The Mitr

Mysterious spiral shells

I have visited one of the most iconic sea forts in Maharashtra – the Sindhudurga at Malvan on the Konkan Coast twice. It is one of the most beautiful sea forts you will witness. The crystal clear waters and blue skies will be imprinted in your memories. The first time I had an analog camera with me, so I couldn’t take as many photos as I would have wanted (one of the analogue snaps is above c. 2001). The second time I had a really nice digital camera with me, so I did take a load of photos (all the ones below).

Maratha navy was a very formidable force on the Konkan coast. The maratha navy under the Angre’s was a force feared by the Portuguese and English alike. They ruled the waters from North of Goa to Colaba Fort (not the same as Colaba in South Bombay) with their capital at Gheria (Vijaydurg), in the later half of 1700s. Maratha navy ships were fastest and very agile in open waters and were absolute terror for the europeans. Anticipating the need for a strong navy as well, Sindhudurga was one of the first sea forts to be built by Shivaji. The fort is built on a rocky outcrop off the coast of Malvan. The fort played an important role after Shivaji as well. It housed Tarabai, the widower queen of younger son Raja Ram during the invasion Mughals. The fort has walls which have stood the test of time and are still standing well even after several centuries against sea water and constant barrage of waves. (though in the second visit we found parts of wall had collapsed). The architecture of the fort is amazing, though we only see the bastions and ramparts now. The fort walls are shaped smoothly when required (in the mathematical sense of continuity), unlike later european forts which are more angular in nature. The front gate is hidden inside a curved pathway between two bastions so that is not visible directly (hence cannot be hit directly with a canon. This mechanism is also seen in several other forts such as Raigad and Janjira. But so much about the fort, coming back to the topic of the post.

On the southern end of the fort there is a small door which opens to a patch of beach. This is one of the best spots on the fort if you are a nature lover. In this small patch there is clear water and beach.

From a bit far
A bit closer to the small door, Sindhudurga Sea Fort Malvan
A bit more close to the small door, Sindhudurga Sea Fort Malvan
Crystal clear waters at Sindhudurga Sea Fort Malvan

The beach sand is coarse, meaning most of it is actually made of seashells. This is a treat to see, myriad shapes, sizes and colours of seashells blended to form the sand. Sand is a fascinating mixture which results from erosion (air, water) and evolution over long time scales. Sand has both inorganic and organic content.

Waves, caustics and crystal clear water…

 

The sea shells are logarithmic in nature, with the nautilus perhaps being seen as exemplary. In many cases of shells the logarithmic spiral is obvious (see the photo), but in other cases as well it shows logarithmic growth.

Close up of the sand shell mix. In the center, Clypidina notata Linne, Size 23-30 mm.
Close up of the sand shell mix. What is this see weed by the way? Upper left white shell is most probably Cardita antiquata Linn  or Cardium sp.
A multishade rock in the sand shell mix
Close up of the sand shell mix, too
Our spiral shell standing out in the sand shell mix
Close up of the sand shell mix. A hermit crab residing in a Trochus radiatus Gmelin (Banded Torchus)
A view of the ramparts and bastion from the small beach
Close up of the sand shell mix, .
Close up of the sand shell mix
Close up of the sand shell mix. This lot needs a lot of identification!
Close up of the sand shell mix
Close up of the sand shell mix. Sundial (Architectonica laevigata Lamarck) with most probably Cardium asiaticum Bruguiere.
A quadrumvirate of hermit crabs in Dwarf turban (Turbo brunneus Röding) with copper legs in a tidal pool near Aguada Fort, Goa)
A company of pebbles (Anjuna beach, Goa)
A company of pebbles (Anjuna beach, Goa)

See a previous post on patterns in nature, this post is a sort of extension of that. For an extensive and excellent treatise on spirals see The Curves of Life by Theodore Cook. On a side note, you should also read Junijo Ito’s spiral themed horror manga Uzumaki.

I did collect a few noteworthy shells from this patch (there were so many to collect!), and these included a few approximately hemispherical ones which were flat on the other side. These shells had a nice logarithmic spiral on the flat side which was also relatively smooth. While the hemispherical side was comparatively rougher.

Now in all other shells, one could easily visualise where would the animal be that created the shell or how the animals used that shell. for example

Connus mutabilis (Reeve) showing excellent spiral structure, from Madh beach
Cantharus spiralis (Gray), from Madh beach
Turritella duplicata (Lamarck) along with a hermit crab, 
Natica didmya (Röding), Revdanda Beach
Burasa tuberculata (Brodip) Tuberculated Frog, Madh Beach
Surcula javana (Linne) Javan Turrid, Nagaon Beach
Mix of organic and mineral sand, Madh Beach
Mix of organic and mineral sand, Madh Beach
Variants of Umbonium vestiarium (Linne), Button shells
Variants of Umbonium vestiarium (Linne), Button shells, Nagaon beach
Variants of Umbonium vestiarium (Linne), Button shells (from Nagaon beach)
Our mysterious spiral shells
Our mysterious spiral shells.
Our mysterious spiral shells
Our mysterious spiral shells

 

But in case of these spirals I could not understand how or where the animal would be using this shell. There was no hole to hide or provide a protection to the animal, as it was solid. It was indeed a puzzle. The shells definitely belonged to an animal, but which one and how did it use it? I asked around but did not get any clear answers. Thus began a journey to unravel the mystery. The image search for spiral shell did not help, though I came to know that these shells are used in astrology oriented rings with silver. And apparently they are seen in jewellery shops.

The first thing we knew for sure was that the shell belonged to a marine animal. But how did that animal use this? It remained an unsolved question for a couple of years. I would ask around to anyone with some knowledge of zoology, but it didn’t get any answers. Then one day someone did faintly recognised, “Isn’t this an operculum?” Now a quick image search with this new term, operculum, I learned gave the answer to this mystery. And commonly it is also known as cat’s eye shell. And the long mystery was solved:

The operculum is attached to the upper surface of the foot and in its most complete state, it serves as a sort of “trapdoor” to close the aperture of the shell when the soft parts of the animal are retracted. The shape of the operculum varies greatly from one family of gastropods to another. It is fairly often circular, or more or less oval in shape. In species where the operculum fits snugly, its outline corresponds exactly to the shape of the aperture of the shell and it serves to seal the entrance of the shell. (wiki)

TODO: Identify all the shells in the photos..

 

References

Deepak Apte – The Book of Indian Shells (BNHS)

 

 

The Mitr

Free Graphics Illustration Resources and Repositories

Finding a good and accurate graphic or illustration for your need is something that we all struggle with. On top of that if the requirement is that the graphic resource has to be free (as in freedom) then it further narrows down the options. Sometimes you see the graphic you want, but its license terms are unknown or are not agreeable to your work as they are not released freely. So what do you do? Either you use an illustration which is not perfect fit or you use one which breaks your resource (in terms of license).  But the problems is also that there are many free resource repositories which are very well not known. I have personally come across great many graphic resources, only to be forgotten after their current need is finished. This post is an attempt to overcome this. This post is a collection of various free graphics and illustration resources and repositories that I have found useful over the years. It is also a sort of personal bookmark list of the these resources if and when I need them in the future. I hope this will be of use to others too. I will keep on updating this list with new resources that I find. If you know of any resources which are missing please do post in the comments.

The Internet Archive Image Search https://archive.org/details/image

Wikimedia Commons https://commons.wikimedia.org/

Open source illustrations kit https://illlustrations.co/

NYPL Public Domain Archive https://nypl.getarchive.net

David Rumsey Map Collection  https://www.davidrumsey.com/  also https://archive.org/details/david-rumsey-map-collection

Metropolitan Museum of Arts Public Domain Images also https://archive.org/details/metropolitanmuseumofart-gallery

Cleveland Museum of Art Open Access also https://archive.org/details/clevelandart

Brooklyn Museum also https://archive.org/details/brooklynmuseum

https://archive.org/details/bibliothequesaintegenevieve_image

Unsplash Free Images

NASA Images

ESO Images

Vintage Ads

Vintage Australian Print Ads

Vintage British Print Ads https://archive.org/details/vintage-british-magazine-ads

Vintage American Print Ads https://archive.org/details/vintage-american-print-ads

 

Vintage Danish Print Ads https://archive.org/details/vintage-danish-print-ads

https://archive.org/details/vintage-new-zealand-print-ads

https://archive.org/details/vintage-canadian-print-ads

https://archive.org/details/vintage-italian-print-ads

 

A library of microorganisms https://archive.org/details/cmpuj

 

National Gallery of Art also https://archive.org/details/national-gallery-of-art-images

Flickr Collections (there are several collections on Flickr which are openly licensed)

Flickr Commons

NOAA Photo Library https://flickr.com/photos/noaaphotolib/

 

 

 

The Mitr

On Stories

What is a story?

A story is a series of imagined sequence of events which may or may not be based on true events. The story takes place somewhen, somewhere. A story may take place in the past, or the present or the future. A story may take place in a real place, or an imaginary one (even the surface of a neutron star!). A story has characters and a narrator (can there be a story without either?). Sometimes the characters tell us the story. Sometimes the narrator tells us the story. Some stories are fun, some are dark, others are just boring. A good story is like a fishing hook. It will compel the listener, and keep them eager for what happens next. It is the anticipation of events, which makes a story interesting. What good would be a story if you knew exactly what is going to happen? They say don’t let facts ruin a good story. But facts themselves can make a great story. Reality is stranger than fiction. Reality is a fractal. More you look deeper you see. The mundane becomes the mysterious. Everyday thing becomes enigmatic.

Timeless stories

There are stories which endure time. They were perhaps told since humans acquired language. Perhaps the enormous capacity of language to express imagination was developed to tell the stories. Stories are culture. Stories give us identity. Stories give us rules and norms. Stories make us who we are. Stories tell us about gods, about demons, about magical beings, about kings, about heaven and hell.

Visual Stories

Not all stories have words. Some stories are told via images. Before the written word, images were the only permanent thing. The spoken word is ephemeral, while images endure. Spoken word is for people who are present then and there, the image can transcend time and distance. The images can be somewhen, somewhere other than where and when it was created. Spoken words live only in the present.

Variations on a Theme

Stories change. When a story is told from one person to another, while retelling the person will inadvertently make changes. Though core idea of the story may remain same, some of the details may change. Thus stories multiply. Language is not a barrier. Same story can be told in different languages. Characters, places may change but the stories remain the same. It is like a person wearing different outfits suitable for the local environment.

Storytellers

But how do stories get made? Who makes them? Well, we make stories.

Each one of us has stories to tell. Some people are good at telling them, some aren’t. The good stories stick around. They pass on. From person to person, from generation to generation. From parents to children. From elders to youth. From friends to friends. From teachers to students. Storytellers form a network. Stories link them to those who came before them and those who come after them.

 

Emotions

Stories evoke emotions. Some make us laugh, some make us cry. Some us happy, some make us sad. Some take us on adventures we will never be able to go. Some make us confront our deepest fears and invoke terror in us. Some make us feel elated, and inspire us to do things. Some possess wisdom, some are just folly. Some stories are about mundane things. Some stories are about mysteries. Some stories you can readily identify with, some are so alien to our experience.

Stories are Everything

Without stories we are nothing. We relate to others via stories in which we are the characters. Everyone has at least one story to tell, their own story. Will your life be a story worth telling others?

The Mitr

Change and hiccups

The site was down for a while due to a technical hiccup while transfering from WP to AWS. But its up and running now, I guess change and transformation is a part of the process… So will start with regular posts, lets see at least once a week!

 

 

PS: thanks to @imsmubarak for helping setup the AWS and @alpesh for sorting the DNS and domain transfer issue…

The Mitr

Obedience and Conformity

Obedience can be understood as “a compliance with an order, request, or law or submission to another’s authority”, while one of the meanings of conformity is “behaviour in accordance with socially accepted convention“. Obedience to authority forms one of the basis of civilisation. Unless the citizens abide by some rules there will be a complete chaos. In early days, this was done by rulers in form of rules and edicts. Now we have the constitution. 

Obedience is essential and must be achieved by cunning…

We usually use obedience and conformity as interchangeable terms or having at least similar connotations. but they are not same. But both are forms of power and control over other humans. Here is how Orwell puts it forth in Nineteen Eighty Four.  

“The real power, the power we have to fight for night and day, is not power over things, but over men.” [O’Brien] paused, and for a moment assumed again his air of a schoolmaster questioning a promising pupil:

“How does one man assert his power over another, Winston?”

Winston thought. “By making him suffer,” he said.

“Exactly. By making him suffer. Obedience is not enough. Unless he is suffering, how can you be sure that he is obeying your will and not his own? Power is in inflicting pain and humiliation. Power is in tearing human minds to pieces and putting them together again in new shapes of your own choosing.”

Stanley Milgram made a distinction between the two and identified four major differences that distinguish the two. This differences tell us about the nature of obedience and conformity. The core of the difference lies in the person who is commanding obedience and the person who is following that in case of conformity usually the person following others is at equal status to them. Where is in the case of obedience there is a hierarchy between the commander and the commanded. This is the first difference: the hierarchy. In a classroom the teacher commands the students to follow certain task or to do some task. This is an example of obedience. When students try to follow each other in terms of the things they do or they dress outside of school or the games that play or the clothes they wear is closer to conformity. The in the first case is the clear distinction and difference in hierarchy in the second case it is not so.

The second major difference is imitation while obedience involves compliance with orders, confirmatory involves imitation and adoption of similar behaviour. If neighbours are beating the plates to make corona go away, so shall I, otherwise we will be seen as socially non-fitting.

The third difference is that in case of obedience the commands are more explicit whereas for conformity is more of an implicit thing. An unspoken rule which everybody around you follows. Conformity is more about following others who are your peers than following explicit commands so in case of confirmative their might be some elbow room to express yourself slightly differently than others. But in case of obedience that is not possible obedience is to the full and no lateral thinking is allowed.

Final difference is the voluntary nature of conformity as opposite to attribution to authority figures. In case of obedience there is no choice but to follow the orders.

Milgram’s experiment on obedience shows how suggestible humans are in presence of authority. 

Would you torture another human because someone in authority tells you to?

The experiment is setup as such. The subject let us call them X is called for doing an experiment in the lab. X is then introduced to the experiment. The experiment involves X teaching another participant Y word association using punishment for wrong answers. If Y does not give correct answer, X has to give Y a punishment in the form of an electric shock. X is given a mild shock to make them experience of the punishment, so that X knows what is the type of punishment and how Y would feel when punished. X has a dial which can control the amount of shock delivered to Y. Overseeing all this is a researcher Z in a white lab coat. A person wearing a white lab coat somehow represents authority figure for most. This is why you see doctors/scientists in advertisements wearing white lab coat: because it is a symbol of authority. 

Now, Y is actually part of the experiment and accomplice of the researcher. X does not know this. The test starts, and Y deliberately chooses to give wrong answers. Now according to the “rules” of the experiment X has to give “punishment” to Y for incorrect answers. The dial for controlling the shock, the “shock generator”, is calibrated with incremental levels of shock, finally going to 450 Volts. (Of course this is make believe, there is no real shock given, but let us keep this secret with us and not tell X.)

With each wrong answer X is supposed to increment the level of shock punishment to Y from the shock generator. Now, X knows that increasing the shock will cause incremental pain to Y. When X hesitates to increase, Z intervenes and tells X authoritatively (remember Z is wearing a white lab coat) that this is the rule: “With each wrong answer punishment must be increased.” This is where the crux of the experiment comes in. It is found that majority of X, under the influence from authority figure Z, obediently inflicts serious punishment to Y. This even when Y cries in agony, asks experiment to stop, but more X continue to punish Y.   

This is highly counterintuitive result. How so you would ask?  Consider the following gedankenexperiment. Suppose someone looking authoritative tells you to go and thrash someone. Would you do it? Most probably the answer would be no. Then why are the participants X so willing to inflict the incrementally harsh punishment? Because they start with something that is seemingly innocent – a mild shock. Once that threshold is crossed rest of the punishment becomes easy. And you adjudge yourself being not at fault as you are “just following the orders or rules”. This has large implications for how we as a society, every now and then, fall for authority figures and do things which we think we will not do. Perhaps in such cases, we think that since authority figure is telling us to do something, it must be for greater good and we let not our puny morals or conscience come in the way. Also, since we are obeying authority, it takes off our own personal responsibilities as such. This is a slippery slope and can lead to genocides and living hell for those Y who are at the suffering end. Just because Z is popular and in authority does not make following them blindly a right thing. But then the aspect of conformity sets in. If we don’t follow the current norms we are seen as outcasts in the society and that leads to further pressuring of conformity.

Moral is we should exercise our own set of morals and conscience as much as possible.  

 

 

Sources

 

Great Ideas in Psychology by Fathali Moghaddam

Nineteen Eighty Four by George Orwell

The Mitr

Nominal expertise

Expertise of Dr. Strangelove
Expertise of Dr. Strangelove

Who is an expert? What qualities in a person defines them as an expert? The dictionary meaning of an expert is

a person who is very knowledgeable about or skilful in a particular area

But how do we know if a person is knowledgeable or skilful in a particular area? A simple way to answer this question is “expert is one who has expertise!”. But this really does not tell us anything (or it does?) about the nature of expertise. One better way to characterise expertise would be if we know by some objective manner that a particular person is an expert. One such way can be to look at the educational qualifications of the person under question. For example, if someone says “You can ask her any question about stars. She has a PhD in stellar astrophysics.” you take on authority of the person telling you and the fact of having a PhD that the person is indeed an “expert”. This is because PhD requires detailed study (at least of a part of the subject area) and we assume that people who have this degree also have a sufficient expertise. PhD holders are highly educated is the claim. Hence most of the experts would be PhD holders in their respective fields. But having a PhD is no guarantee that the person indeed is an expert in the field of study. This is what Frederick Reif has to say about it in his article Interpretation of Scientific Concepts:

Quite a few physics graduate students, and even some physics professors, make mistakes and arrive at wrong answers. Indeed, some experts’ performance resembles that of novices. Such observations indicate that nominal experts (i.e., persons designated as “expert” by virtue of their degrees, A. titles, or positions) can differ very widely in their actual competence. (To paraphrase George Orwell, some experts are much more equal than others). This should be a warning about the interpretation of many cognitive studies where “experts” are selected by purely nominal criteria, without specifying adequately the nature of their actual expertise.

This I feel is a case for normative vs. descriptive dichotomy. The position or degree of a person gives them the virtue of being an expert, but it does not guarantee it. And when we decide our policies based on the expertise of the experts which may not be a true expertise or maybe inherently biased. Perhaps this is one of the reasons that we have flawed policies in the first place. Though, Dr. Strangelove (Dr. suggesting a PhD) was an expert!
But are there experts who do not have a PhD or educational qualifications? Yes! Not all knowledge or skills can be concretised in the form of degrees. Most of the knowledge is tacit in nature, which comes from experience. It doesn’t matter if you have a PhD in theoretical hydrodynamics, fixing that leaking tap requires a different type of skill and knowledge. Cooking is another area where knowledge is tacit. Unless you start cooking, you can’t be called an expert cook!
 
 

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Interesting LaTeX Packages – Drawing the Solar System

Many times we need to quickly illustrate the solar system or the planets. Usually we use photos to illustrate the planets. But sometimes the photos can be an over kill. Also to draw the entire solar system is a typical and can be used for illustrative purposes. Though most of the illustrations of the solar system in the school and other text books are horribly out of scale, (with no indication that the figure is not to scale!). For example,  look at the illustration in the Class 6 Science Textbook from NCERT.

A good visualisation will always present a scale, and/or indicate whether the visualisation is to the scale or not. Though in the visualisation above the distances are given, they are not to scale.

 

Coming back to the topic of our post, a simple way to draw solar system diagram in latex is to use the PStricks package solarsystem. The package can create “Position of the visible planets, projected on the plane of the ecliptic” at a given time and date. This feature might be useful sometimes.

From the package manual:

As we can not represent all the planets in the real proportions, only Mercury, Venus, Earth and Mars are the proportions of the orbits and their relative sizes observed. Saturn and Jupiter are in the right direction, but obviously not at the right distance.
The orbits are shown in solid lines for the portion above the ecliptic and dashed for the portion located below.

The use of the command is very simple, just specify the date of observation with the following parameters, for example:
\SolarSystem[Day=31,Month=12,Year=2020,Hour=23,Minute=59,Second=59]
By default, if no parameter is specified, \SolarSystem gives the configuration day 0 hours to compile.

The resulting output for the above code:

The output also provides the longitude and latitude of the planets at the time given.

Another package that is useful to create free standing planets is the tikz-planets package which we will see next.

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Interesting LaTeX Packages – Bohr and Element – electronic orbits and atomic structure

One of the USPs of using LaTeX is the variety of packages that are available to get things done. Some packages will give you special environments to make your documents better, some will help in typesetting or some will help you create graphics or some just provide you with commands for specific symbols. Of course, all these can be done manually by creating your own command, but why reinvent the wheel? There are hundreds of packages at the Comprehensive TeX Archive Network. I have come across many packages that were useful via browsing the packages at CTAN. In this series of posts we will see some packages that are interesting and might be useful. This series of posts is also a sort of personal bookmarking scheme for me. It has happened in the past that I have discovered some interesting LaTeX package, only to forget its existence when I needed its functionality in a project.
In this first post, we will look at two related packages bohr and elements by Clemens Niederberger. The bohr package provides you with a simple functionality to draw the Bohr diagrams for different elements along with electronic configurations.
Load the bohr package by \usepackage{bohr} in the preamble
To use the package simple type the number of electrons and the element symbol. For example, Lithium \bohr{3}{Li} will simply give you

Similarly for other elements
Lithium \bohr{3}{Li}Oxygen \bohr{8}{O}Carbon \bohr{12}{C}Mercury \bohr{80}{Hg}

Now another very useful option in the vohr package is to print the shell-wise electronic configuration for a given element. For example Oxygen \bohr{8}{O} \elconf{O} will give you

This will be a very useful feature when you are writing chemistry or atomic physics texts. Of course you can change the way the shells look.
\setbohr{
shell-options-add = dashed, shell-options-add = red, shell-dist = .75em, nucleus-options-set = {draw=black,fill=orange,opacity=0.5}, electron-options-set = {color=green}, insert-missing}
Mercury\bohr{80}{Hg} \elconf{Hg}

The insert-missing option will give you either the correct number of electrons when the element symbol is given, or  will give you the element symbol when the number of electrons is given. There are more options to explore in the documentation.
Now let us look at the elements package.

This package provides means for retrieving properties of chemical elements like atomic number, element symbol, element name, electron distribution or isotope number.

The package provides atomic number, symbol, name, main isotope and electronic configuration for elements upto 118. For example, just using the atomic number 35 I can get \elementname{35} \elementsymbol{35} \elconf{35}

Having the data accessible in the form of number can be very useful especially if you want to generate tables. The table below from the package documentation was generated by iteratively looping atomic number and invoking commands

\theelement
\elementsymbol{\arabic{element}}
\elementname{\arabic{element}}
\mainelementisotope{\arabic{element}}
\elconf{\arabic{element}}


 
 

The Mitr

Algorithmic Nature

Such natural beauty! Does mathematics lie at the basis of these diverse and beautiful forms? Photo taken during summer of 2017 in Mumbai. None of these are native to India. On left: The Cannonball tree flower (Couroupita guianensis) is South and Central American, African Tulip Tree (Spathodea campanulata) native to tropical. Africa; On Right: The Cannonball tree flower, and Gulmohar is Madagascan (Delonix regia).

What could be more “organic” and “natural” than looking at a pristine forest with a variety of tree forms and leaf forms of various shapes and shades, with inflorescences of variety of shapes, sizes and colours? Mathematicians and physicists are often accused of being not able to enjoy nature and because mathematics and physical theory is so “abstract” and nature is so “organic”. Organic growth is in the form of variety of morphologies of roots, branches, flowers shapes and arrangement, leaf shapes and arrangements, while mathematics typically is abstract graphs, equations, symbols and numbers. How can these two possibly have anything in common? This has also to do with how biology is traditionally taught. While physics has mathematics at its foundation, the teaching of biology doesn’t acknowledge any need for mathematics – it is mostly descriptive as it was in its early stages a couple of centuries later. This is more so at the school level teaching of biology. So this creates an impression in the students and teachers alike that mathematics is not a part of “biological” nature and it is only reserved for falling bodies and ascending projectiles. 

 

What can be similarities in the two images? One is abstracted representation of motion of a body in algebraic and graphical format and other is organic growth of a plant showing its branching and similar leaves with its pigmentation of chlorophyll.

Of course the variety of forms and their classification is one of the foundations of biology. Linnaeus used the morphological differences and similarities to form his classification system.

 

Linnaean system brought order to seemingly diverse and chaotic forms of natural world. Linnaeus named the different forms. Naming is the first step in studying anything. Naming helps in categorisation, which is one of ways to formation of concepts. This led to further finer classification of the system as whole which now includes both flora and fauna. Then began the programme of finding organisms and classifying them in existing categories with descriptions – or creating new ones when the existing ones did not fit – became the normal way of doing biology in the nineteenth century. Even now finding a new plant or animal species is treated with celebrated as a new discovery. 

Darwin in his thesis about evolution by natural selection used the differences and similarities of the form as one of evidence. He theorised that organisms that have evolved from common ancestors will show similar forms with slight variations. Over long periods of time these slight variations evolve into larger variations which ultimately leads to a completely different species. Fossil records tell us about ancestors and current relatives of organisms.

There is grandeur in this view of life, with its several powers, having been originally breathed by the Creator into a few forms or into one; and that, whilst this planet has gone circling on according to the fixed law of gravity, from so simple a beginning endless forms most beautiful and most wonderful have been, and are being evolved. (emphasis added)

The morphologies tell us about related species, the ancestries and divergences from there. The fossils tell us the ancestors, the missing links. So finding organisms, both extant and extinct, to fit in the jigsaw puzzle of tree of life became the standard programme in biology. This enabled us to construct the tree of life. Ernst Haeckel’s version of the tree, depicted below, is highly anthropocentric which places humans at the apex of evolution. This is rather common misconception about evolution – humans are not at apex of evolution or the prime product of it as some would have us believe – we have co-evolved with all the current extant species. Evolution by natural selection is not anthropocentric, it is indifferent to humans and other organisms alike. Daniel Dennett likens it to universal acid, and makes a point that it is not only applicable to living systems, but applies to any system which fulfil the three required criteria. 

 

Haeckel’s – Pedigree of Man – a version of tree of life which is highly anthropocentric.

But can we make sense of similarities of the form in terms of mathematics? Can we find mathematical algorithms which will generate forms, as they generate trajectories of moving projectiles? Looking at similarities in form, it is Galileo who was one of the first to discuss the problem of scaling and its effect on form.

To illustrate briefly, I have sketched a bone whose natural length has been increased three times and whose thickness has been multiplied until, for a correspondingly large animals, it would perform the same function which the small bone performs for its small animal, From the figures here shown you can see how the proportion of the enlarged bone appears. 

…

Whereas, if the size of a body be diminished, the strength of that body id not diminished in the same proportion; indeed the smaller the body the greater its relative strength. Thus a small dog could probably carry on his back two or three dogs of his own size; but I believe that a horse could not carry even one of his own size.

At the start of twentieth century we had a few  classics which gave a strong mathematical flavour to the study of the biological forms and scaling – The Curves of Life by Theodore Cook (1914), D’arcy Thompson’s  On Growth and Form (1917) and  Julian Huxley’s Problems of Relative Growth (1932). 

The kind of mathematical treatment that entered in study of biology by above classics looked at the mathematical aspects of morphological forms in organisms. The Curves of Life looks at the spiral forms which are found in nature, and also in various human creations – architecture and art. 

The spiral is one of the most easily identifiable mathematical forms in nature.
In many flowers, a double spiral forms the basis of the central pattern. The Fibonacci numbers are easily identifiable with this pattern.
The spiral is found in animals too, most easily identifiable in the shells of various types. They represent logarithmic spirals.

 

Despite tremendous success of Darwin’s theory, physics and mathematics were in a separate compartment from biology. There seemed to be no common elements, while biology became more and more descriptive with focus on the form, but not mathematical. 

The word “form” in this article will refer to the shapes of material objects, the arrangement in space of groups of them, and the arrangement in space of their component parts. Our appreciation of form is partly sensory, but we can be helped by measurement and calculation to gain some confidence that what we perceive is not entirely unconnected with the outside world. (Physical Principles underlying Inorganic Form – S.P.F. Humphreys-Owen)

But this is a folly. Nature and organic growth is as mathematical as is the description of a projectile flying under gravity. Perhaps the mathematical description is 

The sparse branching of Frangipani (Plumeria sp.), a native of central America.
The dense branching of the Acacia (Vachellia nilotica) native to Africa, Middle East and India. Is the branching in Frangipani and Acacia related? What about the branching in grass in the figure at top? Can these be generated from a single mathematical algorithm? And why do only these forms are found and not any other?

In my experience a lot of young children who take to biology do so because they hate mathematics or computations. In India there are even streams at +2 level which allow you to shun mathematics for biological subjects. This utter hatred for mathematics is, IMHO, due to a carelessly designed and too abstracted mathematics curriculum at the school level – a curriculum which takes out the soul of mathematics and puts on a garish display of the cadaver of mathematics with bells and whistles. But this post is not about the problems of mathematics education, I have talked about it elsewhere.

 

The aim of this series of posts is to touch upon the inherent mathematics and algorithms in the natural world. How nature is mathematical especially in living and non-living things. How can algorithms generate natural forms? In the next posts in this series we will explore how the ideas of mathematical models can explain the variety of forms that result from natural selection in environment and possibly why only those forms can be found. 

Note: All photographs were taken by me over the years. Only now I am able to piece a narrative linking them together.

 

 

The Mitr

What develops in children as they grow up?

. . . the most ubiquitous finding in developmental research is that infants show more adult- like performance as they grow older. [1]

The very fact that children grow up and become adults relates to the above sentence. The starting and the ending points of the child’s development are known to us. The main aim of the developmental theories is to find out the ‘paths’ that lead to the change from an infant to an adult. Thus in a way different theories ‘map’ out the regions between the infant and adult. For achieving this, every theory has some tools, processes, structures and concepts. Change and development in each of these parameters results in the overall development of the child. The parameters and the agents of development may be different in the different approaches. We consider each of the major developmental theories with respect to their parameters of development.
The broad outlines for the various developmental approaches presented here follow closely the section What Develops? at the end of each chapter in [4] unless otherwise indicated.

1 Piagetian Approach

The basic paradigm that the Piagetian approach envisages, is the stagewise development of the child and the associated psychological structures or schemes. The stages of child range from an infant at sensorimotor stage to an adolescent in formal operational stage. Associated with each stage is the characteristic structural change in schemes, regulations, functions, and various logico-mathematical structures. So the answer to the question ‘what develops’ according to Piaget would be that the schemes and structures associated with each stage develop, in accordance to characteristic for each stage. This development can be assessed through observations, interviews taken by the experimenter [4] pg. 72.

2 Information Processing Approach

In the information processing approach, the cognitive processing is the measure of development. The increase in cognitive processing means that it becomes efficient, well organized, and the content of information also increases, which results in the overall development. Children acquire ‘rules’, ‘strategies’, ‘scripts’ and more knowledge. The concept of memory is directly related to the cognitive processing, it determines the ‘speed’ of processing as well as the ‘output’. So the increase in the memory capacity results in the overall increase ‘quality’ as well as the ‘quantity’ of the cognitive processing. In case of the connectionist approach the strengthening of connections in terms of number and strengths over time, would represent the development of the particular path of connections related to the input.

3 Vygotskian Approach

In the Vygotskian approach the development of the child has a distinctly social character. Also the development is not just limited to the individual, but is much broader in the outlook; viz. a culture, a species, a child, a cognitive skill. The basic unit of development is the “active-child-in- cultural-context.” This unit is responsible for construction of different cognitive skills, including “system of meaning and its psychological tools.” The ideal end point in development of each culture is dependent of the goals of the particular culture. The goal of the culture is the basic driving force for the development of the child, and the interactions of the child with the society are responsible for this. The psychological tools or the higher mental functions are the parameters of the development of the child. A volitional control, conscious awareness of these higher mental functions represents a final step in the process of development [6] Chapters 5 and 6.

4 Psychoanalytic Approach

In the Freudian or the psycho-analytic approach three structures viz. the id, ego and the superego form the central basis of the theory. The id is the largest portion of the mind, is innate and is responsible for biological needs and desires. The id aims to satisfy the impulses without any delay. The ego which emerges in early infancy, is the conscious part of the personality and is responsible for the completion of id’s impulses in accordance with reality. The superego develops between 3 -6 years and incorporates the values of the society. The emergence, interaction and the struggle between these three structures form the basis of development. [2] pg. 14, [4] pg. 137.

5 Social Learning Theories

The learning theorists provide only a few universal behaviors as the act of learning itself depends on ‘what the environment has to offer.’ Since this theory accounts for development primarily as a quantitative change, one in which the learning episodes accumulate over time; the ability to skillfully learn what is observed or listened from the other people or by attending to symbolic characters or imitation in the society is developed in the children universally [4] pg. 201.

6 Ecological Theories

In the Gibson’s ecological theory child actively learns from experience and environment. The child learns to detect the structure, which specifies the information available to be perceived. Gibson has proposed four parameters for human behavior viz. agency, prospectivity, search for order, and flexibility. Agency “is the self in control, the quality of intentionality in behavior.” We see ourselves as distinct from the environment, and can be agent to cause the change in it. Thus with development our aspect towards this relationship changes. Prospectivity refers to the intentionality, planning and anticipation of the future. This is also seen to develop with the age. The search for order would involve the search for patterns, order and regularity in trying to make the sense of the environment. The aspect of flexibility comes into picture with the adaptation to the environment with whatever ‘skills’ one has. The affordances [“what an environment offers it provides for an organism; they are opportunities for action”] needed for working in another setting are obtained by changing the activities [4] pg. 360.

7 Modularity Nativism

The term modularity nativism refers to a set of approaches that postulate certain innate modules, structures or constraints, each specialized for a particular domain of cognition [3] pg 20. The modules are ‘pre-programmed’ to respond to specific sorts of information. These innate modules require a ‘trigger’ in form of little experiences, with appropriate content, to be activated. The different modules are posited to be relatively independent of each other, such that the development in one does not overflow into another. The developmental changes in thinking are caused by external factors such as maturation [4] pg. 427. This in turn implies that the infant mind is not very different from that of an adult.

8 Theory Theory

The theory theory approach is another domain specific approach to child development, which likens the children’s knowledge to a scientific theory [3] pg. 20. The children are capable of constructing intuitive, folk , everyday na ̈ıve “theories” for a particular domain [4] pg. 423. According to this theory the child has different theories for different domains. In the development process the children ‘test’ these intuitive theories, just like a scientists, in light of their experiences, thus they are like ‘little scientists’. So the answer to the question, What Develops? is that these intuitive na ̈ıve theories develop, with the experience of the children with the real world.

9 Dynamic Systems

The dynamic systems approach to child development addresses change over time in the complex holistic systems, especially self organizing ones [4] pg. 432. The term dynamic system most generally means “simply systems of elements that change over time.” In dynamic systems we have two basic themes for development [5] pg. 563:

  1. Development can only be understood as the the multiple, mutual, and continuous interaction of all the levels of the developing system, from the molecular to the cultural.
  2. Development can only be understood as nested processes that unfold over many time scales, from milliseconds to years.

One of the metaphors that is used to explain the dynamic systems approach is a mountain stream . The behavioral pattern are analogous to the eddies and the ripples of a mountain stream. In mountain stream metaphor “behavioral development is seen as an epigenetic process, that is truly constructed by its own history and system wide activity” [5] pg. 569. Thus development is seen as a process in which new behavioral patterns emerge because of interaction. [5]

References

[1]  Aslin as quoted in [3] pg. 47.
[2]  Berk L., Child Development 3rd Ed. 2001, Prentice Hall of India
[3]  Flavell J. H., Miller P. H., Miller S. A. Cognitive Development 4th Ed. 2001, Prentice Hall
[4]  Miller P. H., Theories of Developmental Psychology 2001, W.H. Freeman
[5]  Thelen E., Smith L. B., “Dynamic Systems Theories” Chapter 10 in Handbook of Child Psychology : Vol. 1. Theoretical Models of Human Development 1998, Wiley
[6]  Vygotsky L. S., Thinking and Speech Ed. Rieber, Carton The Collected Works of L.S. Vygotsky, Vol. 1: Problems of General Psychology 1987 Plenum