A rambling talk on the śaiva mantra tradition

A rambling talk we had given on the śaiva tradition. Talking is easier but less precise than writing. So please be aware of the insufficiency that goes with the domain of any talk while perusing this material.

Part 1; Introduction to mantra-s and pāśupata-s as a MP3 file: https://app.box.com/s/3wx313rfm8wijnpiv42f1q8r8z9zg2vu

Part 1, 1st addendum; early legends and iconography of Rudra: https://app.box.com/s/iqt7g7vye7c9z9tvb314swljh3um9eem

Part 1, 2nd addendum; Deployment of Rudra mantra-s in vaidika rituals; give a brief account of rājagavī-homa the animal sacrifice to rudra in the somayāga, the somārudrā ritual, the abhicāra ritual invoking rudra against enemy’s cattle, apotropaic rituals for protection again rudra’s attack (rudraḥ paśūn/prajāḥ śamāyeta), atharvavedīya mṛgāreṣṭi, and paippalāda tuṃburu-yāga: https://app.box.com/s/zg2v7zr3b36ckps4xo1i6fzavmzow8qb

Part 2; The emergence of a new mantra-śāstra: the beginnings of the tāntrika lore. Touches upon 3 key features that contributed to the emergence of the tāntrika mantra-śātra – 1) the sectarian traditions; 2) the origins of iconic and maṇḍala worship; 3) The emergence of a new mantra languages, i.e. mantra-s without semantic value, prose mantra-s with onomatopoeic elements, and mantra-s emphasizing repetitive action:

Part 3; The śaiva tāntrika tradition; the 3 categories of śaiva tāntrika sādhakas: 1) the siddha-s and vidyādhara-s; 2) the sādhaka-s and ācārya-s; 3) the nīrbījadīkṣita-s and lay followers; the classification of knowledge in the śaiva tradition; the division of the śaiva śāstra into the five srotāṃsi: garuḍa, bhūta, vāma, bhairava and siddhānta; the gāruḍa tradition: Garuḍa, Śarabha and Nīlakaṇṭha; the bhūta tradition: Khaḍgarāvaṇa and his parivāra, Caṇḍāsidhara, Hayagrīva, Ardhanārīśvara and his dhyāna, Krodheśvara and Jvareśvara; the vāma tradition: Tumburu and the catur-bhaginī; the dhyāna of Tumburu in the Candradurdaṇda-maṇḍala; the transmission of vāma tantra-s and their history; special dhyāna-s of the four sisters; vāma mantra-s and yoga:

A very basic introduction to aspects of śaiva iconography with an emphasis on the icons depicted in saiddhāntika sthāpana tradition.


Posted in Heathen thought, History | Tagged ,

Citrabhānu’s cubes

The Hindus unlike their yavana cousins preferred algebra to geometry. Yet on occasions they could indulge in geometric games for demostrating proofs of algebraic relations. We see a bit of this in the Āryabhaṭa school and the great Bhāskara-II, but one of the masters of the self-evident geometric demonstration was Citrabhānu the nambūtiri brāhmaṇa poet, mathematician and astronomer from the cera country (1500s). His student (also that of one of India’s greatest pre-modern scientists Nīlakaṇṭha somayājin) was the ritual assistant Śaṃkara Vāriyār. In his Kriyakramakārī he records some of these self-evident proofs of his teacher. Now one such question Citrabhānu poses is to show without any algebra that:


Of course, this is simple 7th class algebra in our old school system. My father probably taught it to my while I was in 4th class. Yet it is some fun to show without using any algebra at all.


Citrabhānu has several such ‘wordless’ proofs that delve into figurate numbers. Of course this was not a late medieval innovation suddenly dawning on Citrabhānu. It has a long Hindu tradition going back to the Āryabhaṭa school and from there to figurate numbers of the Vedic ritual altar. Thus, it is another example of the continuity of Hindu science with the Vedic tradition of which Citrabhānu, like his coethnics, was a major practitioner. Finally, we may note that Citrabhānu seems to have had a particularly fertile abstract conception of higher dimensional space in the context of figurate numbers and what we show above is only the simplest of algebra.

Posted in art, Heathen thought, Scientific ramblings | Tagged , , , , , ,

Make your own sky map

We live in the age of photo-realism, be it for maps of the world or of the sky – it is a realism that not long ago was beyond the reach of even our vision. That is why even though we have such a photo-realistic sky map just a click away we still have a deep nostalgia for the days before it. This is part of the reason we finally bit the bullet and wrote our own latest iteration of a good quality sky map. There is another reason which we shall come to later but for now we shall wander as part of recording our own memoirs into its biographical precedence. In our early childhood we were introduced to astronomy by our father from book which impressed upon us the great size of the sun relative to earth or even Jupiter and also introduced us to some constellations wherein shone stars that made the sun look small. To our luck the northern horizon was almost completely cleared by the demolition of a building leaving a space that was not built up for ten years due to the typical ways of the municipality in those days. Even before we had the book, we could see Ursa Major from our balcony and we had also internalized the unmistakable Pleiades and Orion. The said book had few more famous constellations like Leo which we learnt thereafter. Those days of halcyon youth were marked by relatively dark skies in our city with the occasional boon of complete power-cuts which brought the even greater darkness that we desired. It was then that it became apparent that a vast starry realm lay above above with more stars than our book ever showed. Beholding that, the idea took root in us to draw our own sky map marking all the stars based purely on visual observation. This endeavor went or for about an year or so after which for reasons unclear to us our interest in astronomy faded and the said book vanished.

But then as we have alluded to before, almost 4 years later our astronomical interests resurfaced with a vengeance and this was in the year before the apparition of the Halley’s comet. Then we obtained another book with slightly better maps and also embarked on making our own telescopes. This showed us that the maps in the book were woefully inadequate and the quest for new ones began. It was then that we found a kindred spirit a schoolmate from the 3rd varṇa from the Lāṭa-Ānarta country who was possessed with all the correct lakṣaṇa-s of his varṇa. He, with his mahādhana, procured a set of deep sky maps of Japanese provenance. To our eyes these charts had a great beauty in addition to their accuracy. Their symbols made a deep impression on our young minds distinguishing variable stars, planetary nebulae, globular clusters, open clusters, galaxies and other nebulosities with their own symbols. Armed with these we were among the first lay people in our city who caught sight of the Halley’s comet when it came out from behind the Sun. We wrote about this to our local news paper and our names and photos appeared in it. But the summer of the apparition of the comet had much more in store for us. Those maps led us to many observations of deep sky objects and variable stars as also in the subsequent year the great supernova of the Tarantula nebula. We caught sight of that on a single day close to the southern horizon when the weather cooperated. It was in this phase that the urge to sky maps surfaced again for our precious photocopies of the Japanese maps did not cover the whole sky. Our parents took us to an astronomer in a nearby city and we showed him our maps of drawn for the observations of the great supernova. He kindly commended the accuracy of our maps and showed those from a well-known American collection of maps for comparison. He wondered how we determined the relative positions of stars and we remarked that we had a fairly good memory of the sky and also used the time-honored instrument the yaṣṭi-yantra for getting various relative positions.

However, that American collection was way too expensive for us to afford and the Japanese copies were wearing away from use and recopying made them only fainter. Simultaneously, the skies were deteriorating in our city and new constructions were coming up in front of our house completely blocking our view. But the Tarantula nebula supernova and our improving mathematics initiated us into a new line of study – one of a more theoretical type. It amazed us that a blue supergiant had gone supernova – this was unlike what the stellar evolution theories of the date had postulated. Hence, we compensated for the loss of the skies with more theoretical pursuits. Yet, as we occasionally taught astronomy to a few people the urge of making star charts remained alive recaptitulating the by now dead Japanese maps. We got poorer substitutes from a planetarium in our birth city but they were still only that. We had also made a planisphere. We then ran into an elderly brāhmaṇa man from the small-town Karṇāṭa country who was an autodidactic astronomer. He was a poor man who tried to make some money by selling planispheres, tops and some other objects he made by himself but hardly anyone bought them. He also remarked that there was hardly any one who was interested in studying astronomy in his midst though he tried hard to popularize it. He had a deep astronomical knowledge, had observed the sky for long, and made his planispheres with some ingenuity but his fate had some emotional impact on us. We saw a grim image of him gazing heavenwards from a dark spot with silhouette of a once glorious but now abandoned Hoysala shrine behind him. He shared with us a drive for astronomical study but it was enormously difficult for him to get new information which he desired too. We had read that the Hipparcos project had been completed and told him about it. Against this backdrop it hit us that there was no use sharing a fate like his, something that easily could have been ours too. We were essentially not of any consequence, even as amateur astronomers, simply outpaced by the instrumentation possessed by the amateurs in richer parts of the world. We had other passions too; hence, we turned towards them away from astronomy, and the urge for making star charts vanished. When our astronomy rose back yet again like the god Vaiśvānara who lay smoldering it was already a thing of the past as we were born again in the era of electronic sky maps.

But a parallel thread ran in our lives. As we were studying the śruti, it was becoming increasingly clear that, much to our Yajurveda teacher’s chagrin, we lacked the conviction in the central pillar of mīmāṃsā – the doctrine of apauruṣeyatva. Instead we became very much intrigued by the question of the age of śruti or the various layers therein. There were numerous obvious astronomical references that were catching our ears, leave alone the less-obvious ones. It was then that we studied the great leader of the Hindus Lokamanya Tilak, the poorly known Kameshvara Ayyar and Shankar Dixit all of whom had used similar techniques to arrive a plausible date for the composition of various vaidika texts. This idea fascinated us and colluded with our star-chart making urge. One of the things that was sorely lacking was a nakṣatra-based map which divided the ecliptic into the nakṣatra-s and showed the sky using a nakṣatra-based ecliptic grid. For this purpose we wrote a primitive program which interfaced either with our paper star charts or in the last 21 years with an existing program to visualize the sky as usual.

Recently, an interlocutor on the internet expressed the wish of incorporating a nakṣatra grid in the open-source program sky map software Stellarium (something we have been using for several years for our regular observing needs) and asked some questions in that regard. It was in this context that our desire to make our own sky map resurfaced and we finally implemented something which recaptitulates the aesthetic of those old Japanese maps. Of course today we have loads of raw material available a click away so the task was nowhere as complex as it might have been before. We outline that below so that anyone with some computer skills can make their own in a straight-forward way:

1) We conceived our map as fitting in a 33×24 sq.inch (83.82 x 60.96 sq.cm) rectangle and covering the whole sky. The idea was to produce it in png, pdf and svg format for use on the web or for printing in large format.

2) We implemented it in the R because it was one of the easiest to use, open-source languages supporting vector operations. It is good for generating graphical outputs such as this and has a preexisting library for computing projections (see below). The code for plotting was written in base R and special fonts were imported and embedded using the R extrafont library. The conversion of coordinates from ecliptic to equatorial were done using a converter function that we wrote based on code written by Arnab Chakraborty in the AstrolibR package.

3) After some experimentation we chose the Eisenlohr projection for our map. This rather remarkable projection was introduced by German mathematician and physicist Friedrich Eisenlohr in 1870 (Incidentally his brother was the chemist and Egyptologist who studied the famous Rhind Papyrus). It has advantages over all the rest but was not used commonly perhaps because its formula was intensive on calculations in the pre-computer era.

Let \lambda \in [-\pi,\pi] be the longitude; \phi \in [-\tfrac{\pi}{2},\tfrac{\pi}{2}] be the latitude and R the radius of the globe which we wish to project. Then the Eisenlohr projection in terms of the x and y coordinates on the x-y plane is given by:


T=\dfrac{\sin\left(\dfrac{\phi}{2}\right)}{\cos\left(\dfrac{\phi}{2}\right)+C_1 \sqrt{2 \cos(\phi)}}



x=(3+\sqrt{8})\cdot R \cdot \left(-2\log(V)+C\left(V-\dfrac{1}{V}\right)\right)

y=(3+\sqrt{8})\cdot R \cdot\left(-2 \arctan(T)+C T \left(V+\dfrac{1}{V} \right)\right)

It has the following notable features: i) it has no singularities so every point on the globe can be simultaneously represented; ii) it is completely conformal, i.e., it preserves angles across the globe completely; iii) it has the narrowest scale range for a conformal projection of 1 : 3+2\sqrt{2}=1 : 5.83; iv) it has a constant scale along the boundary colures. Thus, even though we get some size distortion near the boundaries its preservation of conformality is a useful feature for astronomical depiction especially when we want to highlight the ecliptic which lies close to the equatorial aspect. Given the above formula provided by the US Geographical Survey, we wrote our own function for the Eisenlohr projection but we also implemented projection using the projection function from the R Mapproj library (which has several different projections) in the event we wished to try out alternative projections in the future.

4) For the plotted data we chose the following (the source is: Vizier unless specified otherwise): (i) The venerable Yale Bright Star Catalog with 9096 stars which is for practical purposes complete in its coverage till visual magnitude m_V=7. It also provides variability and double star data; (ii) For the background shading of the Milky Way we chose a catalog of 18693 of the brightest stars from the Milky Way; (iii) For the Milky Way and Magellanic cloud boundaries a set of 1073 positions giving the boundaries and constellation lines, i.e. stars to be connected, made available in electronic format by professor Dan Burton based on his article, which I think appeared in Sky & Telescope; (iv) Boundaries for all 88 constellations defined by the International Astronomical Union; (v) shading in and around the Magellanic clouds obtained from 526 bright electromagnetic sources in those regions; (vi) we extracted 196 deep sky objects from the NGC. It covers all genuine Messier objects and other brighter objects that are accessible to telescopes in the 6-10 inch range (e.g. the Centaurus A galaxy).

5) The sky map distinguishes variable and doubles stars recorded in the YBSC using a bull’s eye symbol and a dash passing through the star respectively, which we inherited from our old Japanese maps. The deep sky objects are shown thus: galaxies are marked with an ellipse; planetary nebulae with a fish eye symbol; globular clusters with a many pronged burst; open clusters with a circle and cross; other nebulosities with a dotted circle; miscellaneous objects with an empty circle. All these symbols are obtained using the unicode Symbola font made freely available by George Douros. Unfortunately, none of the Devanagari fonts render correctly for conjunct akṣara-s other than in the png format. The star-color was computed using the difference of the B and V band magnitudes wherever available.

6) Finally, one of the key motivations for making this map was to have the nakṣatra division on the ecliptic. But how does one get the nakṣatra divisions? While it is clear from the Vedāṅga Jyotiṣa that the early Hindus did conceive the ecliptic as having fixed nakṣatra divisions, these have not come down to us. Over the ages starting from the Vedic period the nakṣatra divisions were named according to yogatārā-s. The root yuj in this term might be taken to mean junction stars i.e., stars marking the divisions of the nakṣatra-s or it might be taken to mean stars united with a given nakṣatra division. Making note of the asterisms defined by the yogatārā-s over the ages, we feel that the latter definition is more likely and sensible. It seems that the yogatārā-s are a remnant of an early observational phase in Hindu astronomy where they were a rough guide to locate the moon within a nakṣatra division. This observation was likely carried out using a rod (yaṣṭi-yantra) which was projected against the cusps of the Moon to see which yogatārā they were aligned with. Based on that the Moon’s nakṣatra division was then assigned. However, as predictive algorithms improved starting with the VJ gradually the lay Hindu practitioner lost touch with the nakṣatra-s as actually asterisms, i.e. the yogatārā-s. The situation was so bad among the astrologers that in when in the 1800s the great historian of Hindu astronomy, Shankar Dixit, asked several astrologers if they could identify the nakShatra-s, not one of them could correctly point even a few out in the sky. This knowledge, however, survived among the rare śrauta ritualists. Dixit encountered one from near Kolaba who was able to correctly point out all of them in the sky. He further cited an incantation from tradition that went as below:

khau kha jā trī gu cu gai co cho bhū yuk
cha hi trī ku cū che ko dvi yuk
ṅau kha chā ṅī ku ghu tri yuk
kha jā ku cū ghe gho

102 112 128 140 153 156 183 196 197
217 232 240 252 266 277 291
305 312 327 345 351 354
12  28  51  66  74  94

This contains the above code in the kaṭapayādi system with the vowels (a, ā, i, ī, u, ū, e, ai, o, au) being 1:9 and 0. To determine the rising point of the ecliptic from the nakṣatra asterism at the meridian at midnight one does the following: the first is Aśvayuji. If it is at meridian then the rising point of ecliptic is 102 degrees on the ecliptic. Thus, the separation of the rising point is given for each nakṣatra (note they are taking Betelguese to be Ārdra and it comes really close to Mṛgaśīrṣa. This illustrates that the Hindus were clearly aware of the positions of the yogatārā-s in ecliptic coordinates and that they were unevenly positioned. However, the nakṣatra divisions themselves are a regular 13 \tfrac{1}{3}^o (800 minutes of an arc) since the removal of abhijit; so the question remains where to start them. One could use the above incantation to calculate one such starting point. That was for the 1800s. Somewhat later, following up on the monumental work of Shankar Dixit, his acquaintance, the great patriot, BG Tilak, proposed that there should be a pan-Indian astro-calendrical reform the use of modern astronomical calculations to determine the pañcāṅga and fixing of nakṣatra divisions in a standard way across Hindudom. This project was taken up after the English tyrants left by the calender committee constituted under the great physicist Meghnad Saha and the jyautiṣa Lahiri. They produced a nakṣatra division recommendation, which while not exactly equivalent to the ancient division, is an entirely usable as a modern substitute. They began with with the premise of having the star Spica (Citra; \alpha \; Virginis) in middle of the citra division of the nakṣtra in ecliptic coordinates for 1956 CE. This resulted in the equinoctial colure passing through uttara-proShTapada. Examining this proposal we find that it has issues with respect to the yogatārā-s at Ārdra, which we have argued was originally Sirius, Uttarāṣāḍa, and Śroṇa but it is clear that some such issues will be there irrespective of the juggling of boundaries one might do. Hence, we took these divisions and converted them to the J2000 coordinates currently used in astronomy. This gives us the start of the Revati division at 9.112253 ^o in decimal ecliptic coordinates. Starting from there we took 27 divisions of 800 minutes of an arc each.

With the nakṣatra divisions in place we created 2 types of maps – one with the nakṣatra points as defining ecliptic longitudes and another with the same defining equatorial longitudes. Finally, one could also use the AstrolibR or any equivalent in your favorite system to get the positions of the Moon and plot it on the map as a visual pañcāṇga if you like.

Finally one may ask why write so much and make such ado about just plotting a bunch of points from coordinate files. Yes, that is all there is to this exercise but do it yourself and if you are a mere mortal like us you might find the process interesting.

The maps in PDF format(view with high magnification 150% or more):
1) Ecliptic longitudes
2) Equatorial longitudes
3) With Full moon(filled) of Sep 6 2017 and New Moon (empty) of Sept 21 2017 plotted on the map.

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

A note on Cepheid variables: reproducing the study of the period-luminosity laws

In the 13th year of our life, when the skies of our city were still tolerably dark, we observed two stars \beta Persei (Algol) and \delta Cephei from our balcony which faced north on every night they were visible. Thus, we reproduced the beginnings of one of the great, but less-appreciated, stories in science that took place in the late 1700s. The story of variable stars, i.e. stars whose brightness changes over time, goes back even earlier. Bullialdus in the late 1600s observed the star o Ceti (Mira) for 6 years and established that it varies dramatically in its brightness (magnitude) over a period of 333 days. Presciently he declared that the variability was because of dark regions on the star that periodically presented themselves to the earth even as the sunspots. Newton in his Principia adopted the same explanation for the variability in Mira and subsequently after further observations Wilhelm Herschel too believed this to be the cause for stellar variability. Today we know that it is not the case for the Mira variables but then star spots do play a role for rotating variables like BY Draconis.

The first evidence for other mechanisms of variability came up over 100 years later when the astronomers Edward Pigott and young, deaf Goodricke studied \beta Persei (Algol) and \delta Cephei. They concluded that the variability of the former might be due to an eclipse by a darker body. Thus, the theory of eclipsing binaries was born and was subsequently confirmed for Algol-type variables. For \delta Cephei Goodricke proposed the starspot mechanism. \eta Aquilae and \delta Cephei discovered by Pigott and Goodricke were the founder members of a whole class of regularly varying stars like it which were subsequently discovered: the Cepheids. The Cepheids, unlike the Miras tended to be of shorter period — several days rather than of an order closer to an year. Further, the Miras, unlike the Cepheids, while having a general periodicity did not strictly adhere to it both in terms of temporal period and actual magnitudes attained at maxima or minima. Like the Algol-type variables the Cepheids instead showed a strict periodicity in their light variation (the light curve) and also the maximum and minimum brightness they attained (amplitude). However, unlike the Algol-type variables but like the Miras they continuously varied in their brightness. This form of variability completely baffled the astronomers and physicists, who nevertheless came to believe in some kind of eclipse mechanism shortly after their discovery based on the strict periodicity they shared with the Algol and \beta Lyrae type eclipsing variables. This idea remained in force for more than a century. In the late 1800s the German physicist August Ritter carried out one of the first detailed studies on gas spheres. In course of that he almost prophetically predicted an alternative mechanism for variability of stars i.e. pulsation; however, it was totally ignored by astronomers.

But at the turn of that century Schwarzschild’s study of the Cepheid \eta Aquilae showed that its visual light variability was also accompanied by color (spectral type) variability. In 1908 Leavitt, while studying variables from the Small Magellanic cloud, discovered that for some variables with a regularly periodic light curves their period was directly correlated with their apparent luminosity (given that all stars in a Magellanic cloud were expected to be at roughly the same distance from our solar system). By 1912 she had established that this period-luminosity law specifically applied to Cepheids. Shortly thereafter Shapley marshalled various lines of evidence to suggest that the variability of Cepheids was likely not due to eclipses but due to pulsations. In the coming years Eddington developed the first physical theory for pulsating Cepheids by modeling them as thermodynamic engines that operate by pulsation, wherein the energy gain and dissipation compensate each other. Subsequent observations of the radial velocity of Cepheid derived from spectroscopy by Baade and others helped confirm the pulsation theory and also refined it further even as the physics of the stars was being better understood. In this context we should point out the key contribution of the sadly forgotten Hindu astronomer Hari Keshab Sen from Prayag, who was also a student of Vedic literature and studied the origin of the Hindu concepts of infinity in the upaniṣad-s. In 1948 while at the Harvard observatory, which was the great center of variable star research, he presented the first refined model for Cepheid variability: He showed that to explain the Cepheid variability one needed to develop a correct model for the internal structure of the star. He showed that for a star the ratio of the central gas density to its mean gas density was of key importance to explain variability. His work showed that this is particularly high Cepheids and using this in the model could explain the asymmetric amplitude of the Cepheid variability.

In years following the discovery of the period-luminosity law it was used by Hertzsprung to calculate the distance of the Magellanic clouds from the solar system. While this attempt had its errors it was the beginning of one of the most important developments in all of science: the use of Cepheids to measure the size of our universe, starting from the realization that the Magellanic clouds lay outside our own galaxy. The first step in correctly applying the period-luminosity law to measure the universe was the realization that the Cepheids were not a monolithic group but included a range of distinct families each obeying their own period-luminosity relationship:
1) W Virginis or type-II Cepheid stars: These are old stars with mass equal to or lower than the Sun but are distended having radii tens of times the Sun. They tend to be of the spectral types from mid F to early K and about -.5:-2.5 in absolute magnitude. They are mostly found in the central galactic spherical component and globular clusters.

2) RV Tauri variables: These tend to be more luminous giant versions of the W Virginis stars with an absolute magnitude in the -4:-5 range and are redder with more G to later K-type spectrum. Their light curves show alternating deep and shallow minima and concomitantly their period-luminosity laws are much more lax than the classic or type-II Cepheids.

3) The RR Lyrae variables: They are sub-giants usually of the A-F spectral class again with a low mass of about half the Sun. They too are old stars which have previously lost gas as red giants before climbing down to just above the main-sequence in the Hertzprung-Russell diagram. They too are present in the spherical center of the galaxy and globular clusters.

4) \delta Scuti variables: These are whitish stars in the A to early F spectral class and are characterized by a low amplitude and short-period pulsation (roughly < 7 hours). They are close to the main-sequence.

5) Classic Cepheids (DCEP): These are also known as type-I Cepheids and are prototyped by the original members like \eta Aquilae and \delta Cephei. They are massive, giant stars: 4-20 Sun masses and with typical absolute magnitudes of -1.75:-6.4, i.e ~500:31,000 times the brightness of the Sun (the brightest being 200,000 times as bright as the Sun). They are young, metal-rich stars which in galaxies like ours are mostly in regular elliptical orbits around the galactic center. Within the classical Cepheids, we might further recognize the DCEPS group which includes representatives like the current Pole Star \alpha Ursae Minoris which show an amplitude of visual magnitude pulsation less than .5 with generally symmetric light curves.

These Cepheid-like stars define a strip across of the Hertzsprung-Russell diagram known as the instability strip which mark the region in it where an evolving star develops period-luminosity law-type pulsation. Like any oscillator a pulsating variable star can pulsate in the fundamental mode of oscillation or as a higher overtone. The mode can be understood thus: if it is pulsating in the fundamental mode then all of the stellar sphere moves in and out at the same time -i.e. all of the star contracts and expands in unison. If it is pulsating the first overtone then we imagine a central shell, the nodal sphere, within the star where the gas is at rest. When the gas inside that shell is contracting then gas outside it expands and vice versa. In the second overtone pulsation we have two concentric nodal spheres with the gas inside the outer and innermost shells contracting when that inside the middle shell expands and vice versa. Both fundamental mode and overtone pulsation are seen across Cepheids types.

In the remainder of this note we shall explore the period-luminosity law for classical Cepheids using the wealth of modern data from ground- and satellite-based observations. To start we take a data-set of 674 Cepheids from the Milky Way for which we have reliable period, mean visual (V band, m_V) magnitude and parallax-derived distance information. Using the distance in parsecs d we calculate the absolute magnitude ( M_V) of the stars using the formula:

We then plot M_V against the base 10 logarithm of the period (in days) of the star to see if we can recapitulate the period-luminosity law (Figure 1). In Figure 1 the regular classic Cepheids are in blue. The DCEPS group is further shown in red. Those DCEP that do have an overtone pulsation are colored violet and those DCEPS with overtone pulsation are colored green.

Cepheids_Figure1Figure 1

What we see in Figure 1 is that there is a general positive period-luminosity correlation. But by no means it can be termed a strong law
with a low correlation, r^2=.22. Nevertheless we can observe a few interesting things: the DCEPS are not found at the high end of the luminosity range but they are dominant at the lower end of the luminosity range. Similarly, the overtone pulsators are more at the lower side of the range. In any case with a correlation like this one can hardly expect to use Cepheids as an accurate measure of distance. So what is the issue here? The answer to this lies in the observation that was first made in the late 1800s that light from a star does not reach us unimpeded: it is scattered by dust in interstellar space. This dust selectively scatters the lower wavelength light towards the blue end of the spectrum more than that from red end. Details of this extinction of starlight were worked out only in the last century. It can be quantified by using the measure E_{B-V} which is the observed excess of the difference between the B band and the V band filter magnitudes relative to the real B-V color index.

We have these E_{B-V} values for a high-quality data-set which is a subset of 383 brighter stars of our starting data-set. For this we additionally have the magnitudes measured in several photometric bands in addition to the V band. Hence, to start we make the same plot as above of M_V computed from distance in parsecs (Figure 2).

Cepheids_Figure2Figure 2

We note that even though we have only half the data as in the first plot and we have lost some of the lower luminosity stars the results are similar. This suggests that the what we observed above is generally consistent across the data: a low correlation r^2=.23.

Now we make use of the E_{B-V} values for these stars applying a multiplication factor of 3.23 and using the below formula:

Now we plot this extinction corrected M_v against base 10 logarithm of the period (Figure 3).

Cepheids_Figure3Figure 3

Now with a correlation of r^2=.92 we get an unambiguous period-luminosity law for the classical Cepheids ranging from -1.75:-6.4 M_v. The tendency of the low-amplitude DCEPS to be excluded from the high end of the magnitude-period range is also clear.

In the above approach we solved the issue of extinction of starlight by interstellar dust on a star-by-star basis using E_{B-V} for each star. To tackle this directly rather than for each case the astronomer Madore defined a new magnitude function called Wesenheit W that is free of the extinction effect. Wesenheit can be calculated by measuring the magnitude in multiple photometric bands e.g. V and and the near-infrared I band (806 nm midpoint). For illustration with our current data-set we shall use the V and I band Wesenheit. For that we first calculate the near-infrared absolute magnitude from the apparent m_I magnitude of the star which has been obtained using the I band filter:

We then couple this with the uncorrected M_V calculated from m_V as above using the below Wesenheit formula:

We then plot this against \log_{10}(P) as before (Figure 4).

Cepheids_Figure4Figure 4

We see that we now get an even better period-Wesenheit law than with the case-by-case extinction correction: r^2=.98. Thus Wesenheit is a very effective means of capturing the Cepheid period-luminosity relationship in an extinction independent manner.

Cepheids_Figure5Figure 5

Finally, if we look at a histogram of extinction-corrected absolute magnitudes for these Milky Way Cepheids (Figure 5) we find an interesting distribution with a clear right skew and hint of bimodality or a thick right tail. The skew is likely in part an artefact of losing the lower luminosity overtone pulsators. There is a clear peak at M_V=-3, second peak or shoulder at around M_V=-3.8 that leads to a fat tail. This suggests that at least in this dataset classical Cepheids themselves might have at two sub-categories based on luminosity.

We next turn to classical Cepheids in the Large Magellanic cloud and the Tarantula nebula. For this we use the data collected by the ground-based Visible and Infrared Survey Telescope for Astronomy with a 4.1 meter mirror operating from Chile and released in 2012. This data-set contains 334 Cepheid stars from the LMC and the Tarantula Nebula with high quality period and magnitude data in the V, I and the K_s (deeper near-infrared: 2190 nm midpoint) bands. It also has information regarding whether the star is pulsating in the fundamental mode or has overtone pulsation. However, there is no direct distance data: the LMC being extra-galactic is out range for direct parallax measurement. Hence, we shall use this data to perform a modern recapitulation of the efforts starting with Hertzsprung. However, in this case like Leavitt did we can assume that all the stars are at a generally similar distance from our solar system as they are from a single external galaxy. Thus, to investigate the period-luminosity law we may directly plot apparent magnitude m_V against \log_{10} period (Figure 6).

Cepheids_Figure6Figure 6

We see some interesting points right away: 1) We get a much better period-luminosity relationship even with no correction for starlight extinction: correlation r^2=.58. The LMC being an external galaxy would be subject to roughly similar extinction across all its stars. Hence, the deviations caused by extinction have less consequence for the basic period-luminosity relationship in this case. 2) As we saw in the case of the Milky Way, the overtone pulsators (plotted as red asterisks) occupy the lower-end of period-luminosity range while fundamental tone pulsators the higher end (blue dots). Notably they share a clear zone of overlap in the middle of the range. 3) Closer examination of the two pulsation modes hints that the two groups of stars might actually lie on separate lines.

We next utilize the K_s magnitude to plot a period-luminosity for it (Figure 7). Being in the deeper near-infrared we expect it to suffer much less extinction than the visual band of the spectrum.

Cepheids_Figure7Figure 7

Sure enough we see that the K_s magnitude shows much less dispersion than the V band magnitude. Strikingly, what we saw as less-clearly with the visual magnitude, comes out plainly in the K_s band: not only are the overtone (red asterisks) and fundamental mode (blue) pulsators separated by the range occupied in the period-luminosity relationship, they also lie on two distinct, roughly parallel lines which appear shifted by an approximately constant factor.

We next utilize the V-I Wesenheit as per the above-stated formula to study the same thing (Figure 8).

Cepheids_Figure8Figure 8

Again, Wesenheit being free of the extinction factor gives a clean period-luminosity relationship and clearly shows the separation of the fundamental and overtone mode pulsators into two nearly parallel period-luminosity relationships.

Now that we have the period-luminosity laws for the LMC Cepheids in place we next use it calculate the distance of the LMC. For this we first need to calculate the absolute magnitude M for these stars from the period. For this we can try out any of a number of formulae:

This formula was based on relatively small sample of Cepheids studied using the Hubble space telescope.

M_V= -2.76(\log_{10}(P)-1)-4.16
This formula using the visual band magnitude was derived by Madore and Freedman in their seminal article on classical Cepheids.

M_I= -3.06(\log_{10}(P)-1)-4.87
This formula was provided for the infrared I band by Madore and Freedman in the same article.

This formula was based on calibration of Milky Way Cepheids using the Hipparchos space observatory’s parallax measurements.

Once we get M_V or M_I with these formulae we next compute the distance modulus:
From the distance modulus we can easily get distance in parsecs as:

Since we saw considerable dispersion of the m_V values (Figure 6) and given that the corresponding M_V computation by these formulae will have the issues of extinction we shall instead resort other formulae that can correct for this using both m_V and m_I to compute distance modulus directly:
This is the formula of the team of the astronomer Sandage. It seems to underestimate \mu.

This is the formula of the Freedman team which is what we shall use for our purpose. Having calculated \mu we get distances for the stars from it and plot the distribution (Figure 9).

Cepheids_Figure9Figure 9

Thus, we get the median distance modulus \mu=18.5 and median distance for the LMC as 50,086.8 parsecs. However, we notice that there is a clear bimodal distribution of the computed distances with peaks around 54,000 and 42,000 parsecs. Why do we see this and which of these is correct? The answer lies in the earlier period-luminosity plots where we saw two distinct laws for fundamental mode and overtone pulsators. This was the key finding detailed by the astronomer Böhm-Vítense who suggested the overtone pulsators are shifted by \Delta \log_{10}(P) \approx .15 with respect to the fundamental mode Cepheids. Hence, to apply a uniform period-luminosity law we have to transform the observed period P of the overtone pulsators to their fundamental mode period P_0 by the formula:

If we use this transformation in the above formula of the Freedman team we get the distance distribution plot seen in Figure 10.

Cepheids_Figure10Figure 10

We now get a sharp, single-peaked distribution with a mean distance modulus of \mu=18.64 and median distance of 53660 parsecs for the LMC. These values are within the range of values published in the last two decades. Most authors these days favor a distance modulus of \mu=18.5. Thus, even though the period-luminosity law looks like an obvious way to measure the distance of the universe and even estimate the expansion rate of the universe in the form of Hubble’s constant, actually doing so presents several challenges some of which we have just seen in the above experiments.

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

The apparition of the Scottish surgeon

On a late summer evening, at the twilight hour, Somakhya accompanied by Lootika met Vidrum outside his home. Vidrum was supposed to show Somakhya something special he had found during a visit to the limestone excavations near his ancestral village. Vidrum did not seem happy on sighting Lootika: “Lootika I was supposed to be showing this only to Somakhya. I had not invited you to come and take a look!”
L: “Why the gruffness? What did I do?”
Vidrum: “Your sister Vrishchika is so rude.”
L: “OK. Now I was not the one who told her to be rude. What happened?”
Vidrum: “I kindly showed her the object to see which is why you have accompanied Somakhya and she laughed much and taunted me in front of the other girls much as you all had done when I showed you my earlier discovery.”
L: “We did not taunt you before other girls. It was just the four of us then.”
Somakhya: “Hey what is behind this little drama between you two? In any case think we should cut the crap and get to business.”
Vidrum: “As I told you I have found evidence for a gigantic human who lived in the prehistoric times in my ancestral village. I have recovered a piece of the skull and another bone of this fossil man! I showed it to Vrishchika as she had come along with some of the other girls to take a look at my discovery. She laughed and told them something like: ‘this Vidrum is anxiously waiting to get admission into Med school. Even if he does get in I think it is not going to be easy for him. How can it when he shows off such things as human skulls?’ Then she added with a smirk looking at everyone in the group: ‘Even Scrotum humanum seemed more sensible.’ I don’t know what exactly that meant nor do I think the rest but it certainly sounded rude and there was much ridicule at my expense.”

L: “Oh Ṛbhu-s! if that was what she said it is rude indeed. I will certainly ask her to apologize the next time she sees you. But while recounting this to us last night over dinner she said that you had rudely shooed her away from studying the fossil. She added that she was merely trying to correct your mistake – namely taking a remarkable tortoise shell to be a giant human calvaria. I did not know she had told it to you in this fashion. I was wondering why you were so brusque as to not let her photograph it. By the way the Scrotum humanum was not a risque reference to you – an old English gentleman had labeled the distal fragment of the femoral head of the dinosaur Megalosaurus as that for he thought it resembled the said human organ. So she was simply striking the analogy with your own conflation.”

Vidrum now felt a bit soothed by Lootika’s presence as most men felt so long as they did not have have to engage in a cerebral tournament with her.

Somakhya: “Alright. Let us see it man!”

Vidrum pulled out his box and proudly displayed his finds saying: “As per your suggestion I recorded the precise coordinates of my finds and have even brought a little piece of limestone from karst where I found. Somakhya: “Good job”. Somakhya and Lootika quickly sized up the fragment of the carapace and remarked that when complete in life it was a giant tortoise with a shell along up to two meters in length. This was confirmed by the stout femur which was the second bone Vidrum had uncovered. Somakhya: “Vidrum, this is a glorious find. It is the great tortoise Megalochelys. We need to be describing this properly if you let us do so.” As they took some further measurements and photographs Vidrum remarked: “You all are the masters of such recondite knowledge. So when you say it is a tortoise shell and not a human skull I should be believing you. I was thoroughly disappointed that it was not a giant fossil man. But from what you say there is something redeeming here.”
Lootika: “Certainly. Do not feel too bad about it. The famous English anatomist Richard Owen, an enemy of Darwin, who was renowned in his age for his anatomical expertise had mistaken a large turtle Meiolania for a lizard. He was subject to even greater embarrassment than what you suffered from Vrishchika’s words when his error was exposed by Darwin’s friend Huxley.”

Vidrum: “That feels better. I never thought turtles could get that big. I thought it was only in legend that we hear of Viṣṇu in the form of a great turtle.”
Somakhya: “I have had this hypothesis that perhaps our Ārya ancestors found remains of Megalochelys which inspired the legend of the great turtle which originally was a form of the great god Indra, Ākūpāra. Later it was conceived as an avatāra of of his brother Viṣṇu.”
Lootika: “Perhaps they found fossils of this reptile in the Siwalik range which served as the inspiration.”
Somakhya: “While a plausible hypothesis, we encounter the Ākūpāra even earlier in the 8th maṇḍala of the Ṛgveda, which was likely composed before they even entered the Siwalik region. So I really don’t know if it was our subcontinental Megalochelys which inspired them.”

Somakhya: “It is late now and the we better get back home especially as I need to cross the Jawaharlal Nehru street before the marūnmatta-s spill out from the Friday sermon. We’ll be back tomorrow to study your fossil at greater depth.”


The next day Somakhya and Lootika again met with Vidrum at the appointed hour. He seemed wan and forlorn: “Friends! it is an utter disaster!”
Lootika: “What?”
Vidrum: “My precious fossils were stolen by someone when I was playing cricket today.”
Somakhya: “Why the hell did you have to take it along to show off. For a little attention from the girls you have now lost something colossally important as this! Who do you think stole it?”
Vidrum: “I, Sharvamanyu and Golashiras tried hard to figure it out for much of the day but we failed. You know the rougher elements of our class like Sphichmukh who stole the infamous compass box or the mystery thief who stole the green-covered chemistry book and the ball-bearing that Somakhya had gifted you Lootika.”

Lootika in a weak voice remarked: “Ah! the green-covered book and ball-bearing. I won’t forget that day. Some of classmates are going to be finished taskara-s even before they are adults.” Somakhya noticed a tear drop beneath the normally unsentimental Lootika’s spectacles. It was a book he had gifted her on the occasion of her birthday the year they had come to know each other. He was internally happy to note that they carried value for her but otherwise their moods had turned somber. Somakhya, who was leaning on his bike, sprang on to it to leave when Vidrum asked them: “Why go so soon. What where you two going to do?” Lootika: “We had set the evening clear to study the fossils. But then … And you?” Vidrum: “Sharvamanyu was to stop by and we were to play some chess. I know you two hate that game but may be you could hang on and we could do something else?”

As Somakhya and Lootika were wondering whether to stay on or go do some sky-gazing Sharvamanyu arrived: “Hey we need not play chess. Since you two are around I suggest we ply the planchette! You have an uncanny ability to get real ghosts into the machine.” Somakhya and Lootika agreed that it might be a good thing for the gloomy evening.

Going up to Vidrum’s terrace they pulled out his Parsi planchette from its case and for a while they plied it the usual way hoping for an interesting specter to catch the pointer. A whole series of nonsense-words were all that seemed to tumble out from the frivolously wandering pointer. They were pretty soon convinced that it was more eidolon than phantom. Vidrum and Sharvmanyu looked at their other two friends quizzically: “Nothing seems to be working here. Why don’t you two do your more magical prayoga-s?”. Lootika: “They usually work only in the cemetery. Do we really want to head there right now?” After some hesitation Somakhya decided to deploy another one right there. They snuffed out the lamp and lit it again and began. It took sometime for Somakhya to get his focus on the dhyāna right. Finally, when he had settled into it and gone through the incantations he uttered the terminal summoning call: “kumbhodharā mahodharā rudrasyānuyāyinaḥ śmaśānvartinam ānaya ānayeheha svāhā |”. For a minute or so, which almost seemed like an hour, nothing at all happened. Vidrum and Sharvamanyu cast confused looks at Somakhya whose visage barely concealed a sense of great triumph and expectation. Even Lootika did not seem to get it and had to restrain herself more than once from saying that something might have gone wrong in his spell. Then like a sudden autumn mist descending upon them the air went absolutely silent and against that backdrop they heard the footfalls and heavy breathing of someone who seemed to be tiredly shuffling in. Vidrum was no stranger to strange happenings and Sharvamanyu had seen their friends pull off the outre more than once; yet, the novelty of it happening never ceased to cause them amazement. The two were even more startled when the lamp on the pointer violently flickered and went off and the pointer rattled a bit on its own. They exclaimed in tense excitement and, though Somakhya and Lootika remained quiet, they nervously asked: “Hey are you there? Tell us your name?” The pointer clattered again as though to answer in the affirmative.

But what happened next made Vidrum and Sharvamanyu almost jump up in shock: an incorporeal voice seemed to emanate from the pointer speaking with a clipped accent almost betraying a stiff upper lip: “I would rather that you show me some respect. I am not to be ordered around”
Sharvamanyu: “But who are you?” The bhūta spoke: “Why should that matter to you.” Then the pointer rattled again and turned towards Somakhya and Lootika: “These two Hindoos here have awoken me from my resting place at my favorite peepul tree in Saharanpur and brought me all the way south here! I never knew someday a native would be talking about my tortoises in the same words as me!”
Lootika: “Pray tell us the story of your tortoises. We are all ears.”
The phantom: “My Hindoo collector had found a large shell as I was prospecting in what your people called the lower steps to the abode of the god Śiva. Having received them I decided to make a detailed anatomical study and placed them in the baggage train which was being plied by my muleteer – a Hindoo from the Punjab. I knew very well that the Hindoo and the Mahometan were fundamentally embittered. Hence, I asked my colleague Cautley to make sure that there was no rupture between the two since he had a couple of Mahometans as assistants. Despite my caution mayhem erupted that night and the two buggers hacked my muleteer to death over some dispute that we never unraveled. They also smashed the great tortoise shell we had obtained. I ordered my Sikh sepoy Chukrum Sing to catch and discipline them. He handed over the Mahometans to the Hindus of the village from which our muleteer hailed and they immolated the poor wretches. Nevertheless, I managed to recover some fragments of the tortoise shell and duly described it upon my return to Britain. It was then that I had a conversation with my dear friend ol’ Charles and the young naturalist Mr. Wallace. As we marveled and meditated upon the anatomy of the great tortoise I informed them that this tortoise was the one that had inspired the legends of of the great tortoise among the Hindoos and the Pythagoreans. Hence, I was stirred in my resting place at Saharanpur when I heard you natives talk about this. I was even more interested to know as to what geological age you would have ascribed that tortoise to be.”

Somakhya: “While remains could have have been reworked from a much older sediment, based on the sediments among which our friend Vidrum found them we would say that they are actually of very recent provenance – from the later part of the Pleistocene – perhaps a mere 50,000 years before present. Sadly we have lost the fossil and would not be able to say more for now.”

The phantom: “That is certainly of some interest if true. Why! it pertains exactly to what I wished tell old Charles a couple of days before I expired.”
Somakhya: “Please make yourself comfortable and if you do not mind kindly tell us what was it that you wished to convey to your friend?”
The phantom: “Ah! It is a long yarn. Due to a persistent illness I left India and returned to Britain. It was then that I met my friend Charles who confided with me his thoughts on the origin of species. It shook me to the core and I was not sure for all its grandeur if it even had a smidgen of truth to it. After all it conflicted thoroughly with my own observations on the races of fossil animals in India. But the more I thought of it the I began seeing the elements of truth in Charles’ thesis. When he published his grand essay on the origin of species I was seeing more from his viewpoint than before. But one thing still bothered me. My studies on the races of fossil mammals had suggested that there were prolonged periods of stasis during which I saw little change unlike what Charles’s theory would imply. Then all of a sudden we would see a spurt of new animal forms. What Charles said made sense with respect to his pigeons and all the little changes we saw with domestication and the like. But in the actual fossil record that was not the way things appeared. Hence, when I sent friend Charles my heavy memoir on the prehistory of elephants I remarked to him that the glaciation was a profound climatic shift which could have selected for anatomical changes of the order of making his dear pigeons into ducks. Yet I saw nothing of that sort with my elephants. I suspected that nothing like that happened with my giant tortoises either. I wished to bring this to his attention and see how his theory and my findings on the extinct races of animals might be reconciled.”

The phantom paused and seemed to breath very heavily and groaned a bit. Then he added: “I knew well that Charles was thinking deeply about this too and trying to reconcile the matters. He wrote to me that weekend inquiring of my health as I felt my body come apart and heard the toll of the funeral bells. Seeing his letter I felt better the next day and hoped to write back regarding the tortoises and also that splendid Jurassic dinosaur Archaeopteryx which I had examined during my ramblings in Germany.”

Somakhya: “That’s most remarkable. Biologists over the ages since your weighty pronouncements have been struggling with that problem of reconciliation. The proximal details have been worked out well since your times but a fine point remains mostly unsaid. I and my friend Lootika here have also been thinking about it intensely I believe we might have a prolegomenon to a solution.”

The phantom: “Someday I might visit again to hear of that. But it is not good for a gentleman to yarn much; hence, I desist from asking you more.”

Lootika: “We would certainly let you return to your lair but please tell us if you might care the story behind your residence in that aśvattha tree in Saharanpur.”
The phantom groaned and emitting the sound of labored breathing said: “My aches still seem to pursue me into my phantomhood. But I should not refuse a charming young lady her request. As I was making a study of the plants in the east of the country, as a surgeon I was impressed by the orderly presentation of information on medicine by the Hindoo of yore. I was convinced that the spring-head of old knowledge in the arts, grammar and the sciences lay in India. It was in your variety of the Caucasian branch of the human family that highest capability for mental improvement had taken root. Hence, to learn more during my second stint in India I made acquaintance of the native physician Mudhoosooduna. In course of our rambling conversations he recounted the many quaint superstitions and fables of your peoples. In all earnestness he once remarked that since I was a bachelor with no issues and a mleccha it was conceivable that I would transmogrify into a bhoot upon my death as there would be no one to offer turpana for me. I laughed off his superstition but in a half-jocular vein remarked that if that were to happen I would like to lead a dryadic existence on the peepul tree I had planted in my groves at Saharanpur. Mudhoosooduna seemed to take it seriously and even approve of it. He remarked something to the tune that it might after all be my ultimate soteriological solution. Hence, the last time we met before I had to return to Britain due to my rapidly deteriorating health, old Mudhoosooduna handed me an amulet and asked me to hold it even as I heard the harsh barking of Kerberos. I took it not so much out of belief as much as adding another anthropological curiosity to my vast collections. On the eve of my expiration for some reason I used it as bookmark while reading friend Charles’ latest work – and lo I passed away from the world of men clutching it. Indeed when the pains of passing had passed I found myself upon that very peepul tree in old Saharanpur. I rarely leave my haunt except on occasion to see old Mudhoosooduna who is doing is time as a brahmuhrakshas on the banks of the Ganges. He was sadly blown to smithereens by one of my former patients from the Bengal army during the great mutiny of 57.”

The same quiet which came upon them when the phantom had come seemed to return and a few seconds latter ebbed away into the cacophony of orthopterans and other insect musicians.

Sharvamanyu: “This was most wondrous! Had it not been for you all even I might have felt some fright. This was unlike anything we ever felt in the cemetery.”
Lootika: “Yes indeed! That was quite unexpected.”
Vidrum: “This was a bhūta which actually spoke! Thankfully the gorā sāhib seems to have left without causing any trouble unlike those who normally prowl here. But you all seemed to know him well.”
Somakhya: “While one of the ākrānta he is a benign chap – in terms of pure knowledge he was perhaps the most knowledgeable biologist of his age! Let me tell you what his friend Charles, whom he kept referring, to remarked upon his death. Somakhya pulled out his tablet and read: “He says ‘What a mountainous mass of admirable and accurate information dies with our dear old friend. I shall miss him, not only personally, but as a scientific man of unflinching and uncompromising integrity…’”
Sharvamanyu: “But who was he?”
Lootika: “I believe we should not be actively revealing his identity anytime soon. But he told us enough that any discerning person can identify him.”


Somakhya came back early from the game of ghaṭika as his mother had told him that Lootika’s mother was to come home for a certain strīkarman along with her four daughters. When their mothers and some other women became busy with the rite, Somakhya and the caturbhaginī huddled into his lab-room. Somakhya allowed little Jhilleeka to use the box of sticks with sockets, which his father had gifted him from a foreign land for making polyhedra. He also gifted her a brown porcelain insulator. Varoli told Somakhya of her visit to the dreadful city of Visphoṭaka and her adventure of the astronomy quiz. Somakhya sensed some deeper excitement in her which certainly could not come from the second place she had won in the quiz – having trained her he knew that she was no less competitive than Lootika or Vrishchika and cared little for a second place in anything. With a sly smile Vrishchika announced to Somakhya that she felt that Varoli had been afflicted by the caprine or the ovine sprite of Kārttikeya. Lootika remarked that it was confirmed because as they fed the old woman’s goat near the bastard poon tree at the foot of the Caṇḍikā shrine on the way to his house it leapt up only to Varoli. Lootika and Somakhya then told the rest about their singular encounter with apparition of the Scottish surgeon.

Vrishchika: “Somakhya, you suspect these tortoises to have persisted until rather recently. That is quite unusual I must say. For given those dates their extinction comes suspiciously close to the early colonization of the subcontinent by our species.”
Somakhya: “Indeed I entertain the possibility that Homo stands accused for the disappearance of Megalochelys from India and elsewhere. Evidence for such extinction is rather strong for another group of giant turtles, the marvelous basal meiolanids, which persisted for at least 60 million years if not more. These turtles appear to have spread from the large Gondwanan landmass of Australia to the remote Pacific islands like those of Vanuatu, New Caledonia, and Fiji where they persisted until the maritime Austronesians of the so called Lapita culture reached those islands and ate them up just 2-3000 years ago.”
Varoli: “I guess the hand of Homo looks quite plausible given that today the giant tortoises are barely clinging on in the islands of Seychelles, Zanzibar and Galapagos where the hand of Homo was rather late in reaching. I just wish the Austronesians had not reached at least one island where we might still see those meiolanids today.”
Lootika: “Moreover, the range of Megalochelys was apparently vast even on the eve of their extinction. With Vidrum’s lamentably lost find we know they stretched throughout the subcontinent. They have been found in Burma and Java across a fairly wide climatic range. This does make me feel less comfortable with an explanation that invokes climate change rather than the hand of Homo. It does seem the passage of Homo through Asia into the Indo-Pacific islands was their death knell.”
Somakhya: “As you know I am wholly sympathetic to this, but just to present the counter-point the giant tortoises seem to have gone down the drain much earlier on much of the African subcontinent – before any representative of Homo or even our more immediate stem-ward predecessors were around. Also the extinction of the Meiolanids on the South American landmass had little to do with Homo. So there could be other factors too – may competition with other herbivores.”

Vrishchika: “The work we are doing with Indrasena suggests there was a fairly prolonged presence of one or two archaic Homo clades in the subcontinent. One of them were the Denisovans or there was just one clade that had mixed a bit with the Denisovan clade. All this action was even before our clade settled in. Based on your latest dates it would seem that the tortoises did manage to survive these archaic Homo.”
Somakhya: “Sadly, our dates are hardly precise enough to say this with certainty. That would need more work both in the field and with instrumentation beyond our reach currently. Yet, there is little doubt that at least archaic Homo and the tortoises overlapped. But generally I am inclined to the idea that the newer varieties of Homo – the clade from which we sprang – were behind these extinctions. In Africa the prolonged coevolution with Homo appears to have allowed much of the fauna to survive until more recently. But elsewhere the sudden appearance of the more modern variant of Homo seems to have been way more destructive than earlier variants like the small-brained Homo from Georgia or some of the other archaic versions.”
Lootika: “The evidence in this regard seems rather strong for a faunal contemporary of the giant tortoise found first by none other than our Scottish phantom when he was still embodied, namely the ostrich. At around 120000 YBP it was present in the Siwaliks along with the tortoises and then spread widely to become pan-subcontinental by 65000 YBP. We encounter it in rock art and eggshell art suggesting that it existed along side Homo for a quite a period. In fact its extinction was only 25-20000 YBP, well after H.sapiens was ensconced in the subcontinent.”
Somakhya: “That rather late date suggests that the arrival of a newer wave of Homo sapiens with projectile weapons or the spread of that technology was perhaps the cause of the demise of the ostrich in our lands.”
Varoli: “I guess the extirpation of the ostrich need not have coincided with the end of the tortoise since the slow-moving behemoth despite his heavy shell could have been much easier to bring to the plate even with a more primitive technology like a braining club.”
Lootika: “In some places the giant flightless birds and tortoises seem to have both collapsed quite quickly with the appearance of Homo but in other places indeed the latter seem to have outlasted the tortoises –like probably in Australia.”
Somakhya: “Ain’t it notable that we see a parallel in the South American continent? There at some point the niche vacated by the meiolanids was taken over by their mammalian equivalents, the xenarthran glyptodonts. These slow-moving armored behemoths show how successful this body plan was against other bipedal attackers until the coming of Homo. There again the rhea seems to have survived despite the existence of rhea-hunting Amerindians.”

Vrishchika: “I am wondering if this anthropogenic extinction is something unprecedented. I can chart up three other types of extinction that clearly have a much wider presence in earth’s history: 1) the background extinction which is always occurring; 2) the extrinsic mass extinctions from extra-terrestrial collisions; 3) massive climatic changes due to geochemical/geological processes resulting in what one may term an intrinsic extinction. But do we see evidence for earlier mass extinctions caused by a single faunal component like Homo?”
Lootika: “I would merely place that as an extreme case of things that did happen before. The land-bridge between the South and North American landmasses triggered extinctions which were evidently due to appearance of faunal elements with which the older isolated faunas could not cope.”
Varoli: “I wonder if the same thing happened to the old Gondwanan fauna when the Indian plate crashed into Asia?”
Somakhya: “That’s a good question. Our paleontological record has not been studied closely enough and our tectonics still remains incompletely understood to reach that conclusion. But it is indeed possible that it is reason why do not have any archaic fauna of the type of Australia. But the Vastan mine fauna suggests that other faunal inputs into India could have also caused such extinctions even earlier – like a faunal exchange via rafting from East Africa. Moreover, it might have happened over a prolonged period. Lemur-like adapiformes survived in the Pāñcanada till around 9 million YBP.”
Lootika: “Ah! Indraloris… Perhaps the longest survival outside of island Madagascar.”
Vrishchika: “And I used some nights I dream that somewhere in the western ghats we might still find an adapisoriculid mammal.”
Lootika: “Despite the occasional find of a Gondwanan amphibian or snail in those forests that’s unlikely to be ever happening at all!”

Jhilleeka: “The final extinction of a late-persisting lineage like the adapiformes brings to mind those simple mathematical models based on tag-systems that you just asked me to write.”
Varoli: “Jhilli, the trilobites and choristoderan reptiles like Lazarussuchus too – they all went out the same way. Hence, I think those causes which you listed out Vrishchika can sometimes conspire together in precipitating extinction – a once speciose clade is numerically dented in a great extinction and then it stochastically peters out to become fully extinct.”
Somakhya: “Perhaps, that is how, as we peter out into extinction, the curtains would come down upon the planet of apes.”

Vrishchika: “There are so many mysteries regarding the prehistory of our own piece of Gondwana real estate. Is it not a great tragedy that none of our people, baring that relatively lackluster attempt by that brāhmaṇa from the Pāñcanada, bothered to follow up on the path of Scottish surgeon? OK, let us even take it that until the English arrived no one was aware of what existed beneath their feet but there is no excuse for what happened after that. Though he the surgeon himself remarks that when the Sarasvati channel was being dug in the days of the Mohammedan tyranny fossils were found. It is truly a failure of the brahma.”
Somakhya: “As I have remarked before to Lootika much more than the failure of Hindu arms in 1857 the failure of Hindus lay in their ability to develop men with a “mountainous mass of admirable and accurate information” in their midst even when confronted by such.”
Lootika: “Even our Scottish surgeon admitted that we were once the spring-head of systematic knowledge. So it is indeed a tragedy, perhaps a symptom of a civilization exhausted from nearly a millennium of conflict with the evils of West Asia. Now indeed our peoples look more like the rank idiots that the gorā sāhib started calling us after the days of the Scottish surgeon who took a more positive view of us.”
Somakhya: “Is the concentration not notable? It is not a matter of coincidence that old Charles sprung up in the midst of our our śatru-s. In addition to our Scottish physician contemporaneously there stood Wallace and Huxley in their midst. Even Owen their detractor and Lyell, both of whom our vistor when still embodied had skewered, were quite a good, and then there were others like Spencer, Hooker and so on. There was enough of good foil for old Charles to whet his edge. In contrast…”
Varoli: “We look homeopathic in our dilution! Just imagine a parallel universe where we were all born separately and never knew that the other existed. Perhaps, plodding away in a little corner of Saharanpur with nothing to inform us that Megalochelys even existed – surrounded by a hundred who want to get a seat in med school without being able say what lies below sulfur in the periodic table.”
Jhilleeka chimed in smirking: “You know, may be it is our ‘belief’ in ghosts. That is one of the reasons my history textbook lists as a cause for the failure of the Marāthā before the English.”
They all laughed adding: “Of course how can we forget that one!”

Just then the caturbhaginī’s mother called out: “Kids we need to go right now. I know well you love talking and will be doing so all day left to yourselves.” Somakhya’s mother added: “As your father would say it is good to talk less and do more.”
Lootika and Somakhya struck up a parting hi-five laughing: “There you go. May be that is the reason after all…”

Posted in art, Life, Scientific ramblings | Tagged , , , , , , , ,

The Meru and Nārāyaṇa’s cows: Words and fractals

The fractals described herein are based on and inspired by the work of the mathematicians Rauzy, Mendes-France, Monnerot and Knuth. Some their works, especially the first of them, are dense with formalism. Here we present in simple terms the means of generating and visualizing these remarkable objects which anyone with high-school mathematics and computer skills can reproduce and enjoy.

The “prehistory” of these objects goes back to their discovery among the Hindus. Old Indo-Aryan hieratic and epic poetry primarily utilized meters which were based on the count of the syllables. Thus, the famous gāyatrī meter had 8 syllables in each of its three feet amounting to a total of 24. Alongside these meters which conserved syllable count there were other meters which conserved total syllable length or duration in temporal terms. These appear to have been primarily used in secular poetry and maxims. Syllables come in two lengths or morae — short (laghu) and long (guru). Thus, in these meters their count of morae had to add up to a constant. Let us denote laghu by 0 (=1 mora) and guru by 1 (=2 morae). If we had a meter/metrical unit of just one mora then it had to be just 0; for two morae units we can have: 1 or 00; for three morae units we have: 10, 01, 000; for four morae units we have: 11, 100, 010, 001, 0000. If we write it the traditional Hindu way we get:
\begin{matrix} & & & & 1 & & & & &\\ & & & 1 & & 00 & & & &\\ & & 10& & 01 & 000 & & & &\\ & & 11&100 & 010& 001 & 0000 & & &\\ 110&011&101&1000 & &0100 & 0010 & 0001 & 00000 &\\ \end{matrix}

This arrangement of the combinations for each mora-length was seen as resembling a mountain and duly termed Meru by the ancient Hindus. They also noted that the total number of permutations for each mora-length forms a sequence (1, 2, 3, 5, 8…) which is the famous Meru-średhi.

This idea can be extended to generate Meru words. One way of doing so is using the following substitution rules: 0 \rightarrow 01; 1\rightarrow 0. To make the word we start with the initial word 1 and recursively apply the above substitution rules. Thus we get:

We notice that length of the Meru word grows as the Meru-średhi towards infinity. If we observe the pattern within each word we notice that it is not periodic yet there is a structure to it. Importantly,

\displaystyle \lim_{n \to \infty}\dfrac{N(0)}{N(1)}=\phi

The ratio of 0s to 1s in the Meru word converges to the Golden ratio. For example for the 22nd Meru word we have 10946 0s and 6765 1s which approximates the \phi correctly to 8 decimal places. But to get a better picture of the structure of the word we perform the following operation:

1) Take a Meru word. Start of by drawing a segment in the horizontal direction.
2) If you encounter a laghu syllable in the word, i.e. a 0 then draw another segment in the same direction as the current one.
3) If instead you encounter a guru syllable in the word i.e. a 1 then check if the syllable is at an even or odd position.
4) If 1 is at an even position then turn counter-clockwise by some angle, say \tfrac{\pi}{2}, and draw the segment.
5) If 1 is at an odd position then turn clockwise by the same angle as above and draw the segment.
6) Continue thus till the end of the word.

Figure 1 shows the result of this operation on the 22nd Meru word of length 17711 with angle of rotation as \pm \tfrac{\pi}{2}.

Meru_simple_ruleFigure 1

The result is a striking fractal curve that is tightly folded but never crosses itself. The aligment of the folds give the impression of “lines” forming kites and arrowheads with angles of \tfrac{\pi}{4}, \tfrac{3\pi}{4}, \tfrac{\pi}{2} passing through the fractal. We can also visualize this as the process of folding a “polymeric” string made up of segment monomers of the length of a Meru number of segments. The monomers either continue in the same direction or turn by a given angle in one direction or the opposite as per the sequence. Instead of turning 90^o we can turn by other angles too.

meru_simple_hexagonFigure 2. Turning by \pm 60^o

This results in fractal that is superficially reminiscent of the famous von Koch curve but is clearly different from it. It has a core “hexagonal” symmetry which can be simulated by drawing hexagons at the vertices of a core hexagon and repeating it.

meru_simple_pentagoFigure 3. Turning by \pm 72^o

This operation is similar to the above one but by virtue of its angle generates a “pentagonal” symmetry. These examples establish the innate fractal structure of the Meru words.

We had seen earlier that given a quadratic surd q we can perform the following operation:
s=\left \lfloor{(n+1)q}\right \rfloor - \left \lfloor{nq}\right \rfloor, \; n \in \mathbb{N}

It generates a sequence which can be represented as a word in a two letter alphabet (i.e in 0 and 1). For a quadratic surd but not transcendental number we can even figure out a substitution rule which generates that pattern. For the Golden ratio the pattern is the same as what we get by the above substitution operation to generate the Meru word. The words for \sqrt{2} or \sqrt{3} when subject to the above drawing mechanism, unlike that for the Golden ratio do not generate a fractal pattern. However, we discovered words generated in a 3-letter alphabet that encode \sqrt{3} or \sin(\tfrac{\pi}{3}) which can result in fractal curves (see below). In any case, sticking to the 2-letter alphabet for now, a substitution rule which generates words where the ratio of 0s to 1s converges to 1+\sqrt{2}, also called the Silver ratio, generates a fractal curve with an underlying bifurcating pattern symmetry (Figure 4).

meru_root2plus1_wordFigure 4. Rule: 0 \rightarrow 001;\; 1\rightarrow 0, Silver ratio.

Notably, a rule which generates 0s and 1s in 1:1 ratio also generates a fractal, which is equivalent to that generated by iterative removal of a rectangle from a half-square isoceles triangle (Figure 5).

Meru_1isto1_ruleFigure 5. Rule: 0 \rightarrow 011;\; 1 \rightarrow 010

We can generate Meru words by yet another method: Start with w_1=0,\; w_2=1. Then concatenate the current word with the previous one to get the next word: w_{n+1}=w_{n}w_{n-1}. This generates the words ( w_1:w_5 shown):

One will notice a relationship of these words with a particular mora-count of the Hindu metrical system (0 and 1 inverted). Meru words generated by this mechanism will have the ratio of 1s to 0s converge to the Golden ratio. Applying the above drawing procedure results in a non-crossing fractal related to, but distinct from, the Meru curve obtained by the above procedure (Figure 6).

Meru_concatFigure 6. Concatenated Meru curve, rule: w_1=0,\; w_2=1

If we start with w_1=0,\; w_2=001 and apply the same procedure then we get words which have ratio of 0s to 1s converging to the square of the Golden ratio \phi^2.

Now instead of a 2-letter alphabet we can next try a 3-letter alphabet (0,1,2). Here, instead of the even-odd evaluation for turning we can instead use 0 as an injunction to continue in the same direction, 1 to turn counter-clockwise by a given angle and 2 to turn clockwise by the same angle. We discovered that a series of rules which generate words where the ratio of 0s to 1s and 2s converges to \sqrt{3} generate a wide range of distinct fractal curves (angle \pm \tfrac{\pi}{2}):

meru_3letter.1Figure 7. Rule 0 \rightarrow 210; \; 1 \rightarrow 020; \; 2 \rightarrow 10

meru_3letter.2Figure 8. Rule 0 \rightarrow 120; 1\rightarrow 020; \; 2\rightarrow 10

meru_3letter.3Figure 9. 0 \rightarrow 120; 1\rightarrow 020; \; 2\rightarrow 01

meru_3letter.4Figure 10. 0 \rightarrow 012; 1\rightarrow 200; \; 2\rightarrow 10

meru_3letter.5Figure 11. 0 \rightarrow 210; 1\rightarrow 020; \; 2\rightarrow 10

There are more of these curves which potentially deserve a more systematic study. We have merely provided some of the visually most interesting examples. The first four of these curves are non-crossing curves.

Such curves might also be generated using an analogy drawn from biochemistry. The sequences nucleic acids are in a 4-letter alphabet. These (specifically that of RNA) is translated by the ribosome into proteins which are in a 20-letter alphabet. To encode a 20-letter alphabet with just a 4 letter alphabet you have to assign a mapping to strings of length greater than 2 in the 4-letter nucleic acid alphabet to the letters in the 20-letter alphabet of proteins. This mapping is the genetic code. We can thus map the Meru word in a 2-letter alphabet on to a 3-letter alphabet of 0,1,2 by making 2-letter strings in the Meru word encode letters in the 3-letter space. Thus, we get translated words. As in the above 3-letter alphabet case while folding the translated word we simply interpret the 0 as a directive to draw a segment in the same direction, 1 as a counter-clockwise turn by a given angle and 2 as a clockwise turn by the same angle. By applying this folding rule we can now generate curves with different translation rules:

meru_dense.1Figure 12. Rule 00 \rightarrow 0; \; 01 \rightarrow 1; \; 10 \rightarrow 2. This simple translation produces an armless svastika-like curve.

meru_dense.2Figure 13. Rule 00 \rightarrow 12; \; 01 \rightarrow 1; \; 10 \rightarrow 2 This translation produces a crossing curve with some resemblance to the loops in the sand/flour alaṃkāra patterns.

meru_dense.3Figure 14. Rule 00 \rightarrow 21; \; 01 \rightarrow 02; \; 10 \rightarrow 10 This a svastika-like non-crossing pattern.

meru_dense.5Figure 15. Rule 00 \rightarrow 210; \; 01 \rightarrow 010; \; 10 \rightarrow 20 This a stepped variant of the basic Meru curve.

Now the question arises as to whether we can generate words corresponding to Nārāyaṇa’s classic dhenu sequence just as we did with the Meru sequence. It turns out that we can generate a set of simple substitution rules in a 3-letter alphabet (we use 1,2,3 here simply for some plotting conveniences) along the lines of the Meru sequence: 1 \rightarrow 12; \; 2 \rightarrow 3; \; 3 \rightarrow 1. Application of these rules on w_0=1 gives us the following set of dhenu words (up to w_6):

We note that the length of these words is the classic dhenu sequence, 1, 2, 3, 4, 6, 9… Further, in these words the ratio of 1s to 2s and 2s to 3s converges to Nārāyaṇa’s convergent N_c=1.46557123. The ratio of 1s to 3s converges to N_c^2.

In the previous article we noted (see Figure 2) that the Hofstadter sequence, which is related to the dhenu sequence, has a fixed bandwidth when rectified. Moreover, the individual oscillations have a range of fixed amplitudes but the pattern of oscillations is not periodic despite showing some pattern. Indeed, the pattern of 1, 2, and 3 in the dhenu words is again not periodic but shows a peculiar pattern which can be captured by making circles of diameter 1, 2 and 3 and plotting them based on the dhenu words (shown below for w_{12}; \; l=88):

nArAyaNa_circlesFigure 16.

Lurking within this pattern is a deep fractal structure that can be unpacked by using Rauzy’s analysis of such words. The first step for this involves construction of a matrix which captures the dhenu word generator. The matrix is a 3\times 3 matrix because we have a 3-letter alphabet with 3 substitution rules — the rows are for the substitutions and the columns for the alphabets.
-The first rule 1 \rightarrow 12 puts one element in the first and one in the second alphabet column.
-the second rule 2 \rightarrow 3 puts one element in the third alphabet column.
-the third rule 3 \rightarrow 1 puts one element in the first alphabet column. Thus we can write out the matrix:

M= \begin{bmatrix} 1 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0\\ \end{bmatrix}

The eigenvalues of our matrix are: 1.465571, -0.23278+0.79255i, -0.23278-0.79255i, the solutions of x^3-x^2-1=0 which we encountered in the previous article in the context of the dhenu sequence. The one real value is N_c. The other two values indicate that N_c is a Pisot-Vijayaraghavan number.

In the next step we find the eigenvectors of the matrix. Recall that if A is a n \times n matrix then (A-\lambda)\vec{v}=0, where \lambda is an eigenvalue of the matrix and \vec{v} is the corresponding eigenvector. For our above matrix we get the three eigenvectors as:
\vec{v_1}=\langle -0.77098, -0.35894, -0.52606\rangle
\vec{v_2} =\langle -0.39162-0.25177i, 0.68232, -0.15883+0.54078i \rangle
\vec{v_3}=\langle -0.39162+0.25177i, 0.68232, -0.15883-0.54078i \rangle

The eigenvector \vec{v_1} has all real-valued components; hence, we drop it. The next two eigenvectors have one real-valued component which is equal to the rectification value of the Hofstader H sequence which we encountered in the previous article. The remaining two values are complex and also conjugates of each other between the two eigenvectors. Hence, we may use either vector for the next step.

The next step involves first taking a long dhenu word w_j (e.g. w_{28}; \; l=39865). We define a sub-word of a w_{j,k}, which means the first k letters of word w_j. We start with the first sub-word which includes all 3 letters of the alphabet. That would be w_{j,3}=123. From there we increment k by 1 until we reach the end of the word w_j. Now for each such sub-word w_{j,k} we calculate the frequency of 1s, 2s and 3s; respectively denoted as f_1,f_2,f_3. Then we multiple these frequencies by the corresponding component of the eigenvector (in our case \vec{v_2} or \vec{v_3} and sum up the three values:

\displaystyle S_k=\sum_{n=1}^3 f_n\cdot v_{2,n} = f_1\cdot v_{2,1}+f_2\cdot v_{2,2}+f_3\cdot v_{2,3}

The same may be done with \vec{v_3} too. The result of each sum will be a complex number and as we traverse the word w_j we will get a set of l(w_j)-3 complex numbers in our case. If the k^{th} letter is a 1 then we assign then we assign color red for the set of complex numbers derived at that letter; if it is a 2 we assign color blue; if it is a 3 we assign color green. The sizes of these three color sets will be in the proportion of N_c and N_c^2 to each other. Remarkably, when we plot each of these three sets we find that they neatly segregate into three similar fractals. Even more remarkably, the three distinctly colored copies define a fractal tiling of the complex plane (Figure 17).


Figure 17.

Seeing this fractal tiling manifest from the dhenu words we were reminded of the ideas of our ancestors on words manifesting form.

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

Hofstadter and Nārāyaṇa: connections across space and time

The scientist-philosopher Douglas Hofstadter presents an interesting single-seeded sequence H in his book ‘Gödel, Escher, Bach: An Eternal Golden Braid’. It is generated by the recurrence relation,

f[n]=n-f[f[f[n-1]]] where f[0]=0 …(1)

Working it out one can see that it takes the form: 0, 1, 1, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 13, 13, 14, 14, 15, 16, 17, 17…

If we connect f[n] \rightarrow n in (1) then we get a tree structure which simulates a specific pattern of succession and duplication (Figure 1).

Hofstader_H_treeFigure 1

As we mentioned before, we only got to read Hofstadter’s book briefly when we first came across it. Hence, we did not have the chance to take in all that was discussed in it. However, it seeded our own explorations along the lines he has proposed in the book. Thus, in that period we discovered for ourselves a two-seeded sequence generated by the recurrence relation,

f[n]=n-f[f[f[n-2]]], where f[1]=f[2]=1 …(2)

This, while similar in from to the above recurrence relation, produces a different sequence: 1, 1, 2, 3, 4, 5, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 13, 14, 14, 14, 15, 16, 17, 18, 18, 18…

Our sequence (2) is related to Hofstadter’s H sequence in that adjacent duplications in his H are replaced by one singleton and a triplication in ours.

As we have described before, a convenient method for visualizing sequences such as this is to ‘rectify’ them: They increase linearly with a constant slope m. Hence, we find that slope and use f[n]-n\cdot m to render the sequence along the x-axis. For many common sequences m=\tfrac{1}{2} works. However, during our experimentation with the sequence (2) (illustrated in Figure 11 in the previous article) it was obvious that m>.5. Hence, we took the arithmetic mean of \tfrac{f[n]}{n} for large n and obtained m \approx 0.682, which served as the required rectification factor. Notably, the same value of m applied for the H-sequence and we could accordingly rectify it (Figure 2)

Hofstader_HFigure 2

Similarly, during our explorations of two other two-seeded sequences (See previous article) we discovered that their rectification factor m=\tfrac{1}{\phi}, where \phi is the Golden Ratio:

f[n]=n-f[f[n-1]], where f[1]=f[2]=1 …(3)
f[n]=f[f[f[n-1]]]+f[n-f[f[n-1]]], where f[1]=f[2]=1 …(4)

Since, (3) in particular resembles H and our above sequence (2) we wondered if we could similarly get a precise expression for their shared m. We noted that in the case of the doubly nested recurrence relation (3) its rectification factor m=\tfrac{1}{\phi} was the first root of the quadratic x^2+x-1=0. For the triply nested recurrence relation (4) we noted that its rectification factor m=\tfrac{1}{\phi} was the root of the cubic x^3-2x+1. Thus, we realized that a connection exists between the rectification factors and algebraic numbers. Armed with this knowledge searched the roots of cubic polynomials to get the rectification factors for H and our sequence (2). The real root of x^3+x-1=0, x=0.6823278 yielded their required rectification factor. In the case of (3) and (4) the rectification factor is the reciprocal of the Golden Ratio, which is the convergent of the famous Meru sequence (known in the west as Fibonacci),
\displaystyle \lim_{n \to \infty} \dfrac{M[n+1]}{M[n]}=\phi

Thus, the rectification factors of the linearly growing two-seeded sequences (3) and (4) were reciprocals of the convergent of a non-linear sequence M. Notably the terms of M appear at each level of the tree of these linear sequences (3) and (4); see Figures 2 and 14 in the previous article. So question arose as to what is the corresponding non-linear sequence to which the rectification factor m=0.6823278 of H and (2) is similarly reciprocally related? The answer to this remarkably leads us to the original cow sequence of the great medieval Hindu mathematician Nārāyaṇa, son of Narasiṃha.

In his Gaṇita-kaumudi, Nārāyaṇa presents one of the earliest studies to identify a discrete formula for the ideal population dynamics of an organism which continually reproduces upon reaching a certain age. He poses the following problem:
prativarṣaṃ gauḥ sūte varṣa-tritayena tarṇakī tasyāḥ |
vidvan viṃśati-varṣaiḥ gor ekasyāś ca santatiṃ kathaya ||
Every year a cow gives birth, from its 3rd year, [and so also] her calves.
O scholar, tell, in 20 years [of reproduction] what will be the clan size from one cow?

He then provides the answer as the following sum of a series:
abdās tarṇy abd[a+ū]onāḥ pṛthak pṛthak yāvad alpatāṃ yānti |
tāni kramaś c[a+e]aikādika-vārāṇāṃ padāni syuḥ ||
Subtract the number of years (when a calf begins giving birth) successively and separately from the number of years till the remainder becomes less than the subtractive. Those are numbers for repeated addition once etc in order. The sum of the summations along with 1 added to the number of years [is the desired number]. Translation as per Ramasubramanian and Sriram’s interpretation.

Let the sequence dh[n], for dhenu (cow), represent the number of cows in the nth year. While Nārāyaṇa gives the direct formula for the nth term, it can be expressed in modern terms rather simply by the below recurrence relation for a triply seeded sequence,

dh[n]=dh[n-1]+dh[n-3], where dh[0]=dh[1]=dh[2]=1 … (5)

This sequence goes as, 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189, 277, 406, 595, 872, 1278, 1873, 2745, 4023, 5896, 8641… Thus, the answer for Nārāyaṇa’s problem with 20 reproductive years is dh[22]=2745.

We note that, just a \phi for the Meru sequence, this sequence too has a convergent which we will call Nārāyaṇa’s convergent N_c,

\displaystyle \lim_{n \to \infty} \dfrac{dh[n+1]}{dh[n]}= N_c

Investigating this value, we found it to be the only real root of the cubic equation x^3-x^2-1=0, i.e. N_c=1.4655712319. This result yields the relationship between the rectification factor m for the Hofstadter H sequence and our sequence (2) on one hand and the cow sequence of Nārāyaṇa on the other: m=\tfrac{1}{N_c}. Thus, the same reciprocal relationship, which we saw between the Golden ratio, the convergent of the nonlinear Meru sequence, and the rectification factors of the linear sequences (3) and (4), is obtained for N_c, the convergent of the dhenu sequence (5), and the linear sequences H and (2).

Since the Hofstadter H sequence includes all Natural Numbers we ask if there is any pattern to the occurrence of the dhenu numbers in it. This relation turns out to be,

f[dh[n]] = dh[n-1]

The same relation also holds for our sequence (2) except that alternately the values are either dh[n-1] or dh[n-1]+1. A further interesting observation emerges as we examine these sequences more closely. The sequence (3) with rectification factor m=\tfrac{1}{\phi} has a fixed bandwidth oscillation (see figure 5 in previous article). The H sequence and sequence (2) have a similar type of fixed bandwidth oscillation. Consistent with this, their recurrence relation and that of sequence (3) resemble each other with one of the terms being directly n itself. In contrast, sequence (4), which also has the rectification factor m=\tfrac{1}{\phi}, shows increasingly larger loops of the size of the Meru numbers and a fractal structure. Hence, we asked if there is a similar fractal sequence with m same as H and sequence (2). Given that in the recurrence relation (4) the two terms recursively call the sequence with one having a subtraction, n-f[f[n-1]], we looked for similar sequences and found that the sequence discovered by Mallows has a comparable structure,

f[n]=f[f[n-2]]+f[n-f[n-2]], where f[1]=f[2]=1 …(6)

Duly, we found that m=\tfrac{1}{N_c} serves as a rectification factor for it (Figure 3) consistent with what we had experimentally determined for the sequence.

Mallows_sequenceFigure 3

N_c and \tfrac{1}{N_c}, unlike \phi, are not constructible by standard compass and straight-edge construction. Rather, they need a construction of the type used for the Delian altar of Apollo, i.e. doubling the cube. It goes thus (Figure 4):

1) Draw a parabola with focus at (.5, 0) and directrix as y=-\tfrac{1}{2}. This parabola has equation y=x^2-x.
2) Draw a rectangular hyperbola with the x- and y-axes as its asymptotes and (1,1) and (-1,-1) as its vertices. This hyperbola has the equation y=\tfrac{1}{x}.
3) The point of intersection of the two conics, X, gives our desired constants: (N_c, \tfrac{1}{N_c}).
4) Using this and the two conics we can construct a rectangle ABCD comparable to the Golden rectangle.
5) Dissecting it using the square of side 1 allows us to construct two further rectangles: ABCD \sim GCXF \sim BEFX. Together these furnish the various powers of N_c as shown in Figure 4.

nArAyaNa_convergentFigure 4

This finally leads to the question of whether this observation regarding the algebraic number emerging as a convergent of a Nārāyaṇa-type series and it reciprocal as the rectification factor of a Hofstadter-like sequence is a more general one. In course of our explorations of Hofstadter-like sequences we discovered a fractal sequence that we termed the seahorse sequence (Figure 12 in the previous article). This is given by the recurrence relation,

f[n]=f[f[f[n-1]]]+f[n-f[f[n-2]]-1] …(7)

For this sequence we experimentally established a rectification factor m=.45. Using the above-described principle we then identified its proper form as the real root of the cubic equation x^3+2x-1=0. Thus, m=0.4533976515. Then we asked if there was a corresponding Nārāyaṇa-like sequence whose convergent is the reciprocal of m. Our search yielded the following triply seeded Nārāyaṇa-like sequence nl provided by the recurrence relation:

nl[n]=2nl[n-1]+nl[n-3], where nl[0]=nl[1]=nl[2]=1 …(8)

It goes as 1, 1, 1, 3, 7, 15, 33, 73, 161, 355, 783, 1727, 3809, 8401, 18529, 40867, 90135, 198799, 438465, 967065, 2132929, 4704323… We found its convergent,

N_1= \displaystyle \lim_{n \to \infty} \dfrac{nl[n+1]}{nl[n]} \approx 2.205

This allowed us to establish it as the real root of x^3-2x^2-1=0, thus N_1=2.2055694304006. Here again for sequence (7) we get m=\tfrac{1}{N_1}. In addition to these examples there is the trivial case of sequences shown in the previous article where the rectification factor is m=0.5. Its reciprocal c=2 corresponds to the trivial duplication sequence: 1, 2, 4, 8, 16, 32… Together these four show us that the form of the polynomial equations for the convergent and its reciprocal are also notable in their parity:
2: \; 2x^2-3x-2=0
0.5: \;2x^2+3x-2

\phi: \; x^2-x-1=0
\dfrac{1}{\phi}: \; x^2+x-1=0

N_c:\; x^3-x^2-1=0
\dfrac{1}{N_c}: \; x^3+x-1=0

N_1: \; x^3-2x^2-1=0
\dfrac{1}{N_1}: \; x^3+2x-1=0

This leads to a conjecture: The reciprocal of an algebraic number which is the convergent of a non-linear Nārāyaṇa (Meru/dhenu)-like sequence serves as a rectification factor for a Hofstadter-like sequence. We have not attempted to prove this formally but the mathematically minded might be interested in doing so.

As we saw above, for \phi and N_c their reciprocals are the rectification factors for two types of Hofstadter-like sequences, namely one with a fixed bandwidth oscillation and another with fractal loops of increasing size. In the case of N_1 its reciprocal rectifies the seahorse sequence (7) which is a fractal sequence. We have not thus far found a fixed bandwidth sequence rectified by \tfrac{1}{N_1}. If such a sequence exists then more generally we might speculate that for each rectification factor there are both fixed bandwidth and fractal sequences.

In conclusion, we find a remarkable link (to us) between the medieval mathematics of Nārāyaṇa and the modern mathematics of Douglas Hofstadter. This yields some interesting results some of which to our knowledge remains unexplored and unproven in terms of formal proofs. Notably, unlike the Golden ratio which appears in many places in mathematics and even in nature, we have not found equivalent occurrences for N_c, N_1 and their reciprocals. It almost appears as if nature has a predilection for things constructible by compass and straight-edge. However, we may note in passing that while writing this article we saw a recent paper proving a theorem that a number which is related to our N_1, precisely N_1-1, appears in the convergents of the random Fibonacci series of Divakar Viswanath.

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