form. However, an article published in the Nature magazine “The new face of US science” by Misty L. Heggeness, Kearney T. W. Gunsalus, José Pacas and Gary McDowell (03 January 2017) allows us to get approximate figures needed for such a normalization. This article informs us that in 2014 (which is in the time range of the data under consideration and also given that a typical graduate student/postdoctoral career in about 4-5 years) that there were about 69000 biomedical researchers in the US. Of these about 40020 were of European descent (excluding Hispanics) and 23500 with ancestry in various Asian countries. That primarily includes Indians, Chinese, Koreans and Japanese. To obtain the breakdown we can use two methods: 1) Sample a random set of 100 biomedical research publications and see what proportion of these Asians are found in them; 2) Use the proportions of graduate students from these nations in American biomedical research programs to get a sense of the breakdown. Based on this we can estimate the approximate maximum number of biomedical researchers of these ethnicities to be: 4236 Indians, 12236 Chinese, 3928 Koreans and 3059 Japanese. For West Asians it is harder to get an estimate but based on the biggest contributing group of these the Israelis from a publication in the Jewish newspaper Haaretz we an infer this number to be \approx 1725. This allows a more specific normalization, which is shown in column 5 of Table 1 as the number of fraudsters per 1000 researchers.

The Indians continue to remain the most prone to fraud even after this more realistic normalization. If the 60 fraudsters were evenly distributed among the 69000 biomedical researchers then the probability of finding a fraudster at random would be 0.00087. Given this and their numbers in the biomedical workforce, the probability that at least 20 Indians are frauds in the federal register by chance alone is 4.44 \times 10^{-10} or they are \sim 8 times more likely than other populations to commit biomedical research fraud. From the names of the Indian fraudsters in the federal registry we can infer that in all likelihood all of them come from the “forward castes”. Up to 1/3 of them are likely of brāhmaṇa descent. This means they are drawn from the Indian elite. As can be seen in Table 2 they are drawn from all over India with a particular over-representation of individuals from South Indian states.

Table 2
\begin{tabular}{lr}  \hline  Region & Count \\  \hline  Tamil Nad/Kerala & 7 \\  Telugu states & 4 \\  Bengal & 2\\  Sindh/Panjab & 2 \\  ``Hindi belt'' & 3 \\  Maharashtra & 1 \\  Karnataka & 1 \\  \hline  \end{tabular}

Further, since entry into the US biomedical work force typically involves an IQ test (administered either for admission to an American graduate school or to an Indian institute), we are dealing with people most likely with IQ \ge 123. Thus, what we are seeing is not per say a problem of cognitive capacity but a problem of “ethics” or “corruption” in the Indian cognitive elite.

One could point out that there is some bias against Hindus in the US academia; hence, they might be specially picking on the Indians as opposed to the European origin majority or other foreigners, though all commit fraud to a similar degree. Undoubtedly there is bias — powerful fraudsters of European or Jewish descent are more often “rehabilitated” or overlooked than those of Indian descent. Nevertheless, that is unlikely to have been the primary cause for at least the cases in the federal register. Several of the Indian cases in the register were considered “golden boys” by the American institutions or had vanity articles about their fraudulent research in American outlets. Thus, it does not appear that at least in these cases they were being specially targeted. Thus, we posit that there is a real problem.

A part of this problem is a general one. The Euro-American biomedical research (which sets the trend in most of the world except to a degree in Japan) is beset with several serious problems:
* There is very little attention paid in biology education to the theoretical foundations of the science. We would go as far as to say that less than 50 \% of the practitioners in biomedical research have good grasp of the foundations of biology. To give an analogy of how bad this is, imagine more than half the physicists and real engineers plying their trade without knowing classical mechanics in any serious sense (e.g. having not much of a clue of how to set up a Hamiltonian or a Lagrangian of a system). As a result poor hypotheses abound, which in turn spawn a glut of bad ideas.

* The Euro-American scientific system has an unhealthy model of competition and the fetish of peer-review, which favor both an urge to cheat to get an article published as well as rich dividends for nuanced plagiarism -i.e. plagiarism of ideas without citation rather than outright copying of text. To put it bluntly we have seen some form a plagiarism of our work with total impunity almost every other month in the past few years. Plagiarism also contributes to confirmation bias and fake reproduction of bad ideas. A part of this competition is fostered by big labs in several Euro-American institutions, where powerful principal investigators run the show like industrial sweat shops. These environments also do not allow for proper oversight — I am aware of cases where the graduate student or postdoc did not see the PI for more than a month at a stretch. Further, most interaction was limited to sanitized presentations in lab meeting rather than direct oversight by the PI at the bench.

* The magazine culture: Euro-American biomedical research assessement and funding agencies place enormous emphasis on publication in the two famous British and American science tabloids or the journal Cell, their many offspring and upstarts like eLife, PLOS Biology etc. In some of these venues, especially the tabloids and Cell articles are subject to insane review processes with time lags of 6-12 months from submission to acceptance for publication. These venues are thus high stakes venues that increase the urge for plagiarism and chicanery.

These are general causes that affect both Indians and others. However, as the scientific system in India increasingly emulates these practices in some form, it increases the incentive for Indians in India to commit fraud rather than do good science. Now let us consider some factors that might predispose Indian origin researchers to commit fraud more often than others:
* Some of the fraudsters who are in the federal register come from labs run but Indian origin PIs in American institutions. Their labs are reported as having an unhealthy environment — high pressure to produce results at short notice without adequate mentoring or oversight. We have evidence that this was the case even in the fraud cases which are not in the register. As noted above this is a major recipe for malpractice. Thus, it appears that there is some tendency for Indian origin PIs to be less than professional in managing their labs.

* Indians face major immigration constraints in the US. This can be used as an anvil both by American and Indian origin PIs to pressurize their students, who typically have no other avenues for escape or alternative employment due to sword of deportation hanging above their necks. Hence, science takes a back seat to survival and the incentives are slanted towards getting ahead by means of malpractice.

* Training in Indian schools and colleges does not emphasize aspects of honest scientific practice. Students often manipulate laboratory experiments to get results that their instructor expects. Little training is given in the statistics of variation and experimental error in school and college. For example, in my first semester in college I vividly remember the instructor conducting physics lab demanding that we exactly get g=9.8\tfrac{m}{s^2} in single trial experiment!

* Importantly, most students in the science stream in India have no real interest in science per say but merely see it as a means for obtaining a seat in engineering or medical school. Those who do not make it typically enter the sciences and gradually drift their way through a B.Sc. and then a M.Sc. to finally reach a graduate program. Most of them are not from the cream of the educational system and are often not suited for cutting-edge science. Thus, when they make it to graduate school in the US they come ill-equipped for science and when subject to pressure might have some incentive to “game the system” just as they gamed the exams in India to get ahead.

* After the near-death encounters with Islam and Christianity, Hindu civilization is a shadow of its former self. There is serious decline of the internal system known as dharma. The decline of dharma shows up in the form of loss of discernment regarding the pursuit of knowledge and the pursuit of gain. In the past when a person committed himself to the pursuit of knowledge, there was a strong demand on him to observe certain ethics. These were enforced by the gate-keepers of the system and quacks would be punished. Indeed, father Manu the law-giver has a long list of criminals engaging in fraud like palmistry or future-prognostication, quacks, scams, and fellows claiming qualifications which they do not possess. He recommends that such be caught by the Rājan using investigators sent to keep an eye on such and if their crimes are proven to fine them. However, multiple repeat offenders could be put to death. Similar the legal tradition of Yājñavalkya has various provisions for the punishment of the quack physician.

In conclusion, we do not find any pleasure in presenting this. It personally only affects us in a negative way given that it has brought a bad name for Indian researchers, which is not going to go away in the near future. That is why I tell people that it is better to be a small man doing some low key but real science rather than professor Big who appears in the newspapers but does fake science. The approach of fake it till you make needs to be adopted with greatest care and does not apply to every aspect of human activity. These observations might have much deeper, unpleasant implications. It is relatively clear that the Hindus have not performed too well for being an old nation with clearly visible past achievements after they saw off the English tyrants more than 70 years ago. There are many reasons offered like the effects of the medieval Mohammedan incubus, the English tyranny, and Gandhi and Nehru. The former factors certainly have had their effect. But the past 70 odd years since independence have seen no major revival of scientific pursuit. There have been no major successors to many of the exemplary solitary Hindu researchers from the pre-Independence or early post-Independence era. We suspect that one cause for this is a terrible culture of knowledge generation among Indians in recent times. An offshoot of this is this tendency of dishonesty that we are seeing among Indian biomedical researchers. It shows in terms of tangible technology too: despite having an big need for the aeronautical engine or a proper assault rifle, Indians have had considerable difficulty in successfully mastering these technologies. Facets of this are also seen in other areas of Indian creative expression, e.g. journalism and cinema. I do not watch cinema but I am reliably informed by someone who does that there is some tendency for plagiarism from the occident. As with science, this tendency in journalism along with unthinking adoption of occidental memes are damaging for the nation. In journalism the idea is merely to produce uncritical stuff that the pay-master (e.g. the mleccha) likes. Hence, we feel that this data should be presented so that our people make take deep look at their problems and consider their science policy implementations accordingly.

See also:
Big man’s story
The fake scientist

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