Mongolica: Knowledge preservation and generation, Bolad Aqa and the like

We had earlier written an essay on the preservation and production of synthetic knowledge by the Chingizid Mongols. Here we discuss a few additional points in that regard.

It is clear that throughout the Chingizid clan there was a certain proto-scientific interest right from the beginning. Astronomy was one area that appears to have interested many of them. It perhaps relates to the fact that the full expanse of the sky leaves a profound impression on the observer on the vast openness of the steppe. Indeed, in Mongol shamanic oral tradition there are accounts such as Ursa Major being the banner of the tngri-s. Thus, starting from Chingiz Khan we see considerable interest in patronizing astronomers. In 1218 CE the Chingiz Khan recruited the gigantic, long-bearded Yelü Chucai (said to have been ~2.05 m in height), who came from the earlier Mongolic branch, the Khitan. While Yelü Chucai was an all-round scholar, the great Khan was particularly interested in his knowledge of meteorology and astronomy. The Khan used his knowledge to predict snow late in year on the steppes during his Khwārizm campaign. He also predicted eclipses for the Mongols and showed the Khan a comet towards the end of the campaign in Turkestan, which is likely to have been the 1222 CE apparition of Halley’s comet. Not surprisingly these scientific observations of Yelü Chucai were mixed with a form of Sino-Mongolic astrology. After the conquest of Samarqand, Chingiz Khan set up an observatory there and appointed a cīna astronomer Li, likely at Yelü Chucai’s behest. In course of the Turkestan campaign Yelü Chucai obtained astronomical works of the Hui-ho [from the Mohammedan world] and studied them closely. As a result he detected errors in the Chinese method of calculations and created a revised method for the Mongol calender and astronomical tables.

The next great Khan Ögödei, the second son of Chingiz, after completing the destruction of the Jin was quick to repair the astronomical observatory near Zhongdu in 1236 CE and establish regular observations. After the line of Ögödei was displaced, the title of great Khan came to the line of the fourth son of Chingiz, Tolui, with the latter’s first son Möngke. He had the geometry of Euclid brought to the Mongols from west and had it explained to him. Rāshīd ad-dīn mentions how thereafter Möngke proved several geometric theorems by himself – this might constitute the earliest geometric study in Euclidean geometry in east Asia. Khan Möngke’s brother Qubilai, even while crown prince, had geographers and astronomers from the Hui-hui land [Mohammedan world] and others brought to his court. The work of the Japanese researcher Kodo Tasaka shows that he had them make a terrestrial globe and an armillary sphere for the Mongols. Qubilai’s observatory at Dadu had a wide collection of instruments like these and also sundials and astrolabes. Their next brother, the Il Khan Hülegü, had an astronomical observatory built on the top of a hill at Marāghah in Iran. Here he appointed Nāssir ad-dīn al-Ṭūsī , the Mohammedan mathematician and scientist, whom he had captured during the destruction of the Hashishin-s. Nāssir ad-dīn al-Ṭūsī had internalized a lot of the knowledge of the West Asian “freethinkers”, who in turn had acquired the foundation of their knowledge from the Greek and Hindu worlds. This work lead to one of the best pre-Copernican planetary models in the western world. Nāssir ad-dīn also prepared an “edition” of Euclid that was influential in transmitting Hellenistic geometric knowledge widely. Again when Hülegü found that the Mohammedans with Egyptian ties in his kingdom were plotting an insurrection in league with the Mamluqs, he had all of them killed except the astronomers.

The governor of Tibet in the mid-1300s, Barandnā (Skt: Prajña), the descendant of Qubilai Khan, also commissioned extensive astronomical works deriving from Hindu and Hellenistic traditions that led to Mongolian commentaries being prepared on them. This Hindu astronomy was transmitted in part via the last great tantra of the nāstika-s, the Kālacakra, which had elements modeled after the now lost eponymous saura tantra. Some of this material survives in the ritual manual preserved Sangwar, a Mongol official in the Manchu times from Boro Balgasun in Ordos. Here we see a legendary Khan receiving India calenderical knowledge from goddess Vimā, the daughter of the great Indra. There is a table preserving the old Indian nakṣatra system: “naghšidar-un ner-e odun-u togh-a bui düri ba maqabud ene bu”: The nakṣatra names, number of stars, figure and element. We also see the incidental transmission of ritual of ultimately Hindu origin in the process such as: “basa nigen eketü Mahašuvari tngri”: Another topic concerning the god Maheśvara; “Maqagala baghuqu edür anu”: As for the day Mahākāla descends; “Okin tngri baghuqu edür kemebesü”: As for the day the goddess Śri descends.

Outside of the “Golden family” there were some Mongol intellectuals who played an important role in this knowledge venture of the Khans. One extraordinary individual who was involved in this process was Bolad Aqa. We wished to detail his history in our earlier note centered on Rāshīd ad-dīn but wanted to read more sources pertaining to him; so we left out that part of the story then. More over, a good review on his history role was also provided by TT Allsen. In this part of the note we shall offer a brief summary of his history. In the Secret History the origin myth of the Chingizid Mongols states that they were descendants of the wolf Börte Chino and the doe Qo’a-maral at the holy slopes of the Burqan-qaldun by the Onon river. From them eleventh in descent were the two brothers Du’a Soqor and Dobun Mergen. Du’a Soqor was said to have a single cyclopean eye by which he could see a great distance. He found a vast slab of nephrite which is said to have been his throne [Later Ulugh Beg is said to have obtained the same from Mongolia and embellished Timur’s grave with it]. From the top of Burqan-qaldun he is said to have seen a beautiful woman Alan Qo’a in a camp at a great distance and asked his brother Dobun Mergen to seek her as his wife. Alan Qo’a became the legendary ancestress of the Chingizid Mongols. But her sons are said to have had a rift with the four sons of Du’a Soqor and they parted ways. While Chingiz descended from the former, the latter are said to have formed the four clans, the Dörben irgen. They were ranged against Temujin during his rise. Aligning with the Tayichi’ud they fought him but were forced flee when he destroyed that alliance. Then the Dörben joined Jamuqa against Temujin but he decisively demolished that alliance in 1202 CE. Then they joined the tayang Khan of the Naiman Turks against Temujin but even that alliance was smashed by him in 1204 CE. At this point they surrendered to him and joined his ranks.

One of them, Yürki won Chingiz Khan’s trust and he was appointed as the ba’urchi (chef) of the Khan’s main wife Börte. In addition he served a military role as commander of a hundred men in a force of 1000 that was directly commanded by the Khan himself. His son was Bolad Aqa, who inherited his position as a royal chef under Qubilai. The prestigious position as the ba’urchi allowed him to receive education along side Qubilai’s sons. In course of this, his intelligence became apparent and he was found to have great skill at translating extemporaneously between Chinese and Mongolian. Thus, he was given the additional responsibility of dealing with the Chinese officials who were being absorbed into the Mongol administration. He also likely knew some Sanskrit for he is said to have liaised with Kashmirian Bauddha Tāntrika-s and trained some of them for joining the Qubilai’s personal corps. Bolad was also given the task of adapting cīna rituals as part of the ceremonies for the coronation of Qubilai as the Kha’Khan. As the ba’urchi he also performed the four animal-sacrifices (like the Indo-Aryan śamitṛ in the Vedic animal sacrifice), which the Mongol Khans conducted to prepare the meat offerings to the tngri-s and ancestors. He also supervised the preparation of beverages for the royal family.

Subsequently, Bolad served as minister of agriculture to manage the farming and grain distribution in the Chinese conquests and the minister planning the construction of hydraulic works. In course of these he introduced the practice of letting the Mongol animals feed on the farmland after the autumn harvest so that they could make use of the crop remains and recycle manure into the fields. On the military front Bolad assisted the veteran general Bayan in completing the conquest of the Sung. He also fought in the north near the Irtysh along with Nomoghan the son of Qubilai and the general Toghan of the Jalayir clan against counter-Khanate of Qaidu and Du’a. One of his last official actions in the east was the investigation of the enormous monetary scam by the Mohammedan official Ahmed who had siphoned considerable amounts from the treasury while Qubilai completely was unaware and took him to be honest. When Bolad finally intimated the Khan of the scale of this scam, he executed several Mohammedans who were part of the operation. A Chinese official who had made a fake of Bolad’s seal was also executed. In 1286 CE Bolad went from China as the ambassador and advisor to the Il-Khan Arghun. While he probably returned once to Beijing he went back and remained the prime minister of the Il-Khans till his death in 1313 CE while on course of a campaign to defend Azerbaijan. In this period he not only performed several administrative duties but also fought some battles for Il-Khans. He was probably 73 years when he died with a life spanning much of Asia and the reign of several Chingizids all over the the empire. Among other things, in west, he also appears to have introduced the use of fingerprints for identity as in the east, what Rāshīd ad-dīn calls the khaṭṭ-i angusht

In the east itself, at the age of ~33 years, Bolad was called upon by the Khan to set up the imperial library. He along with the Chinese assistant Liu Ping-chung put together a body of archivists, historians and other collectors of information. TT Allsen suggests that this library collected a body of books, maps and pictures. Such a collection of information appears to have been the basis of the parallel project conducted by Bolad in the west along with Rāshīd ad-dīn and others. This project might be summarized in the words of Rāshīd ad-dīn thus:

Today, thanks to God and in consequence of him, the extremities of the inhabited earth are under the dominion of the house of Chingiz Khan and philosophers, astronomers, scholars and historians from North and South China, India, Kashmir, Tibet, [the lands] of the Uighurs, other Turkic tribes, the Arabs and Franks, [all] belonging to [different] religions and sects, are united in large numbers in the service of majestic heaven [translation in to Persian of Mongolian Möngke Tngri]. And each one has manuscripts on the chronology, history and articles of faith of his own people and [each] has knowledge of some aspect of this. Wisdom, [which] decorates the world, demands that there should be prepared from the details of these chronicles and narratives an abridgement, but essentially complete [work] which will bear our august name.”

[cited from translation of Rāshīd ad-dīn provided by TT Allsen based on the Moscow edition]

This effort of bringing together knowledge form different parts of Asia on part of the Mongols had different consequences on different people. In the west, it led to the reintroduction of Hellenistic knowledge via the Mohammedan sources to the Europeans, thereby greatly contributing to their scientific revolution. In the Mohammedan world itself this knowledge showed patches of revival and survival when not smothered by the hand of Mohammedanism – e.g. the case of Ulugh Beg. The cīna-s, while participating in this process when under Mongol rule never seemed to have realized the value of Hellenistic knowledge and internalized it until closer to our times. Among the Mongols themselves, upon the fall of their empire such interest appears to have mostly waned with apparently only limited revival in later times based on Hindu astronomy filtering via Tibet, among other things via revival of the Kālacakra tradition.

For comparison, we shall take a detour to consider the case of the old Indo-Greek interactions. This interaction was not carried out under any active royal supervision. We have some evidence that Alexander despite his antipathy towards brāhmaṇa-s, who had galvanized the kṣatriya-s to wipe the yavana invaders in Bhārata, had some personal interest in Indian philosophers and scientists and conveyed some of the know-how of the later to Greece (e.g. how Ptolemaios was sedated during a surgery by Sarpagandha). Subsequently, there was clearly an interaction between the two traditions. On the Hindu side, the Hindu astronomers honestly state that they closely studied Greek astronomical works that they could lay their hands on and produced multiple commentaries and Sanskrit reworkings of them. On the Greek side it is less clear from their own sources as to what, if anything they received from the Hindus. Hence, a wide range of opinions have been proffered by modern Euro-Americans in particular. At one end of the spectrum we have Peter Green who thinks that while the Hindus and yavana-s met in years following the Macedonian invasion nothing much really came of it in terms of knowledge transmission. At best both sides poorly understood or forgot the other. On the other end we have the archetypal white Indologist Pingree and followers (“Hindus as idiots”) who believe that everything meaningful in Hindu science came from the yavana-s, so much so that there was no real Hindu mathematics of note before being enlightened by the yavana-s. Others like McEvilley have accepted that Hindu philosophical and perhaps medical knowledge might have been transmitted to the yavana but mostly for mathematics it was the other way around.

After considering the evidence ourselves we feel that all kinds of scientific knowledge was transmitted both ways but it was not entirely incorporated into respective systems. For instance, in the case of astronomy and mathematics the Hindus while studying and reworking the yavana works into Sanskrit never incorporated the material into their core models. Āryabhaṭa’s heliocentric background for the model is unlike that of any Greek work despite the fact he was aware of them and he and his students studied them. Hence, some Euro-American workers try to claim that it should have come from a lost Greek model without present evidence for any such in the Greek world. That simply suggests that it represents a purely Hindu development. The real core of Greek mathematics in the form of Euclidean geometry was never adopted by Hindus, but the pseudo-scientific material of yavana astrology made a deep impact on Hindu astrology. On the Greek side they of course adopted several aspects of Hindu medicine and philosophy but what about mathematics. We hold (as some early Europeans did in more innocent days) that they acquired a bit of Hindu algebra but on their part never really understood its spirit and remained limited in its development in Heron’s and Diophantos’ works. Thus, the two had rather distinct mathematical frameworks that were apparently not able incorporate material from the other at any deep level perhaps in part because the deepest of this work did not travel widely between the civilizations unlike in the case of astronomy. In the case of the Mongol empire while the patrons brought together very distinct knowledge systems it appears that the synthesis did not filter down in a big way in China and Korea, though it might have made more of an impact, even if indirect in the west.

Posted in History | Tagged , , , , , , , , , ,

Civilizational collapse, complexity, innovation and neomania

On these pages we had provided some commentary on the work of the German thinker Oswald Spengler and his ideas of civilizational development. We had also provided other perspectives on this matter derived from analogies stemming from simple mathematical systems. We originally wished to detail other thoughts in this direction, particularly relating to civilizational collapse, within the conversational framework we have often used on this pages. However, not everything the frail mortal proposes is achieved, as the mighty gods, who show reveal glimpses of deeper nature of existence to him, also shower afflictions that dispose of his plans. Hence, we simply decided to place a few words relating to this topic here rather than work it into a piece in the series. In part this was triggered by the “neomania” I heard from people reaching a crescendo, all enthusiastic about the solution to deep human problems by new technologies that were just around the corner. Here, I am using neomania in a sense I have seen the Lebanese author NN Taleb use it – an unbridled sense of desire and hope for new technology as a panacea. It was ironic that two of the biological technologies that were talked about involved me as a discoverer – all I got from them was a sense of pleasure of uncovering a hidden insight at the time I stumbled upon those discoveries – beyond that they have not solved any of my own human problems. Hence, perhaps one could call me biased in this regard. However, I should clarify that even I would welcome some of the neo-technologies and hope they improve my life but I remain a skeptic regarding them ultimately solving any fundamental problems or being better *on an average* from less-glamorous solutions. All of this led to some thinking about the book by the American historian Tainter, which is one of the more detailed accounts of the endgame or civilizational collapse (Collapse of Complex Societies).

He cites Casson regarding the aftermath of the termination of Pax Romana in Britain: “From A.D. 100 to 400 all Britain except in the north was as pleasant and peaceful a countryside as it is to-day … But by 500 A.D. it had all vanished and the country had reverted to a condition which it had, perhaps, never seen before. There was no longer a trace of public safety, no houses of size, dwindling townships and all the villas and most of the Roman cities burnt, abandoned, looted and left the habitation of ghosts

This kind of end is not unique. Tainter collects several examples: 1) the western Chou empire in China; 2) Harappan civilization in India; 3) The Mesopotamian kingdoms; 4) Egyptian Old and Middle kingdoms; 5) The Hittite empire; 6) Minoan Civilization; 7) Mycenaean Greek civilization; 8) The Roman Empire; 9) The Olmec in Mexico; 10) The Teotihuacan civilization in Mesoamerica; 11) Casas Grandes near the Mexico/USA border, the Chacoans of New Mexico and the Hohokam of Arizona (these three adjacent native American cultures probably had some interactions but it is not clear as what the exact relationship between them was); 12) the native American culture of the Cahokia mound complex in the middle Mississippian zone; 13) Huari and Tiahuanaco kingdoms of Andean South America.

One can debate some of the cases of collapses collected by Tainter. One may also say his list is heavy on the old Americans – not surprising given that he is an American historian. For example, right in Asia there are other notable examples, like the Khmer civilization, which showed the classic pattern of efflorescence followed by collapse. It is also pretty weak on sub-Egyptian Africa. Immediately south of Egypt, the Meroëtic civilization can be mentioned as an independent case. The sub-Saharan Great Zimbabwe can also be considered another valid African example. We could also consider the Benin empire, although one could place the primary blame for it is collapse on European Christian contact. Nevertheless, we do feel Tainter’s is generally a good list. Further, the Americo-centricity is not entirely a bad thing thing because the Americas developed for long without significant contacts with Eurasia after the initial peoplings, Thus, they were a good independent laboratory to study the development of human organizational complexity. In these examples Tainter makes the case for development from a relatively simple organizational state to a complex state that was characterized by increasing organizational complexity followed by a “golden age” and then a collapse, where most features of organizational and technological complexity are lost and the population returns to a basic subsistence mode.

One may ask how does one measure complexity of a civilization? We would say a relatively simple device, inspired by devices we have effectively used in our work in biology, can capture social complexity: Let us denote each distinct occupation performed as primary source of livelihood by an individual in a society as a character. Then per household we can create a string of such characters concatenating the character representing each individual in the household. Then by aggregating those across all households in the human habitation or civilization under consideration we get an occupation string for the entire habitation/civilization. We then calculate the Shannon entropy for this string. This will be a reasonable measure of complexity. Such a measure does suggest that as a civilization or culture develops it rises in complexity reaching a peak value. While this peak value might have considerably differed between the Eurasian+Egyptian-zone civilizations and those of deep Africa and the Americas the trend in its value is the same in course the development of a culture or civilization followed by a collapse.

As another dimension we may also consider the recent work by TA Kohler and colleagues (Greater post-Neolithic wealth disparities in Eurasia than in North America and Mesoamerica by Kohler TA et al). They used archaeologically determinable house-size distribution to compute Gini coefficients. The Gini coefficient is a measure of the inequality in the values comprising a frequency distribution (g=0, total equality; g=1, one owns all and rest nothing) and is typically used to measure the inequality in income distribution in a society or habitation. They showed that the Gini coefficients increased with the domestication of plants and animals in the Neolithic and subsequently with the emergence of the Indo-European model of wide-ranging conquests over Eurasia. In their limited analysis they found that the disparity indicated by the Gini coefficient was more in parts of Europe and Asia they sampled than in the parts of the Americas they studied. The average Gini coefficient was markedly higher in cultures with classic states than those that remained restricted to the local habitations.

An implication of all this is that complex societies are intrinsically associated with a non-egalitarian structure. A corollary to this is the need for a degree of centralization or hubs to maintain any degree of complexity beyond the ordinary. We see the same constraint imposed on several biological networks, such as transcriptional networks. Here, there are small number of transcription factors, “hubs”, that control a very large number of genes and a large number of transcription factors that control a small number of genes. In such a network structure, where the degree-distribution of the nodes follows a power-law distribution, an attack that randomly targets nodes has a low probability of breaking up the network. However, an attack that specifically targets the hubs will surely bring it down. However, such networks do fine with regime change, i.e., nodes which are hubs may be freely lost as long as new hubs with statistically similar properties as the lost hubs concomitantly emerge in the network. Hence, we may say that any defect that can propagate up to the hubs so as to irreplaceably destroy them cause a collapse of the system.

In our youth the Rus, who had implemented the Marxian delusion, plied us with cheap books with subtle delivery of their cherished doctrines: 1) Egalitarianism; 2) Communism; 3) triumph of technology over religion and nature. Our meditations on such issues in the social domain were initially sparked by our trying to apprehend the first of these – the superficially alluring egalitarianism of the Marxian doctrine. At that time many around us had bought it into it. At the face of it seemed like channel for rebellion against authority – being a svacchanda we were not fond of authority ourselves. Yet our closest informers had warned us about the dangers of the Marxian doctrine. While playing all these issues in our mind, we soon realized that it was a figment of the Abrahamistically deluded imagination. Thereafter we soon arrived at the interlocking of complexity, structure, functionality and inequality. That interlocking acted like a conservation principle you could increase one side of this without reducing the other – essentially you cannot have your cake and eat it too. Thus, social complexity, intra-social structure and effective multifunctionality went hand in hand with increased inequality. Egalitarianism simply meant that such things had to devolve towards simplification or entirely collapse, since the climb-down is never a mirror image of the climb-up. If the climb-up were to happen by the mode of preferential attachment of new nodes to certain nodes of the network, the climb-down it often proceeds through an attack on hubs of a network with properties approximation a power-law distribution resulting in collapse. In course of our life we have witnessed several examples of such collapse under Marxian or semi-Marxian implementations.

A more subtle understanding of the social role of technology and the production of community goods came from our studies on biological conflict and cooperation, which are but the proverbial two sides of the same coin. There two broad roads to biological cooperation: 1) when you form a multi-individual assemblage with your kin then you can contribute to the “commons” even at some expense to your own fitness if that contribution bumps up the net included fitness which you gain from your associated kin who benefit from it. 2) If you are forming an assemblage with non-kin – like say a bacterial mixed biofilm or a mixed flock of birds – then you develop cooperation via mutualism and potentially functional differentiation. Both these scenarios sustain a certain type of technological innovation. First, the mutualism, “frees” you from performing certain tasks. Hence, you can devote that surplus to otherwise unaffordable innovations that might contribute to your fitness. At the very basic level, these innovations might relate to biological conflict, i.e. weaponry to prevent predatory/parasitic entities from taking your gains. Second, in any cooperation, both between kin and between non-kin, there is a strong incentive for “black sheep”, i.e. cheaters, who gain a fitness advantage by not contributing to the commons but only using the communal goods. Indeed, we see their emergence routinely in biological assemblages, often even via single point mutations. For any cooperation to survive we hence need policing. This takes the form of weaponry similar to that used in inter-group biological conflicts. These scenarios spark off arms-race like situation which are the strongest drivers of “technological” innovation in biology. These might then lead to “peace-time” use of these innovations in other biological contexts. For example, we have shown that much of the eukaryotic epigenetic-information-encoding apparatus has emerged from components that originally evolved as part of the offensive/defensive weaponry in biological conflicts in the prokaryotic world. We came to realize that this parallel of technological innovation broadly applies even to human technologies. In the “big-picture”, technologies that are not ultimately contributing to fitness in some fundamental way are unlikely to lead to technological utopia as some of its proponents hold; rather they might simply lead in the direction of civilizational collapse.

This brings us to the concluding part of this discussion: Among those doctrinaire Soviet books we leafed through in our youth, there was at least one which was inclined towards futurology. If my memory serves me right in 2018 CE we were predicted as not too far away from what was already a technological utopia. If this was the Soviets, one of the very few science-fiction works I read (by some American) also had rather grandiose predictions for this date. None of that has come to pass. Yet, in parts of my admittedly small circle there is a sense of anticipation – a feeling that we might be close to an unprecedented time where a great revolution manifests from new technology. For some it does have shades of a utopia. For others it raises the specter or the joyous possibility of the times of the right to a salary without the need to do any work. Yet others talk of the freedom from human limitations. Yet, we may go back to Tainter’s list and ask if we might join that too. One point which might be noted is that, whether it was in the metal-limited Americas, or Eurasia, or Africa, the rise of civilizational complexity also went hand in hand with technological innovation. Sometimes this innovation was rather complex as in the Greco-Roman world or India. Yet those civilizations collapsed with all or much of that technology entirely vanishing in the successor populations. The votaries of the coming technological golden age say that this time around it is different, whereas Spengler did not see it as being different. If the technology is not enhancing its fitness then we may ask if it can in any way stave off a collapse in the future.

Tainter lists four points which he sees as being major factors pertain to collapse:
l. human societies are problem-solving organizations;
2. sociopolitical systems require energy for their maintenance;
3. increased complexity carries with it increased costs per capita; and
4. investment in sociopolitical complexity as a problem-solving response often reaches a point of declining marginal returns.

We would add that unless the costs of increased complexity in someway translated into a benefit for fitness (and here measuring fitness is complicated) it is likely to hit a road block. The neomaniacs think that for point 4 new technology might provide a way out. Here again we remain skeptical that it would escape his “declining marginal returns” clause especially when weighed in terms of its contribution to fitness. Hence, we say: 1) There is likely a good reason dolphins have retained high “IQ” for a long time but only developed limited technology; 2) The so-called Fermi’s paradox is a real thing for most intelligent organisms are either leading the lives of dolphins and ravens or are repeatedly undergoing collapse rather than taking hold of the galaxy in the technological utopia of science-fiction.

Posted in History, Life | Tagged , , ,

Astronomical experiments: a preliminary look at globular clusters

While Alexander von Humboldt was exploring South America he observed a fuzzy object in the constellation of Centaurus and thought that it was a new comet. However, it turned out that it was the globular cluster \omega Centauri. It had already been noticed long before his times by Edmond Halley of comet fame with his primitive telescope as a fuzzy object which was not a star. Long before Halley, Ptolemaios the yavana astronomer observing from Egypt thought it was a star in Centaurus. Halley soon discovered a second such object in the constellation of Hercules, which was to be later known as M13. Charles Messier following up on these observations discovered that there were several more such objects and obtained the first hints that they were made up of stars. This was confirmed finally by Wilhelm Herschel, who was a prolific discoverer of globular clusters (GCs).

Globular_figure1Figure 1 \omega Centauri

We obtained one of the clearest views of \omega Centauri from a fort of the erstwhile Marāṭha admiral Angre on an exceptionally clear night — it was truly one of those sights which cause a romaharṣaṇa in the beholder. We have had several fruitful nights of GC sighting using various instruments from a 20×75 binoculars to a 12 inch telescope. Indeed, GCs remain objects of great fascination to everyone from the casual observer to the astronomers at the edge of their science. They have ramifications for everything from the age of the universe, the origin of galaxies, the detection of dark matter to gas physics.

A GC is unmistakable for anyone who has seen one – a dense globular conglomeration of stars with up to a several million stars packed in a close oblate spheroid of few parsecs (1 parsec \approx 3.26 light years). We had the desire of illustrating some basic features of them using the data from 147 GCs of the Milky Way. This number is not far from the total number of GCs in our Galaxy. The brightest in our set is of course \omega Centauri with integrated visual magnitude, m_V=3.68 and the faintest is UKS 1 with m_V=17.29, which is beyond any telescope we have personally operated. \omega Centauri is also the brightest in absolute visual magnitude M_V=-10.24, whereas the absolutely faintest GC in our set is AM 4 with M_V=-1.55.

Globular_Figure2Figure 2

One of the first things an observer notices is that GCs are not uniformly distributed in the sky. Even the brightest GCs, those from the Messier catalog, make this apparent — there are 7 each in Ophiuchus and Sagittarius; thus, they seem to be clustered close to the center of the Milky Way which lies in Sagittarius. If we take all the GCs in our data and convert their coordinates to galactic coordinates using some spherical trigonometry we can plot them with respect to the galactic equator. Figure 2 shows such a plot using the Eisenlohr projection with the Milky Way outline in green. The GCs are colored red for higher than average metallicity an blue for lower than average metallicity (see below for discussion on metallicity). One can clearly see that they are clustered in just one part of the sky with the galactic center as the focal point. This observation establishes their presence around the core of the Galaxy in the halo.

Globular_Figure3Figure 3

We can also visualize their distribution by directly plotting their inferred distance from the galactic center (Figure 3). Their density rapidly falls off with distance and \tfrac{2}{3} of the total number of GCs in the Milky Way lie within 10 kiloparsecs(Kpc) of the galactic center.

Globular_Figure4Figure 4

We next examine the distribution of the absolute magnitudes M_V of the GCs (Figure 4). We find a clear central tendency; \mu=-7.022 (vertical line), \sigma=1.54631. The observed distribution has a sharp peak close to the mean. Thus, the GCs seem to come with a clearly preferred M_V. This has been exploited to use the magnitude of GCs in other galaxies as a standard candle to measure their distances.

Globular_Figure5Figure 5

The radii of GCs is measured in two ways. First, it has been observed that luminosity of a globular drops off when one moves from the center of the GC toward the periphery. The radius at which luminosity drops by half of what it is in the center is defined as the core radius R_c. Another measure of radius is based on gravity: The distance from the center of the GC beyond which the gravity of the galaxy has a greater influence than that of the GC itself is termed the tidal radius R_t. Based on these radii the central concentration index is defined as c=log_10(\tfrac{R_t}{R_c}). The distribution of c is shown in Figure 5. Its most interesting feature is the situation above c=2.0 — we see a significant over-representation at the very end of the range c\approx2.5, which is greater than even that at the mean. What does this signify? As one moves towards the center of the GC in a typical GC they luminosity rises and then plateaus. However, about 20 percent of the GCs which show c=2.5 behave differently. In these high c GCs the brightness just keeps rising right to the center. Hence, they are described as having undergone core collapse wherein stars clump more closely at the center than what is seen in typical GCs. This is believed to happen due to the dynamics of the cluster wherein sorting of stars due to close encounters reduces the central kinetic energy and favors the closer clumping of stars as gravity dominates.

Globular_Figure6Figure 6

We next examine a key property of GCs, their metallicity (Figure 6). The early universe was almost solely made up of Hydrogen and Helium, which still remain the dominant elements. But since then stellar nucleosynthesis and supernovae have generated other heavier elements. The fraction of elements other than H and He is absolute metallicity of an astronomical object. In practice it is measured using spectroscopy as the ratio of Iron to Hydrogen. Typically, the metallicity of an astronomical object as relative to the sun:


Where N is the number of atoms of Iron or Hydrogen inferred from the spectrum. Thus, stars with [Fe/H]>0 are more metallic than the sun and those with [Fe/H]<0 are less metallic than the sun. The stars with greater metallicity tend to be redder and those with lower metallicity are bluer, all other things being equal. All GCs in the Milky Way, barring Liller 1 ( [Fe/H]=0.22), have lower metallicity than the sun. The distribution of GC metallicities however interestingly shows a bimodal pattern (Figure 6). One peak, the dominant one, is seen at [Fe/H]=-1.5 and another at [Fe/H]=-0.7. Notably, in the past 2 decades studies on several other galaxies have recapitulated this bimodal distribution we see in the Milky Way. The dominant peak in this distribution points to a prevalence of low metallicity GCs, which in turns implies that they are predominated by population-II stars or those that formed early in the Galaxy's history. The second peak could represent the formation of later clusters during galactic merger/cannabalism events. Indeed, the Hubble space telescope observations suggest that large-scale gas clumping in galactic collisions might allow formation of new GCs. A differential history of the GCs with different metallicities is supported by their spatial distribution (Figure 1). One observes that clusters with lower than average metallicity are more widely distributed and dominant at higher galactic latitudes. This suggests that they are indeed likely to be "fossils" from the early era of star formation in the galactic halo and contain some of the earliest surviving stars in our galaxy. Consistent with this, more recent studies show that galaxies with bigger central bulges have more GCs.

Globular_Figure7Figure 7

Now we can look at some correlations between the between the above discussed variables. First, we look at the correlation between metallicity and distance from the galactic center (Figure 7) as a follow up to the above discussion on Figure 1. Each GC in Figure 7 is color-coded according to its metallicity as in Figure 6 and the distribution of the metallicity on the y-axis is indicated by the rug. We observe a trend of decreasing metallicity at greater distances from the center of the galaxy. Importantly, there are hardly any GCs with [Fe/H] > -1 at distances greater than 8 kpc from the galatic center.

Globular_Figure8Figure 8

If we discount the GCs which have undergone core collapse, we also observe a positive correlation between central concentration c of the GC and its absolute magnitude M_V (Figure 8). This suggests that as long as a GC does not undergo core collapse its brightness increases with the central concentration. Notably, this plot also shows that there are few GCs, which have undergone core collapse which are brighter than M_V=-7.5. Astronomers propose that this might mean that the time required for the most luminous GCs to undergo core collapse is greater than the current age of the universe. Notably, we also see that in a plot of c versus distance from the galactic center (Figure 9), nearly all GCs, which have undergone core collapse, occur closer to the center of the Galaxy ( <8 kpc).

Globular_Figure9Figure 9

This preliminary glance of at GCs is just to obtain a flavor of the mysteries they hold regarding the origin and evolution of the universe. Recent studies (e.g. that of WE Harris published last year) have shown a particular relationship between the luminosity of the host galaxy and number of GCs they have. While the Milky Way has close to the 147 GCs we have explored here, giant galaxies like the great Virgo galaxy of M87 have over 10,000 GCs. Over the years astronomers have also obtained evidenced for another interesting relationship. Let M_h=M_b+M_d (where M_h is the total mass of a galaxy, which is the sum of the total mass in baryonic matter M_b and the total dark matter M_d), then \eta=\frac{N_{GC}}{M_h} (where N_{GC} is the total number of GCs associated with the galaxy). This \eta has been found to be nearly constant for galaxies ranging from dwarfs to giants. Why is this the case remains a mystery.

Recent studies with the Hubble space telescope have show other types of associations of GCs. In the great galactic assemblage Abell 1689, there is swarm of approximately 160,000 GCs a sphere of diameter of 2.4 million light-years close to center of this galaxy cluster. These are apparently not GCs associated with one galaxy but seem to be a common pool of GCs from the entire galactic cluster that have aggregated close to its center, which is rich in the mysterious dark matter. It such indirect clues regarding dark matter that give us faint hints regarding one the greatest mysteries of the universe. What exactly is the link between GCs formation and dark matter and how did they form in the first place. These are questions which show that as ever astronomy leads the way to where the frontiers of physics lie.

Posted in Scientific ramblings | Tagged , , , ,

Frustrations and ramblings ensuing from Cretaceous amber

Time and again I have been frustrated by the inability of Hindus to make the most of the riches that are available in their own land or right next to them. One such case is that of Cretaceous amber from Myanmar, which has recently become the focus of an enormous scientific exploration. It is telling us about arthropod natural history like never before: Many insect lineages were already close to their modern form by the time window of 98.7-108 million years ago to which this amber is dated. For example we see ants and termites were already on the rise and had established their caste structure. This gives a firm evidence for the enormous antiquity and resilience of the caste-structure in social insects, thereby supporting its adaptive value. It is giving us the first picture of the ticks that sucked dinosaur blood as well as dermestid beetle larvae, which were probably nest symbionts that fought these ticks with their specialized structures called hastisetae. We are also getting the rare dinosaur chicks, which were unfortunate enough to get sloshed in this resin and be preserved for us to understand their anatomy in some detail. How is all this being studied? The researchers appear to simply go to Myanmar and buy amber and study its inclusions. However, I do not see one Hindu involved in this process despite India having close ties to Myanmar and knowing of this resource for centuries. This is research where the Hindus could have beaten the cīna-s and mleccha-s by a stretch but failed to even notice the wealth in their neighborhood.

This set off a train of thoughts that go a long way back in time. Pliny the elder, was one of those charismatic figures in history with few parallels – a general in both the Roman navy and land army and a naturalist, who apparently died during the eruption of mount Vesuvius. Of him his nephew writes:

“For my part I deem those blessed to whom, by favor of the gods, it has been granted either to do what is worth writing of, or to write what is worth reading; above measure blessed those on whom both gifts have been conferred. In the latter number will be my uncle, by virtue of his own and of your [Tacitus of Germania fame] composition.”

The composition his nephew is talking about is the Naturalis Historia. This interesting work has a whole chapter on amber, and mentions the Indian amber on three occasions. He mentions that the yavana Nicias had noted that amber was used in place of guggulu or aguru as a high-end incense. He also mentions that the yavana physician Ktesias, employed by the Iranian emperor, noted that the Hindus obtained amber from a river called Hypobarus that flowed from the north to the eastern ocean. The river flows by a mountain where trees called “siptachorae” produce amber and discharge it into the river. Then Pliny states that Archaelaus, a chief of Cappadocia, brought amber from India with bark still adhering to it and processed it to get the shiny polish. It was said to contain ants, gnats and lizards in it by means of which Pliny inferred that amber must have once been liquid.

These ancient Greek accounts would suggest a pre-Mauryan knowledge of amber in India. Importantly, the river flowing into the eastern ocean brings to mind the tributaries of the Irāvati river in Myanmar in whose valleys amber is found. However, it does appear that the accounts might have in part conflated real amber, i.e. the fossil resin, with more recent resin. Nevertheless, that real amber was used as jewelry by the Hindus is confirmed by the cīna-s, who during the great Tibet-Tang-Hindu drama mention that the rājā of Nepal had jewels of amber.

We have often thought that the great Hindu naturalist Varāhamihira was a kindred soul of Pliny. In his natural history, the Bṛhat-saṃhita, he has a few chapters on gemstones. Turning to those we find a curiously named gem called saugandhika, which literally means good-smelling. This is consistent with Nicias account of Indian amber being used for incense. Hence, we believe this was an old name for amber. Varāhamihira classifies this gemstone in the padmarāga category. He says the gems of this class are of three types in the beginning of chapter 81:

saugandhika-kuruvinda-sphaṭikebhyaḥ padmarāga-sambhūtiḥ |
The padmarāga-s are formed from saugandhika (resin), corundum and crystals

saugandhikajā bhramara-añjana-abja-jambūrasa-dyutayaḥ ||
That originating from resin have the shimmer of bees, collyrium, clouds and the rose-apple juice.

kuruvinda-bhavāḥ śabalā manda-dyutayaś ca dhātubhir viddhāḥ |
Those originating from corundum are dappled, with a dim glow and permeated by mineral inclusions.

sphaṭika-bhavā dyutimanto nānā-varṇā viśuddhāś ca ||
Those originating from crystals are lustrous, with multi-colored [sparkle] and clear.

There are few points of note here: One may wonder why Varāhamihira groups these together as padmarāga-s. First, several samples of Burmese amber can have a reddish tint (he specifically mentions jambūrasa) placing them together with the other reddish gems he describes in this category. Second, Burma also produces rubies and corundums, which might have led to a geographic association between these stones; in any case Varāhamihira does recognize their different “geological” origins. Finally, one may wonder what the saugandhikajā’s bhramara-dyuti would mean. Most simply take it to be bee-colored i.e. yellow and black. However, we are tempted to see it as an acknowledgment of the insect inclusions in amber. Varāhamihira tells us of a nāstika author Buddhabhaṭa (“Buddha’s soldier”) who had composed treatise on gems to which he refers the readers. May be we would have gotten more information from that text had it come down to us. Perhaps, due to declining scientific interests of the bauddha-s over time or the Mohammedan book-burning this text has not survived.

The Greeks and Romans called amber elektron after its yellowish luster. This word has give rise to the modern words like electricity and electron. This is due to the static electricity generated in amber, i.e. triboelectric charging, when rubbed with a woolen cloth. This property has given rise to the second name for it in Sanskrit tṛṇamaṇi or tṛṇagraha. This literally means the straw-gem or the straw-seizer. This property of amber is used by the great Kashmirian kavi Kalhaṇa in a beautiful simile, where he describes the acts of a warrior Rilhaṇa:

tat khaḍgasya ghnataḥ khaḍgāñ jīvair jālac chalād dhruvam |
utthāya lagnaṃ śatrūṇāṃ tṛṇais tṛṇamaṇer iva ||

As his[Rilhaṇa’s] sword’s blows struck down their swords,
the enemies’ lives [were stuck firmly] as a web [to his sword],
even like straw-blades rise up to stick to amber.

This clearly indicates that not just amber but its propensity for static electricity was well-known even in the Hindu world. Indeed, another Kashmirian kavi Bhallaṭa also talks of the same property but his verse indicates that by the beginning of the last century of the first millennium of the common era amber’s value was declining in the Indian mind. He says:

cintāmaṇes tṛṇamaṇeś ca kṛtaṃ vidhātrā
kenobhayor api maṇitvam adaḥ samānam |
naiko’rthitāni dadad arthijanāya khinno
gṛhṇañ jarat-tṛṇalavaṃ tu na lajjate’nyaḥ ||

The cintāmaṇi and amber, both made by Vidhātṛ,
now then why should the gem-hood of both [be considered] equal?
One has undesired [qualities], giving sadness to one desiring it,
attracting dry pieces of straw, but is not ashamed by the other.

Bhallaṭa’s anthology is filled with verses with a suggestive satire. Here, he compares amber to the fabulous gem cintāmaṇi and says that while both are called gems one (i.e. amber) is in reality rather worthless compared to the other member in the same category. This verse reminded me of the category “biologist”. While many scientists are placed in that category the difference between actual exemplars is like that between the cintāmaṇi and the tṛṇmaṇi of Bhallaṭa. While they sit in high seats and publish papers in magazines considered prestigious their knowledge and understanding of the science is shockingly abysmal. Yet the mleccha system allows this charade to continue much in the manner of things lumped in the gemstone category of Bhallaṭa.

However, we would say this decline of amber in the Hindu mind was not a good thing, bringing us back to where we started this note. For a paleontologist the Cretaceous amber, the tṛṇmaṇi of yore is indeed nothing short of a cintāmaṇi: pariṇāmavadibhyas tṛṇamaṇiś cintāmaṇir eva |

Posted in Heathen thought, History, Life, Scientific ramblings | Tagged , , , , , , , , ,

Of lives of men; of times of men-III

Of lives of men; of times of men-II

As they were talking, they saw a sallow-complexioned youth pass by them some distance away carrying a bat on his shoulder. Vidrum waved out to him and he responded similarly and after some delay so did Sharvamanyu.
Lootika: “Who is he?”
Sharvamanyu: “He’s our old schoolmate Mudgar.”
Vidrum: “You don’t remember him? He was the cricket champ.”
Lootika: “As you guys I know I never cared for that most bizarre of games! I believe my parents rightly advised me and my sisters to stay far away from guys who fritter away all their time at the kandūka-krīḍā.”
Sh: “Hey, but I know Somakhya is quite a fan.”
L: “That does not mean I should be a fan too.”
Somakhya: “Did not recognize him. But now I recall seeing Mudgar last on the day the results of our school-leaving certificate were declared. He was in a funereal state having the pulled the plug in more than one subject. Sadly, his prowess with the bat and the ball helped little in that contest. But he still seems to be keeping to his daṇḍa-bhañjana-krīḍā.”
Vi: “Somakhya, evidently Lootika’s company has made you lose your edge with the game. Forget about the school-leaving certificate – he indeed never cleared it – but don’t you know he is a famous man now?”
So: “Famous?”
Sh: “He apparently smashes big sixes in one of the lesser leagues and is slated to make it to the Indian league this summer to play alongside the stars.”
So: “Ah! the gods have been favorable to him.”

Sh: “That’s indeed true. His father was a noted player for the first division in his days. He hoped his time that some day he might be called up for the national trophy matches. But those days they did not play so much cricket. So, like Mudgar, when he pulled plug in his exams he had no real means of employment. Based on his facility with the game, they gave him a job as peon in railways and he continued to play for them. But he never made much money and Mudgar’s youth was one of those with a low economic status. When you come from such a background, even your school curriculum can be a stiff uphill climb.”
L: “That’s why I say that sports should never be taken so seriously as to think that it can be your profession.”
Sh: “Whatever you might think, they put the little money they had into his cricket and it might pay its dividends now. For all you know, in an year or two, he could be earning much more than any of us.”
V: “Lootika, despite your condescending attitude towards the noble game, I am sure even you have heard of our great hero Musal Gandulkar? He is one of the richest men in the country.”
L: “But that is the whole point. For every Gandulkar, I am sure a million other lesser kids, who thought that they too might become Gandulkar-s, ended up like Mudgar’s father. But on the other hand, Vidrum, even if you are far from being the Caraka of our yuga, you will still make a great contribution to society for having put in heroic effort into your MBBS rather than in kandūka-krīḍā. Moreover, think about this – what is so heroic in Gandulkar? His existence or lack thereof would make a negligible difference to the good of the society. On the other hand, upon graduating I am sure you will make a bigger difference to some people’s lives than this Gandulkar.”

Vidrum: “It feels good to hear that from your mouth, paṇḍitā. But to bring the focus back to why we wanted a long chat with you two – can you provide arguments for why sportsmen should not be accorded special respect or admiration. After all, I could point out that scientists and mathematicians pursuing obscure knowledge for knowledge’s sake are probably as useless to the social good, which you seem to take as an important criterion, as a supremely entertaining cricketer as Gandulkar. At least he contributes to the mental health of the masses by way of entertainment. The scientist in the rarefied realms of inquiry cannot be understood by anyone but a minuscule minority.”
L: “Vidrum, while there is something to what you say, I think you have shifted the goal posts in the mean time. Remember, that, while proximal reasons might be difficult to discern, the normal distribution describes quantitative human traits, including success in sports. A Gandulkar is far to the right of the distribution. I need not remind you that the despite many of you guys having great facility with the game in our school circles, Mudgar notwithstanding, you all are still closer to the mean in this regard, where a great mass from our nation is positioned. Hence, my statement that it is not a good idea to invest in it with such seriousness as one might for a proper profession. Coming to science, I agree that science too is not suitable at all as a career for most people but those closer to the right extreme in scientific ability. I even warned my dear sisters of this even though I trained them in science. Indeed, when our junior college physics teacher asked me to speak to the girls in her class to motivate them to consider science as a career I did something she did not expect. I told them just this it is something for the far-extreme and not for everyone. So they should simply focus on being useful to their families and society.”

Sharvamanyu: “Lootika, note that the times have changed. That’s what I was pointing out. In Mudgar’s father’s day, indeed, only the far-right of the distribution could dream of making the cut – after all only 11 people and few extras can play for the country. But what has happened in the mean time is that the game changed and became potently monetized. This allowed a much larger fraction of individuals to be able to make living out of it.”
Somakhya: “While I don’t disagree with that particular point, let us not forget that still only a very small percentage of our vast population can really make the cut in even the current hyper-monetized cricket circuit. In that regard Lootika has a point. On the other hand I don’t think people are exactly massing to consider cricket as a profession and failing in the attempt to lead more useful lives. There may even be some advantage, as Vidrum suggested, to cricket or more generally other spectator sports because they are a palliatives for the masses – it is better that the masses are engaged in cricket rather than films, which are often better vehicles for sneaking in parasitic memes into our unsuspecting population. Finally, I have to say that, while science and mathematics practiced by its cutting edge practitioners might be understood by very few, even of their own ilk, it is not a bad thing for the general population to strive for as deep a knowledge of these fields as they can achieve. Such knowledge in the population will always be more useful to society than cricket. Thus, in the long run, acquiring a few lessons from JC Maxwell are going to be of much greater consequence to a society than those from a WG Grace. At the same time, it is important to distinguish acquisition of such knowledge from the worship of science or mathematics as opposed to its actual acquisition as propagated by the Marxian liberals. That can be devastating to society.”

Lootika: “It is not for nothing that our ancients, while nor prohibiting the entertainers in anyway, placed the likes of a naṭī, nartakī, śailuṣa, māgadha, jhalla and malla at the level of puṃścalī-s rather than at the pinnacle of society. They were employed and experienced adequately-provided lives but these avocations were not at all seen as an aspiration for the central mass of society. It is also notable that other than the boxer/fighter, which are outlets for the natural and ancestral male aggression, none of the many other sports of the ancients were seen as avenues for professional pursuit.”
Sh: “Ha! Vidrum you may recall our classmate Manjukeshi’s vehement calls for the complete ban on violent sports such as boxing. I wonder what she might think of Lootika’s words.”
Vidrum: “She would have termed it biological essentialism. I might have leaned towards such things in the past when spending lot of time with her and Samikaran but I think I am coming around more and more to see the biologically informed points Somakhya and Lootika have made ever since I have known them. However, it is interesting that Lootika mentions the exception made in our old tradition for these sports relating to male aggression. Why would that have emerged when you have the ever-available, universally more honorable profession of the military?”
Sh: “I think the military is serious stuff – a matter of survival of a nation, while the other thing is entertainment, much like our other sports. That’s why the jhalla or the malla is not placed in the league of the rājanya who is at the top of the social ladder. Yet I am sure our friends might have some deeper biological reason for this.”

Somakhya: “The anatomy of the skull of Homo hints that it might have undergone some selection for over-engineering to survive momentum transfer from blows delivered by the hand. It is reasonable to posit that the emergence of bipedalism in the Homo lineage freed the hands for combat. Indeed, this is a common trait that we see convergently evolving in other bipedal species. Many anseriformes (waterfowl) use their arms in combat (mostly male-male) much like Homo. Thus, the legitimization of the jhalla and malla is likely the result of a deep-rooted social role for such face-to-face hand combat. I think we should indeed distinguish that as a remnant of the old intra-group male aggression relating to dominance and mates as opposed to the inter-group aggression which relates to the emergence of the military profession.”
Lootika: “Indeed, their link to the intra-group male is suggested by the fact that humans clearly recognize its counterparts in non-human animals and considered those as equivalent forms of entertainment. For example, watching the combat between human males is comparable to the specter of intra-specific conflict between males of other animal species. Not surprisingly, among the yukti-s of the old Hindus we see alongside the malla and the jhalla, head-butting of rams and goats, and conflict between various male birds from galloanserae as related entertainments.”

Sh: “Our discussion thus far clarifies the role of intra-group male-aggression and the emergence of sports based on that. But the important question of the emergence of the military profession which Vidrum mentioned remains. That also seems have a male bias. What is its relationship to ancient biological struggles?”
S: “First, there needs to be some tendency for sociality – i.e. tendency for aggregation of members of the same species as opposed to the tendency of existing as free-ranging individuals with no interactions with others from the same species beyond mating. Such sociality can emerge convergently across organisms. Take for example the lion, it is a social cat, in which sociality has emerged very recently, given that closely related cats are solitary in their behavior. Such incipient sociality can emerge quite easily from aggregation offering improved predator-survival or simply persistence of birth associations between siblings or sibling clusters at communal nesting sites. Second, once you have incipient sociality, there is good evidence that traits favoring the ability of a social group to capture resources from a competing group are likely to be selected for. This is the origin of inter-group conflict and the emergence of what we term military today. In our cousins, the chimpanzees, such inter-group conflicts are seen. While males already play a dominant part in chimp conflict, the inter-group raids also feature some females. This situation is persists in some human groups too – for example among the Mongols we hear of some female participation in actual combat activity. But we have keep in mind that females are a reproductive bottle-neck for a social mammal like us. Loss of females is a real loss of fitness for the social group, while loss of even a fraction of the males will not change seriously change fitness as long as the number of females are intact. Thus, inter-group conflict is likely to eventually evolve a strong bias towards male only participation, which will be further exacerbated in favor of males when there is strong sexual dimorphism in mean size, speed and strength between the two sexes.
Lootika interjected: “Of course it must be stressed that biology of the social species in consideration plays a big role in the sex bias of combatants in inter-group conflict. As Somakhya clarified, in mammals, the females bearing small number of live young after prolonged gestation and a clear sexual dimorphism in mean size means soldier-formation will be male-biased. However, in insects of the cockroach clade, wherein termites evolved, the sexual dimorphism generally manifests in the form of bigger females. Thus, the soldier caste in termites shows a clear bias towards females. Similarly, the gene dosage from diploidy, underlying size dimorphism, and relationship bias from haplo-diploidy (sisters being more related than brothers) makes the soldier caste of hymenopterans predominantly female only. There are only rare exceptions where both sexes might participate in soldiering like embryonic soldiers of the the encyrtid wasp Copidosoma and we can explain that. This only reinforces the biological foundations of the sex-bias of the primary participants in inter-group conflicts in different species. That’s why I think this whole talk of sending girls to fight in the military clashes with some fundamentals.”

Vidrum: “This brings us back to a topic we had discussed few years ago when I was swayed by Samikaran. Why a specialized caste, like in our tradition, arose to perform the military role. If the whole population fought you have a numerical advantage; so why set aside a caste for that. What are the biological precedents for this?
Lootika: “Vidrum, I guess we have already explained the gist of this to you in the attempt to bring you out of your classmate Samikaran-the-maniac’s māyā.”
Somakhya: “But now that Vidrum has come away from those fancies I think we should try to unpack that again a bit and also in case Sharva is interested. It is indeed true that having the whole population fighting provides numerical advantage. It appears in chimps there is no particular soldier caste. As we saw in them there is even some female participation. Given the bias against female involvement in lethal combat in mammalian societies like ours, which we have just explained, we could still posit an advantage in the involvement of all males. Now certain ‘Kriegstaat’-societies indeed take that route. For example, the Mongols were one such. Chingiz Khan sent out his officials to conduct detailed censuses of all conquered territories so that the males could be recorded and organized into tümen-s which could be called up for wars. Thus, the Mongols saw all males a potential military man power. Now a society which has alternative structure when under stress will transform into this Kriegstaat pattern. We saw that with Mahārāṇa Pratāpa. We saw it on even larger scales with the Marāthā-s. There, in addition to V1 and V2s, traditional castes involved in various activities like agriculture and livestock-rearing, and even fine specializations like making cow-dung-pats and tapping palm-sap transformed into the warrior caste – that’s how we have a Karhāṭaka brāhmaṇa, the Hoḷkar, the Śinde and the Gāikvāḍ becoming Rājā-s (and Rajñī-s). But in the long run it is a more diversified economy that allows the effective conduct of war. Thus, it means other castes performing their specialized roles and channeling the fruits of the diversified economy to the war machine, which itself is primarily performed a dedicated caste. At a basic level this might mean farmers who can produce food, a essential to field large armies but it can involve various other specialized sub-groups. When the Marāthā-s transformed into a Kriegstaat, that seems to have drawn people away from these specialized guilds. That is where the English won. They could still maintain a large body of seemingly ‘useless’ knowledge-producers, like a Darwin and a Maxwell, for each of whom there were lesser tinkerers who could ultimately supply key technological innovations to the system that kept edge on the English war-machine. The success of this type of specialization in nature is simply evident in the world conquest of the ants, bees and termites.”

Sh: “That brings us back to the normal distribution which Lootika mentioned earlier. I guess that would also predicate that castes are likely to form when intra-group specialization enhances survival of the group.”
Vidrum: “That is fine. While I am no longer convinced by Samikaran’s Marxian uniformity I am still bothered by inequality which caste engenders.”
Lootika: “Over the years Somakhya has impressed upon me that 3 distributions more-or-less help us understand much of what we see around us. One is of course the normal distribution. Be it IQ, height, strength all of these are thus distributed and there is bound to be a consequence from that. Then paradoxically there is the power law distribution. This is the pattern seen in as disparate things as the numbers of genes in genome controlled by a given transcription factor, the sizes of human settlements, and the sizes of grains of sands. It is often called Pareto principle after an Italian who saw that in his country 80 percent of the land is owed by 20 percent of the population. Then there are so-called pathological distributions violating the central-limit theorem like the so called Cauchy distribution, the implications of which Somakhya mentioned sometime ago. Now, coming back to the second of these, the power-law, we can crudely state that the ‘rich get richer’. Thus, a lineage which is good at one thing tends to amass that trait in themselves. Hence, you are bound see inequality with a relatively small number who are really good at somethings and a large fraction being close to incapable. I can see why this can cause resentment in the have-nots. I think the way our tradition tried to resolve this is by specialization so that every group has something to be good at, thus limiting competition in a single track. When this system broke down under the assault of modernity we are seeing all these resentments bubble back. Hence, if at all we are going to find some means of mitigating it we need understand the force of these natural distributions rather than deny them. ”

Posted in Cricket, Life, Politics | Tagged , , ,

Pattern formation in coupled map lattices with the circle map, tanh map, and Chebyshev map

The coupled map lattices (CMLs), first defined by Kunihiko Kaneko around the same time Wolfram was beginning to explore cellular automata, combine features of cellular automata with chaotic maps. The simplest CMLs are defined on a one dimensional lattice with n cells. The value of the j^{th} cell in the lattice generated by a combination of the action of a chaotic mapping function f(x) and coupling of that value with the values of adjacent cells. Imagine a CML where the j^{th} cell is coupled with the two cells on either side j-1 and j+1 with a coupling fraction of \epsilon \in [0,1]. Then, the value of this cell in the next generation (n+1) (indicated as a superscript) of the CML is given by:

x_j^{n+1} = (1-\epsilon)f\left(x_j^n\right)+\dfrac{\epsilon}{2}\left(f\left(x_{j-1}^n\right)+f\left(x_{j+1}^n\right)\right)

Thus, the CML adds a further level of complexity coming from the chaotic behavior of the map defined by f(x) to the basic one dimensional cellular automaton principle.

One map of interest that can be played on a CML is the circle map discovered by the famous Russian mathematician Vladimir Arnold, which he proposed a simple model for oscillations such as the beating of the heart. It essentially performs the operation of mapping a circle onto itself:

x_{n+1}=x_n+\Omega-\dfrac{K}{2\pi}\sin(2\pi \cdot x_n)

Figure 1 shows the the iterates of x_0=\tfrac{1}{3} for K=.9 and \Omega \in [0,1]. We observe that there are several regions where the circle map generates chaotic behavior and other bands where it is mostly non-chaotic. Also visible more subtly are regions of less-preferred values.

Circle_mapFigure 1

We then play the circle map on a CML of 101 cells by keeping K=0.9 and varying \Omega to take multiple values. We take the coupling fraction \epsilon=0.5, which we found experimentally to give interesting results. We initialize the CML by setting the value of cell 51 to 0.5, and setting the 50 flanking cells on either side of it to the value of 0 in generation n=1. We then let it evolve such that if the cell on either edge of the lattice are neighbors of each other — thus the CML here is in reality plays out on a cylinder. Each value of \Omega results in a different kind of behavior of the circle map (The left panels in Figure 2). The corresponding evolution of the CML is shown in Figure 2, for 500 generations going from left to right.

kaneko_circle_1Figure 2

1) In the first case one can see that the map converges to a single value after a brief initial fluctuation. Correspondingly, when played on the CML it results in the seed anisotropy quickly dying off and the CML settles into a constant state.
2) In the second case the circle map shows an oscillation with a gradual concave rise and a sharp fall. While the oscillations are roughly similar in shape they are not identical. This results in the CML rapidly evolving into a complex pattern. The triangular elements seen in the pattern are reminiscent of those which emerge in cellular automata.
3) In this case the circle map generates sharp approximately regular pattern of oscillations, with rapid, abrupt changes in values. The corresponding CML evolves into a basic pattern of waves. Central seed sets up a pattern that develops into a fairly fixed width pattern the keeps propagating independent of the background waves.
4) Here, the circle map generates oscillations similar to above but slightly less-abrupt and has a more convex descending branch. This results in a more complex pattern developing from the central cell that stands out more clearly from the background waves. It gradually grows in width and shows a central band and flanking elements.
5) In this case the circle map generates oscillations with an abrupt rise and gradual, convex fall. This again, like case 2, rapidly generates a complex pattern.

To investigate the effect of other types of chaotic oscillations applied to the CML, we next considered the tanh map which is based in the hyperbolic tangent function. It is defined thus:

If x_n<0, x_{n+1}=\dfrac{2}{\tanh(r)}\tanh(r(x_n+1))-1,

else, x_{n+1}=\dfrac{2}{\tanh(r)}\tanh(-r(x_n-1))-1

This maps x_n \in [-1,1] \rightarrow [-1,1]. Figure 3 shows a plot of iterations of x_n for the parameter r \in (0,10]. For r<1 the distribution of x_n is all over the place. For r>1 the distribution of x_n becomes more and more U-shaped with a preference for values closer to 1 or -1.

tanh_chaosFigure 3

We then play the above tanh map on the CML with 101 cells initialized with a central cell (51) x_{51}^1=.5 for 500 generations. The coupling fraction is chosen as \epsilon=0.1. The experiment is run for different values of the parameter r. These results are shown in figure 4.

Kaneko_tanh2Figure 4

The first two values of r are in that part of the parameter space of the tanh map that produces highly chaotic oscillations. This results in the CML quickly evolving into nearly random continuous variation.

In the next three cases the effects of the U-shaped distribution of the iterates kicks in and we have predominantly abrupt up-down oscillations of the tanh map. However, the subtle difference in the oscillations causes clearly distinct results, albeit with some common features. In the third example, surprisingly, the CML quickly converges and freezes into several tracks of distinct periodic patterns. In the fourth and fifth case, we see tracks with patterns similar to those seen in the above case. However, they do not freeze, at least in the 500 generations we ran them. Rather, the tracks persist for different number of generations and then become extinct or evolve into other patterns after persisting for even 100 or more generations. These more regular patterns play out in a more irregular rapidly changing background.

In the last experiment presented here we consider the effect of the coupling fraction \epsilon on long-term dynamics. For this purpose we use the Chebyshev map, which is related to the eponymous polynomials of that famous Russian mathematician.

x_{n+1}=\cos(a \cdot \arccos(x_n))

This surprisingly simple map produces extreme chaos with a distribution similar to the tanh map for values of the parameter a>1.5. Values of a=1.5:2 produce interesting behavior in CMLs. In our experiment the we keep the Chebyshev map itself the same for all runs with a=1.8. Figure 5 shows the chaotic pulsations produced by this Chebyshev map.

Chebyshev_mapFigure 5

In this case the CML was run for 5000 generations and every 10th generation was plotted. It was initialized with the central cell x_{51}^1=0.1 and each of the flanking 50 cells on either side were set to x_j^1=-0.75. Here, the \epsilon value is varied to establish the effects of coupling on the behavior of the CML. This is shown in figure 6

Kaneko_ChebyshevFigure 6

The behavior is rather interesting:
1) At \epsilon=0.05 we observe that randomness permeates the entire evolution of the CML.
2) At \epsilon=0.075 there is a fall in randomness with repeated emergence of lines of persistence, lasting for several generations before going extinct. Some times they reappear several 100s of generations later. Within, each line, while it lasts, we see some fluctuations in intensity.
3) At \epsilon=0.085, the randomness mostly dies out by one fourth of the total number of generations of the CML’s evolution. Thereafter, barring the fringes, the lines of persistence alone remain over the rest of the evolution, albeit with some fluctuations of intensity within each line.
4) Interestingly, at \epsilon=0.095 we observe the emergence of “wandering” chaotic lines that emerge from old or spawn new lines of persistence.
5) At \epsilon=0.105, 0.115 we observe that most of the CML rapidly settles down in to strong unchanging lines of persistence.
6) The \epsilon=0.13 shows similar behavior to the above, except that certain lines of persistence display a periodic variation within them like a regular wave.
6) At \epsilon=0.16, we interesting see the return of chaos with repeated episodes of chaotic behavior breaking up old lines of persistence followed by emergence of new lines of persistence.
7) Finally, in the last two runs we see a return to the predominantly random pattern. However, this is qualitatively different from the first case in that it shows some short lines of persistence, which establish small domains of local structure.

Thus, the degree of coupling between the cells of the lattice affect the long term evolution of the system for same initial conditions and driving chaotic oscillator. In the range of \epsilon explored above we see an optimal point for freezing of persistent patterns with randomness dominating in the extremes of the range. However, within the more “orderly” zone we may see outbreaks of mixed chaos and pattern-persistence.

Simple CMLs are computational elementary and conceptually easy-to-understand as simple cellular automata. In some ways they captures natural situations more closely than cellular automata. But on the other hand extracting interesting behavior from them appears to be more difficult. Importantly, they are unique in providing a tractable model for how the local chaotic oscillations couples with other such oscillators. This is seen in many biological systems — networks of neurons, interacting bacterial cells in a colony, colonial amoebozoans and heteroloboseans — all are good natural systems for real-life CMLs to play out. We see chaotic oscillatory patterns in individual cells, which if coupled appropriately, can result in regularized patterns after some generations or rounds of interactions. Both nature of the underlying chaotic oscillator and the degree of coupling will determine whether randomness, frozen patterns, or dynamic but not entirely random patterns dominate. This gives an opening for an important force, namely natural selection, that is often neglected in such dynamical systems-based approaches. Selection is required for setting up the oscillator and its parameters as also the coupling fraction. Further, like CAs, CMLs also have potential as historical models, where local oscillations in populations and their interactions could be captured by the coupling of the chaotic oscillators.

Posted in Scientific ramblings | Tagged , , , , , ,

Some novel observations concerning quadratic roots and fractal sequences

Disclaimer: To our knowledge we have not found the material presented here laid out here presented in completeness elsewhere. However, we should state that we do not follow the mathematical literature as a professional and could have missed stuff.

\sqrt{2} has captivated human imagination for a long time. Perhaps, its earliest mention is seen in the tradition of the Yajurveda, which provides an approximation for the number in the form of the convergent \tfrac{577}{408} for construction of diagonals of squares in the vedi (altar) for the soma ritual. Yet, it has secrets that continue to reveal themselves over the ages. Here, we shall describe one such, which we stumbled upon in course of our study of sequences inspired by Nārāyaṇa paṇḍita, Douglas Hofstadter and Stephen Wolfram’s work.

A fractional number h lends itself to an interesting operation (the floor-difference sequence; we had earlier described it here; an operation studied by Wolfram),
f_0[n]=\lfloor (n+1) \cdot h \rfloor -\lfloor n\cdot h \rfloor
Here the integer sequence f_0[n] is defined by performing the above operation. If we use h=\sqrt{2} results in the sequence,
1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1…
This pattern of 1 and 2 is not periodic. Nevertheless, it has defined pattern. Wolfram showed that it can be produced by a substitution system entirely independently of \sqrt{2}, namely,
1 \rightarrow 1,2 and 2 \rightarrow 1,2,1
Notably, the ratio of the number of 1s to 2s in the string produced by the floor-difference operation (or equivalently the substitution system) converges to \sqrt{2}. Thus, the numbers of 1s and 2s in the sequence f_0 generated by the above process results in convergents that are like the partial sums of the continued fraction expression of \sqrt{2},

\sqrt{2}= 1+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\dots}}}}

Thus, in f_0[1:70], the number of 1s is 41 and the number of 2s is 29. This gives us a convergent for \sqrt{2} as \tfrac{41}{29}=1.413793 which is the 4th partial sum of the above continued fraction.

Case-1: Summation sequences defined on the floor-difference operation

Next we define a second integer sequence f_1 based f_0 thus,
f_1[n]=\displaystyle \sum_{k=1}^n f_0[k]==1,

i.e. we take the sum of all 1s present till position n in sequence f_0. Thus, for the above 20 terms of f_0 the corresponding terms of f_1 are,
1, 1, 2, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 8, 9, 9, 10, 11, 11, 12…
The basic idea for this procedure is inspired by Hofstadter sequences and the process to generate the tiling fractals described by Rauzy. We notice right away that the value of f_1[n] increases with n in a step-wise fashion along a linear growth line. But what is the constant of this linear growth?

We can derive this thus: Let x be the number of 1s and y be the number of 2s in a sub-sequence of f_0 of length n. From above we know that the ratio \lim_{n \to \infty} \tfrac{x}{y}=\sqrt{2}. Hence, we may write,
y=\dfrac{x}{\sqrt{2}}\\ x+y=n \; \therefore x + \dfrac{x}{\sqrt{2}} =n\\ x=\dfrac{n\sqrt{2}}{1+\sqrt{2}} =(2-\sqrt{2})n

With this constant 2-\sqrt{2}, we can now “rectify” the sequence f_1 i.e. remove its linear growth by straightening it along the x-axis and capture only its true oscillatory variation along the y-axis (see this earlier account for this). Thus, we get the rectified sequence,
Figure 2 shows the first 500 terms of this sequence.

root2_f2Figure 1

We observe that while f_2 takes a wide-range of positive and negative values they are all contained within a fixed bandwidth of 1. However, the values of f_2 are not symmetrically distributed about 0. The highest positive value is 2-\sqrt{2} and the lowest negative value is 1-\sqrt{2}.

We next perform a serial summation operation on f_2 along the sequence. Given the above asymmetry in f_2 with respect to negative and positive value take by it, we again get a sequence oscillating about a linear growth line. This time we can rectify by taking the midpoint of the bandwidth of f_2, i.e.,
\textrm{Midpoint}(2-\sqrt{2}, 1-\sqrt{2})=\dfrac{3-2\sqrt{2}}{2}

Thus, we defined the rectified sequence f_3 as:
f_3[n]=\displaystyle \sum_{k=1}^n f_2[k]-n \left( \dfrac{3-2\sqrt{2}}{2} \right)

Figure 2 shows a plot of f_3[1:n] up to different values of n. Figure 3 shows the same for a large cycle, n=33435 (see below).

root2_towersFigure 2

root2_towers_bigFigure 3

We see that f_3 has an intricate fractal structure resembling rising gopura-s around a central shrine. A closer examination reveals that the fractal structure of f_3 has cycles of increasing lengths, i.e. the same structure re-occurs with greater intricacy at the cycle of the next length (Figure 2, 3). We determined that the lengths of the cycles centered on the highest successive values of f_3 are 27, 167, 983, 5739, 33435… This led us to establish that ratio of successive cycle lengths converges to 3+2 \sqrt{2}. This number is the larger root of the quadratic equation x^2-6x+1=0.

We can do the same thing with the Golden Ratio \phi which has the continued fraction expression,
\phi= 1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\dots}}}}

In this case, the sequence f_0 can be generated by the well-known substitution rule,
1 \rightarrow 2; \; 2 \rightarrow 2,1
Here the ratio of 2s to 1s in f_0 converges to \phi. We can likewise construct f_1 by counting the number of 2s as we walk along f_0 up to a given n. As with \sqrt{2}, we can rectify f_1 to get f_2[n]=f_1[n]-n(\tfrac{1}{\phi}). Here again, the bandwidth of f_2 is 1 but the values it takes are asymmetrically distributed about 0 with a maximum of \phi-1 and minimum of \phi-2. This gives us the rectification to obtain f_3 for \phi,
f_3[n]=\displaystyle \sum_{k=1}^n f_2[k]-n \left( \dfrac{2\phi-3}{2} \right)

Figure 4 shows the fractal structure of f_3 for \phi comparable to that which we obtained for \sqrt{2}

phi_towersFigure 4

We then established that the fractal cycles for f_3 of \phi are of lengths: 32, 87, 231, 608, 1595, 4179, 10944… when defined on the basis of the successive highest values attained by f_3. Thus, the ratio of successive cycle lengths converges to 1+\phi in the case of the Golden Ratio fractal. 1+\phi is the root of the quadratic equation x^2-3x+1=0. This shows a similarity to the above convergent of the cycles of the \sqrt{2} fractal. Further, while that convergent can be expressed as \left (1+\sqrt{2} \right )^2, this one for the Golden Ratio can be similarly expressed as \left (1+\tfrac{1}{\phi} \right)^2

This leads to the conjecture that all such fractals generated from floor-difference-derived sequences of quadratic roots have as convergents such roots of quadratic equations with a relationship like the above ones to the original root.

root2_normal_likeFigure 5

There are some notable features of the distribution of the values of f_3:
1) The number of values >0 is always more than those <0 for a given fractal cycle. This markedly more for the f_3 of \sqrt{2} as opposed to that of \phi.
2) The distribution of the values taken by f_3 is approximately normal (Figure 5; shown for f_3 of \sqrt{2}).
3) Most notably, the f_3 fractal displays structures with quasi-mirror symmetry (figure 2, 3, 4), when we consider the distribution of values around given central points. For the \sqrt{2} case, convenient central points can be easily found in the form of the highest values reached in successive cycles (also the values on which we centered our cycles). To illustrate this quasi-mirror symmetry we show below 10 values on either side of f_3[2869], the central point of the cycle of length 5739:
f_3[2859:2868]: 1.038574, 0.603576, 0.582792, 0.976221, 0.783863, 1.005719, 0.641789, 0.692073, 1.156569, 1.03528
f_3[2869]: 1.328204
f_3[2870:2879]: 1.035341, 1.156693, 0.692257, 0.642036, 1.006027, 0.784233, 0.976652, 0.583284, 0.60413, 1.03919
We notice that the corresponding mirrored values are not equal on either side but very close. Further, the difference is systematic, i.e. the values on one side are consistently higher than their counterparts on the other side. The pair closest to the central point (1.03528, 1.035341) differs by 6.158394 \times 10^5. The next pair by twice that amount, the next by thrice, the next by 4 times and so on. Thus, as one moves away from the center there is a linear increase in the asymmetry by a constant amount until one reaches the ends of the cycle. By the end of a cycle the difference between the quasi-mirror symmetric pairs reaches a maximum of \approx 0.17. Thus, the minimum difference, i.e., the difference between members of the pair closest to the center-point is \approx \tfrac{0.17}{l}, where l is the length of that cycle. Hence, as the cycles get larger the symmetry increases closer to the central point (Can be seen visually in above figures). Similarly, for the f_3 of \phi we can establish the axis of mirror-symmetry as the being the central point of a cycle. Here too, the same dynamics as reported above for \sqrt{2} are observed, but the maximum difference of a pair for a cycle is \approx 0.22 and accordingly for a given cycle of length l the minimum difference of the quasi-mirror symmetric pairs is \tfrac{0.22}{l}. We have not been able to figure out the significance of these maximum difference values for either sequence and remains an open problem. Moreover, this structure of f_3 is of some interest because it seems asymmetry (or randomness) or perfect symmetry are way more common than quasi-symmetry which we encounter here.

Case-2: Product-division floor-difference
Indeed, contrasting real symmetry is obtained in a related class of sequences that we discovered. We shall describe their properties in the final part of this article. Instead of the floor-difference described above, we use a related kind of operation using irrational square roots of integers define the following sequence:
f_0[n]=\left \lfloor n \sqrt{2}\right \rfloor -2 \left \lfloor \dfrac{n}{\sqrt{2}}\right \rfloor

This is a sequence of 0s and 1s: 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0…

Based on f_0 we can define, along the lines of what we did above, another sequence thus:
f_1[n]=\displaystyle \sum_{k=1}^n f_0[k]==1
It is the count of the number of 1 up to the nth term of sequence f_0. It is an integer sequence of the form: 1, 1, 1, 2, 3, 3, 4, 5, 5, 5, 6, 6, 6, 7, 8, 8, 8, 9, 9, 9, 10, 11, 11, 12, 13, 13, 13, 14, 15, 15…

As with the above cases, we observe that the value of f_1[n] grows in a step-wise linear fashion with n. Thus we can again rectify it by determining its constant of this linear growth. We observe that in this case the f_0 as an equal number of 0s and 1s for a given length. Hence, we get the rectification constant as \tfrac{1}{2}. Thus, we can define a further sequence,
We multiple by 2 instead of using \tfrac{1}{2}n for rectification because we can that way keep f_2 an integer sequence.

Then, we define the next sequence based on f_2 thus:
f_3[n]=\displaystyle \sum_{k=1}^n f_2[k]

Since the distribution of f_2 in this case is symmetric about 0 we do not need any further rectification in defining f_3 and it remains an integer sequence. In this case the values of f_3 define a symmetric fractal with a bifid peak-like appearance (Figure 6).

sqrt2_double_floor2Figure 6

Here again, the fractal repeats itself at each cycle, with increasing detail as the length of the cycle increases. However, at every cycle the fractal remains perfectly symmetrical unlike the above-discussed cases (Figure 7). We can define the length of each cycle for this fractal based on the palindromic structure of f_3 for each cycle: Each cycle begins and ends in the sub-sequence: 1, 1, 0, 0, 1, 1

sqrt2_double_floorFigure 7

We determined that the cycle-lengths show the progression: 26, 166, 982, 5738, 33458, 195022…
Strikingly, the maximum value reached by f_3 for each of these cycles shows the progression: 6, 35, 204, 1189, 6930, 40391…
Thus, the ratio of both successive cycle-lengths and the maximum height reached in successive cycles, remarkably, converges to 3+2 \sqrt{2} — this is the same as the convergent for the above \sqrt{2} fractal derived from the floor-difference operation.
Notably, the successive partial sums of the continued fraction for 3+2 \sqrt{2} are,

6, \dfrac{29}{5}, \dfrac{35}{6}, \dfrac{169}{29}, \dfrac{204}{35}, \dfrac{985}{169}, \dfrac{1189}{204}, \dfrac{5741}{985}, \dfrac{6930}{1189}, \dfrac{33461}{5741}, \dfrac{40391}{6930}, \dfrac{195025}{33461}

We notice that the maximum value reached in each cycle is captured by the denominator and numerator of every 1, 3, 5, 7… 2n-1^{th} partial sum. The numerator minus 3 of every 2, 4, 6 … 2n^{th} sum captures the cycle-length: the reduction by 3 is evidently because we defined the cycle based on the re-occurrence of the palindrome.

This kind of sequence derived from the product and division by an irrational square root of an integer can be generated from such square roots too. Using \sqrt{3} yields a fractal with a single peak (Figure 8).

sqrt3_double_floor2Figure 8

Here, the cycle-lengths and maximum value attained by the f_3 converges to \left(2+\sqrt{3}\right)^2=7+4\sqrt{3}. In the case of \sqrt{3} we also have minimum values of f_3, which are <0 (Figure 8); interestingly, the ratio of minimum values from successive cycles also converges to 7+4\sqrt{3}. This number is the root of the quadratic equation x^2-14x+1=0

In conclusion, we find that two different operations of the floor function on irrational square roots or roots of quadratic equations yield fractals, whose cycle-lengths are convergents for roots of quadratic equations, which can be constructed based on the original root. The formal proof of this might be of interest to mathematicians. In the second case this also relates to the maximum value attained by the sequence f_3. Finally, it is notable that in the first case the values of f_3 show a certain quasi-mirror symmetry and an approximately normal distribution. Despite this overall distribution, the actually values are arranged in precise manner as to generate a fractal structure. This might yield an analogy to natural situations where a normally distributed population could organize into a highly, structured pattern.

Posted in Scientific ramblings | Tagged , , , , , , ,