loka-nīti-carcā

•January 22, 2017 • Leave a Comment

loka-nIticharchA

vijaya-nāma mahā-mlecchānām bahuprajāvān bahupatnīvāṃś ca vyāpārī gṛha-krayāc chailūṣa-pradarśanāc ca mahādhany abhavat । sa marūnmattair abhibhṛtāṃ pūrvatana-mleccha-rāja-patnīm atikrāntvā rāja-nirvācanam ajayata । so ‘bhavat mlecchādhipatiḥ । virodhakās tasya+anekāḥ । tasya vijayasya ca paṭṭābhiṣekasyānantaraṃ vṛṣṭy-ante puttikā ivo(u)tplavante bahavaḥ saṃkṣobhakāḥ । saṃkṣobhakeṣu gaṇeṣu santi śailūṣa-puṃścalyaḥ klībāḥ strīvādinī-striya āpaṇa-bhañjakā niyamollaṅgha-vadino ‘pasavya-mārgādhyāpakā vārttā-vikrayikāś ca (preto rākṣasa-putrasya+anuṣaṅginaḥ sadṛśā āsan) । te sarve sva-tantreṇa alpa-vivekinaḥ । tatas te kaiś codayanti ca vidhīyante ce(i)ti praśnaḥ । kecid vadanti so’sti soro-nāma-mahādhanī+iti । sa bejho-nāma+antar-jāla-mahāvyāpārī+ity anye vadanti । etāny uttarāṇi prayeṇa samucitāni kiṃ tu gūḍhaṃ viṣayaṃ na spṛśanti ॥

iyam asya viṣayasya mīmāṃsā । vijayopanāma-vyāpārī keṣāṃ pratinidhiḥ ? sva-dhana-vardhanañ ca sva-sāmarthya-bhogaś ca tasya pradhāne lakṣye । paraṃ tu pradhānā anuyāyinas tasya santi śveta-jāty-ādhipatyavādinaś ca vyatiropitāḥ śveta-karmakarāś ca viśiṣṭāḥ pretasādhakāś ca । kasmāt kāraṇāt te sarve vijayaṃ vyāpāriṇaṃ pratisevante? mlecchānāṃ madhye vartate ekaika-rākṣasa-nāma-saṃpradāyaḥ । prathamā anuyāyinas tasya+āṅgla-deśasya holadeśasya mulaikarākṣasavādinaś ca sāmarthya-sādhakāḥ । mahāmlecchavarṣasya vistārānantaraṃ yūropāt tasmin deśe jagmuḥ । te jana-bhāvebhyo rāṣṭra-sīmābhyo deśānusarebhyaś ca na cintayanti vyavasāyino ‘rthasādhakāḥ । teṣām ekarākṣasvatvaṃ matavihīna-veṣaṃ vā (āṅglānusāreṇa “secularism” iti) apasavya-mārga-veṣaṃ vā gṛhītvā janānāṃ madhye praviśati । etasya matasya bhāṇakānāṃ karmabhyaḥ sādhāraṇā janāḥ kupyanti । te dhana-kṣayaṃ kuṭumba-kṣatiṃ samāja-bhaṅgaṃ mūlya-nāśam anubhavanti । tasmāt pradhānāt ekaikarākṣasatva-viśiṣṭaikarākṣasatvayo raṇam ajāyata (viśiṣṭaikarākṣasatveṣu purāṇā bhedās santi) । tatas te prati-yoddhāraṃ mṛgyanti । vijayo vyāpāriṇo bhavena janasya kopaṃ sulabham abhijñātvā naya-yojanāṃ prāyuṅkta । anayañ janā vijayaṃ jayāya ॥

sanātana-deva-sādhakebhyo kiṃ mahattvam eṣa-viṣayasya ? ekaika-rākṣasatvam anyāny ekarākṣasāni+iva deva-dharmasya ripuḥ । “Secularism” iti mārgeṇa dharmasya nāśam anviṣati । sa sarvonmāda-samyukta-matavihīna-veṣa-bhṛt-bhūtaḥ । eṣa hi mārgo dharma-virodhāya mleccha-marūnmattābhi-sandheḥ ॥

paśya paśya sarvonmāda-samāyoga-puruṣaḥ । tasya prathamonmādaiva śiraḥ । bilmy anabrahmaṇā ca amahāmadena ca guptaḥ । tatra vartate abrahmā-nāmonmatta-janakasya duṣṭaḥ siddhāntaḥ । pretonmāda+ādadhāti tasya niguḍhaṃ dharma-niṣūdyamānaṃ kalevaram । rākṣasonmādas tasya bāhū dāruṇav asi-dhāriṇau । mahāmadasya vākyāni bhavanti tasya hastaghnāv aṅgulitre ca । rudhironmādo dṛḍhaḥ kavaco dātra-mudgarābhyām bhūṣitaḥ । adhobhāgas tasya janānāṃ mandatā । upahata-jana-vṛddhis tasya mahāmuṣkau । tasmin rakte vahati mūṣa-nāmonmattasya kutarka-pūrṇo durāgamaḥ । sa piparti navīkaroti ca sarvonmādāṃś ca teṣān tviṣaś ca । ayam eva deva-dharmasya ripur mahān । yady etañ jānāti tarhi sahasrāṇāṃ varṣāṇām ajñānaṃ layati ॥

Deliberations on richness and beauty: discovery of some multi-parameter iterative maps

•January 17, 2017 • Leave a Comment

As we have explained in the earlier notes (1, 2, 3), the second major factor in our exploration of 2D strange attractors maps, IFS and other fractals was the aesthetic experience they produced. Around that time we came across a curious statement of Plato in the Timaeus: “Now, the one which we maintain to be the most beautiful of all the many figures of triangles (and we need not speak of the others) is that of which the double forms a third, the equilateral triangle.”(translation by Jowett via Birkhoff). Thus, for Plato the most beautiful of triangles is the so-called \frac{\pi}{6}-\frac{\pi}{3}-\frac{\pi}{2} (a.k.a. 30-60-90) triangle. As we meditated upon this statement we realized that this triangle can be recursively used to make many aesthetic polygons beyond the equilateral triangle, like a rectangle, parallelogram, kite (all 2 \times), square, rhombus (both 4 \times), hexagon ( 6 \times). In doing so we realized what Birkhoff had noticed in the 1930s. To us it also illustrated the second of the two basic principles behind aesthetic experience that had dawned on us namely, symmetry and recursion.

Starting with our study of the Henon and Lozi maps, we soon realized that there is more to this: All fractal objects produce a profound aesthetic experience in some people. I happen to be one of them, though I am cognizant of the fact that this is not universal — many people are neither wonder-struck nor aesthetically moved when I show them such objects. Nevertheless, the fascination for such objects is widespread across human cultures: In Hindu tradition, temple architecture increasingly converged towards fractional dimensions before its expression was terminated by the coming of the Meccan demons. We had briefly alluded to this earlier, pointing to a relationship between the floor plan of the central spire and the boundary curve produced by an IFS fractal using simple rotations. Other forms of fractal structures were also depicted in Hindu temple art, such as “vegetal motifs”. Similarly, Wolfram has documented examples of fractal objects in medieval Western art. Tendencies towards fractality might also be noted in Japanese Ukiyo-e, like in the famous “The Great Wave off Kanagawa”. All this suggested to us that complexity might have an important role in aesthetics.

An attempt to understand the role of complexity in aesthetic experience was made by in the 1930s by Birkhoff. For an aesthetic object he defined two values, “order” (O) and “complexity” (C), which led to the aesthetic measure, M=\frac{O}{C}. However, his M measures for various polygons did not have a strong correlation with aesthetic experience they produced in us. This made us suspicious of the value of Birkhoff’s measure. The pioneer in the study of fractals, Mandelbrot, suggested that fractality of an object might be related to the aesthetic experience it produces. Exploration of fractal maps for aesthetics emerged from pioneering productions of physicists/mathematicians like Mira, Gumowski, Sprott, Pickover and Abraham among others. Sprott and Abraham carried out analyses of the aesthetic experience from fractal objects attempting to relate it to their fractal dimension. Based on their experiments they suggested that a fractal dimension in the middle of the range 1..2 was probably a sweet spot for the best aesthetic experience in 2D maps. This generally corresponds to our own aesthetic evaluation of fractal objects.

Thus aesthetic experience was a major driver for us in the exploration of new maps for fractal objects. Thus, we studied the work of Sprott and Pickover among others, reproducing and exploring many of the strange attractors they had discovered using iterative maps. Our own experiments led us to the discovery of the map that produces the “butterfly attractor“, which we had described earlier. After obtaining it we realized that a certain Martin had also discovered a comparable class of maps which produced an attractor with a considerable richness of structure and aesthetic variety (also called hopalong or chip maps).

Martin’s maps are surprisingly simple but produces remarkable beautiful and complex structures. The simplest of these maps the “hopalong” is:
x_{n+1} = y_n - sign(x_n) \sqrt{|b*x_n-c|}
y_{n+1} = a - x_n

Where a, b, c are three constants. An examples are shown in Figure 1 and the map is robust over a wide parameter range

figure1_smallFigure 1

We modified versions of the Martin map into a more complex set of maps. Map-1 works thus. The first mapping is similar to the Martin process:
x_{n+1} = y_n - sign(x_n) \cos \left( \dfrac{\sqrt{|b x_n - c|} d}{2 \pi} \right) \arctan \left ( \left (log_l \left (|c x_n - b| \right ) \right)^2 \right)
y_{n+1} = a - x_n
In place of \cos() in the first x-mapping we could also have \sin().
Then we subject (x_{n+1},y_{n+1}) to a further “affine” rotation before plotting it:
x_{n+1}=x_{n+1} \cos(k) - y_{n+1} \sin(k)
y_{n+1}=x_{n+1} \sin(k) + y_{n+1} \cos(k)

Here we thus have 6 constants: a, b, c, d, l, k

Map-2 works thus:
x_{n+1}=y_n-sign(x_n) \cos \left ( \dfrac {log_{l1}(|b x_n-c|) d} {2 \pi} \right) \arctan \left(\left(log_{l2}\left(|c x_n-b|\right)\right)^2\right)
y_{n+1}=a-x_n
In place of \cos() in the first x-mapping we could also have \sin().
Here again we subject the map to an “affine rotation” before plotting:
x_{n+1}=x_{n+1} \cos(k) - y_{n+1} \sin(k)
y_{n+1}=x_{n+1} \sin(k) + y_{n+1} \cos(k)

The constants here are similar to those in the above map but we have one extra one because we have two distinct logarithm terms. Finally in both these maps replacing \arctan() by the hyperbolic function \tanh() also produces interesting maps. Figures 2 and 3 respectively show examples of each of the above maps.

figure2_smallFigure 2 (First two with \cos() and second two with \sin()

figure3_smallFigure 3 (all maps use \tanh() in place of \arctan() in Map-2; the third example uses \sin().

These maps produce new forms of great beauty, diversity and richness of structure with considerable robustness — like what the old Hindus would term “nava-nava-camatkāra” or “muhur-muhur-āścaryāya kāraṇam”. Indeed the beauty in these maps relates to features beyond simple symmetry. They have a degree of rotational symmetry but it is not perfect. What is striking in however the fractional dimension and the tendency for doublings or multiplications (an element see in the evolution of the Hindu temple too). These features deeply touch the heart of beauty. Perhaps the order within chaos is the most beautiful of all.

The maps displayed here along with some additional ones can be see in this PDF file.

“Like the vidyādhara’s sword”

•January 15, 2017 • Leave a Comment

In old Hindu tradition a man who attained siddhi in his mantra practice was believed to become a vidyādhara whose might was manifest in the form his beautiful female partner who flew beside him embodying the power of fertility and the sword he held in his hand embodying the essence of might itself. We had remarked to a clump of modern Hindus that unless his might and that of his nation is manifest like the vidyādhara’s sword which lays opponents low all their opulence will come to naught and not be set in history. They did not get anything of what we had said. We said: “Never mind. Nothing matters. Like the what happened to the archosaurs of the Mesozoic everything comes to an end so why care for anything at all?”

◊◊◊◊

A clump of Hindus were seated at a table on a somber afternoon in the big mleccha-land. The meal which was being consumed seemed so unremarkable that most would have not even realized that they had finished lunch. Yet most seemed to be quite contended. We do not know why that was the case for the rest but at least in our case it was simply because we do not think too much into the future and at that moment the gods had kept us free of pain. Some where positively exultant, talking of their successes with grants or businesses. Others were triumphantly talking of the monetary success story of the Hindus as a group in mleccha-land. They were duly comparing themselves with the prathamaikarāksasavādin-s. Then someone brought up the topic of politics in mleccha-land. Suddenly there was a bit of shift in the mood. Most expressed surprise and anguish over the crowning of the new mlecchādhipati. They went about the Russian conspiracy as though they were senior agents of the mleccha-spaśālaya. Some of them started the discussion of whether the outgoing ardhakṛṣṇa-mlecchendra was one of the or the greatest mleccheśvara-s. With three exceptions most in the clump seemed to settle on him being at least one of the greatest mlecchendra-s. Basking the pleasant warmth of that feeling their calm was restored a bit as though in the gentle twilight glow of the setting sun of the mlecchapa. Soothed, they began to talk again of the great monetary achievements of the Hindus of mleccha-land.

Some of them noticed our silence and asked what we had to say on the matter. After some attempts at deflection we simply presented our view of the reality. Most found it utterly unpalatable and were unable to come to terms with it. The only two in the room who had voted for the jayitṛ rather than the favored candidate turned to us and said: “You have a point. We have learned this the hard way after stumbling through the mleccha-maze for 35 years. How did you get there? We have always thought of you as not being a man of the world lost in impractical arcana.” One of them continued: “We have always felt bad for you given what people would think of you for your pursuit of the recondite. This vision you present is deeply depressing to some of us. How could live with it without going insane?” We simply smiled and said if it were to make them go insane then it was better they disregarded what we had said and move on. As for us we told them perhaps a bit too bluntly that their supposed commiseration was of utterly no use.

In the background decadent Hindi film music was gently playing. Lulled by it and the unmemorable meal drawing blood to the belly, we thought they would return to their state contentment. The only word we registered from the wafting song was ‘halcal’ and it was a ‘halcal’ our words seem to have set off a in their midst. One of them went to shut off the music. nM’s perpetual state of self-certainty in the service of the mleccha seemed to have been challenged. Barbarin who had survived certain crises by the skin of his teeth seemed to be torn between two opposites. Unlike before, the crises had taught him that he was not pratiṣṭhāvān in the midst of the mleccha as he thought he was. He wondered about the real significance of the fall of tailakeśa, and was probably apprehending faint flashes of the truth. Another brought up the case of the professor who was defrauded in toto by his pāṇḍūrikā indologist former wife, all while purchasing expensive antiques Hindu images only to be used as props and doorstops in her study. Suddenly, some were talking about the mysterious “injustice” they had faced in life. Repressed and unspeakable tales of liberal allies turning against them spurted out of the mouth of one or the other – how the super-nice friend John or colleague Dick or boss Harry had turned against them when “push had come to shove”. Some murmurings passed around of sons or daughters finding mleccha sexual partners and losing all vestiges of “values” and “culture”.

But others started vaguely fighting back against the current which was sweeping the room. Yet they could not apprehend precise targets. One began with a windy denouncement of our insistence on using cousin primates, rodents, isopterans and hymenopterans as models to understand the human condition, and concluded with the reassuring statement: “We are humans with a free will to do what we want in a free country that protects our rights by law. Its all in our hands. Only losers go about blaming others for their failures.” Someone followed up speaking of William “semiconductor” Shockley though we did not get the drift of why they had gone all the way there. In all this nM found his footing again and set forth to defend various “truths” that were suddenly being doubted. He went on to narrate how he has given had received a niṣka at the hallowed hands of the ardhakṛṣṇeśvara himself. Then palitatvac chimed in saying how after long years of service to a great mleccha lord he was given a mudra too. Defensively he added that there was nothing wrong in transmogrifying his name with a Sanskrit root into meaningless garbage for we have to be understanding of the mleccha-s who cannot pronounce “complex” names. Perhaps pricked by the smirk on our face he turned to us and condolingly added us: “You were always smarter than us from school days. But you are not getting ahead because you have been disrespectful of the mleccha-s who matter, just like to the teachers in school and college. Remember in our culture we respect the guru-s.” Bringing back the focus to the financial someone quipped that it was not long before “a Bill Gates, an Elon Musk or those google guys emerged in our midst.” Another said: “arrey! It has already happened. See google and Microsoft!” But then Krodhavaśa who seemed to have flipped retorted: “I do think we should distinguish working for an established company or starting yet another floppy start-up from what those guys have done. We have some way to go. So may be there is [a point to what we had said].” Like with passing moon the tides eventually ebbed and a different kind of uneasy quite prevailed in the room.

◊◊◊◊

To palaver about what we have said in many ways on these pages, even as our ancestors were said to speak of the same sat in many ways.
-Most of these Hindus do not get it that they will never become a śveta-tvaca-mleccha even though their chief desire seems to be to earn respect from such.

-Whatever great deeds the Hindu achieves on the academic front he is not going to be acknowledged for those and his conquests will be attributed to the mleccha. So if he is pursuing such conquests primarily to be recognized and awarded by the mleccha system he has little hope of getting there. This leads to tremendous frustration among those who do not get it and simultaneously an inability appreciate the genuine conquests of their own people because they are constantly using false yardsticks.

-In a subset it sparks the temptation to get into the “cartel” by cheating on matters of substance which usually results in even bigger damage.

-However good the Hindu is *on an average* he is going to be paid less and given fewer resources than a mleccha with lower or equal capacity to him. Unless he puts this in proper perspective, there will always be discontent and misunderstand regarding why he has not yet entered the “club”. This is because he has internalized the mleccha framework as the truth and is struck by why things are not working as enunciated within it. It is like tackling a problem in non-Euclidean geometry within a framework assuming the 5th postulate as true.

-Those who gloat over monetary success do not realize that king Vaiśravaṇa’s nidhi without his million-slaying antardhānāstra is of no consequence. The Hindu does not put his money into things that strengthen his memetic ramparts against the other. Instead, he pours it as āhuti-s of Triśiras Tvāṣṭra for the dānava-s and hopes that the deva-s would lift him up. The strengthening of the *Hindu* nation is very far from their minds. They content with their monetary success or of theirs as a group for the sake of boasting but do not translate that into efforts that will actually strengthen them as a nation. Instead they expend it either on mleccha causes or in plain hedonistic pursuits: do you really need a roomful of unused electronics etc etc…

-So is all this the whimpering of the losers? So let us take a look at the winners – may be we can learn something from them. There they are seated on their resplendent vāhana-s bearing niṣka-s and mudra-s conferred by great mleccha lords. How did they get there and what have the achieved? They got there by faithfully serving the mleccha or marrying a mleccha and the proud niṣka-s they sport are merely the biscuit Tim has tossed to his cur Tom. They might pass their whole life in great ease and in a cocoon of recognition from the mleccha. But at the end will they have set themselves in history? Not at all. The best they would be is a footnote in small print. What about their demography? As a part of the great “becoming” they will now be merged with mleccha and be swept away by the dysgenic memetic infections they eagerly inoculate themselves.

-Only he is an abhijit in the world of men who is backed by the possession of the mighty sword like that of that of the vidyādhara.

-ity alaṃ vistāreṇa

A strange Soviet construction

•January 14, 2017 • Leave a Comment

in our college days we used to visit the lāl-pustak-bhaṇḍār in our city where Soviet books on science and mathematics were sold at a low price (alongside Marxian literature). They were a great resource that enormously contributed to our intellectual development. Among the books were some which contained nasty problems in mathematics and physics that were used for boasting rights in our circle. Some of the more curricularly oriented students in my class apparently used those books for entrance exams to certain Indian undergraduate institutes that we had no interest in joining. We were reminded of  the Soviet penchant for such problems recently by an acquaintance. She brought to my attention a Euclidean problem which was allegedly part of a set of difficult mathematical questions which were used by the Soviet authorities to prevent Jews and other “undesirables” from entering the Moscow State University (She learned of those from a paper containing a whole set of such problems and their solutions which was posted several years ago by Tanya Khovanova and Alexey Radul and was recently highlighted by Pickover on his well-known site). I am simply recording it here for I found it interesting and spent some time on it. I took a while to solve it but once achieved it looked so trivial that I felt like a fool. The yavana-s of yore and other ancients might have liked it.

Problem: “Construct with ruler and compass a square given one point from each side.”

soviet_square

1) Let A, B,C,D be the 4 points each from one side of the square.
2) Construct parallelogram ACDE using 3 of the points (A, C, D). Thus \overleftrightarrow{DE} would be parallel to AC.
3) Drop a perpendicular from point D to the side of the parallelogram containing the other two points \overline{AC}.
4) Draw circle with center D and radius \overline{DE} to cut above perpendicular at point F.
5) Draw \overleftrightarrow{BF} and drop a perpendicular to \overleftrightarrow{BF} from A to cut it at point J.
6) Drop a perpendicular to \overleftrightarrow{AJ} from point D to cut it at point G.
7) Draw \overleftrightarrow{GD} and drop a perpendicular to from point C to cut it at point H.
8) Draw \overleftrightarrow{HC} and complete the desired square GHIJ via obtaining the intersection I between the above line and \overleftrightarrow{BF}.

A square is formed by intersection of two orthogonal, equidistant pairs of parallel lines. Thus, a segment formed by points lying on two opposite sides of a square if rotated by 90^o using one of the remaining two points as the pivot would define two points on the other pair of opposite sides. This is the principle enacted by the above construction to get the desired square.

We adduce below several other Soviet problems with the respective solutions without any detailed explanation.

*Given two intersecting lines on a plane find locus of points D such that the sum of the distances from D to each line is equal to a given number.

soviet_lines

*Can you put six points on a plane, so that the distance between any two of them is an integer, and no three are colinear?

soviet_integral_hexagon

*Given two parallel segments (AB, CD) divide one of them (AB) into six equal parts using just a straight edge.

soviet_segment_6parts

*Given a triangle ABC, construct, using a straight edge and compass, a point M
on AC and a point K on BC, such that
\overline{AM} = \overline{MK} = \overline{BK}

soviet_triangle_segments

Matters of religion: “he becomes Naravāhanadatta”

•January 2, 2017 • Leave a Comment

Somakhya’s mother (SM) and Lootika’s mother (LM) ran into each other during their visit to the shrine of Rudra beside the river on a Monday evening. They sat at the platform below the vast aśvattha tree beside the subsidiary Viṣṇu shrine to chat for some time.

LM: “How is it going with all the quiet at your place now that Somakhya has left?”
SM: “Well, I’ve returned to teaching the Mahābhārata and have been contemplating on the features of early Indo-Aryan. By the way I saw you brilliant daughter last evening at the clothes shop.”
LM: “Ah! You mean Vrishchika; have been worrying about her.”
SM: “Why? I heard that she has been invited to give a talk at some famous human genomics meeting regarding the paper she has just published. She seems to be following her stellar elder sister with this paper while just in the fourth year of med-school.”
LM: “That’s exactly the matter of worry. The conference is at Kṣayadrājanagara, a big bad city, where I am sure you heard that just a week ago there was a major attack at the train station by the marupiśāca-s in which several were killed. Vrishchika has set her mind on going and you know when that happens there’s no easy way to change it.”

SM: “You should let her go. After it will bring her yaśas. It would seem she is on course to out doing even Lootika. Moreover, you let Lootika go on multiple trips, which included that really scary day of the Uniform Civil Code riots. You also let Varoli go to the dreadful Visphotaka, which is an even worse city than Kshayadrajanagara.”
LM: “O dear, you over-estimate Vrishchika. Knowing my daughters well I can say that that Lootika is to the rest of them like Maghavan among the gods. Thus, even on that day, while I feared for her greatly, I knew deep within she’ll be back home: lūtikā bahuyutikā vā.”
SM: “But some day Vrishchika will be on her own. You can’t just keep her protected at home and unexposed to the big world. My brother lives in Kshayadrajanagara. I could give you his number. May be his daughter Saumanasa or son Mandara could pick Vrishchika up at the station and make sure she’s alright.”

LM: “That’s very kind of you. I will certainly take their number, it will be useful in more than one way. However, I understand that Somakhya’s friend, that kid Indrasena, who is her co-author and co-speaker in that conference, also lives there. I know she has gotten very pally with him and he would pick her up at the station. But at the same time I fear the two of them might be up to some mischief. Vrishchika let it slip that after their talk they were going to skip part of the meeting to go roaming in the city and make an excursion to Devaparvata.”
SM: “Ah, if Indrasena is there I’m sure he’ll ensure she’s OK. As for the mischief, I could give you his parents’ number. Call them and express your concerns; they would make sure that the two don’t do something too extravagant. But Vrishchika has to find her way in life: educate her but let her have her fun within limits. Let her meet my niece Saumanasa too: I am sure Saumanasa would benefit a lot from advice from Vrishchika.”

Just then they saw a mongoose scampering away into overgrowth adjacent to the northern wall of the temple. It paused for while giving them a good “darśana” and then vanished. LM : “That looks almost like an adbhuta to me. What might it mean?”
SM: “Well, since you saw it first possibly you are going to be favored by the mighty Kāmeśvara.”
LM : “Hope both of us are…”

◊◊◊◊◊

Vrishchika’s father had boarded her onto the train to Kshayadrajanagara, which soon got moving. Vrishchika felt tremendous happiness as the train started coursing on its way towards its destination – for the first time she was travelling alone, unlike her sister Lootika, and sensed that she had become an adult. She also felt nice to be in solitude – she remembered Lootika’s words on the importance of solitude: “When you have to spend all your time in close contact with mundane people you start thinking like them and soon lose your ability to see the parokṣa by which that which mystifies or even kills them can be apprehended. Hence, it is useful to have some time off in silent contemplation.” After a while she started preparing for her presentation. Here again she followed her sister’s method of carefully preparing her talk: laying out the time for each slide, planning what she was going say and when, and anticipate various questions and keep answers for them ready. At that point she wondered how their friend Somakhya had this uncanny knack of giving rather disparate talks back to back without preparing at all. She thought to herself: “Perhaps, that is tattvāveśa”. But then Lootika’s words came back to her that a good experimentalist must be methodical and well-planned in their approach to anything.

After a while, even as she was preparing to sleep, a woman passed by Vrishchika to her seat and as she did so dropped notes amounting to a few thousand rūpaka-s and her passport on the floor. Vrishchika called out to her and but she had her ears plugged with a headset and walked on oblivious of her lost property. Vrishchika gingerly gathered the stuff and walked up and gave it to her. She was very happy and gave Vrishchika a note as a reward. Vrishchika smiled saying that she did not need it as: “Dhaneśvara keeps me well!” The woman responded: “whoever that Dhaneśvara is may he continue to be good to you”, even as Vrishchika walked back to her seat.

◊◊◊◊◊

Indrasena and Vrishchika were rather disappointed that the former’s parents had poured water on their plan of roaming through certain parts of the city that night after the meeting was over. As they were dropping them off at the conference center they said: “Indra, if it were just you we would not even ask where you have been. But we have promised this charming young lady’s parents to keep her out of all trouble. While we have some confidence in your fighting skills, and even if you were to take your pistol with you, it is a wholly different matter of defending oneself and defending a young lady by your side. You know well that mahāmada’s prowlers and other assorted dasyu-s are constantly looking out for girls in places you want to go to.” Instead, they proposed that would pick them up at the conference center and bring them home for dinner.

As they were walking in Indrasena said somewhat sharply: “Gautamī, you should have been more careful with keeping secrets. If you want to have adventure you cannot announce everything to your parents.”
Vrishchika simpered and said: “I know. I just accidentally let it slip under pointed interrogation. But after all we might get to talk about the rahasya-s that I have been long wanting to know more about.”
Indrasena: “OK. Indeed, there are many exciting rahasya-s to talk about but remember that with such rahasya-s it is even more imperative to shut ones trap with the general public.”
Vrishchika: “I know, I know. I can be quite a Kunti when it comes to such matters but you know how it is when you are interrogated.”

Later that evening Indrasena’s parents picked them up and took them home for dinner. After dinner Indrasena took Vrishchika to his room. He offered Vrishchika a cushion as a seat and as she looked around his room for the first time taking in the sights she felt a strange sense of déjà vu. She saw a large painting of Naravāhana on the wall with his three wives Śriyā, Ṛddhi and Bhadrā hugging him and his son Nalakūbara seated beside him. The mantra “namo dhanadāya ca dhanasvāmine ca ||” was inscribed around the painting. Below it hung a Japanese painting with several figures of which she could only recognize one as being a yakṣa. Below that on a separate stool was a well-bound copy of the bṛhatkathā.

Vrishchika: “O ātreya, I see them all, the signs of Yakṣarāṭ everywhere. This is the indeed the sign of knower of rahasya-s. Who else today can even contemplate being on the path of Naravāhana-datta the son of Udayana. How was that most powerful path revealed to you? Most only sadly wish for it as losers do.”
Indrasena: “O Gautamī it is a long story but short of it goes thus. In my childhood as I was learning the Taittirīya-śruti I had dreams on many continuous nights of the great peak of Kailasa-parvata. Above it I saw hovering a great airplane, the Puṣpaka. By myself I began uttering the mantra-s starting with tirodhā-bhūḥ svāhā| Other mysterious mantra-s of Vaiśravaṇa started coming to my mind as the old Hindus would say as if from a past birth. But they were incomplete and I had wait till I could apprehend most of them due to the teachings of the vañga-siṃha.”

Vrishchika: “Who is this teacher – is it not remarkable that he obtained the Kauberī vidyā that lies concealed?”
Indrasena: “My father has a friend. He was known as the vaṅga-siṃho rājarāja-mata kaṇṭhīravaḥ: a vipra who was a lion among the vaṅga-s. He could be truly described as ‘dhanaṃ meru-tulyam’ and possessed of many kinds of vidyā-s. People looked at him and wondered ‘how could he be endowed with all of this wealth, health, beauty, intelligence, wife and children? He never seemed to age!’ After all it is almost a truism that: “indreṇa sarvāni śubhāny ekasmin puruṣe saṃyuktaṃ na sthāpitāni |’ Seeing him people got the hint that after all the advaitin-s were simply being losers when they asked ‘tataḥ kim?’ when confronted with dhanaṃ meru-tulyam| As though guided by Rājarāja, with youthful impropriety, I once jousted with him on a scientific topic very familiar to the two of us, as also your agrajā and Somakhya of quick discernment. I will tell you about that at a different point in time. While he did not openly admit it, despite being a widely acknowledged professor, he felt he had been defeated by a mere youth like me in that debate. It was then that it slipped out of his mouth: ‘Perhaps, this why the ācārya had said the totality of Dhanada-siddhi is not on me. It must be due to the mantra of the Bhṛgu-s I don’t have.’ I queried him on the Dhaneśvara-siddhi and he agreed to help me apprehend the Kaubera-śāsana whose mantra-s had briefly flashed before me saying if one mastered it ‘he becomes Naravāhanadatta.”

Vrishchika: “Most interesting, I see signs of a genuine Jambhala-vidvān in you. Indeed, why talk about the imaginary ‘eko brahmaṇa ānandaḥ’ when you cannot achieve ‘eko mānuṣa ānandaḥ’ in this world. It is more like wishful thinking coming from not applying oneself vigorously to the path of Dhaneśvara. But since there is no gain without pain I have long suspected that this path must have be arduous. What is the significance of that Nipponic painting below that of the Rājarāja-parivāra? It reminds me of the incident I have told you from childhood when I captured as a khārkhoḍa the bhūta of a prācya from those regions.”
Indrasena: “Ah that one is important to me as a reminder that, even though arduous, one who applies himself to the Vitteśa-siddhānta can be like a Naravāhanadatta even if he might be from the far-off land of the pītavarṇa-s”
Vrishchika: “How did you get it? Who are those figures depicted in it?”
Indrasena: “My father obtained it when visiting the prācya-s on matters relating to his work. He managed to with some difficulty befriend the phlegmatic prācya “deśika’’ of the temple on mount Shigi who eventually gave him this painting on mulberry paper. The figure you see to the left is a Nipponic mantra-vādin known as Myoren with immense Kaubera-siddhi. The figure to the right is Daigo the emperor of Japan. He was afflicted by an incurable disease. Myoren performed a Vitteśa-sādhanā and as result a guhyaka appeared – he is the central figure with the sword – the yakṣa then cured the emperor.”
Vrishchika: “Remarkable!”
Indrasena then walked up to the painting and turned it around: “Here on the back you can see the images of the great Vaiśravaṇa along with his wife Śriyā and son Nalakūbara. This a painting of their idols installed on Mt Shigi.”

Vrishchika: “O Atri, that painting of a sword with the siddham script also seems to be Nipponic. It too seems to be connected with śrī Naravāhana. It looks like the sword of a vidyādhara upon attaining siddhi. Could you please tell me more about it?”
Indrasena: “Gautamī that is the painting of the sword received by Sakanoue no Tamuramaro from Yakṣarāṭ before going to war with the Ainus. There is a long back story here. It is said that when Xuanzong, the Tang emperor of the cīna-s, was in deep fear of his enemies he was aided by Vaiśravaṇa and Nalakūbara due to their mantra-s deployed by Amoghavajra. A little later when the Silla Koreans backed by the cīna-s threatened the Nipponians the latter similarly invoked Vaiśravaṇa through an elaborate ritual that gave them total immunity from those mainland cousins of theirs. At that time a Nipponic mantravādin had a dream wherein a horse with a jeweled saddle led him to a holy site, Kuramadera, to the north of Kyoto, where he found a svāyambhuva idol of Kubera and installed it in a great temple there. Thereafter in the late 700s of the Common Era, the warrior Sakanoue no Tamuramaro worshiped Dhanada at that temple, attained siddhi much like our Naravāhanadatta, and obtained a sword from the god. He used this sword in his campaign against the Ainus and conquered their land. There he built a copy of the temple at Kuramadera depicting Yakṣarāṭ as being borne by guhyaka-s known as nara-s exactly as prescribed in our tradition.”
Vrishchika: “Thank you for the most interesting narrative. What is written in the siddham script on that painting?”
Indrasena: “It is the mantra Sakanoue used: oṃ vaiśravaṇa sahaparivāreṇa samāja jaḥ huṃ vaṃ hoḥ oṃ ve svāhā ||

Vrishchika: “Indra, I have so many things to talk to you about. But since time is short right now I would like to ask you regarding the Puṣpaka-vimāna yantra which is mentioned in the śruti of the Taittirīyaka-s as being engineered by the god Tvaṣṭṛ for Kubera. Could you please give me a darśana of the Puṣpaka-vimāna-yantra you worship and lead me to the Puṣpaka-vimāna-sādhanā?”
Indrasena: “Alinī. That’s a rahasya-prayoga. You would need to engage in much Kaubera-yogābhyāsa for any degree of success in that direction. Nevertheless let me show the Puṣpaka to you.” Indrasena led her to a closet in his room and opened the door to reveal the image of the vimāna in the center of which were the seated images of Rājarāja, Ṛddhi and Nalakūbara. His antardhānāstra was placed in front of him. In front of the vimāna was the image of yakṣa Maṅkanaka the gatekeeper of Kubera. There was also an image of the cow Sarvakāmadughā. There was a pot of the five-metal alloy with water and a glass pot with honey in it on the vimāna-pratimā. Indrasena: “Alinikā utter the following mantra-s after me gazing single-mindedly at the Puṣpaka-vimāna:
oṃ haṃ jambhalāya vaiśravaṇāya maṇibhadrāya pāṅcikāya pūrṇabhadrāya nalakūbaraya yakṣebhyo yakṣāṇāṃ patye namaḥ ।

Then he gave her some honey with a silver spoon and asked her to eat it with the mantra:
idaṃ jambhalasya madhu maghonaṃ madhunā prajāvatī payasvatī dhanavatī dhīmatī ayuṣmatī bhūyāsam |

Then with another spoon he gave Vrishchika the water with a small amount of cedar nut oil: “Apply this on your eyelids with the following mantra:
kuberasyodakam idam tenādṛṣṭaṃ dṛṣṭam bhavati huṃ nakulahastāya svāhā |

Vrishchika: “Ah this is the pratikṛti of the water sent by Viteśa to help Rāma and Lakṣmaṇa penetrate the māyā of Meghanāda!”

Indrasena: “Indeed. Then you shall close your eyes, visualize the great cave in the Himalayan heights where Kubera has placed the pot of golden kauberaka honey and meditate upon it doing japa of the mantra:
oṃ hrīṃ vaiśravaṇa! dhanaṃ puṣṭiṃ dehi me svāhā |
When you have become sthiramati emerge from your meditation uttering oṃ vaiśravaṇa arthatamom। 3 X huṃ phaṭ|

After Vrishchika did so she emerged from her dhyāna uttering: “apaśyaṃ tvā yakṣam ugraṃ tvāṣṭrīṃ vimānarūḍhaṃ puruścakrāṇi sahasra-vandhurāṇi । sa vaiśravaṇo rājarājā mahato mahīyān yasya pratigraheṇāpsyāmi dhanaṃ meru-tulyam । kāmam pratigacchāmi । dhanam pratigacchāmi । prajāḥ pratigacchami । īśvareṇa mahado3 o3 oṃ ||

Indrasena: “That’s good it appears you are on the path of īśvara-siddhi. I know you already have accomplishments as a mantravādinī but before proceeding I need to know if you have studied your earlier siddhānta-s well. Utter and expound the dvādaśa-nāmāni.”
Vrishchika: “I do know them from my sarvādhikāra-dīkṣa:
dhanadaś ca yakṣapatir vitteśo nidhipālakaḥ ।
rākṣasādhipatiś caiva piṅgalākṣo vimānagaḥ ॥
rudrasakhā kuberaś ca guhyakānāṃ patis tathā ।
vaiśravaṇeśvaraś caiva yakṣendraḥ parikīrtitaḥ ||

Dhanada: the giver of wealth; Yakṣapati: the lord of the yakṣa-s; Vitteśa: the lord of wealth; Nidhipālaka: the guardian of wealth; Rākṣasādhipati: the lord of the Rakṣa-s, this has been already stated in the Yajuṣ and the Atharvaṇa-śruti-s; Piṅgalākṣa: he whose eyes are of a golden tint; Vimānaga: one who can go anywhere on his Puṣpaka airplane; Rudrasakhā: friend of Rudra, this has been explained in our national epic; Rudra with his family often resides in the space station of Kubera; Kubera: One who has a frightful form; Guhyakānāṃ pati: lord of the hidden yakṣa-s who in the śruti are known as tirodhā; Vaiśravaṇa: the lord of the northern garden-land known as Viśravas; īśvara: the great lord; Yakṣendra: who is like Indra among the Yakṣa-s.”

Indrasena: “Great. At some future point when we are united after passing beyond the place known as Mlecchadigdvāravṛtti we shall perform together the great yāga as enjoined in the Taittirīya-śruti concluding with the mantra rājādhirājāya… By that time I would have mastered the full rahasya-s of this very mysterious mantra of the Bhṛgu-s from the Atharvaveda:

yad indrādo dāśarājñe
mānuṣaṃ vi gāhathāḥ |
virūpaḥ sarvasmā āsīt
saha yakṣāya kalpate ||

But for now you may do puraścaraṇa of the antardhānāstra-mantra. It will provide us with the fury needed withstand the assault of the ekarākṣasavādin-s who would seek to place in the museum in that great clash in the future.”

Vrishchika: “That sounds frightening. Tell me more of how one approaches this rahasya?”
Indrasena: “The full vidhi goes thus: One observes fast on the dvitīya. Then on tṛtīya one performs the rite having broken ones fast. One places an image of the gadā in a golden or silver vessel with ghee on which has been inscribed the 12 names of the yakṣa. One invokes Kubera with the following mantra enjoined by the Vaikhānasa-s:
rāyas-poṣāya āyuṣe prajāyai nīdhīśam āvāhayami ।
(For increasing prosperity, life and offspring I invoke the lord of wealth)

Then he does the visualization of Kubera thus:
atha dhyānam:
yakṣa-rākṣasa-sainyena guhyakānāṃ gaṇair api |
yuktaś ca śaṅkha-padmābhyāṃ nidhīnām adhipaḥ prabhuḥ ||
rājarājeśvaraḥ śrīmān gadāpāṇir dhaneśvaraḥ |
vimāna-yodhī dhanado vimāne puṣpake sthitaḥ ||
sa rājarājaḥ suśubhe yuddhārthī naravāhanaḥ |
prekṣamāṇaḥ śivasakhaḥ sākṣād iva śivaḥ svayam ||

(The lord of wealth, the overlord, is united with yakṣa, rakṣa-s and guhyaka hosts, as also Śaṅkha and Padmā. The lord of the king of kings, the wealthy one, the lord of riches holds a mace in his hand. The airplane-warrior, the giver of wealth is stationed in his Puṣpaka airplane. He the king of kings, residing in great auspiciousness, eager in combat, is borne by yakṣa-s known as Nara-s. Watching on, the friend of Śiva is himself like a second Śiva)

Then as ordained by the Vaikhānasa-s one offers the pūrvārghya of water in a receptacle with the gāyatrī:
rāja-rājāya vidmahe dhanādhyakṣyāya dhīmahi|
tan no yakṣaḥ pracodayāt ||

Then one performs japa and/or homa of the mantra:
asya mantrasya vadanya ṛṣiḥ । virāṭ chandaḥ । antardhāna-dhārin-ugra-kubero devatā ।
oṃ chaṇḍograyakṣāya huṃ phaṭ tirodhehi sapatnān naḥ svāhā ||

There after one concludes with the incantation of the Śānkhāyana-s:

nainaṃ rakṣo na piśāco hinasti na jambhako nāpy asuro na yakṣaḥ ||
(Neither rakṣa-s nor piśāca-s, nor jambhaka-s, i.e. Kubera’s agents, nor asura-s nor yakṣa-s harm him)

Then one offers the madhyamārghya as above with the mantra:
rudra-sakhāya vidmahe vaiśravaṇāya dhīmahi |
tan naḥ kuberaḥ pracodayāt ||

Then one recites the stuti:
dhanasya kāmasya praṇāyakas tvaṃ ।
sukhasya lābhasya pradāyakas tvaṃ ॥
vimānagas tvaṃ lokeśvaras tvaṃ ।
nāmāmi jiṣṇuṃ caṇḍogra-yakṣaṃ ॥
(You are the leader in wealth and desire;
You are the giver of happiness and profit;
You are the airplane-rider the lord of the world
I salute the conquering fierce formidable yakṣa)

Then one does tarpaṇa with the 12 names in the accusative case.
Additionally he also offers tarpaṇa to the parivāra:
ṛddhiṃ tarpayāmi ।
śriyāṃ tarpayāmi ।
bhadrāṃ tarpayāmi ।
nalakūbaraṃ tarpayāmi ।
maṇibhadraṃ tarpayāmi ।
śaṅkhaṃ tarpayāmi ।
padmaṃ tarpayāmi ।
maṅkanakaṃ tarpayāmi ।
yakṣa-pārṣadāṃs tarpayāmi ।
yakṣa-pārṣadīṃs tarpayāmi ॥

One then offers the prasannārghya with the mantra:
yakṣeśvarāya vidmahe gadā-hastāya dhīmahi |
tan no yakṣaḥ pracodayāt ||

One then makes offering of pure cooked food as bali with mantra ordained by the Vaikhānasa-s:
rāja-rājo dhanādhyakṣaḥ kubero viśravas-patiḥ |
prīyatāṃ nidhi-saṃyukta īśvarasya sakhā prabhuḥ ||
(The king of kings, the administrator of wealth, Kubera, the lord of the northern land of Viśravas, possessed of wealth, the friend of Rudra, may the lord be pleased)

Then he does upasthāna:
parivāreṇa saha kubero nakulahastaḥ suprīto suprasanno varado yathā sthānaṃ tiṣṭhatu ||
(With his retinue may Kubera, holding a mongoose, pleased and happy, boon-giving remain in his own region)”

Indrasena then continued: “One performs this vrata for an year observing restraints of not consuming alcohol and eating pure food and maintaining one’s body with discipline.”
Vrishchika: “I express my profuse gratitude to you for revealing the vidhi. I am hoping to perform the vidhi even as you have described it. But the only issue is how I might be able to obtain the gadā?”
Indrasena: “I will be duly sending you one. Again most only attain partial īśvarasiddhi. Only those born of good families for multiple generations attain that complete siddhi. In Gomūtra-nagara lived a man who saw all these vidyā-s but his discipline was incomplete so his lābha was also incomplete. Likewise with the vaṅgasiṃha. But fearlessly perform the puraścaraṇa. Someday all will come together.”
Vrishchika: “We will not be able see each other from some time in the coming future as our paths diverge but in the future we shall be united again and shall perform the great yāga.”

Some reminiscences of our study of chaotic maps-2

•December 26, 2016 • Leave a Comment

Continued from part-1

The second two dimensional map we studied in our early days was that of Lozi:
x_{n+1}=1-a|x_n|+by_n\\ y_{n+1}=x_n
where a and b are constants.

It becomes immediately evident that this map is conceptually similar to the Henon map, using the absolute value operator instead of the squaring operator. Both generate a positive signed value causing the observed change in direction of the curve. However, the squaring operator produces a smooth parabola, while the absolute value operator produces a sharp inflection. In 1 dimensional chaotic maps these correspond to the situation seen with the logistic map and the tent map: the former corresponding to the Henon map and the latter to the Lozi map.

At a=1.7;\;b=.5; \; (x_0,y_0)=(0,0) this map produces a “tent” equivalent of the parabola-like Henon strange attractor.

lozi_tentFigure 1

However, this map produces richer behavior than the Henon attractor in terms of interesting forms of attractors. One such attractor is seen at a=1.5 and b=1, (x_0,y_0)=(0,0), where the Lozi map produces the striking “isosceles triangle” attractor with vertices at (0,-2); (2,0); (-2,2). Thus the base angles of the triangle are \arctan(3) and the two sides form angles of \arctan(2) and \arctan(1)=\frac{\pi}{4} with the x-axis — a configuration which results in the famous \arctan(1)+\arctan(2)+\arctan(3)=\pi expression.

Lozi_1.5.1.pngFigure 2

Incidentally, this triangle and its bounding square also presents the creases for an origami base, which can be used to prove the above identity in a self-evident way (Also see our earlier note for trigonometric proof of same). It was used by a man from Japan to make a simple grebe that even a beginner can make. Inside this isosceles triangle the points are chaotically distributed except for fractal exclusion zones. The two biggest exclusion zones are ellipses placed orthogonal to each other and along the vertical axis of the isosceles triangle; they have a major axis to minor axis ratio of 3 which indicates that they are defined by two triangles fused at their bases, similar to the outer isosceles triangle. Also along the axis of the main isosceles triangle are small, nearly regular, pentagonal zones of exclusion — perhaps a reflection of the characteristic angle \arctan(3) being close to the pentagonal angle \frac{2\pi}{5}=72^o. As one moves from the axis of the main triangle to the sides one gets further and further distorted elliptical and pentagonal zones of exclusion.

lozi_analysisFigure 3

When a=-1 and b=-1, (x_0,y_0)=(4,-2.1) the Lozi map produces the famous “Gingerbread man” attractor.

gingerbreadFigure 4

Giving at little “nudge” to the parameters (a=-1.0001;b=-.9999) for the same (x_0,y_0) results in the map wandering all over the Gingerbread man before eventually (somewhere between 5 \times 10^4..10^5 iterations) to start spirally converging to five stable points:( (-1,-1);\; (3,-1);\; (5,1);\;(3,5);\;(-1,1)).

gingerbread_5convergenceFigure 5

When one increases a in the Gingerbread man configuration slightly, e.g. a=-1.001;\;b=-1, we get a Gingerbread man with a somewhat smooth “aura”.

gingerbread_auraFigure 6

Further increases in the a result in larger and larger strange attractors where the Gingerbread man acquires increasing number of “legs” and a multi-spiked aura.

lozi_big_gingerFigure 7: The attractor for a=-1.48, b=-1 with a tripodal Gingerbread man.

Lozi_big_ginger2.pngFigure 8: The attractor for a=-1.7523, b=-1 with a pentapodal Gingerbread man.

Just as with the Henon attractor, we also explored the escape plot of the Lozi attractors where we plot regions in a different color depending on the number of iterations in which they escape towards \infty. When we do this for a=-1, b=-1, i.e. the standard Gingerbread man parameters we get a interesting fractal region of entrapment (for the number of iterations we used to test escape n=100) at the middle of which lies the Gingerbread man attractor we plotted above.

Lozi_Julia.pngFigure 9

Outside of this basin of entrapment is a collage-like pattern of different escape zones. These are bounded from the outermost escape zone by a cardiomorph boundary. We wondered as to why this boundary arises and what might be the form of the function which specifies it? The answer to lies in the generating equations of the Lozi map itself. In order to compute the potential for escape to infinity we use the radius from the center of the plot region. This, with the Lozi map leads to the equation:
\left(1-a\left|x\right|+by\right)^2+x^2=1
https://www.desmos.com/calculator/psvopnbtq7

Indeed the shape of the cardiomorph depends on the (a,b) values as can be see from the below version of the escape plot for a=.2, b=1.02) where the attractors are two points in the two entrapment regions or they escape to \infty along the central line between them.

lozi_julia_02Figure 10

This study in our youth of the Lozi map then lead us to discover another related attractor, which we arrived at by a rather simple modification of the Lozi map:

x_{n+1}=1-a \sqrt{|x_n|}+by_n\\ y_{n+1}=x_n
where a and b are constants.

At positive a \approx 1.25 \;..\; 1.9 and b \approx -.1567 \;..\; .1567 it leads to an incomplete deltoid-like attractor that is at border of chaos and regular convergence. When b=-1 this map yields a reasonably rich and aesthetically fairly pleasing set of strange attractors. We term these the butterfly attractors because most them have a vaguely four-winged form with a bilateral symmetry along the x=y line. They appear when b=-1.

lozi_sqrt_attractorsFigure 11

They are fairly stable over a large range of a values with the attractor increasing in size with increasing a.

lozi_sqrt_attractors03Figure 12

Some elementary lessons from iterative fractal maps

•December 25, 2016 • Leave a Comment

The famous Sierpinski gasket was one of the first fractals we wrote code for when we got access to a computer. It impressed us enormously that an intricate object with self-similarity over all scales could be generated by a rather simple process:
1) Take three points that would define a triangle.
2) Take a fourth starting point.
3) Randomly pick one of the three vertices of the triangle. Plot the midpoint between the fourth starting point and the chosen vertex.
4) Repeat this process with that midpoint for a large number of iterations.
5) The points thus generated would converge to an attractor which is the Sierpinski gasket.

This process inspired us to use this procedure more extensively to develop other such fractal objects. We found an answer for this shortly thereafter in a book by the mathematician Barnsley precisely about this topic. In it Barnsley said something to the tune that we would stop looking objects in nature the same way once we see the fractals such as those for which he provided a recipe for construction. This had indeed already started happening to us with our encounter with the Sierpinski gasket. Notably, around that time we recorded a statement of Johannes Kepler (which we read somewhere), which was earlier expression of the same idea:
I believe the geometric proportion served the creator as an idea when he introduced the continuous generation of similar objects from similar objects.

The Hindu enacts such a concept when he performs śrauta rituals with complex rituals developing recursively with some modification from a simple “prakṛti”.

Coming back to Barnsley, he called these fractals Iterated Function Systems (IFS) and they are generated thus:
\begin{bmatrix} x_{n+1} \\ y_{n+1} \end{bmatrix} = \begin{bmatrix} a & b\\ c & d\end{bmatrix}_i \begin{bmatrix} x_{n} \\ y_{n} \end{bmatrix} + \begin{bmatrix} e \\ f \end{bmatrix}_i; \; p_i
where i=1..n

As one can see this map defines a set of affine transformations that are recursively applied to the starting coordinates (x_n,y_n). However, there can be a set of n such of matrices 1..n any one of which i can be applied to the point at a time with a probability of p_i. For each mapping of the point we have 7 numbers, 6 from the transformation matrices (a..f) and the 7th being the probability with which it is applied p_i.

Barnsley’s most famous fractal map was the fern leaf. Several variants of this have been generated (some by us and others whose creators are either Barnsley himself or others not known to us).

fernleaves

Figure 1: the first one is Barnsley’s original fern leaf

Given that a relatively small number of values to can generate a rather complex object by this procedure, Barnsley developed this as a method for compression of images. To us the IFS immediately struck us as being a mechanism for illustrating evolution: In the simplest scenario, the values in the above matrices and the probability of its choice can be seen as ‘genes’ which are prone to mutations with quantitative effects on traits. Thus, one can allow the matrices to mutate while we act as the proxy for natural selection.

The same is quite well-illustrated by another of the IFS fractals the leaf. In this leaf we have 4 matrices which gives us a total of 24+4=28 ‘genes’ which might be mutated. If we impose the conditions of continuity of the attractor region (a very real biological one) and reasonable bilateral symmetry then only 10 of those ‘genes’ can take mutations. Now, by letting them undergo mutation and selection we can see evolution occur in silico and spawn a variety of leaf forms close to what we actually see in nature.

ifs_leafFigure 2

One might learn two things from this:
1) If we give a person with no knowledge of how the matrices were constructed for generating a given shape, e.g. the leaf, he can mutate the ‘genes’ randomly and figure out how it works. After some mutagenesis he would soon discover that different ‘genes’ have different morphological roles and there is even a degree of hierarchy in terms of their effect on form. He would soon understand the difference between null mutations and change of function mutations. Thus, he can put together a developmental network for the role of the ‘genes’ in the development of the in silico leaf. In doing this on a very small scale he would have reproduced what geneticists have done to unravel the genetic networks for the development of real organisms like Caenorhabditis, Drosophila and the Danio (e.g. Nüsslein-Volhard and Wieschaus et al). Such efforts entirely dominated developmental biology and genetics for almost two decades.

2) Beyond doubt these efforts of the developmental geneticists contributed fundamentally to our understanding of biology. But in the end it spawned the field that is often irritatingly referred to as “evo-devo” — an attempt to marry the foundation of biology in the form of the evolutionary theory with developmental genetics. In our youth, even as we were discovering the beauty of IFS for ourselves, we were aligned towards a future as an evolutionary developmental geneticist. We even thought we should apprentice ourselves as a slave in one of those powerhouse developmental genetics labs, where we could bring our knowledge of evolution which was much ahead of theirs in those days. It was in this context the IFS experiments offered us an important negative lesson. We saw that the so-called evo-devo field was proceeding in a rather pedestrian direction. It was one where the protein or RNA products of the genes and were mostly faceless blobs (they would often literally be illustrated by researchers as such), with the primary expression of the studies being simple genome-demographic summaries from angle of genes they knew best and organism-specific variations of the maps of gene-interactions in development. To us this was not qualitatively telling us anything new about evolution of development beyond what we could glean from in silico experiments with systems such as IFS. Instead, we realized that we had to take a path that got directly to heart of what is unique to biology, i.e. the biochemistry of those gene products. This resonated with where we first began our science the chemistry of biological molecules. It was placing that within the framework of the evolutionary theory, which was to lead us closer to the foundations of biology than any of this other stuff.

In our childhood we had spent a lot of time simply gazing through the kaleidoscope. It brought home to us that one principle of aesthetics was symmetry. When we learned of IFS we saw how simple recursion was a second principle of aesthetics. We found that a very simple IFS could result objects of great beauty especially when combined with first principle symmetry. This led us to play with a simple map, namely one to generate the limit curve of a bifurcating tree. This can be done by choosing the below three matrices with equal probability:

\begin{bmatrix} x_{n+1} \\ y_{n+1} \end{bmatrix} = \begin{bmatrix} s \cos(\theta) & -s\sin(\theta)\\ s \sin(\theta) & s \cos(\theta)\end{bmatrix} \begin{bmatrix} x_{n} \\ y_{n} \end{bmatrix} + \begin{bmatrix} 0 \\ 1 \end{bmatrix}; \; p_1=\frac{1}{3}

\begin{bmatrix} x_{n+1} \\ y_{n+1} \end{bmatrix} = \begin{bmatrix} s \cos(\theta) & s\sin(\theta)\\ -s \sin(\theta) & s \cos(\theta)\end{bmatrix} \begin{bmatrix} x_{n} \\ y_{n} \end{bmatrix} + \begin{bmatrix} 0 \\ 1 \end{bmatrix}; \; p_2=\frac{1}{3}

\begin{bmatrix} x_{n+1} \\ y_{n+1} \end{bmatrix} = \begin{bmatrix} 1 & 0\\ 0 & 1\end{bmatrix} \begin{bmatrix} x_{n} \\ y_{n} \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \end{bmatrix}; \; p_3=\frac{1}{3}

Here s is the scaling factor whereas \theta is the angle by which the branches of the bifurcating tree are rotated. Finally, we can rotate each fractal a fixed number of times to obtain a particular kaleidoscopic symmetry. In the first example below we have an overall symmetry of 6 with various \theta values and s=\frac{1}{\phi} the Golden ratio.

ifs_angle_curveFigure 3

In the second example we have a symmetry of 4 with similar rotations and scaling as above. One may note that the first case in this figure where \theta=\frac{\pi}{4} we get a structure that is immediately reminiscent of the floor plans towards which the central spire of the great Hindu temples (e.g. at Khajuraho) converged to before the coming down of the green curtain.

ifs_angle_curve_4Figure 4

 
%d bloggers like this: