Paper folding, Sundara Rao and geometrical constructions

Paper folding, which we shall hereinafter refer to as origami without further discussion on the correctness of the usage, is believed to have had a long history in Japan. Some believe that some primitive form of origami might have been transmitted by the bauddha-s from the 600s of the common era but we have no strong evidence for such. Real evidence for methodical folds only comes from the charms known as noshi used by the samurai from the last millennium. In this regard some have speculated that the word ‘kami’ in origami might also be homophonic play on ‘kami’ as a god, with paper having an association with offerings to the gods in the shrines. The first clear references to folds that are part of modern origami only come from the period between the 1600-1700s. In Saikaku Ihara’s story from the late 1600s he talks of his hero distinguishing himself at the age of seven by folding butterflies and flowers with great dexterity. The butterflies and the flowers would imply the presence of the famous waterbomb base assuming they were similar to what has been provided by Yoshizawa Akira, the father of origami in the modern era. In Akisato Rito’s “Sembazuru Orikata” from the late 1700s we encounter the famous bird base for folding a crane.

Our own introduction to origami came from some folds that were known to my late grandfather (see below). Then our parents gave us a book of Robert Harbin. His work which was influenced by Yoshizawa Akira’s made us aware of several simple folds but many of his models were frustratingly unclear. We then ran into a man who had spent some time in Japan and learned folding from a successor of Yoshizawa Akira. He showed the real complexities that can be achieved by folding. We folded models of that level of complexity only when we encountered the diagrams of the virtuoso mathematician-folder John Montroll. Somewhere in Montroll prodigious output lay the highest level of origami we ever reached. Montroll and the physicist Robert Lang can be considered the extant masters of “scientific origami”, which emerges from a combination of a deep understanding of the underlying mathematics with aesthetics. Their models display extraordinary complexity and naturalism.

The encounter with origami quickly kindled our interest in using it as means to study Euclidean geometry. In our youth we discovered for ourselves some trivial constructions for the octagon, the square pyramid and the point of intersection of two lines using such techniques. It was only thereafter that we discovered the pioneering work of a remarkable Hindu mathematician named T. Sundara Rao, who should be regarded the father of the geometry of paper folding.

Figure 1: T. Sundara Rao

Since he has been largely forgotten we shall here reiterate his biography. Sundara Rao was the son of Gopala Rao and grandson of Bava Pandita. The latter came from a family of Mahārāṣṭrī deśastha brāhmaṇa-s who had settled in the Thanjavur in the days of Shivaji’s conquest of the south and had been appointed as advisers to the marāṭhā rājan of Thanjavur. Gopala Rao was born at the Gaṇapati agrahāra on the banks of the Kaveri in 1832 CE. At home he with the rest of his family was conversant in Sanskrit but he largely taught himself English and rose to be a brilliant educator at Kumbhaghona. Even Englishmen were supposed to attend his classes for the special insights he provided in his classes on mathematics, history and English. He was also a Marathi poet but I have little knowledge of his literary achievements in that direction. In 1853 CE his first son Sundara Rao was born. At the age of 21 Sundara was awarded a BA degree standing second in mathematics in the whole Madras Presidency and was also said to be well-versed in Sanskrit. To earn a better wage he joined the civil service and became a deputy collector serving at Tiruchirappalli. He was an inventive man and had made the following: 1) Boat for his own use. 2) An ink pen which he used throughout his life. 3) An ink which was widely used for cyclostyling documents. 4) A version of the Hitchcock lamp that had a much longer life. Throughout his life he kept doing his researches in mathematics and published two books on the subject. Of these the one on paper folding termed “Geometrical Exercises in paper folding” survives to date. This was mainly because it caught the attention of his famous contemporary, German mathematician Felix Klein, who referred to it in his work. As a result two American mathematicians, Wooster Beman from University of Michigan and David Smith from Columbia University made an American edition of his book, resulting in the survival of this remarkable work. Sundara was also a good horseman and photographer and appears to have died sometime after 1923. In many ways his fate was like that of other early modern Hindu scientists and mathematicians — a bright meteor in the intellectual firmament with no successors continuing the school. Indeed, there was nothing at all done in India to build on Rao’s work in geometry by folding, and most of the people who were able to appreciate and continue his work were either from Europe or USA. This was despite the fact that there were several roughly contemporaneous luminaries in India like the great Srinivasa Ramanujan who came from the same town as Sundara Rao, which is not far from the hamlet of one strand of our ancestors.

Rao’s work was inspired Friedrich Fröbel who introduced paper folding as a part of Kindergarten education in the west. But as Rao notes there was also an indigenous tradition of paper folding in India whose history has been much neglected. This tradition produced folds like the boat, catamaran, ink-bottle, briefs, flying boat, lotus bud and the like. These folds were known to my late grandfather who had no contact with western folding. The lotus bud which appears to have a relationship with the Japanese flower and the waterbomb might imply that these designs using the waterbomb base came to India from Japan. However, the other folds noted above are unique and likely to have developed independently of the Japanese and western origami traditions by the 1800s in India. Rao also states that folding was used to maintain register when writing Sanskrit or Marathi on paper.

Rao’s constructions span the entire range from the trivial to the fairly complex but throughout they display a deep familiarity with geometry and opened a new type of understanding for old problems.

Figure 2: Equilateral triangle

One of the simplest of his constructions is the folding of an equilateral triangle from a square sheet of paper:
1) Fold the square in half by getting corners A to B and D to C to produce crease $\overline{EF}$.
2) Fold through corner A to make corner B lie on $\overline{EF}$. Similarly fold through corner B to make A lie on $\overline{EF}$. In each case crease along the folded $\overline{EF}$ to get creases $\overline{BI}$ and $\overline{AJ}$.
3) The point of intersection of above two creases ($B'$) defines the equilateral $\triangle ABB'$.
4) Once you have an equilateral triangle we can determine its center and use that to fold a hexagon (all described by Rao).

Figure 3: Golden Ratio and folding a pentagon

A notable construction of Rao is to obtain the Golden ratio by folding a square piece of paper and use this to fold a regular pentagon:
1) Bisect the square as above to produce crease $EF$. This in turn produces rectangle $EFCD$.
2) Fold through corners of this rectangle E and C to obtain $\overline{EC}$. This crease is the diagonal of the said rectangle.
3) Fold through E to bring corner D to lie on $\overline{EC}$ at point G. As a result $\overline{DH}$ will now lie along $\overline{HG}$
4) Fold along the edge of the paper $\overline{DH}$ to produce crease HI.
5) Fold through corner C to bring side $DC$ of the paper to lie along crease $EC$.
6) Keeping the fold mentioned in the above step use crease $HI$ and again fold along it. This will produce crease $KG'$. Extend this crease vertically folding through it to produce crease $G'L$.
7) This will Golden section $\overline{DC}$. $\therefore \overline{G'C}:\overline{G'D}=\phi$.

As an exercise one can use this construction to mark out two linked squares whose sides are in Golden ratio. This can be used fold a Golden version Akisato Rito’s linked cranes from each of the squares, perhaps in the memory of the tragic tale of Sadako Sasaki a victim of the mass murder by the Americans at Hiroshima.

To construct a regular pentagon from this square piece of paper do the following:
1) Bisect $\overline{AL}$ to get crease $MM'$.
2) One will note that $\overline{AM}=\overline{BI}$.
3) Fold through point M so that point I comes to lie on edge $AD$ of the paper. Similarly fold through point I so that point M comes to lie of side $BC$ of the paper. This will determine points N and O. Fold along $\overline{MN}$ and $\overline{IO}$ to get the first three sides of the pentagon.
4) Fold through point N so that point M comes to lie on the vertical bisector of the square and crease the paper along $\overline{MN}$. Do the same with point O so that point I lies on the vertical bisector of the square. This operation will establish point P. Folding along $\overline{NP}$ and $\overline{OP}$ will give us the required pentagon.
5) One will note that the pentagon has sides of length $\dfrac{1}{\phi}$ and diagonals of unit length, where the unit is the side of the starting square paper.

Nearly a century after Rao’s folding of the pentagon using the Golden ratio, non-compass and straight-edge polygons like the heptagon have been folded by Robert Geretschläger using pure origami. These folds reminds a Hindu of the noshi carried as charms by the samurai. The polygonal folds could be similarly adopted as yantra-s: e.g. the pentagram in the yantra of the yoginī Śyāmala

Figure 4: Various polygons folded from square paper and a couple adapted as yantra-s

Rao boldly advanced folding as a method to construct conic sections. Below we shall describe his construction of the parabola:

Figure 5: Parabola

1) Take a square or rectangular piece of paper. One side of it $AB$ shall be the directrix of the parabola.
2) Fold a vertically to crease rectangle ABIE. Use this rectangle to crease rectangle $IGHE$ such that $\overline{BI}=\overline{IG}$.
3) Fold the paper horizontally in half to get bisecting crease. Where this crease intersects crease $GH$ will be the focus of the parabola $F$.
4) Fold the top half of the paper into $n$ horizontal parallel creases.
5) Fold the point of intersection of each of these parallel creases with the directrix edge of the paper such that it comes to lie on point $F$.
6) This will create the creases shown in green. Repeat the same for the bottom half.
7) These creases are tangents to the parabola and their envelop with mark out the parabola on the paper. One can see that this simulates via folding the construction of the parabola we have described earlier.

This construction and our other experiments with envelop constructions inspired us to fold a square paper into successive mountain and valley folds which creates a 3D paraboloid surface. One can see that it is a curve of the form: $z=xy$ – a hyperbolic paraboloid. We later realized that such origami is being used by Erik Demaine to make a variety of more complex shapes. A similar technique can also be used to fold a double helix structure like a cartoon of double-stranded nucleic acid.

Figure 6: Hyperbolic paraboloid

Most of Rao’s constructions were origami equivalents of constructions that can be done using a compass and a straight edge. For example, he was unable to double the cube using a square piece of paper and origami. However, with subsequent developments in this field several problems beyond the reach of classic compass-straight edge methods have fallen to origami. We had seen above that the heptagon can be folded. Below we shall illustrate two of the classic problems of the old yavana-s, namely doubling the cube (the Delian problem) and trisecting any angle.

Doubling the cube
This involves folding $\sqrt[3]{2}$ which was first achieved by Margherita Beloch 1936 CE. We first illustrate how this can be done and the proof of it using elementary geometry.

Figure 7: Doubling the cube

In the below protocol fold and unfold so as to create creases. The first process involves dividing the square into 3 equal rectangles:
1) Fold along diagonal $\overline{f}$
2) Bisect the square by folding along $\overline{g}$ by bringing D to A and C to B.
3) Fold along $\overline{h}$, which is the diagonal of the rectangle AEFB that was formed by the previous crease.
4) Fold through point G where creases $f$ and $h$ intersect creating crease $j$ parallel to the sides of the square.
5) Fold along the segment where $\overline{AB}$ lies as a result of the above fold to create crease $a'$. This trisects the square into 3 equal rectangles.

One can see that if corner A is at origin then $\overline{f}$ has equation $y=-x+l$ where $l$ is the side of the square. Similarly $\overline{h}$ has equation $y=\dfrac{1}{2}x$. Their intersection would hence be $\left (\dfrac{2}{3},\dfrac{1}{3} \right)$. This is the rationale for the trisection of the square.

Next we double the cube:
1) Fold the paper such that the end $I$ of the first trisecting crease $j$ comes to lie on the second trisecting crease $a'$ created above. This is point $I'$.
2) The corner $B$ of the paper should lie along edge $AD$ so that it marks point $B_2$ on it.
3) Then point $B_2$ partitions $\overline{AD}$ in the ratio $1:\sqrt[3]{2}$.
4) Once we have the starting square’s side partitioned in the ratio $1:\sqrt[3]{2}$ we can then fold two squares with sides equal to each partition. Each square is then separated and folded into a cube using the Japanese waterbomb method. This will result in cubes whose volumes are in the ratio $1:2$.

Figure 8: Cubes doubled

We need to show that $\dfrac{\overline{DB_2}}{\overline{AB_2}}=\sqrt[3]{2}$. Hence for ease of calculation we take $\overline{DB_2}=x$ and $\overline{AB_2}=1$. Then we have: $\overline{A'B_2}=x-\left(\dfrac{x+1}{3}\right)=\dfrac{2x-1}{3}$; $\overline{I'B_2}=\dfrac{x+1}{3}$.
In $\triangle AJB2$ let $\overline{AJ}=y$. $\overline{JB2}=x+1-y$. Thus, by the bhujā-koṭi-karṇa-nyāya we have: $1+y^2=(x+1-y)^2 \; \therefore y=\dfrac{x^2+2x}{2x+2}$.

We can see that $\triangle AJB_2 \sim \triangle A'I'B_2$.
Thus, $\overline{AJ} \equiv \overline{A'B_2}$ and $\overline{I'B_2} \equiv \overline{JB_2}$.

$\therefore \dfrac{\dfrac{2x-1}{3}}{\dfrac{x+1}{3}}=\dfrac{\dfrac{x^2+2x}{2x+2}}{\dfrac{(2x+2)(x+1)-(x^2+2x)}{2x+2}}\\[10pt] \dfrac{2x-1}{x+1}=\dfrac{x^2+2x}{x^2+2x+2}\\[10pt] (2x-1)(x^2+2x+2)=x^3+3x^2+2x\\ 2x^3+3x^2+2x-2=x^3+3x^2+2x\\ x^3=2 \; \therefore x=\sqrt[3]{2}$

Trisecting any angle

Figure 9: trisection of any angle

This classical yavana construction can be achieved by folding using the above diagram. It is detailed in the below video.

The professor’s war

It was the end of his work week. Vidrum came to the office of Vrishchika of long tresses; she had just started her lab at the medical school. Vrishchika put on her gloves and handed Vidrum a box: “Here is the kit with the vials of recombinant hypocretin-1 that my technician purified from the human hypothalamic fusion cell culture I had developed. Use it on your patient as nasal spray using the buffer which is also inside. You can get the instructions from my internal website for which I have already issued you a password. If you might have any issues about preparing the spray let me me know make sure to get it into the cold storage ASAP.”
Vidrum: “Thank you very much. I will ready it for him during the next appointment but I am just curious have you tried it out on yourself?”
Vrishchika smiled knowingly and said: “I had made these clones even when we were in med-school and all I’ll say is I was certainly less depressed than most female colleagues during the internship. By the way if you need hypocretin receptor inhibitors I have made a bunch with my sister Varoli, which are specific to either receptor or both.”
Vidrum smiled back and said: “I get it. I’ll certainly obtain the vidyā of the inhibitors from you whenever you have the time. Anyhow you have good evening – I guess you are going to be busy working.”

Vrishchika: “Actually, I am leaving right now. Not everything is ready in my lab here so I will be doing some things this evening in my husband’s lab.”
Vidrum: “I can drop at main campus if you don’t mind.”
Vrishchika: “Thank you for offering a ride but I fear Meghana hates me as much as she hates Lootika and my presence is likely spark unnecessary tension.”
Vidrum: “Of course I realize that and would not have even asked you if I were picking her up too. But I must tell you that something strange happened. Meghana has mysteriously gone totally out of contact for the past month. Her phone number does not work and she has even vacated her apartment.”
Vrishchika: “That is odd indeed. But does no one else in your circle have any information?”
Vidrum: “Frankly, this is embarrassing to tell anyone else. It has left me in a bit of a daze and you are the first person I am telling this. Let me put the kit into storage. We will meet in about 15 minutes at the parking lot and I’ll drop you off at your husband’s department.”
Vrishchika: “But I don’t know if your vehicle can fit my bike.”
Vidrum: “Sure it can. Let’s meet in 15 minutes”

As they were driving to the main campus Vrishchika said to Vidrum: “I know you may not want to talk anymore about this but do you think Meghana might have contracted marūnmāda?”
Vidrum: “That’s a very strange question. Why?”
Vidrum: “I know you’re never the person to just ask something – is it because of the hate for Meghana that your sister has passed on to you or something less frivolous.”
Vrishchika: “Vidrum, as you have seen over all these years we have tried our best not let the issues with Meghana come in the way of our interactions with you. So why would I say something like this just to hurt you. I don’t have a good recollection of all my seniors from school and their life stories but you should check to see if a girl from your school class was in yesterday’s news. May be it will give you some clue regarding this mystery that has hit you.”
Vidrum: “What a strange thing to say. Could you tell me more.”
Vrishchika: “As I said I really don’t know too much and am unsure if I recognize my seniors after all these years. Moreover, I don’t want to wade into your personal matters and into things I poorly understand. But I do recommend you scan the news when you get a chance to confirm or dispel what I said. I must say, I didn’t register details of the news item I am thinking about to be more precise in my inference – just a bit of deduction that could be all wrong.”
Vrishchika then went quiet.

Vidrum realized that it might not be something she was very keen to say more about and broke the silence by asking her: “So are you all doing some fun things this weekend?”
Vrishchika: “Fun or not fun is all relative. I am in the midst of a battle now – an investigation closely contested with a competitor. Hence, I will be busy at work over the weekend.”
Vidrum: “For once I don’t envy you guys.”
Vrishchika: “It is all relative Vidrum. Everyone has their own worries. ”
Vidrum: “Hope Indrasena is well? Say hi to him.”
Vrishchika: “He is good, but as luck would have it he and Somakhya are in the midst of a battle of their own with their bhrātṛvya-s. You know being in science needs you to be something of a warrior. What are you doing this weekend?”
Vidrum: “Good luck to you scorpion-girl – sting well. I am having lunch with Sharvamanyu and Abhirosha tomorrow and then we are going rock-climbing with friends. Were you all not so busy you all could have joined us.”

By then they had reached the school of life sciences and Vidrum dropped off Vrishchika. Vrishchika bade him good bye and taking her bike scampered off: “Thanks a lot Vidrum, good night, do check the news and be careful with everything.”

Vidrum was puzzled by Vrishchika’s remarks and back home he started checking the news. Before anything notable caught his eye he was distracted by the interview of a famous cricketer. Just then he realized that it was getting late for his exercise session and he dropped the news to get his body worked. After dinner his sleep deficit and exhaustion caught up with him and he was soon fast asleep.

◊◊◊◊◊

The next morning Vidrum woke up and used some of the hypocretin-1 he had aliquoted on himself. After a while he felt he was feeling really good and alert like he had never felt in sometime. He remarked to himself: “That hypocretin-1 sure works.”

Sharvamanyu had already reached the restaurant and was waiting for Vidrum sipping his glass of water. Seeing Vidrum nowhere in sight he started reading the news on his tablet. He was jolted out of it when Vidrum seated himself at the table: “Hey, sorry I had to attend to an emergency with one of my patients.”
Sharvamanyu: “No problem. Even Abhirosha said she would be late as she had to do something at work.”
Vidrum: “But you seemed to be quite lost in what you are reading – something interesting?”
Sharvamanyu: “Tucked away from all the usual news on the T20s and other masālā is this strange bit of news: ‘Tamanna Sharif vanishes from custody.’ ”
Vidrum: “What? You mean the girl who was in our class by that name?”
Sharvamanyu: “I believe it is actually the very same woman. She seems to be keeping with her distinguished family as a good marūnmattā.”
Vidrum: “Come on, why do you say so; not all Moslems are terrorists. Meghana had even brought her to our house with her husband Abu Hilal. They seemed like really nice, cultured people with a taste for fine poetry and a passion for justice in society.”
Sharvamanyu: “Well, nice and cultured means nothing to me. I have personally seen her people doing all things marūnmatta-s do.”
Vidrum: “What happened?”
Sharvamanyu: “Well you may remember the arson at Dakṣiṇaśālā which happened when we were young.”
Vidrum: “Yes, I do recall that. I thought it was the radical Moslems from the Mohammadwadi dargah. What does it have to do with Tamanna?”
Sharvamanyu: “Let me refresh you. I was returning that day from the Sanskrit language class since I could not attend the morning batch with you that day. Just then I got the news from my darker friends, whom you guys don’t know, that the marūnmatta-s had set fire to many shops in at the Dakṣiṇaśālā and where pelting stones at all those who were trying to get out of the inferno. I made it to the parking lot where I had parked my bike and was joined by my two friends who were also well-armed. Just then Saif ad-din Sharif with four other dāḍīvāle arrived at the parking lot with knives, petrol bombs and fire brands to set fire to the vehicles. We were just three and they five but we walloped the hell out of them and could have easily sent them to naraka that day but stopped with handing them over to the māmū-s to take care of the rest.”
Vidrum: “Wow, you can be violent. But they fully deserved it. But I still don’t get what all this has to do with Tamanna ?”

Sharvamanyu: “The article says she had been arrested a couple of days back at Mumbai while she was on route to London to talk at a human rights meeting. The Intelligence Bureau had good evidence that she was involved in espionage and might have been in the process of transmitting critical information to an agent in Laṇḍapura. Hence, they nabbed her. She had been placed in custody for interrogation. Her employers from an NGO which is funded by the Guillaume Glympton Human Rights Fund are said to have sent a top lawyer to represent her. But the news today is that she has mysteriously escaped from custody. The IB agents suspect Tamanna Sharif some how got off when her lawyer visited her and has flown off to Laṇḍapura using the passport of another woman who worked with the GG foundation. Saif ad-din Sharif the mastermind of that arson was none other than her brother. ”
Vidrum’s face turned pale in a visibly quivering voice he asked: “Sharva, really?”
Sharvamanyu sensed the sudden change in Vidrum’s face and voice: “Why, any problem?”
Vidrum: “I have something really bad to tell you.”
Vidrum then conveyed to Sharvamanyu the strange situation regarding Meghana
Sharvamanyu: “What! that is disturbing indeed! Did Meghana not have something to do with the GG foundation?”
Vidrum: “Well that’s why I find this all disturbing. She was indeed a legal assistant for the said foundation. Though I had no clue what the foundation exactly did. But I must further tell you that Vrishchika made some strange statements to me yesterday. I simply did not understand them then but from what I hear now she seemed to be linking Meghana’s disappearance with that of Ms. Sharif.
Sharvamanyu: “Vrishchika as ever is a sharp kid. From all we know of this case she seems to have made a potentially right connection. Intriguing indeed!”
Saying so Sharvamanyu too went quiet even as he tried to wrap his mind around all that was known. Vidrum simply stared at emptily at the menu card even as he turned all the worst-case scenarios in his head.

Just then Sharvamanyu’s fiancee Abhirosha glided in and having hugged him sat beside him.
Sharvamanyu: “Got some work done sweetie?”
Abhirosha: “Yes had a fairly productive morning. But everything OK with you’ll? You seem to have rather this rather perplexed look?”
Sharvamanyu: “Nothing; we were just puzzled over this news item regarding an old classmate being up to some bad mischief.” Saying so he passed Abhirosha his tablet.
Upon glancing through it Abhirosha remarked: “Ah, our daṇḍika-s seem pretty inept to let her go!”
Just then she realized that Vidrum had hardly registered her arrival and noticing the look on his face asked: “Everything OK Vidrum?”
Vidrum: “Not really. But Sharva can fill you in later. Let us get moving with lunch. I would just have something light my appetite seems to have gone.”
Abhirosha looked at both of them and sensing the tension in Vidrum asked: “Why, what’s wrong. Is there something we can help with?”
Vidrum however insisted they get moving with the starters and as they started savoring them he gradually reiterated everything, partly, speaking out aloud and partly to fill in Abhirosha about all that had happened.
Abhirosha: “What you say about Meghana is really sad. But I must tell you this woman Ms Sharif may not be as innocent as she seems.”
Sharvamanyu: “Did you know her?”
Abhirosha: “Not really but one incident comes to mind which I have told you before – the past life of Abu Hilal – all keeping with what we know of the Sharif clan.”
Sharvamanyu: “Yes.”
Vidrum: “What’s that?”
Sharvamanyu: “Vidrum, don’t worry and eat well. We have good day’s worth of climbing ahead of us.”
Vidrum: “But I hope you realize this hits much closer to home for me.”
Sharvamanyu: “I do realize it. But it is water under the bridge now. Meghana has disappeared without telling you – that tells you she is not someone to spend your life with. Moreover, it is best you dissociate yourself from her. If you were in the big police state of the mlecchas you would have been by now subject to some Bhāgurāyaṇa treatment by their over-armed māmū-s. You have done well in training yourself as a physician and have bright career ahead of you. So with help from your friends you can certainly move on in life and fulfill the incomplete matters sooner or later. The gods never give all goods things to one person.”

Abhirosha and Sharvamanyu tried to change the topic repeatedly but they found Vidrum to be quite affected by the events. After lunch they took the train to go the station where Gardabh and Mahish and two others were going to meet them for their climb. While on the train Vidrum again asked them about the incident Abhirosha had alluded to. Seeing his anxiety she decided to tell him the story:

Hearing the story Vidrum felt a bit relieved that it had not direct impact on the situation at hand on the other hand it only reinforced his concerns about Meghana’s entanglement with them. He nervously remarked: “I don’t know why he had to convert to marry her.”
Abhirosha: “Well a marūnmatta might use anything in jihad be it a solid object like a stone or a mere abstraction like romance.”
Vidrum: “I wonder what Abu Hilal might be up to in all this.”
Sharvamanyu: “I was wondering about that too. But not something we can do much about. He is probably enjoying himself at Oxford debating about Islam versus Islamism with the famous atheist professor Dave Cockburn.”
Just then the train was pulling into to station they were headed for. Abhirosha again remarked: “Now let’s drop all of this and get going with our climbing. After all what can inconsequential little people like us do about this complicated geopolitics. As part of my job I so often send notices to the Finance Department of our state about all kinds of accounting chicanery of these mleccha NGOs. They just don’t seem to be bothered.”

◊◊◊◊◊

For Yang He the escape of Tamanna Sharif mattered much more than it did to Vidrum. He was known as the Chinese physician and public health expert who was helping the terrorist state with with a health survey for a prevention program at Gwadar. He was buzzed in early in the morning in his sleep by an emergency message, which upon decrypting read:
“Both the ISI and the MI6 are likely to be upon you if they have already not already crossed your path. You need to abort and get out ASAP to save all other assets.” He quickly took his key data and scampered to access his secret boat that he had kept ready for this escape.

It mattered nearly as much to the engineer Sven Focke who had come from Norway to help the Pakistanis with some high tech installations at Gwadar. Focke had just received an encrypted message from England which read:
“Yang He is the man. He is said to have stuff which the Research Department says would be of great interest to you. It might be in a well-protected safe, which he probably will take with him in case the Hindoo bastards manage to tip him off. They are surprised these Hindoo crooks have managed to pull off something like this. There must be some mastermind in their midst and we need to get to the bottom this. By the way Abu Hilal expects us to do him a good turn by snuffing out He once you get what we need.”

Focke remarked to himself: “These Hindoos seem to have gotten rather smart but I was quite on the right scent it seems. I must strike the blow for against the the deceit of the wily Brahmin and the Baniyan.” Thinking so he loaded his weapons, suitably concealed them, and looked at his device for the position of all the foreign and ISI agents he had bugged using the remarkable technology they had developed at the Research Department of the MI6. Yang He was not locatable any more. Sven Focke now swung into action. He messaged his “mother ship” of the coast of Oman to deploy a couple of drones to sweep the route that he inferred Yang He might be taking to reach the Indian coast. He himself boarded his boat and set off to position himself suitably to perform an interception.

Unsurprisingly, beyond a handful of people this death and drama on the high seas remained unknown to the rest of humanity.

◊◊◊◊◊

It was a somber winter afternoon when Vrishchika heard a knock on her office door and saw her senior colleague Prof. Jay Budhoo enter. She ushered him to a chair in front of her desk.
JB: “Is this time alright for a brief chat?”
Vrishchika: “Sure, Dr. Budhoo”
JB: “You would certainly know Prof. Steven Harrison from Ukṣatīrtha the noted neurologist, who I am extremely delighted to say is doing a sabbatical with me. He is working on improving cognition in the elderly and has been doing some remarkable basic research on the same and more generally on longevity. He would like to meet you one of these days for he wants to discuss something that he believes would be of great mutual interest.”
Vrishchika: “Sure, I would be glad to meet him any day this week in the afternoon. He can drop me a mail before stopping by just to make sure I am not in the middle of an experiment.”
JB: “Thank you.”
Vrishchika: “If you don’t mind Dr. Budhoo, just out of curiosity, how do you happen to know Harrison?”
JB: “Steve is a great indophile and has always been interested in progress of the Indian peoples and diaspora. He used to be on the Oxford Thought Leadership Program and had selected me to do my medical research as a graduate student funded by that scheme. He has always kept in good touch with the labs of his students and the work they have been doing for literally saving the world.”
Vrishchika: “That’s great. Thank you.”
JB: “Vrishchika, you must try to impress him. If Steve is happy with you he could even recommend you for funding from the Guillaume Glympton Foundation, which has just announced that they would be funding interesting medical research.”

Duly Harrison and Vrishchika had set up an appointment for the meeting. On the said day Vrishchika was in her office checking the time wondering why Harrison had not yet showed up. As she did so she thought to herself: “These mleccha-s! I am pretty sure it was the same Harrison who tried to hold up our paper for nearly an year to try to reproduce our work and pass it off as his. I wonder what he wants from me. I am sure this urge to meet me cannot be innocent. Moreover, it is really strange that he is doing a sabbatical with Budhoo, who has hardly anything to offer him…” More than half an hour had passed since the appointed time and he had not yet shown up. Vrishchika called up Jay Budhoo to check what was going on. He spoke to his students and said that they had not seen Harrison the entire day. By that night Budhoo was already contacting the police over the case of the missing Prof. Harrison. The police knowingly told him that they were on the case and he should quietly wait.

A few days later officials from the Intelligence Bureau were at Vrishchika’s office asking her about Harrison. She told them all that had happened. Then they specifically asked her if he had tried to obtain information from her about the stuff she had trained the Research and Analysis agent Dev Buragohain in. She again clarified to them that beyond the e-mail which they had exchanged there was no further communication between them and that he had mysteriously not turned up for the meeting they were to have.

That evening as Vrishchika was riding home she was telling Indrasena about her questioning by the IB officials and wondered aloud what it might have to do with the agent with whom they had interacted, Dev Buragohain, who was trained as a physician. Indrasena: “That interesting dear. I just saw a news today that the IB has arrested a guy called Capt Virendra Chauhan from the Military Medical College in connection to leaking secrets that compromised some of the most important intelligence operations.”
Vrishchika: “We never heard anything ever again from Buragohain. Wonder if they every deployed our stuff?”
Indrasena: “Indeed, your trials were very promising”
Vrishchika: “They were. That’s why I recommended them to deploy it and trained Buragohain. Wonder if he was compromised in any way.”

◊◊◊◊◊

It was an evening of the navarātrī festival. The caturbhaginī, their consorts, and children had gathered at Sharvamanyu and Abhirosha’s house for a feast.
Abhirosha: “Lootaa, this book, a memoir of a retired English scientist and agent is rather interesting. He was a biomedical researcher – I believe you would find it interesting in more than one way.”
Lootika: “As you have known well – I seldom read such literature. But how come you got interested in it?”
Sharvamanyu: “I brought it to her attention: as you know I keep an eye on this intelligence stuff.”
Abhirosha: “Lootaa, this actually might be of considerable interest in clearing many things.”
Lootika: “Why? What’s his name?”
Abhirosha: “Douglas Fieldman”.
Vrishchika: “Ah! He was a close collaborator with Steven Harrison on studies to reverse aging. I believe he had undue ‘interest’ in the work we had done with Somakhya and Lootika on the nucleotides in ADP-ribose system.”
Varoli: “Had run into Fieldman at a conference – he apparently started off as an engineer. Mentioned a nasty bit of his autobiography. He had apparently worked with Maoist terrorists in Chattisgad.”
Abhirosha: “Exactly, that is true. He mentions it in his memoir. That’s why I said it is interesting.”
Indrasena: “Sharva and Abhi could you tell us more. I am just beginning to wonder if it might have something to do with mysterious incidents around Harrison’s unexpected return to England and Chauhan’s arrest.”
Sharvamanyu and Abhirosha: “We believe reading his memoirs we have uncovered a tangled skein the runs through so many things that has puzzled over the years.”
Mitrayu: “Why ‘am I not surprised. I guess at its heart lies the mleccha-marūnmattābaddham that the discerning have the eyes to see. Whatever the case, you seem to have hit a gold mine; we would indeed like to hear the details.”
Jhilleeka: “Perhaps it also ties together the mysterious computer incidents. Remember those Prachetas?”
Sharvamanyu: “Vipra-s and vipra-patnī-s why don’t you all seat yourselves comfortably. We will lay out everything to you all in order. But there are some holes that we believe you all can best fill in for us. Abhi, could you start by reading out that interesting opening paragraph from Fieldman’s account of his passage to India?”

Vrishchika: “That plainly makes an intersection with our sphere. Do you recall this Hilal and Sharif made a dramatic escape to Laṇḍapura years ago.”
Sharvamanyu: “How could we forget it? It also marked the disappearance of Vidrum’s girl Meghana, whose fate also become clear in the memoir. Abhi could you read that part?”
Abhirosha read on: “The major operation could have all come to naught if the young lady Meghana, who was so strongly committed to the cause of social justice in India had not extricated Tamanna at great personal risk… In the subsequent years Meghana worked closely with the GG foundation and my trust on several human rights cases in the Middle East. Most unfortunately, this bright star in the fight for social justice was snuffed out young when she was shot by an Israeli agent while working on the legal defense of the Palestinian activist Mohammad Sumbel.

Vrishchika: “Did you tell Vidrum about that?”
Sharvamanyu: “Why rake up old wounds. I believe he has moved on…”

Prachetas: “But you seem to have skipped something what was the major operation in which he is talking about?”
Abhirosha: “He does not provide the details but says that at one point he was called upon by the British Army chief to ‘use his assets to bring to conclusion a key operation in Pakistan’. It was of such critical importance because had he not pulled it off all English assets in Pakistan might have been ruined once and for all and the very existence of this important client state in the subcontinent jeopardized! He goes on to add that he carried out the operation with the highest level of cooperation from Tamanna and Abu who created a mole in the Indian intelligence head quarters. This allowed him to discover and neutralize the sophisticated Indian agent who had been operating as the Chinese physician Yang He at Gwadar!”

Indrasena: “Vrishchika, that explains what happened to Dev Buragohain!”
Sharvamanyu: “That was indeed a gap which we never understood could you please fill that in.”
Vrishchika: “Since the idea began with Somakhya and then took shape in Lootika’s hands I will let them tell the first part of the story. Please fill everyone in on the mysterious Erk5 UMPylating enzyme.”

Somakhya: “Years ago during my study on the toxins involved inter-organismal conflicts systems I discovered a novel family of nucleotidyltransferases of the Polymerase Beta superfamily, one of which was from an isolate of the bacterium Burkholderia pyrrocinia. I had passed it on to Lootika to identify its substrates. She found that one of its targets in humans was the MAP kinase ERK5 and showed it to be UMPylated by the enzyme. Thereafter the story passes back to your hands Vrishchika, so continue…”
Vrishchika: “The inspiration for it was really tangential. I was at a conference where I saw two researchers. One of Hindu origin and the other a prathamonmatta. The two were well-matched in IQ and technical skills, yet the latter was a winner while the former a loser. While there are many factors that could collectively explain it, I wondered if at least one of those might have a genetic basis. I found support for such a genetic foundation in the polymorphisms of the oxytocin system. I wanted to do some experiments in this regard when I got chatting about it with my agrajā and Somakhya who mentioned I could fiddle with the ERK5 signaling in neurons using their nucleotidyltransferase. It was around that time Somakhya and Indrasena discovered another class of toxins and found that it was delivered into eukaryotic cells using an N-terminal ubiquitin-like domain. I engineered a protein with the Burkholderia nucleotidyltransferase and a ubiquitin-like domain from the toxin of a fungus Mortierella and was soon able to easily get it inside neuronal cells. To cut a long story short I had a means of playing with oxytocin signaling in human neurons and even figured out a way to get it into live animals and manipulate their behavior, especially trust behavior. This was what we eventually developed as a potential mind-altering tool where the UMPylation of ERK5 in the neurons could be used to manipulate the oxytocin system and make people extremely trusting and thus reveal secrets. The military was super-interested in this and we trained Dev Buragohain who had east Asian ancestry to infiltrate the marūnmatta-s and deploy this agent. It was seen to greatly aid in intelligence gathering but the project mysteriously ended.

Somakhya: “By putting together what came out in the public of the Virendra Chauhan case we can infer the following: Abu Hilal used Tamanna Sharif to set up a honey-trap operation to lure Virendra Chauhan at the Central Command Medical College who was liaising with Yang He, i.e., Dev Buragohain. He spilled the beans to them and blew the cover of this operation. They were to make their way to the mleccha island with their intelligence – Hilal got through but our māmū-s captured Sharif. The mleccha-s used Meghana to smuggle Sharif out using money and resources of the famous NGO the GG foundation for which she worked. Fieldman was the one who was in TSP and put an end to Buragohain. He came to know of our whole oxytocin system and knew that only people who could be at the center of this is our gang. So he sent his colleague Harrison to get us and our secrets. Among other things Harrison was personally interested in the nucleotide we had devised and synthesized which binds inactive Nudix domains and to improve aspects of physiology while aging. But for some reason that was abruptly aborted…”

Indrasena: “The reason it got aborted was he was going to pay agents to attack our computers or us or our labs physically. The night before he put that plan to action the government had suddenly demonetized the large currency notes. He was now in trouble because all his notes were useless. So he sent his agent to the bank to convert them to valid denominations and send them over to Chauhan from whom he was to collect it. This was when the IB caught scent of them. He escaped with help from the English embassy, but Chauhan fell into the trap and has since then probably been enjoying Bhāgurāyaṇopacāraḥ. We know this because a retired general who joined Rajiv Jaisval Pratibhraṣṭācār party routinely complains about the gross human rights violation arising from the torture of army physician Chauhan.”

Prachetas: “It did not end with Harrison’s departure. There were many sophisticated break ins into Somakhya and Vrishchika’s machines. Luckily the inner core was not breached due to Jhilleeka’s devices. However, many of the military machines were broken in. When she brought this to my attention I counter-attacked and traced them to Laṇḍapura, while shoring up our defenses.”

Sharvamanyu: “Nor has it ended. In fact we remain deeply infiltrated and might see a grim clash in future where only side would remain standing. A section from the conclusion of Fieldman’s memoir sums it up. Dear Abhi …”

Abhirosha read from Fieldman’s memoir: “Earlier I delighted in the spurts of adrenalin which came from fighting all over the world for justice and equality through the smoking end of the barrel. Thereafter, the thrill that high espionage gave me cannot be compared with anything else. However, as we have learned from comrade Che Guevara’s life and from my own experience ultimate success lies in a different kind of work. In my youth I thought that the great British empire was a symbol of oppression. But the more I thought about it, I learned that it was the opposite. Real oppression is what you saw in the rule of the Nazis, or the south American despots or the brahminical system of caste in India. We in contrast were bringing the rule of law to replace tramplings by elephants, the wonders of science to replace obscurantism, and above all an egalitarian social message that every woman and man should have a equal and fair share of the nations wealth. Indeed, this is appreciated by those who have transcended the narrow and artificial straits of nationalism. My friend, professor Saptagiri Raman from Oxford, said to me that had it not been for Britain’s civilizing influence the Indians would still be insisting that the sun goes round the earth. There might have been some excesses that went with the age during the British rule, but certainly the vast subcontinent of India could not have been civilized without the values we brought to them.

Hence, I believe that it is these values which we need to spread even more than ever before as the Hindu fascists threaten to consolidate their rule over the subcontinent. Not surprisingly, they hate Islam which can be seen as an earlier attempt to bring a rational discourse to mankind along with the attendant concepts of egalitarianism. For these reasons my colleague Harrison and I have developed and supported this program where we have been creating the harbingers for tomorrow’s India. Trained in liberal thought and charged with a zeal for bringing justice to their people they would return to their country and transform it. In this context I must acknowledge Justices Thapar and Karim formerly of the Indian Supreme Court who had been so helpful to me as I was being harassed by the rowdy elements passing off as police in India. They had told me that they had a vision for India as a secular nation that has shed its meaningless shibboleths and approached life and law in rational way. They had an appreciation for my work with Dr. Assolkar in rooting out irrationality and said that this India could only be born if we could create thought-leaders who would spread among the masses ushering an age of reason in the midst.

Some lessons we learned from 3-color totalistic cellular automata

Cellular automata (CA) have attracted people’s attention to different degrees over the past several decades since the early work of pioneers like Ulam and von Neumann. Remarkably von Neumann played with his earliest versions of CA using a graph paper and pencil rather than a computer. Subsequently, the man behind the modern computer and high-level programming languages, Zuse, realized their great importance as a fundamental model for existence. Further notable developments were the game of life of Conway and the systematic study of CA by Wolfram. Our first acquaintance with these entities occurred in our youth first from Conway’s work and a paper of Michael Frame.

This immediately induced us to start pursuing CA. At that time we did not really have the chance to study Conway and Wolfram’s work on CAs in any great detail beyond obtaining the basic facts of how they worked. Hence, we explored them on own with our first computer and consequently obtained some qualitative insights and aesthetic satisfaction both of which seemed remarkable to us. Here we talk mainly of the “3-color totalistic CA” which were among the first we explored. Some of the rules graphically presented here are those which Wolfram has studied in great detail. The account here is simply our personal journey through them and what we found interesting in their behavior.

The definition of the 3-color totalistic CA:
1) They are an one-dimensional array of cells with each cell having a starting state (represented by their numerical value or color). In a three color automaton they have 3 colors. We have chosen triplets like (white, deepskyblue, navy), (white, cyan3, darkblue) or (white, orange, red) for aesthetic reasons (they could be any other 3 colors). Correspondingly, the states of the cells are assigned numerical values $[0,1,2]$.

2) The state of the cells changes from one generation to the next depending on the state of 3 cells in the current generation: their own state and that of their neighbors to their immediate left and right. For a totalistic automaton it is the “total state value” across the 3 cells that matters. Thus, we can have the following $7$ values:
$[2,2,2] = 6; [2,2,1] = 5; [2,1,1] = 4; [1,1,1] = 3; \\[5pt] [1,1,0] = 2; [1,0,0] = 1; [0,0,0] = 0$.
Each of 7 total state values specify a one of three states for the given cell in the next generation. These specifications are termed the rule of the CA. For example, we can have the following rule:
$6 \rightarrow 0; 5 \rightarrow 0; 4 \rightarrow 2; 3 \rightarrow 0; 2 \rightarrow 1; 1 \rightarrow 2; 0 \rightarrow 0$.
Since total state value for the 3 cells is always listed from $6:0$, we can represent the rule by merely providing the vector of cell state values specified by the $7$ total state values in that order, e.g. in the above case: $[0,0,2,0,1,2,0]$ or simply the string $0020120$. Thus, one can see that there are total of $3^7=2187$ possible rules. Wolfram further converts this string to a decimal number that represents the rule by taking the string to be a ternary number (i.e. base $3$).

3) The starting values of the cells are plotted as the first row. These are then updated based on the rule and plotted as the second row. The rule is then applied to this row to get the next and so on. These are successively plotted below the initial row to get a visual representation of the CA.

4) In principle the CAs were seen as operating on an infinite array of cells. However, for practical purposes we plot them on a on cylinder. Thus, the image presented is the cylinder rolled out as sheet with the first and the last column being neighbors on the cylinder.

5) The CAs can either be initiated with a single cell with a non-0 value or with a random sampling of values for the cells in the initiating array. In the first figure all are initiated with single cell with value 1. In the figures used here the array consists of 100 or 101 cells which evolves for 100 generations before being plotted.

Figure 1

Several general kinds of behavior can be observed in the CAs in a rule-dependent fashion as depicted in Figure 1

1) Sterility: An example of this rule $1110020$ which after 4 generations becomes an entirely 0-valued array (Figure 1, top panel 1). In these cases all cells converge to a single state with no further evolution.

2) Simple alternation: Here all cells in the array alternate between two state over successive generations. See below.

3) Simple linear repetitive patterns: An example is rule $2112020$ where from generation 5 onwards the cells converge to a repeating pattern that spans 7 generations. The maximum width of the non-0 cells remains constant in these patterns (Figure 1, top panel 2).

4) Complex linear repetitive patterns: These patterns are similar to the above but attain greater complexity in terms of the structure and number of generations spanned by the repeating unit. Rule $1211020$ is a good example with a long repeating pattern of 78 generations, such that we see only one unit here. Wolfram has shown that this is the longest repeat for these CAs (Figure 1, top panel 3).

5) Expanding repetitive patterns: These patterns are repetitive like the above but do not have a constant width; instead they keep expanding (e.g. rule $2120210$; Figure 1, bottom panel 1).

6) Fractal patterns: These converge to a fractal-like form, typically a version of the Sierpinski’s gasket. Geometrically this can be obtained by forming triangles connecting the midpoints of the three sides of a starting triangle. In the case of the CAs they grow from a minimal unit outwards (e.g. rule $0110220$; Figure 1, bottom panel 2).

7) Complex chaotic patterns: These keep growing in width but never have a repeating pattern. Local patterns sometimes reappear elsewhere but they do not come in the same context as the previous occurrence (e.g. rule $2210120$; Figure 1, bottom panel 3).

Figure 2

Similar looking rules do not mean similar behavior. As an example we can look at the 9 CAs with successive rules from $1102120$ to $1102212$ in ternary notation which correspond to the Wolframian $1041..1049$ in decimal (Figure 2). Mostly, the simpler patterns of convergence tend to dominate. Here 1/9 rules produces a sterile pattern; 3/9 rules converge to simple alternation; 2/9 rules converge to simple linear repetitive patterns. Of these $1102121$ is notable in that it initially produces 32 generations of complex non-repetitive growth and then suddenly collapses to a simple linear pattern. 2/9 rules produce expanding repetitive patterns. 1/9 rules ($1102120$) produces an apparently complex chaotic pattern that keeps growing.

Figure 3

If we instead start with an initial array of cells each randomly showing one of the three values we get generally similar behavior but with some interesting features that are not seen in the single cell initiations (Figure 3):

1) A striking example is $0121020$ which shows a sterile pattern in a single cell initiation that goes extinct after 4 generations. However, when it is initiated with a random array of the three values it develops a complex pattern with several branches as well as runs of linear repetitive patterns. Thus, a singly sterile rule when given the opportunity to interact with other initiations as a result of the random initiation develops into a complex system. We term this symbiotic development.

2) Rule $0022010$ shows some initial complex patterns but all of these are quickly channelized into simple linear repetitive patterns. Likewise, rules $0111020$ and $1011020$ initially evolve complex branching which then goes extinct and leaving just one or two dominant complex linear repetitive strands. In rule $1012020$ the initial complexity is quickly channelized into both simple and complex linear repetitive patterns.

3) Rule $1022010$ produces expanding repetitive patterns which are interspersed with runs of simple linear patterns.

4) Rules like $0022210$, $1001210$ and $1020210$ quickly produce chaotic structures but these have certain stereotypic sub-structures.

5) Rules like $0020120$ and $0120120$ rapidly converge to even more chaotic structures than the above, spawning what look closer to completely random patterns.

Obvious analogical insights gained from these CA
CA provided the analogy for world view first developed by the computer science pioneer Zuse in his “Rechnender Raum” that the universe based on physics is actually computed on a system of CA. For Zuse this was more than just an analogy with the structure of space itself perhaps providing the canvas on which these CA played out with the “image” forming as a result being all of existence. This intuition itself goes back to Thomas Huxley the early Darwinist. While not having classic CA, he had such an general analogy in mind for the working of nature:
The chess-board is the world; the pieces are the phenomena of the universe; the rules of the game are what we call the laws of Nature.

For us, primarily being a biological naturalist, the first encounter with CA provided an analogy for how the processes of life were executed. The fundamental unit of biological systems being the cell made this a natural analogy for development. Indeed, on first seeing CA, the most direct connection to certain biological forms appeared before our eyes. For example, the patterns formed on the shells of Conus snails can be quite explained by some form of totalistic CA as those illustrated here (e.g. rule $0022210$ and $1020210$ can be favorably compared with actual snail shell patterns illustrated by Gong et al). The totalistic rules can be explained in a biological context as the concentration of a diffusible (like one of the many signaling molecules that are secreted by cells) or cell-surface morphogenetic signal. This concentration is the determinant of cell fate and it can literally play out on a growing sheet of cells as seen here.

Gong et al  E234–E241, doi: 10.1073/pnas.1119859109

However, CA also provided us with a more abstract biological insight. Here, the CA are not to be equated with the actual biological cells but used as analogs for other one-dimensional systems like transcription factors binding to a linear DNA molecule or RNA-binding proteins binding to a RNA molecule. Alternatively, they might be seen as covalent modifications with “epigenetic” consequences on a linear string like nucleic acids or the histone octamers on DNA. This kind of state differences on the linear biological entity can determine the state of a region in the next iteration. Continuing with this picture, at the highest level of abstraction we can visualize CA representing serial states of an auto-regulatory biological network, whose current state of inputs determine the next state of the network, and those in turn the next state and so on. This provided a powerful means of analogically visualizing the emergence of various behaviors ranging from simple oscillations to chaotic emergence of states in biological systems. The advantage of visualizing them as emerging from underlying CA is that we now have a discrete simple mechanism, which can be easily equated with molecular interactions, behind various state manifestations in biology.

Less obvious analogical insights gained from these CA
Finally, such CA provided us with intuition in an area which had looked quite baffling and not amenable to mechanistic penetration, i.e. a macroscopic view of history of civilizations. In history there are several tendencies that have been qualitatively described: 1) “History repeats itself” (rule $2112020$); 2) “Long cycles” (rule $1211020$); 3) “Rise and fall of a civilization” (rule $1110020$); 4) “Triumph of groups over units” (rule $1110020$); 5) “Great diversity at the base converging to a standard type” (rule $1011020$); 6) “A flowering followed by loss of creativity” (rule $1102121$); 7) “Parallelism” (rule $1012020$); 8) “A many branched tree” (rules $1102120$); 9)”Waxing and waning periods” (rule $1022010$) 10) “Chaotic development” (rule $0120120$); 11) “navo-navaś camatkāraḥ (rule $1020210$).”

One could ask: “Alright, one can draw general parallels between these patterns in history and the evolution of CA, but is there anything more to it than just some vague similitude? Is it even useful?” We believe the answer is in the affirmative, at least in the analogical realm, even if not in the modelling sense. One could conceptualize civilization as having certain minimal units (cells) which occupy a certain space over which they can expand (the array on which the CA play out). The civilizational units themselves can differentiate to have few distinct intrinsic states while interacting with themselves in the near vicinity and their environment (the null cells). The total of all these interactions can be captured relatively simply as a single value, but the fine gradations of this value matter in terms of the development of the units in the subsequent generation.

Some might object to this as an overly simplistic reduction but studies in other areas suggest that this is not necessarily a wrong thing. At a fundamental level the statistical operation of obtaining the mean of a data set can be seen as simplifying or reducing the complexity of the data. Yet in many cases, the things we can do with, or understand from a mean are much more than with the mass of data. Indeed, the mean or some such statistical measure of central tendency can be sufficient to infer the *general* behavior of the mass more easily than having just the mass. As a practical example, while intelligence emerges from a complex interaction between the products of many genes and the DNA containing the genes themselves, it can still be simply captured as a number, which is useful for several purposes. Likewise, the totalistic procedure can be seen as sufficient to usefully specify the development of the system but its fine gradations matter.

Thus, melding of intrinsic factors like genetics, social interaction terms, and the environment of a civilizational unit into simple measure can be seen as being sufficient to sustain a wide range of evolutionary patterns that characterize history. A particularly notable case for us was what is seen in rule $0121020$ which when initiated with a single cell with value 1 dies out quickly but when initiated with multiple cells random bearing an equal number of 0,1 and 2 states shows complex development. This is a good analogy for how single civilizational units might die if they are isolated on their own but if they have many neighboring copies they can synergistically develop complex behavior. Indeed rule $0121020$ can combine many different types of behavior described above when initiated from different sets of multiple random cells (Figure 4).

Figure 4

Syllable, number and rules in the ideal realm

This note is neither meant to be complete exposition of this matter nor a complete view of all what we have realized in this regard. Nor can it be completely understood by those who are not insiders of the tradition.

Syllable in the primal realm
In below verse, from what is perhaps the most famous sūkta of the Ṛgveda, the Aṅgira Dirghatamas Auchathya, the founder of the Gotama clan says:

ṛco akṣare parame vyoman
yasmin devā adhi viśve niṣeduḥ |
yas tan na veda kim ṛcā kariṣyati
ya it tad vidus ta ime sam āsate || RV 1.164.39

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

He continues:
gaurīr mimāya salilāni takṣaty
sahasrākṣarā parame vyoman || RV 1.164.41

The female gaur [Bos gaurus] having measured out fashions the waters,
she is one footed, two-footed, four-footed,
becoming eight-footed and nine-footed,
she is a thousand syllables in the highest realm.

In the Atharvāṅgirasa-śruti Kutsa āṅgirasa says:
ekacakraṃ vartata ekanemi
sahasrākṣaraṃ pra puro ni paścā |
ardhena viśvaṃ bhuvanaṃ jajāna

One-wheeled and one-rimmed it spins,
[with a] thousand syllables north, east, south and west,
With half it gave rise to the all the universe,
That which is the other half, what has become of it?

The Atharvaveda has several more mysteries pertain to this primal realm of syllables but we shall only touch upon some here. In the long Rohita recitation we hear:

sahasrākṣarā bhuvanasya paṅktis
tasyāḥ samudrā adhi vi kṣaranti || AV-vulgate 13.1.42

She is one footed, two-footed, four-footed,
becoming eight-footed and nine-footed,
she of thousand syllables [emits] the series of worlds;
from her the oceans flow forth.

The above ṛc-s have much in them that can be expounded but what we wish to stress here is the repeated allusion to the syllables, “akṣara-s” which are:
1) constituents of the spinning wheel from which the universe originated.
2) residents of the “parame vyoman” which can be understood not just as the highest world in some physical sense but also as the highest in terms of “ideals” and also in terms of time as the primordial world from which all has sprung.
3) constituents or the very form of highest realm that is the seat or the dwelling of the gods.
4) associated with counting or numbers in a sequence.

Indeed, this understanding of the syllables or the primal syllable is the secret behind the Vaidika ritualist receiving the teaching of the 4th foot “paro rajase ‘sāvadoṃ” upon having completed his basic initiation into the sāvitrī.

Now moving many centuries down in time we come across the rudra-yāmala tantra, which teaches:
a-mūlā tat kramā jñeyā kṣāntā sṛṣṭir udāhṛtā ।
sarveṣāṃ caiva mantrāṇāṃ vidyānāṃ ca yaśasvini ।
iyaṃ yoniḥ samākhyātā sarvatantreṣu sarvadā ॥
The process of emission [of existence], whose root is ‘a’ [and] in order known to have ‘kṣa’ as the end has been expounded. O glorious one, this is declared in all tantra-s as the source of everything, of mantra-s and vidyā-s, the giver of all.

This persistent idea of the syllables being the root of the universe, even in the physical sense, and the very manifestation of the gods of the Veda or the tantra might seem strange to one who has not grasped its secret. The way it might be understood is by the conception of the Sanskrit language by its tradition of grammarians who were intimately linked to the apprehension and transmission of Vedic knowledge. Pāṇini and his brilliant successors like Kātyāyana and Patañjali see the starting premise as the Sanskrit syllabary which might be arranged in different ways the: the regular periodic-table like structure or as the Māheśvara-sūtrāṇi. On this their rules operate to emit the whole language. Now we hold that Pāṇini did not invent this out of thin air. He was merely following upon a very old tradition that went back to more than 2000 years before him that conceived this process of rules operating on syllables to generate intricate form as manifest by the Vedic incantations. That was the art of the kavi or the vipra by which he crafted his mantra-s – it relates to the old Indo-European tradition of Chandas with a fixed syllable-count. Now these kavi-s extended this logic to the universe. There in the “parame vyoman” there exist these “thousand syllables” from which the universe was emitted by a series of rules (“vrata-s”) even as the Sanskrit language is emitted from its syllabary. This idea persisted with the tāntrika mantra theory which succeeded the Vaidika prototype.

This vision of generating complex structure from a simple syllabary, reflected in the work of Pāṇini, was increasingly important in the evolution of Sanskrit, where rules allowed for unambiguous (or purposely ambiguous) sense while still allowing for enormous structural complexity. This came at what might be called a “marker cost” i.e. marking the elements precisely: nominals precisely inflect – right in the first sūkta of the RV we see in succession inflections of the name of the great god (agnim, agniḥ, agninā, agne) as though to announce the importance of unambiguous markers of sense. Likewise, the verb assumes an enormous range of forms to convey very precise temporal and modal textures. For this very reason, the traditional grammarians split hairs and go into great depths with samāsa-vigraha or dissection of compounds, precisely because the markers are mostly lost in samāsa. For instance, you are left with the puzzle of whether aśvaśiras is the horse’s head or he who has a head like a horse or whether a lokanatha is the lord of the world or one who is lorded over by the whole world. While it makes Sanskrit a difficult language to master and might generate undue pride upon its operational mastery (caricatured by the advaitin-s as the ineffectuality of Pāṇini’s ḍukṛñ), it has had a profound effect on Hindu thought, the modern implications of which have not entirely been fathomed. Curiously, one facet of it which is central to our current discursion is the fact that in Hindu tradition the number is often subservient to the syllable. There are many ways in which numbers might be linguistically represented: the syllabic contraption of the great āryabhaṭa or the #katapayādi system or the system of using markers like: 1= candramas/Indra; 2=pakṣau; 3=vaḥnayaḥ; 4=vedāḥ; 5=bāṇāḥ; 6=ṛtavaḥ/skandaḥ; 7=munyaḥ/abdhayaḥ; 8=nāgāḥ/vasavaḥ; 9=grahāḥ; 10=dik.

“The extraordinary effectiveness of mathematics”
In our times we see a great thinker of the age Roger Penrose talk of three worlds: 1) The “real world” made of particles and energy including forms of matter-energy we do not yet understand. 2) The world of consciousness or first person experience, what in the western philosophical language would be called qualia. 3) The “Platonic world” or ideal mathematical entities both geometric shapes and numbers and the relationships which they contain (what I think mathematicians call theorems). Penrose points out that a great mystery is the fact these objects of the 3rd or the “Platonic” world are the ones that are used to govern the making and the working of the real world. Penrose believes that ultimately there is only one world but it appears three-fold because we do not understand its mysterious unity. That apart, the mathematical world of Penrose is thus comparable to the world of akṣara-s and the vrata-s which operate on them like Pāṇini’s rules in our tradition, which is placed at the root of the world.

Now the existence of such a pure mathematical world is rather easily perceived by anyone who has played with some mathematics, irrespective of whether he really understands its rigorous formulations or not. Its actual manifestation is what Eugene Wigner called the “extraordinary effectiveness of mathematics” or more recently extensively discussed by astronomer Mario Livio as “Is god (in singular emphasizing the certainty of Abrahamistic credo) a mathematician?” (In a sense following a similar Greek statement by Karl Gauss). One can fill a whole volume and more with examples of this mystery of mathematics and we have narrated some of our personal journeys through such on these pages; yet, we shall offer a few here for it is always worth savoring:
1) A trivial case is the Hindu predilection for huge numbers. At the time the Hindus named them people had little reason to count that much. But today we know the various measures at the extremities of the universe fall in the orders magnitude of many of those big numbers of our ancestors. Thus, long after the fact of their discovery in the “mathematical world” we see them as very real entities in the real world. For instance, Avogadro’s number would be approximately ṣaṣṭhi-vṛndāni.

2) This one is more personal: may be a couple of years after we had learned to construct a hyperbola we learned that this curve is describes how enzyme reaction rate changes with respect to substrate concentration. It was the first time we had the personal experience of the mystery of how a curve discovered in a purely mathematical realm in relation to the Delian problem by the yavana-s appeared in the real world of biochemistry.

3) Another personal example: around the 14th year of our life we constructed a curve known as the witch (of Agnesi), which is rather trivial:

* Draw the generating circle which is tangent to the x-axis at origin O and radius $a$.
* Draw the diameter along the y-axis $\overline{OA}$ and a tangent to it at point A.
* Let B be a point moving on the generating circle. Draw $\overleftrightarrow{OB}$ which cuts the above tangent at point C.
* Draw a line perpendicular $\overleftrightarrow{AC}$ at point C and a line parallel to $\overleftrightarrow{AC}$ through point B. These two lines intersect at point D.
* The locus of D as a B moves on the generating circle is the witch which has the equation: $y=\dfrac{8a^3}{4a^2+x^2}$.

Looking at the witch in those days of our youth we thought that it looked like a good function for a statistical distribution and wondered if anything in nature might be thus distributed. Only several years later we learned that the great mathematician Augustin-Louis Cauchy or the statistician Poisson had discovered a probability distribution of this shape which has the interesting feature of being “fat-tailed”. Closer to our times it emerged mysteriously in the real world of physics as the shape of broadening of certain spectral lines with molecular collisions. When Pierre de Fermat discovered this curve in the 1600s in the purely mathematical world he certainly could not have had an inkling that it would appear in the real world of physics centuries later, just like the conics of the yavana-s. Thus, over and over again this objects discovered in that “Platonic” realm were seen to govern the working of the real world.

This triumph of mathematics in the structure of the laws of physics gives them the feel of being true because mathematical truths are quite unlike scientific models – while much of the science of ancients has been quite superseded by new science their mathematics still stands firms and it was their mathematical teachings that allowed all the new science to emerge. The planetary models of the ancients might have died but their bhūjā-koṭi-karṇa-nyāya remains as true today as when, to their wonder, they first discovered it. This mathematical foundation perhaps gives a sense of inviolability to the laws or in the least it gives them the sense of being enforced (see more on that below). Indeed, the Hindus of yore did record this sense of inviolability in the laws in connection of the gods seated in that same “parame vyoman”. For instance the Arcanānas Ātreya states:

The two upholders (Mitrā-varuṇā) of the natural law (ṛta), you two of inviolable laws (dharman) stand in your chariot in the primal realm.

An ideal realm with a syllabary?
As noted above this mysterious intrusion of mathematics from its ideal realm into the real world is rather palpable even for someone with relatively limited knowledge but can the same be said for world of syllables postulated by the Hindus? At the face of it, to most it is less apparent – indeed, to some it even seems to bring up images of the well-known mleccha indological trope of “Hindus as idiots”.

When we look at physics we find the mathematics directly present in the structure of its laws – it is very apparent, even if entirely mysterious. Now, when we look at biology we find that same geometry acting as an enforcer that channels all variety by eliminating what falls outside the allowed geometries. This is what is termed natural selection. This is the foundation of the commonly observed phenomenon of convergent evolution – over and over again synapsid carnivores evolved the same saber-toothed morphologies, archosaur carnivores evolved generally similar skulls with serrated teeth (ziphodont morphology), tetrapods returning to the water assumed fish-like morphs like those of their ancestors. One part of this channeling is from the geometric structure resident in the underlying physics – the shapes that are best suited to fly in the sky or swim within the waters. But a part of it stems from deeper phenomenon that becomes apparent only when we descend down to the molecular level. At the microscopic level the three dimensional geometries of the molecules of life are ultimately encoded in one dimensional strings. The information in these strings is read in many ways, the operationally most important aspect of which is the reading of genetic code. That world is where the encodings occurs as letters quite literally akṣara-s and these can specify many different types of words and phrases by a slew of different rules that run the system from which emerges the 3D geometry. These “linguistic” entities are not just the genetic code but also seen in architectures of domains of proteins. Our studies over decades have shown that these have distinct syntaxes, with these syntaxes being the enforcers of what is seen as natural selection. Bad grammar is relentlessly purged like king Bhoja dismissing the bad grammar of his rivals. At the same time “creativity” of domain architectures within the allowed grammar, including the use of rare domains that conform to rules like a Sanskrit versifier’s use of unusual words, is rampant. Different cellular molecular ecologies have allowed for different styles – great creativity in the bacteria, robust simplicity in the archaea, and unexpected reuse of phrases and words in new contexts like a prolix hack in eukarya. Thus, the biological world offers us a glimpse of an ideal realm which contains as the ideals a set of akṣara-s and rules which operate on them – it is indeed quite a menagerie of rules, like the aṣṭādhyāyī.

Are these “akṣara-s” in addition to or in place of the mathematics present in the ideal realm? We have an intuition that it is more fundamental than the mathematics and the mathematics emerges as a limb of it. However, being of limited intelligence and knowledge, we by ourselves currently have no way of establishing this to be the case. However, it appears to us that the view of one of the eminent scientist-mathematicians of our age Stephen Wolfram bears a relationship to such an idea. While it is not clear if he postulates a ideal realm with computer programs, he does propose the existence of simple computational mechanisms comprised of simple rules acting on a limited set of characters as a powerful alternative to purely mathematical mechanisms in generating enormous complexity.

Measuring the real realm with mumbers
While we have spoken above of the privileging of the syllable over the mathematical abstraction of number and geometry in the primal realm in Hindu thought, this should not be construed as an absence of importance for these. Indeed, the main constituents of ordinary modern mathematics can be seen as numbers and geometry, which respectively stand on that bed-rock of mathematical tradition provided by the two great branches of the Indo-European world, the ārya-s and yavana-s. Any discerning student realizes that the ordinary secular mathematics of today is a successor of an ancient tradition of the Indo-European world, which emerged in the context of ritual constructions going back to at least the ancestral Greco-Aryan period. Such a role for mathematics in the religious process might be seen in many other civilizations – with close parallels in Egypt or much later in time in the temple geometry of Nippon. Ultimately all of this might have a direct connection going back to our shared ancestry with other apes like the chimpanzee. The piling of bricks to construct the altar (literally citi or piled in Aryan parlance) may be compared to recently reported ritual of piling of stones in tree hollows by the troglodyte – a operation with allows for counting in the least.

Geometry of the ritual altars was a key feature that can be traced back to the ancestral Greco-Aryan tradition. The cubical altar, which is at the center of the famous yavana Delian problem, can be seen as the cognate of the near cubical śāmitra altar of the animal sacrifice of the ārya-s. The Delian problem itself, which involves doubling of the cubical altar, might be compared to its planar equivalent found in the Yajurvaidika tradition of the eka-śatavidhā ritual where a square altar of Agni is increased from a single unit to 101 units through successive increments without ever changing the square shape. It was this ancestral tradition that divergently evolved to give rise to the Pythagorean and then Platonic world view among the yavana-s which privileged Euclidean geometry and the discovery of conics and other plane curves. In the midst of the ārya-s it evolved in a more numerical and algebraic form which privileged measurement as the primary feature. This feature emerges early in the tradition of our ancestors: one of the primeval preserved memories of our clan is that of our hoary ancestor Cyavāna Bhārgava measuring out the ritual altar with a depth-measuring device: “cyavānaḥ sūdair amimīta vedim |”. Right in the Ṛgveda we hear the great astronomer ṛṣi Atri Bhauma, who saved our ancestor Ṛcīka Aurva, say this as he praises the great Asura:

imām ū ṣv āsurasya śrutasya
mahīm māyāṃ varuṇasya pra vocam ।
māneneva tasthivāṃ antarikṣe
vi yo mame pṛthivīṃ sūryeṇa ॥ (RV 5.085.05)

I proclaim this great māyā
of the famed Asura Varuṇa,
who, standing in the atmosphere as if with a ruler,
measured out the earth with the sun.

Thus, the physical action of that great Asura Varuṇa who upholds the natural laws (ṛta) is carried out by measurement.

The Aṅgira sage Bharadvāja Bārhaspatya elaborates on this theme in that sūkta which can only be apprehended by those who see the rahasya-s of the heavenly Agni Vaiśvānara and not one who mistakes the ritual fire in front of him to be just a fire.

vaiśvānarasya vimitāni cakṣasā
sānūni divo amṛtasya ketunā ।
tasyed u viśvā bhuvanādhi mūrdhani
vayā iva ruruhuḥ sapta visruhaḥ ॥ RV 6.7.6

By the eye of Vaiśvānara heaven’s heights have been measured out,
by the ray of the immortal one.
Indeed on his head are all the worlds;
like branches his seven tongues have grown.

vi yo rajāṃsy amimīta sukratur
vaiśvānaro vi divo rocanā kaviḥ ।
pari yo viśvā bhuvanāni paprathe
‘dabdho gopā amṛtasya rakṣitā ॥ RV 6.7.7

The skillful one, who measured out the atmospheric realms,
is the sage Vaiśvānara who [measured] out the starry heaven
who spread around all the worlds
the irresistible guardian, the protector of immortality.

Notably, that Gauri who was described in the RV and AV (see above) as the embodiment of the syllables in the primal world also needs to be measured out and this can be done using a god as the measuring rod. This is described Kurusuti Kāṇva:

navasraktim ṛtaspṛśam ।
indrāt pari tanvam mame ॥ RV 8.76.12

The syllabary, with eight-feet and nine vertices
is embedded in the natural law, (literally in contact with the natural law)
I have measured out its body by means of Indra.

The same Gauri is also embodied in ritual as a cow who represents the measure of heaven. In Taittirīya saṃhitā 7.1 we have:

sahásra-sammitaḥ suvargó lokáḥ|…

The celestial world is measured in thousands…

While the holy cow stands in front of him the ritualist offers with this incantation:

ubhā́ jigyathur ná párā jayethe
ná párā jigye kataráś canáinoḥ |
índraś ca viṣṇo yád ápaspṛdhethāṃ
tredhā́ sahásraṃ ví tád airayethām íti ||

You two have conquered, you two are not conquered;
Neither of the two of them have been defeated;
Indra and Viṣṇu when you two contested,
you had divided the thousand into three. Thus, [he recites].

tredhā-vibhaktáṃ vái trirātré sahásram | sāhasrīm eváināṃ karoti | sahásrasyaiváinām mātrāṃ karoti |

Thousand is indeed divided into three parts at the three-night [ritual]; he makes her [the cow] a [symbol] of thousand. He thus makes her the measure of a thousand.

rūpāṇi juhoti | rūpáir eváināṁ sám ardhayati | tásyā upotthāya kárṇam ā́ japet |

He makes oblations to her forms. He thus furnishes her with her forms. Rising up he mutters in her ear:

íḍe ránté ‘dite sárasvati príye préyasi máhi víśruty etā́ni te aghniye nāmāni |

“Iḍe, Ranti, Aditi, Sarasvati, Priyā, Preyasi, Mahi, Viśruti”, these, O unassailable one, are your names (Thus, the ritual cow is identified with the goddesses among others embodying the syllabary of the primal word).

sukṛtam mā devéṣu brūtād íti | devébhya eváinam ā́ vedayati | ánv enaṃ devā́ budhyante ||

Declare me as a doer of good among the gods. She indeed lets [this] to be known to the gods. The gods take note of this.

The world of the heavens is much vaster than this one hence it is symbolic represented as measuring in a bigger unit, the thousand. For the yajamāna to let the gods know that he is a doer of good, by which he can conquer that celestial realm, he needs a measuring unit of the thousand. For that he makes the ritual cow a representative of the goddesses who embody that thousand. Thus, for the ārya in ritual other than geometry the number played an important role. Indeed, he encompassed the base number of all gods of the śruti numbering 33 (12 Āditya-s, 11 Rudra-s, 8 Vasu-s and 2 Aśvins) by measuring out the altar in the form of successive squares formed of square bricks from 1 to 289. This embodies the relationship:

$\displaystyle \sum_{n=0}^{16} (2n+1) = (n+1)^2= 289$

Thus the above sequence goes from $1,3,5..33$ (spelled out the camaka praśna of the Yajurveda) while the sum of sequence goes from $1^2..17^2$. Thus, the first square is one brick. Adding 3 bricks the next in the sequence on its 3 sides give you $2^2$. Adding 5 bricks to sides of this new square gives you $3^2$ so on till you get the square of 17. Thus, geometry for the ārya was closely linked with number, and measurement with a cord and ruler. It was the philosophical consequence of this that marked a subtle point of departure of the Hindus from the Platonic realm of the yavana-s: one hears of Plato disapproving of the measured constructions – feature that dominated yavana geometry thereafter. Thus, while geometry and other Pythagorean mathematical traditions moved into the realm of the ideals as absolute-measure-free entities among the yavana-s, they were firmly as part of real world in ārya-s emulating the measurements of the universe performed by the great gods. Based on what the Kāṇva says even the primal syllabary has to be measured out for the construction of the real world.

A different way of looking at numbers
We first encountered a new class of systems, the iterative systems, with the 3/2 floor map. This we originally explored, to our great wonder, in the old way with a hand-held calculator and graph paper. The 3/2 floor map works thus:
$x_{n+1}=\dfrac{3}{2}x_n-\left \lfloor \dfrac{3}{2}x_n \right \rfloor$
where the starting value $0

The 3/2 floor map for two close starting values for 300 iterations

The remarkable feature about it is how we get chaotic behavior from this very simple system. We can see spiking and bursts of spikes as has been observed with the membrane potential of neurons without any differential equations but just a simple iterative system. We realized that the while the behavior is chaotic it is not entirely random in a statistical sense. For different starting $x_0$ values we get a generally similar frequency distribution of the values visited by $x_n$ over $n$ iterations though there might be very subtle differences for different starting values (shown in the figure for 50,000 iterations for six different starting values). Over all the values closest to 0 are 3 times more frequent than those closest to 1. Moreover, there are two clear steps in the distribution at 0.5 and 0.75 which are line with how the 3/2 floor map operates.

The distribution of values of $x_n$ visited by the 3/2 floor map for different starting values

Strikingly, even though the bulk distribution is generally similar for all starting values the actual behavior of $x_n$ is very sensitive to the starting value. For example in the figure we illustrate the evolution of 0.5 versus 0.50001 over 300 iterations. One can notice that beyond the 20th iteration the two sequences completely diverge that they have nothing in common despite similar bulk statistics.

This opened our eyes to the fact that numbers can be looked at in way different from how they behave in the system of classical calculus. In the method of calculus, the key aspect is that numbers close to each other behave similarly so that we can smoothly transition from one to another in a continuous sequence of ever close numbers. In iterative systems even a small change in the number can result in a very different end result – something which goes against the classic operation of calculus. Moreover, whether such behavior is produced or not depends on nature of iteration rule or the “map”.

The importance of another such kind of behavior was further driven home to us by the floor-difference sequence. This sequence is generated thus:

$x_n=\left \lfloor (n+1)h \right \rfloor -\left \lfloor nh \right \rfloor$
where $n=1,2,3...\infty$ and $h$ is a constant fractional number like say $\pi$ or $\sqrt{2}$.

The floor-difference sequence as step graph for various numbers

The floor-difference sequence which results from the above rule can be represented as shown in the figure – as a step graph – a sequence of up and down values. This set of up and down values can hence be also represented as a binary string. When $h$ is a rational fraction like 3/2 then we get a regularly repetitive graph which translates into a regular binary string like $10101...$. Now if $h$ is instead an irrational number then we do not get a repetitive graph. Instead, there are more complex patterns which have their own structure. For example, Stephen Wolfram, the pioneer in the study of such systems, has shown that if $h$ is a root of a quadratic equation then the pattern of the sequence is obtained by an automaton with simple substitution rules. Thus, he has shown the following substitution rules:

$\sqrt{2}:\; 0 \rightarrow 0,1; \; 1 \rightarrow 0,1,0\\ \sqrt{3}:\; 0 \rightarrow 1,1,0; \; 1 \rightarrow 1,1,0,1\\ \phi: 0 \rightarrow 1; \; 1 \rightarrow 1,0$
These can be easily verified with the figure.

However, for other irrational numbers the rulers are more complex or subtle. The case of $h= e$ or $h= \pi$ is rather notable. For $e$ we see that the step graph looks quasi-regular: it can represented as repeats of the binary string $1101110$. This goes fine till we reach position 32 at which place a we get an unexpected string $11101110$. Then it resumes again as usual with the repeat of the earlier string till position 71 where we again get $11101110$. From this we see that first deviation occurs at 32 and second at 39 from the point of the first deviation. From then on this pattern repeats with successive insertions of the string $11101110$ occurring alternately at 32 and 39 positions from the previous one. Thus, we can predict and see the string $11101110$ at: 32, 71, 103, 142, 174, 213, 245, 284, 316, 355, 387, 426, 458, 497=458+39. But at that point instead of seeing the next occurrence of the string at 529 (497+32) we instead see it at 536 (497+39). Thereafter, from 536 the sequence of the points of insertion of the string are as usual in the 32, 39 alternate pattern: 536, 568, 607, 639, 678, 710, 749, 781, 820, 852, 891, 923, 962, 994, 1033=994+39. But then instead of getting 1065 (1033+32) next we get 1072=1033+39. Thereafter, from 1072 the sequence of appearance of the string continues as usual with the alternating 32, 39 as: 1072, 1104, 1143, 1175 …

To understand the secret behind this interesting pattern we have to turn to the continued fraction representation of $e$, which was developed by the great Leonhard Euler. By working out the fractions to successive terms we get increasingly better approximations of $e$, which are listed below starting from the 0th :
$(0) \; 2 \; (1) \; 3\; (2) \;\dfrac{8}{3}\; (3) \;\dfrac{11}{4} (4) \;\dfrac{19}{7}\; (5) \; \dfrac{87}{32}\; (6) \;\dfrac{106}{39}\; (7) \;\dfrac{193}{71}\\[10pt] (8) \;\dfrac{1264}{465}\; (9) \;\dfrac{1457}{536}\; (10) \;\dfrac{2721}{1001}\; (11)\; \;\dfrac{23225}{8544}$

Now the floor-difference sequence for the first fractional approximation $\dfrac{8}{3}$ produces a repeat of the string $110$; the next approximation $\dfrac{11}{4}$ yields the repeats of string $1110$; the next approximation $\dfrac{19}{7}$ yields the string $1101110$. Thus, the first three approximations set up the basic pattern which forms the floor-difference sequence of $e$. We then note that the appearance of the deviant pattern $11101110$, corresponding to positions alternately separated by 32 and 39 (71=32+39), is related to the denominators of the next three approximations. Strikingly, the further points where we observed break of the preexisting pattern occur at or upon crossing of 536 and 1001 which define the denominators of the 9th and 10th cycles of approximation.

Now when we look at the step graph for the floor-difference sequence of $\pi$ we see that it shows a strikingly regular pattern which can represented as repeats of the string $0000001$. We see 15 such repeats till we reach the position 106 where we see a deviant string $00000001$. Then it resumes with the usual repeat until position 219 where we again see the deviant string as above. This deviant string then reappears regularly at 332, 445, 558, 671, 784, 897, 1010, 1123… i.e. every 113th position. Now the secret of this pattern becomes clear when we look at the continued fraction derived approximations of $\pi$ starting from the 0th cycle:

$(0) \; 3 \; (1) \;\dfrac{22}{7}\;(2) \;\dfrac{333}{106}\;(3) \; \dfrac{355}{113}\\[10pt] (4) \; \dfrac{103993}{33102}\; (5) \; \dfrac{104348}{33215}$

The 0th approximation 3 is the ancient one which is already alluded to as *approximation* of $\pi$ even the RV. The 1st approximation $\dfrac{22}{7}$ is what Archimedes arrived at. Its floor-difference step graph corresponds to a sequence starting with the string $000001$ followed by never-ending repeats of $0000001$. This later string sets up the basic pattern of the $\pi$ sequence. Then we see that as with $e$ the denominator of the 2nd approximation yields the point where the first deviant string appears (106). Then the next copy of that deviant string appears 113 positions away from that point, 113 being the denominator of the next approximation. The next irregularity is would appear in the vicinity of the denominator of the next approximation, which is way beyond the number of iterations to which we have calculated the sequence. That explains why $\dfrac{355}{113}$ serves as great approximation for $\pi$ and was so remarkably captured by Srinivasa Ramanujan in his first construction to approximately square the circle.

Thus, here we see that the application of a procedure (the floor-difference operator) converts an irrational number with uniformly distributed set of digits after the decimal point into an orderly structure – a structure which can be generated by relatively simple rules similar to those of Pāṇini operating on a syllabic string.

As the final example in this section we shall consider the remarkable curlicue curves which were first discovered and described by the mathematician Mendes France. Different types of these curves may be generated using two related algorithms and in addition to a deep aesthetic value have rich implications for various natural systems including optics and the folding of polymers. However, for this discussion we shall restrict ourselves to first of these algorithms which generates these curves thus:
1) Let the seed $s$ be a fractional or irrational number like $\pi, \sqrt(2), \phi$.
2) Start with an initial point $x_0$ say $(0,0)$.
3) Then for n iterations do the following:
$dt_{n+1}=dt_n+2\pi \times s\\ t_{n+1}=t_n+dt_{n+1}\\ x_{n+1}=x_n+\cos(t_{n+1})\\ y_{n+1}=y_n+\sin(t_{n+1})$
4) Join successive points $(x_n,y_n)$ to get the desired curlicue. The figure shows curlicues generated using various seeds run for 2000 iterations. Top row from left: 1) $\pi$; 2) $\sqrt{10}$; 3) $\dfrac{355}{113}$; 4) $\phi$ Bottom row from left: 5) $e$; 6) $\log(2)$; 7) $\pi \times \phi$.

Curlicue curves with various numbers

First, one can see several aesthetically pleasing interesting patterns: $\phi$ curve produces fir-tree like patterns. The $\log(2)$ curve produces several ‘S’ shaped patterns. The $\pi \times \phi$ produces a basic pattern of highly folded “beads” connected by unfolded linkers. Together they are organized into the path of multiple spiraling curves that resemble Johann Bernoulli’s clothoid which emerges in optics (where it is called a Cornu spirals). When a beam from a distant source of light encounters a slit of size comparable to its wavelength we get a typical diffraction pattern with a central peak of high intensity followed by increasingly damped peaks of much lower intensity on its flanks. Now, if the light source is close to the slit then the analysis of the diffraction patterns becomes very complicated. It is in such a case, where a nearby linear light source produces a cylindrical wave front, that this spiral curve can be used to predict the diffraction pattern.

Second, nearness of numbers does not necessarily translate into the similar pattern. We know that $\sqrt{10}$ was used often in early Hindu mathematics as an approximation for $\pi$ correct to one decimal place. However, this number produces a wandering curve that is not at all similar to that of $\pi$. Then we see the famous approximation used by Ramanujan $\dfrac{355}{113}$ correct to six decimal places. It being a rational number does not produce a fractal pattern but a linear sequence of highly folded “beads” connected by a linker element. This resembles the unfolded nucleosome-DNA complex in archaea and eukaryotes. The real $\pi$ in contrast has a highly folded bead similar to the former but the linker gradually curls making curve take an overall spiral form.

In conclusion, here we see numbers playing a different role: different from their role as the backdrop of the smooth continuity of curves emerging as solutions to differential equations but as abstractions similar to syllabic strings, which can be acted upon by simple rules to generate complexity. Here, the actual strings of the digits matter. Seeming uniformly distributed sequences of digits can produce complex patterns that are far from random when sent through the appropriate rules. It would almost seem as though they hide something within them which is not at all apparent until they are made to reveal them by procedures like the floor-difference operation or the curlicue algorithm. This complexity can capture the forms and behaviors we see in nature and produce grammar-like features – some of which also emerges from more complex mathematical descriptions of natural systems.

The possibility of a unified vision
Finally, we intuit the possibility of a unified vision – by no means a complete one or rigorously hammered out. First, we do see the presence of the an ideal realm, one might call it “Platonic”. In our view this is how the “parame vyoman” and “śuddha-bhuvanādhvan-s” of the tantra-s should be understood. We also hold that the correct way of understanding the mīmāṃsaka nityatvam of the word is in this sense rather than the physical śruti composed by our ārya ancestors starting from the early Indo-European period. In this realm reside the numbers or syllables, which are comparable entities in that they are operands of a collection of operations or rules, which in themselves might be simple. However, they are capable of generating great all the complexity we see in the world and mathematics. When this ideal world “shapes” the world of matter-energy we see its expression as a different type of number, the number of measurement. It is this number that appears directly in the sciences. The attempt to apprehend the structure residing in these numbers leads to the discovery of mathematics.

In a biographical sense all this unfolded gradually over time – a story which we might narrate at some other point with different illustrations and more detailed acknowledgments of our influences. The first phase of our life was marked by our experiments in Euclidean geometry and plane curves which culminated around our 14th year in the penetration of calculus. It was the high-point of our personal mathematical attainment in that direction which allowed us to advance into the quantum theory and relativity in physics to the extent we could. As we were fighting our way through Dirac’s book on bra and ket vectors in the summer of our 14th year, on one hand we felt great pleasure with what we had managed to understand but on the other the insurmountable mathematical complexities that underpinned an even deeper conquest of physics put us in place. But our experience was one which nevertheless resounded with the triumph of “ordinary” mathematics – a culmination of the mathematical traditions of the ārya-s and yavana-s in one sense. The existence of the ideal realm came into our focus then and it was a very mathematical one filled with curves. It was around this time we came across an article on the work of Benoit Mandelbrot in a science magazine. This immediately opened the doors to a new world, marking the second phase of our life. However, not having easy or continuous access to a computer this exploration began slowly. But it advanced over the subsequent years with the completion of our transition from the hand-held calculator to the computer and the ideas of Turing and Wolfram become a pratyakṣa. By then we saw a new picture, one converging what has been articulated here.

Chaos in the iterative Hindu square root method of the gaṇaka-rāja

For Hindus big numbers always mattered and our mathematics is quite reflection of this fascination. Since the earliest times, Hindus devised various methods to obtain square roots of numbers, especially approximations of irrational roots correct to multiple decimal places. The earliest of these methods involving a series of terms is seen encoded in the altars for the Soma rituals specified in the saṃhitā-s of the Yajurveda and explicitly spelled out in their the śulbasūtra-s. Indeed, we have evidence that development of these methods continued in the Yajurvaidika tradition as indicated by Rāma dīkṣita’s commentary on Kātyāyana where he provides a tradition regarding a further term to the approximation to get $\sqrt{2}$ correct to 7 decimal places. A similar improvement was likely used in the procedure preserved by Sundararāja dīkṣita in the Āpastamba tradition for an approximate squaring of the circle based on $\sqrt{2}$.

By the last few centuries before the common era the Hindus had already discovered a method similar to what is today known in the west as the first term Newton-Raphson approximation. We also see the exact algorithm for both square roots and cube roots of ācārya Āryabhaṭa further explained for the lay by Bhāskara-I. But the high point of the Hindu tradition of iterative methods is seen in the text of the brāḥmaṇa Chajjaka-putra gaṇaka-rāja probably from Mārtikāvati (unfortunately named Bakshali manuscript: BM), which gives a glimpse of just what Hindu knowledge has been lost over the ages. While this method was misunderstood by the earlier white indological translator of the BM, the sophistication of the gaṇaka-rāja’s method has only more recently become clear. This has been explained and commented upon in detail by the computer scientists Bailey and Borwein in their excellent work on the same. We shall here comment upon an interesting aspect we discovered of the functions involved in the method .

While the method has already been discussed in detail by Bailey and Borwein, we shall go over it here for introducing the system. In order the find the square root of a number $q$ the BM suggests the following procedure:
Take some starting number: $x_n = x_0$
$x_{n+1}=\dfrac{q-x_n^2}{2x_n}$
$y_n=x_n+x_{n+1}-\dfrac{x_{n+1}^2}{2(x_n+x_{n+1})}$

Then $y_n \approx \sqrt{q}$. Now if we take $x_n=y_n$ and iterate the above procedure we get increasingly accurate approximations of $\sqrt{q}$.

As a example let us take $q = 5$ and $x_0=0.1$. Then we have the following:
$1)\; 12.6248003992015985\\ 2) \; 3.6392111847990769\\ 3) \; 2.2506636482615887\\ 4) \; 2.2360679780006203\\ 5) \; 2.2360679774997898$
Thus, in iteration 3 the value of $\sqrt{5}$ correct to 1 decimal place, in iteration 4 it is correct to 8 decimal places and in iteration 5 it is correct to at least 16 decimal places, in line with the Hindu love for big numbers.

Now if we instead take $x_0=2$ because we know that $\sqrt{5}$ should lie somewhere in the vicinity of 2 then we get:
$1) \; 2.2361111111111112\\ 2) \; 2.2360679774997898$
Thus, with this close value right in the first iteration we get it correct to 3 decimal places and in the second to at least 16 decimal places! As Bailey and Borwein had shown it quartically converges on the square root. Now if we take a negative number for $x_0$ it then converges similarly to $-\sqrt{q}$.

Figure 1

Now consider the following alternative procedure where instead of plugging $x_n=y_n$ we plug $x_n= x_{n+1}$ and thus generate for each iteration $(x_{n+1},y_n)$. On plotting the map of $(x_{n+1},y_n)$ we see the points fall on an interesting curve (Figure 1). This curve has two boat-shaped branches which are respectively tangential to the lines $y= \pm \sqrt{q}$. The region of tangency is peculiar in that the curve lingers in the proximity of $y= \pm \sqrt{q}$ over a wide x-interval.

Figure 2

The actual map of the points obtained by the above procedure displays an interesting feature: they are spread all over the two branches of the curve above but fall most frequently in the vicinity of the two root lines. They notably decrease in frequency as one moves away from those lines but we do get to see extreme points far away from the two root lines. Thus, $\pm \sqrt{q}$ serve as the peaks (Figure 2) for the distribution of $y_n$ with a clear decline for greater and lesser allowed values respectively. However, the tails of their distribution are prominent enough that we seen multiple extreme values. The median value for the negative side of the distribution is $\approx -2.3$ and for the positive side is $\approx 2.3$, illustrating the dominance of the values close to $\pm \sqrt{q}$. In line with this, for a large enough number of iterations with a given $x_0$ the overall median value of $y_n$ comes out as $\pm \sqrt{q}$. However, below is an examination of the extreme values reached by $y_n$ for a run initiated with $q=5$, $x_0=0.1$ for 2000 iterations:
Minimum: $-438.98149$
Maximum: $133.19996$
This shows that $y_n$ explores values over 50-100 times the median values in course of the iterations.

To understand this map better let us look at it geometrically (Figure 3). The two expressions that are deployed successively by Chajjaka-putra to get the square root represent the below functions:
$f(t)= \dfrac{q-t^2}{2t}\\ g\left(t\right)=t+f\left(t\right)-\dfrac{f\left(t\right)^2}{2\left(t+f\left(t\right)\right)}\\[10pt] \therefore g\left(t\right)=\dfrac{q^2+6qt^2+t^4}{4qt+4t^3}$

Figure 3

We see that $f(t)$ is a hyperbola with the y-axis as one of its asymptotes. $g(t)$ is a quartic curve, which has $y= \pm \sqrt{q}$ as the as its tangents with the points of tangency being $(\sqrt{q},\sqrt{q})$ and $(-\sqrt{q},-\sqrt{q})$. This curve has a very “flat” type of tangency, i.e. it lingers in the proximity of $y= \pm \sqrt{q}$ over an extended x-range. This is the secret of the gaṇaka-rāja’s method firmly “pulling” things to the vicinity of required square root. Thus, the parametric curve $(f(t),g(t))$ is the one on which the points of the above-described map based on gaṇaka-rāja’s two expressions lie (Figure 1, 3). This curve as noted from the above map has $y= \pm \sqrt{q}$ as its tangents with the point of tangency at $(0,0)$, but like $g(t)$ it lingers over a wide x-range close to the point of tangency. This explains why the map tends to concentrate the points in the vicinity of $y=\pm \sqrt{q}$.

Now, if we look at the positions of actual points of the map on $(f(t),g(t))$ an interesting observation becomes apparent: while tending to cluster in the vicinity of $y= \pm \sqrt{q}$, successive points are not necessarily proximal to each other on the curve. Rather they jump about the curve in a chaotic fashion (Figure 3) either on the same branch or between the two branches. This becomes even rather apparent if we plot a run of $y_n$ against iteration number $n=1..2000$ (Figure 4; The gaps in the plot are where the $y_n$ jumped outside the range of $\pm 50$ which we set for good visualization). What is notable is that the rarer values in addition to appearing chaotically are also rather extreme: the sum of positive $y_n$ for 2000 iterations with $x_0=0.1$ is $4578.39$; of this just 12 values add up to $1005.48$, which is $\approx 22$ percent of the total sum. Each of these 12 extreme values is over 20 times the median value of positive $y_n$. The picture is roughly symmetric for the negative values.

Figure 4

Importantly, this chaotic behavior of $y_n$ is very sensitive to the initial values of $x_n$ with which we start the map. This is dramatically illustrated by two points close to root of $q=5$, $2.2$ and $2.21$ (Figure 5): while for the first 9 iterations the evolution of the two initial values is the same in direction though not magnitude, iteration 10 and beyond they go completely out of synchrony in both direction and magnitude.

Figure 5

In conclusion this map provides an analogy to think about certain processes in nature and historical events. First, it provides a potential model for foraging behavior of variety of organisms. In this model the clustering around the root values represents what might be called the base-line or ordinary foraging and the extreme jumps represent the drastic forays away from their local patch to distant locales. Such behavior may be seen in animals among herbivores moving to new feeding grounds far from their usual feeding areas or certain carnivores like sharks seeking new hunting waters far from their their current zone. This kind of behavior is also seen in certain ciliates like Halteria in the microscopic realm. In the world of protein sequences we see a similar tendency to keep to a tightly constrained space of diversity under purifying selection within which there is a low-radius exploration under neutral drift. This is punctuated by huge saltations that result from strong positive selection for new functional niches. This might happen within a family of proteins or in the proteome of an organism with some proteins showing big saltations in sequence space.

The great mathematician Benoit Mandelbrot, the pioneer in the study of chaos, has brought home the importance of distributions with rare events with extreme values. The role of such events in systems like financial markets has been recently explained at length by Nassim Taleb. In this context, the above-described map provides an analogy for one type of historical evolution of systems. Even with same basic parameter ( $\sqrt{q}$), clearly predictable bulk statistics (e.g. median and range of most frequent values) and similar starting values we see: 1) clear differences in long term evolution with non-overlapping chaotic extreme events different in both magnitude and direction. 2) Extreme events that can disproportionately contribute to the total numerical measure of the events in the series. A historical system evolving under such a model shows us how with very similar starting material and bulk behavior we can have a great difference in actual events and outcomes. This might be similar to actually observed phenomena like the fall empires or the sudden extinction of long-lasting lineages. This is a theme which we might explore further with examples of some other such systems which have been studied for their chaos.

Ramanujan’s second construction for the approximate squaring of a circle

To experience the greatness of great men one has to relive or redo some acts of theirs to the best of ones ability. In ones youth such enactments might inspire one to make a bid for greatness. Whether this happens or not is mostly up to your genetics. Nevertheless, through the enactments one can at least savor the experience of what it takes to get there. If there was one man in our midst who could have lived up to be a Gauss or an Euler it was Srinivasa Ramanujan.

By redoing some of his acts that are within the grasp of our limited intellect we experienced the monument that he was. He gave two constructions for the approximate squaring of a circle using a compass and a straight-edge. We had earlier described the first and more widely known of those. The second appears in his paper titled “Modular equations and approximations to $\pi$“. In this paper in addition to remarkable approximations for the perimeter of the ellipse, which we had also alluded to before, he gives several series for $\pi=3.141592653589793...$. One of these series with just the first term leads to the below approximation:
$\pi \approx \dfrac{99^2}{2*\sqrt{2}*1103} = 3.141592730013305$ which is correct down to 6 decimal places. It is this kind of accuracy he captures in his first construction for the quadrature of the circle. In the midst of the dizzying series he conjures in a very Hindu style of mathematics, he says that he came up with an empirical approximation which leads to the below construction for the approximate squaring of the circle:

1) Draw circle to be squared with center O.
2) Draw its diameter $\overline{AB}$.
3) Trisect its radius $\overline{AO}$ to get a third of it as $\overline{AF}$.
4) Bisect the semicircle AB to get point C.
5) Draw $\overline{BC}$.
6) On $\overline{BC}$ mark $\overline{CG}= \overline{GH}=\overline{AF}$.
7) Join point A to point H to get $\overline{AH}$ and to point G to get $\overline{AG}$
8) With radius as $\overline{AG}$ cut $\overline{AH}$ to get point I.
9) Draw a line parallel to $\overline{GH}$ through point I to cut $\overline{AG}$ at point J.
10) Join points O and J to get $\overline{OJ}$.
11) Draw a line parallel to $\overline{OJ}$ through point F to cut $\overline{AG}$ at point K.
12) Draw the tangent to the circle at point A and cut it with radius as $\overline{AK}$ to get point L.
13) Draw $\overrightarrow{OL}$ to cut circle at point M.
14) Draw semicircle LM and perpendicular from point O to cut this semicircle at point N.
15) Triplicate $\overline{ON}$ to get $\overline{OQ}=3 \times \overline{ON}$.
16) Produce $\overline{OQ}$ in the opposite direction to cut circle at point R.
17) Draw semicircle RQ and a perpendicular from point O to cut it at point S.
18) Thus, we have $\overline{OS}$ as the side of the square OSTU which has approximately the same area the starting circle.

Ramanujan tells us that his earlier construction gave an “ordinary” value $\pi \approx \dfrac{355}{113}=3.141592920353982$, which is correct to six decimal places. This one, however, gives us the value:
$\pi \approx \left (9^2+\dfrac{19^2}{22}\right)^{\frac{1}{4}}= 3.141592652582646$
This is correct to a whopping eight decimal places keeping with the Hindu love for big numbers.

Some meanderings among golden stuff-2

Related stuff:
Golden Ratio-0
Golden Ratio-1

If the golden ratio can fascinate erudite men of high IQ then what to say of simpletons like us. Hence, we shall here talk about some more trivia in this regard. The golden ratio is associated with a sequence, the method for generating which was provided in our tradition first by Piṅgala (the Meru-prastha) and was spelt out in full by Virahāṅka in his Vṛtta-jāti-samuccaya. Hence, we may call it the Meru-średhī ( $M$ with elements $M(n)$; $n= -\infty ...-2,-1,0,1,2,3... \infty$; among the mleccha-s it is commonly known as Fibonacci’s sequence).
$\lim_{n\to\infty} \dfrac{M(n+1)}{M(n)}=\phi \approx 1.61803398875$, which provides the relationship to the golden ratio.

Figure 1

In modern parlance this sequence can be seen as integer values emerging from the function (Figure 1):
$y=\dfrac{1}{2\phi-1}\left(\phi^x-\dfrac{\cos(\pi x)}{\phi^x}\right)$
This function oscillates with ever-increasing amplitude for negative $x$, hits $0$ at $x=0$, remains stable between $x=1..2$, then almost linearly climbs to $2$ at $x=3$ and then explodes nearly exponentially.

The sequence can be obtained from this function thusly:
$M(n)=\lfloor y \rfloor$ for $x=n=-\infty...-2,-1,0,1,2,3...\infty$.
$M \rightarrow ...34,-21, 13,-8,5,-3,2,-1,1,0,1,1,2,3,5,8,13,21,34...$
Thus having obtained the sequence, a well-known property of it, apparently discovered by Kepler, becomes apparent:
$M(n)^2=M(n-1) \times M(n+1)- (-1)^n$.

Figure 2

Now this relationship might be linked to an interesting geometrical procedure of dissection of a square which can performed only using the golden ratio (Figure 2). The procedure goes thus:
1) Let there be a square of side $a$.
2) The square is first partitioned into two rectangles one of sides $a$ and $\dfrac{a}{x}$ and other of sides $a$ and $a-\dfrac{a}{x}$.
3) Partition the first rectangle into two congruent right-angled trapezia with $base=\dfrac{a}{x}, height=\dfrac{a}{x}, top=a-\dfrac{a}{x}$. Partition the second rectangle along its diagonal into two right triangles with $base=a, height=a-\dfrac{a}{x}$.
4) Rearrange the the four pieces of the square thus obtained such that they reconstitute a rectangle of equal area as the starting square. This is done by taking two opposites sides of the rectangle to be the sides of the two trapezia $\dfrac{a}{x}$ and the other two opposite sides made by laying the shorter edge of the right triangle against the top of the trapezium. Thus, this side is $a+\dfrac{a}{x}$.
5) Thus, we have:
$a^2=\dfrac{a}{x}\left (a+\dfrac{a}{x}\right)\\[10 pt] \therefore 1= \dfrac{1}{x}+\dfrac{1}{x^2}\\ [10 pt] \therefore x^2-x-1=0$

The positive root of above is $x=\phi$. Thus, only by using the golden ratio can we convert a square into a rectangle as described above. For any other ratio you will have a parallelogram area in the middle which is either in excess or less than the area of the reconstituted rectangle.

Figure 3

Now instead of the golden sectioning of the square let us use a square of side $M(n)$ and divide it as above using $M(n-1)$ and $M(n-2)$ to make the four pieces (Figure 3; e.g. $8, 5, 3$). If we then reconstitute them as a rectangle we will get its sides as $M(n-1)$ and $M(n+1)$ (e.g. $5, 13$). Then by the above relationship regarding these elements of series $M$ this reconstituted figure will have a long thin parallelogram of excess or insufficient area relative to the reconstituted rectangle by 1 square unit. Thus, if we make a cardboard cutout of the above one could “hide away” that parallelogram worth of area difference along a roughly cut diagonal of the reconstituted rectangle. This can produce to paradox of square of side $8$ yielding a rectangle of sides $5,13$. Many years ago we had seen precisely such a “puzzle”, may be made of wood or cardboard. We must confess that then it took a little while before we realized that it stemmed from the golden dissection of the square. We recently learned via Mario Livio that this puzzle was invented by a guy called Loyd.