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
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.
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.
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 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:
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.
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
