## Doubling the cube with ellipses

The problem of doubling of the cube which emerged in the context of the doubling of the cubical altar of the great god Apollo cannot be solved using just a straight-edge and a compass. It needs one to construct a segment of length of $\sqrt[3]{2}$. The yavana sages of yore solved the problem using various special curves: Menaechmus and Dinostratus using parabolas and hyperbolas; Eudoxus using the kampyle; Nicomedes using the conchoid; Diocles using the cissoid. But to my knowledge no yavana used the other conic, ellipse, to solve this problem. Who did it first whether it was the French savant Descartes or some other modern mathematicians it is not entirely clear. However, it can be done using segments which can be constructed just using a compass and straight-edge i.e. rational fractions and square roots (which can be drawn using the geometric mean theorem), and ellipses drawn using those lengths. To draw an ellipse we need to know its semimajor and semiminor axes. Additionally, we need to know its center to locate it based on a particular point and the direction of the axes. Below is the procedure to do the same:

Figure 1

1) Let $\overline{OA}$ be the side of the cube to be doubled. All lengths/distances from now on will be expressed in $\overline{OA}$ units.
2) Draw a line by extending $\overline{OA}$. Draw two lines perpendicular to it at points J and K at distances $\frac{1}{3}$ and $\frac{1}{2}$ respectively from point O.
3) On the line through J mark point B such that $\overline{JB} = \frac{1}{2} + \sqrt{2}$. On the line through K mark point C such that $\overline{KC} = \sqrt{2}$
4) With B as a center to draw an ellipse with semimajor and semiminor axes $a_1=\frac{1}{2}\cdot\sqrt{\frac{31}{3}}; \; b_1=\frac{1}{2}\cdot\frac{\sqrt{31}}{3}$ and major axis along $\overline{JB}$.
5) With C as a center to draw an ellipse with semimajor and semiminor axes $a_2=\frac{1}{2}\cdot\sqrt{6}; \; b_2=\frac{1}{2}\cdot\sqrt{3}$ and major axis along $\overline{KC}$.
6) Obtain the two points of intersection of the ellipses: D and E. Drop a perpendicular from D to $\overleftrightarrow{OA}$ to cut it at point F.
7) Draw $\overline{OF}=\sqrt[3]{2}\cdot \overline{OA}$.

The secret behind this rather simple-looking method needs some algebra to simultaneously solve the equations of the two ellipses choosing coordinates and axes appropriately which in the pre-computer era could have been difficult for the lay person (Remember mathematicians like Descartes handled such calculations long before computers though). Below are the double cubes as an offering for Apollo or should we say our equivalent deity Rudra.

Figure 2

## A superficial look at national population density and some life history features

Over the years we have repeatedly checked out various collections of data pertaining to the human condition in the same manner as we attempted to apprehend scientific data. We have wondered whether to write on them in any detail. We desisted for the examination was rather superficial and these are matters that are subject to weighty study and superior illustration by those directly involved in such studies. Yet, we finally decided to do so because sometimes even simple illustrations and only partly correct hunches when done by yourself might spark useful lines of thinking and contribute to your understanding of the world. This is the caveat for any such note.

Below is one such examination of the population density of nations of the world. We have a dataset of 242 nations, territories and nation-like entities most derived via the United Nations data repository which forms the basis of this analysis. The mean population density for these 242 is about 400 people/square km and the median is about 84. The first panel of the first graph is the histogram of log(population density) of these 242 nation/nation-like entities. It is pretty normally distributed and might be even more tightly distributed (boxplot in second panel) so but for 1) unusual city states: Saint Martin, Bermuda, Malta, Maldives, Bahrain, Gibraltar, Hong Kong, Singapore, Monaco, Macau (one can see them as the long tail of outliers in the boxplot below); 2) sparse countries like Greenland, Mongolia, Namibia, Australia.

Given the relatively tight distribution of the rest we plotted a scatter of population versus area in square km for all these nations both on the log-scale.

As can be see from the figure they are color-coded by continent: Asia, Africa, Europe, Oceania, North America, South America, Australia, Central America, America (Rest). The two countries in Asia, India and Japan, with high population densities but very different structure (one a subcontinent one a large island group) are circled on the plot. The plot is pretty linear and has a decent $r^2= 0.7024$, holding across continents and a wide range of area and population magnitudes. This suggests that there is an intrinsic limit likely from carrying capacity of land. The high-density nations like India and Japan show how even with very different structures similar population densities can be achieved over large magnitude range of both population and area.

The next plot has mean life-expectancy by continent (with mean for all continents). Mean life-expectancy for the African continent significantly deviates from the mean life-expectancy for all other continents ($p= 2.5 \times 10^{-11} .. 1.5 \times 10^{-14}$ by t-tests).

The life-expectancy is shown in the next plot on the world map. Since humans originated in Africa one might speculate that this lower life expectancy reflects a state closer to the condition of the early humans. But then it could also be due to disruption of the traditional ways of life in Africa due to the clash of civilizations. The clash of civilizations might also be behind another feature that becomes obvious from the map – the Western European nations and their leukospheric settlements enjoy some of the highest life-expectancy in contrast to old civilizations like India, China, Iran (now under Moslem occupation). This disparity might reflects the monopolization of world wealth and resources by the former during the period of their ascendancy often at the expense of the latter leaving them broken.

We next plot life-expectancy versus population density (both on the log-scale). One may ask why these are being plotted. One could reason that higher population densities mean lower sanitary conditions, easier spread of diseases, greater population, lower resources, all resulting lowered life-expectancy.

However, the plot makes it clear that there is hardly any correlation between the two ($r^2=0.0209$). This means humans have long crossed that time when high population densities could not be sustained for reasons as above. But what comes out in the plot again is the separation of Africa (orange) versus the rest irrespective of population density. Africa (especially sub-Saharan Africa) is special in this regard because it fares worse than any of the other tropical nations. Thus, it is not just tropical diseases but their special prevalence in Africa and other factors that lead to the observed life-expectancy. If biological diseases can result in low life-expectancy then the same can arise from memetic diseases and damage due to them. There is only one nation outside sub-Saharan Africa which has life-expectancy in that range – Afghanistan which was once home to a brilliant expression of the Hindu civilization. So why is it there? The answer quite plainly is the memetic disease known as the ekarākṣasonmāda. In Asia, Japan and India present extremes for similar population densities in terms of life-expectancy (both circled again). The Asian city-states enjoy a similar life-expectancy as Japan. We hold that India too is damaged by the same memetic diseases. Given India’s human biodiversity and tropical position we may not be able to reach Japan or the Asian city states in terms of life expectancy. However, Sri Lanka shows us that with similar population density, without the unmāda-s having any power, what can be in the least achieved in principle. I am sure people will think I am trying to connect unrelated things and shift the blame on the unmāda-s for India’s intrinsic problems. But those who are discerning enough will see why this is not far-fetched and the comparison is valid.

It is likely that this lower-life expectancy has had profound impact on other aspects human life-history and strategies. In that context we plot below average national IQ (where available) versus life-expectancy (color-coded by continent; here Africa is green).

The average national IQ values are controversial and have been disputed. Yet, for general trends they might have some value. The correlation between IQ and life-expectancy is reasonable ($r^2=0.61$ for 174 countries). People have argued for IQ being a predictor of life-expectancy in individuals and this holds even when averaged for countries. Their conjecture runs along the lines that higher IQ favors better education and understanding of complex relationships. This might in turn inform against risky behaviors and enable foretelling less-apparent dangers. But it is not entirely clear if the arrow of causation does run that way on the country-level. It is possible that emergence of higher life-expectancy allowed for higher IQ to make a real difference, thus selecting for it. In the lower-life expectancy regime it might not have mattered as much as death could strike irrespective of that.

## Some biological analogies for certain sociopolitical issues

In Hindu society we often see certain relatively straightforward sociopolitical issues endlessly debated. A person with relatively commonplace IQ should in principle easily arrive a correct apprehension of these issues by applying correct analogy and/or logic. However, due to emotionalism and other factors, as discussed in this fictional account, they do not grasp the matter for some reason. On other hand an excuse that is commonly used is that we do not have a yard stick for properly positioning a geopolitical stance. Here is where, even for very simple matters, biological analogies can be very useful. They are particularly useful because the adverse edge of natural selection is most unforgiving – there is no place for fancies and only what works finally makes it past the hatchet. Likewise, the role of chance and absence of constraints are also placed in proper perspective.

To build the analogy for the matters under consideration we start by a brief description of certain basic biological facts for the less-aware reader of these pages. In any biological system be it a living organism or a virus or some other nucleic acid parasite like a plasmid the biochemical functionality is divided into two basic categories: 1) Core functions and 2) peripherals.

The core functions pertain to the essence of life. They are evolutionary more strongly conserved than the peripherals suggesting that there are much stronger constraints on them – purifying selection is the stronger force active on them. Now, within the core functions there are three broad domains of biochemical activity: 1) Replication of the genetic nucleic acid. 2) Transcription of the genetic nucleic acid into RNA for templating proteins or for action by itself. 3) Translation: the process of making proteins based on the genetic information encoded in the messenger RNA. Of the three process the maximal evolutionary constraints are seen on the translation. The translation system is what is universal conserved across the three domains of life. Its basic components, proteins (ribosomal proteins and translation factors) and RNA, are easily recognizable as cognate between the three superkingdoms of life. Their phylogeny has a strong vertical signal i.e. they have been inherited from parent to offspring for billions of years from the last universal common ancestor. There is lower lateral transfer i.e. exchange of the genes coding for these components between distantly related organisms. Even when that does happen it works alright because the system is so conserved. After the translation system the transcription system is the next highly conserved – in the form of the RNA polymerase subunits. Finally, the least conserved of the core systems are the replication apparatus proteins starting from the DNA polymerase down to other components like the replication enzymes like ligases and helicases.

What this tells us is that in general diversity is not tolerated in these core systems. Even more tellingly the strongest constraints being on translation and then transcription indicates that in an organism the language and apparatus for communicating and interpreting the genetic information tolerates even lesser diversity than the apparatus for copying the genetic information for inter-generational transmission. Furthermore, organisms are constantly locked in biological conflicts with each other and one of the most frequent targets for attack across all levels of life’s organization is the translation apparatus. This is not just because it tends to be conserved but also because crippling it is most damaging for an organism (just think ricin). Despite of all of this the bottom-line is that the common language of translation remains in place and all attacks against it are countered by increasing fidelity in face of attacks rather than just evolving away from the attack by diverging. It is indeed this common language that has made intimate symbiosis possible. Two distantly related bacteria Sulcia (a bacteroidetes) and Hodgkinia (an alphaproteobacterium) reside as endosymbionts in insects known as cicadas. They have individually degenerated to such an extent that neither of them encodes a complete set of ribosomal proteins or tRNA synthetases. They instead make it up by complementing these components. Also they share a common $\sigma$ subunit for their RNA polymerases (encoded by Sulcia). This would not have been possible if the core system was not so well conserved between such distant organisms. Likewise eukaryotic organisms like us have evolved only because of this common language in the core systems – the eukaryote is a symbiont of an alphaproteobacterium and an asgardarchaeon. A now extinct bacteroidetes was also likely part of this endosymbiosis. The mRNAs of bacterial origin can be translated in the archaeal-derived translation system for use in the bacterial endosymbiont (the mitochondrion). This again is only possible due to common language and the ability for it to be deciphered the same way between the distantly related organisms. Indeed close to the origin of life with which we are monophyletic might have involved such a process. First, we notice that even today there is some surviving diversity in the types of distinct RNA polymerases catalyzing transcription and the DNA polymerases catalyzing replication but we see only one ancient translation system that can produce genuine proteins. Why might this be the case. It is likely because in the earliest phases of life’s evolution those systems that shared a common code and code-reader could exchange genes more effective and as a group have a fitness advantage over those with unique codes and translation systems that could not be read by others. Thus, the common language helps the group resources to come together more effectively.

Peripherals include many other biochemical functionalities. Some of the most important being: 1) energy production; 2) metabolite biosynthesis; 3) sensing and signaling external and internal environmental changes; 4) weaponry for biological conflicts. These, in stark contrast to the core components show enormous diversity between organisms, often even between closely related ones. Why? Because here there tends to be more of positive selection which might diversify information rather than purifying selection which favors stasis. This is easy to understand: being able to use different kinds of energy production mechanism or metabolite biosynthetic mechanisms help exploit different kinds of niches. Likewise, sensing and signaling needs also changes with the local niche which the organism adapts to: an organism capable of anaerobic growth needs to sense oxidation potential in its environs. An organism that uses light needs light sensors while one living deep in a cave does not. Thus, these systems are constantly under selection for diversification. Biological conflicts involve cycles of evolutionary arms races thus diversity is often an asset in these systems. Interestingly, our studies indicated that genes involved in these more peripheral aspects of life like exploiting unusual compounds or resistance against xenobiotics tend to be more “noisily”expressed in a population of cells. So the cells appear to be hedging their bets via a diversified strategy even as the cliché, which your financial adviser might use regarding a your investment strategy.

Thus, in conclusion there are some systems where there is a strong selection for conservatism and preservation of a common language and “sense” while in others there is simultaneous selection for diversification – all explainable by the same process of natural selection i.e. survival of the fittest. How and where does all this basic biology serve as an analogy for sociopolitical issues.

For that let us first look at the development of the Hindu nation in the Indian subcontinent. Prior to the coming of the Indo-Aryans, there was probably already some degree of unity established by the neolithic and early metal age agricultural cultures. We still see echoes of that unity in our agriculture-/rural-economy- related words from an extinct language (e.g. kustumburu, karkaṭi, kulāla, kīnaśa etc). The coming of the Indo-Aryans sometime during the Harappan period or after it provided a strong unifying force in the form of a common language (old Indo-Aryan aka Sanskrit in its living expression) and religion (the extant Hindu dharma in its living expression). As the Indo-Aryans and Aryanized peoples spread over the vast subcontinent over nearly a millennium this factor played a major role in the formation of a unified system in the subcontinent. Certain mobile groups played a over-sized role in this unification but the more sedentary groups played along rather than oppose them. However, at the same time, the longitudinal extent of the subcontinent and major terrain variations implied that the phenomenon of local diversification in terms of linguistics to a greater extent and religion to a smaller extent was also going to take place. As a result in the coming two-three thousand years we had a peculiar situation where both diversification and unity simultaneously existed – a cliché Indians oft use.

All this was fine as long as India remained heathen. But starting with the irruptions of Mohammad bin Qasim and culminating in the conquests of the Turkic Mohammedans this heathen unity was mostly broken due to the direct attacks on the centers of the Hindu religion and the unifying Sanskrit language (despite protestations of white supporters of the Mohammedans to the contrary). However, the Mohammedan tyranny in India did not great break the diversified local linguistic formations and to some extant not even the local (“folk”) expressions of religion. This had a negative effect on the unity of the Hindu system. The Hindu reconquest of the subcontinent was neither complete nor prolonged enough to restore the situation to the old state when English conquered the subcontinent from them. While the English imposed their language, which we continue to use today, that too did not deeply penetrate the Indian masses. In the end when the English departed and we had partial restoration of Hindudom under a secular state we were left with the main unifying undergirding in the form of the Hindu religion and Sanskrit language in shambles. In contrast, the diversified regional languages and the identity associated with them remained relatively untouched. This was only strengthened with the secular state caving in to the demands of local identity by instituting linguistic states and calling for Hindi (whatever its quite serious merits) rather than Sanskrit as the national language.

Thus, today we are faced with a scenario where a major component of India’s masses have their local languages rather than nationalism or national issues as the biggest animating factor! Indeed it is rather palpable that it forms the core of their identity much more than the Hindu religion or the Indian nation – you commonly see a man typically call himself first a Tamil or Marathi or something like that rather than a Hindu. This subnational identity was perhaps fine in world where there were no serious conflicts with extrinsic doctrines like Abrahamism. Some of my interlocutors have objected that local languages have stymied the progress of Abrahamism because they preserved local non-Abrahamistic cultural elements. They have bellowed against Hindi as purveyor of Abrahamism via its Mohammedanized register, more correctly the Urdu. But our own observations indicate the opposite – an unhealthy local language identity transcending religion is widely seen – Panjabi Hindus sympathizing with Mohammedan speakers of that language (often across the modern border); Bengalis defining themselves as that rather than as Hindus; the Malayali identity which wraps in Mohammedans and Christians alike and so on.

Hence, we posit we that when confronted with large groups unified by their urge to place heathens like us in the grave or the museum there is not much to be gained from the local linguistic diversity and an identity based on it. It will simply result in the eventual ruin of each group by itself. Hence, for our survival we need to function as a larger unified group with a natural unity coming from the pre-Mohammedan past where Sanskrit and the Hindu dharma played the role of the unifying framework. This unified super-group of Hindus would have much greater chance of survival against the Abrahamisms and be able to better command the resources of India and abroad. Here is where the biological analogy buttresses things. For this symbiosis as a super-group we need a unified language and religion which forms the basis of our identity. That common language with a common script for it would be like the translation and transcription apparatus. The analogy gives a decisive basis for whether linguistic and associated script diversity is good or bad diversity – diversity here will not be advantageous by any means and will not increase robustness in the system. Hence, the only hope for Hindus is to surrender their local linguistic identities for a national unitary linguistic identity. The religion might be compared to the replication apparatus it allows greater diversity than the translation or transcription apparatus but is still mostly part of the conserved core. So we can have some diversity there but again unity dominates.

So let us be resoundingly clear about this thing – not all diversity is good or a strength – there are systems where tolerance for it has to be low. Now that said the biological analogy should also show you where diversity can be useful. Putting the nations future on IT or making just doctors and low-grade engineers is unlikely to be a recipe for success. However, since something as simple as this is not easily understood by people, it is clear that, like with many other issues, the going is not going to be smooth for our people – and that is putting it mildly.

## The magic of the deva-ogdoad

Classical Hindu tradition holds that the ogdoad of deva-s corresponding to their directions is:
Indra: East; Agni: Southeast; Yama: South; Nirṛti: Southwest; Varuṇa: West; Vāyu: Northwest; Kubera: North; Īśāna: Northeast.
The central position might be occupied in certain traditions by Prajāpati, Viṣṇu, Savitṛ or the Puruṣa-Prakṛti continuum from which the gods differentiated.

Now, a famous verse, quoted by the great scientist-mathematician Bhāskara in his Līlāvatī, says:

indro vāyur yamaś caiva nairṛto madhyamas tathā |
īśānaś ca kuberaś ca agnir varuṇa eva ca ||

Now using the positions of the gods as in the classical ogdoad we get the below matrix:

$M=\begin{bmatrix} vAyu & kubera & ishAna\\ varuNa & madhyama & indra\\ nirR^iti & yama & agni\\ \end{bmatrix}$

We then plug values for each of the elements of the matrix based on the positions of the gods in the above verse. Indra= 1; Vāyu= 2; Yama= 3; Nirṛti= 4; madhyama, the middle or mean= 5; Īśāna=6; Kubera= 7; Agni= 8; Varuṇa= 9
When we do so we get the famed magic square (sarvatobhadra) of Hindu tradition known as the ubhaya-pañcādaśaka

$M=\begin{bmatrix} 2 & 7 & 6\\ 9 & 5 & 1\\ 4 & 3 & 8\\ \end{bmatrix}$

This magic square is found widely in Hindu tradition. It is one of the magic squares that is worshiped by the physician while deploying bheṣaja-s pertaining to child-birth along with a mantra to Agni, Vāyu, Sūrya and Indra. There it is specified using partial bhūtasaṃkhya notation as:

vasu-guṇābdhy-eka-bāṇa-nava-ṣaṭ-sapta-yugaiḥ kramāt |
yantraṃ tu pañcadaśakaṃ likhyate nava-koṣṭhake ||

8, 3, 4; 1, 5, 9; 6, 7, 2 in order are to be written in the 9 cells for the yantra with sum of 15.

It is also recorded in śaiva traditions as a yantra. Indeed tradition holds that the mathematics of magic squares (bhadragaṇita) was taught by the god Rudra to the yakṣarāṭ Kubera or his ectype Maṇibhadra (e.g. see Narāyaṇa paṇḍita’s chapter on bhadragaṇita).

There is literally much mathematical magic in the bhadra square:
1) If you add the rows, columns and diagonals you get 15, which is what gives it name. Thus the magic constant $\sum(M)=15$

2) If you reflect the square about any of the sides or transpose the matrix i.e. make rows into columns then the resultant square is also magic.

3) It contains the first 9 positive integers and is a $3 \times 3$ magic square; hence, its order $n=3$ and its elements are $1:n^2$. For such a magic square (known as a normal or fundamental magic square) the $\sum(M)=\frac{n^3+n}{2}=\frac{27+3}{2}=15$

4) If you put any sequence of consecutive positive integers serially in the same positions as the occurrence of 1:9 in the square then you get another magic square. Thus you can generate infinite magic squares. Tradition holds that the ancient astronomer Garga described this additive derivation of a sequence of magic squares. He states that sun is represented by the above deva square; by adding 1 successively the sequence of squares associated with other planetary bodies. Adding 19 to each element generates the Kaubera-sarvatobhadra:

$M_k=\begin{bmatrix} 21 & 26 & 25 \\ 28 & 24 & 20 \\ 23 & 22 & 27 \\ \end{bmatrix}$

Its magic constant is $\sum(M_k)=72$

5) For the deva (fundamental) magic square and all magic squares sharing the general form with it (see below for general form) the arithmetic mean of the elements of the matrix equals the central element or the madhyama as it is termed in Hindu tradition: $\mu(M)=M[2,2]$. It is both central positionally and in the sense of being the mean. Further, for the deva square and all its derivatives obtained as above $\sum(M)=\mu(M)\times n$, where $n=3$ the order of the matrix. Thus for the deva square it is $3\times 5=15$ and for the Kaubera square it is $3\times 24=72$

More generally we have the siddhānta attributed to Narāyaṇa paṇḍita which generates these magic squares using the madhyama ($a$) and 2 other numbers $b,c$ thus:

$M_g=\begin{bmatrix} a+c & a-b-c & a+b \\ a+b-c & a & a-b+c \\ a-b & a+b+c & a-c \\ \end{bmatrix}$

Thus we get the magic constant as $\sum(M)=3 \times a$. We get the deva square by plugging $a = 5, b = 1, c = -3$. This general form of the square can be used to prove other results as those presented below pertaining to such squares. By plugging $a=10, b=-2, c=6$ we get the ubhayatriṃśaka seen in śaiva and medical works:

$M_{ut}=\begin{bmatrix} 16 & 6 & 8\\ 2 & 10 & 18\\ 12 & 14 & 4\\ \end{bmatrix}$

Its magic constant is $\sum(M_{ut})=30$ the idealized number of days in the lunar month and its dot-product constant (see below) is 364 which is 1 short of the idealized number of days in the year.

6) Whereas the off-diagonals or the wrapped diagonals do not add up to $\sum(M)$ their sums $\sum(M_{od})$ do show a regular relationship: $\sum(M_{od})=n \times Cr(M)$ where $Cr(M)\Rightarrow M[1,1], M[1,3], M[3,1], M[3,3]$ i.e. the corner elements. Thus for the four off-diagonals of the deva-square we have:

$2+1+3=3 \times 2=6\\ 4+1+7=3 \times 4=12\\ 6+9+3=3 \times 6=18\\ 8+9+7=3 \times 8=24$

One immediately notices that the mean of the sum of these off-diagonals is: $\frac{6+12+18+24}{4}=15$. This property is a general feature that holds good for all the magic squares derived from the same general form as the deva square such as the Kaubera square.

7) Remarkably if we write out each row, column and diagonals (including the off-diagonals) as 3 numbers forward and reverse then the following relationships hold:

$276+951+438=672+159+834=\\ 294+753+618=492+357+816=\\ 258+714+693=852+396+417=\\ 654+798+213=456+312+897=1665$

Now if one does the same with any other additively derived magic square we get the same. But we need to keep in mind the places thus for the Kaubera square we get:

$212625+282420+232227=252621+202428+272223=\\ 212823+262422+252027=232821+222426+272025=\\ 212427+262023+252822=272421+232026+222825=\\ 252423+262827+212022=232425+272826+222021=727272$

Thus this summation property is a general one for such magic squares. Further if we denote this sum as $S$ then its ratio to the magic constant $\frac{S}{\sum(M)}$ is a mirror number of the form 111, 10101, 1001001 etc. For the deva square made up of single digit numbers it its 111. For magic squares with double digit numbers it is 10101. For magic squares with triple digit numbers it is 1001001, so on.

8) Notably, as above if we write out each row, column and diagonals (including the off-diagonals) as 3 numbers forward and reverse then the following relationship holds for their squares:

$276^2+951^2+438^2=672^2+159^2+834^2$
$294^2+753^2+618^2=492^2+357^2+816^2$
$258^2+714^2+693^2=852^2+396^2+417^2$
$654^2+798^2+213^2=456^2+312^2+897^2$

This is also generally true for generalizations of the deva square.

9) The dot product rules: Let $R_j, C_j$ be the rows and columns of such magic squares. For our 3rd order squares they define three element vectors. The dot product of these vectors would be a scalar e.g.: $R_1 \cdot R_2=M_{11}M_{21}+M_{12}M_{22}+M_{13}M_{23}$. Then for the deva-square and its generalization we have:
$R_j\cdot C_j=K$; where $K$ is the dot product constant of the magic square.
$R_1 \cdot R_2= R_2 \cdot R_3$ and $C_1 \cdot C_2=C_2 \cdot C_3$

10) We can create a square matrix $D$ by the process of cyclic difference of rows or columns: $D_R=[R_1-R_2, R_2-R_3, R_3-R_1]$ or $D_C=[C_1-C_2, C_2-C_3, C_3-C_1]$. $D_R, D_C$ are zero-sum semi-magic squares in that the rows and columns add up to zero but the diagonals do not. This applies for the deva square and its generalization. Thus for the deva square we have:

$D_R=\begin{bmatrix} -7 & 2 & 5\\ 5 & 2 & -7\\ 2 & -4 & 2\\ \end{bmatrix}$

$D_C=\begin{bmatrix} -5 & 4 & 1\\ 1 & 4 & -5\\ 4 & -8 & 4\\ \end{bmatrix}$

11) For the deva square and its additive derivatives we can define the eigenvalues $\lambda$ thus:
$(M-\lambda I)\overrightarrow{v}=0$; where $\overrightarrow{v}$ is the eigenvector and I is the identity matrix:
$M_k=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}$

Then for the deva-square $\lambda$ is obtained by solving the cubic equation:
$|M-\lambda I|= (\lambda-15)(\lambda^2-24)=0$
Strikingly, the eigenvalues are $15,-2\sqrt{6},2\sqrt{6}$. Thus, the largest eigenvalue square is its magic constant. More generally for all additive derivatives of the deva square the eigenvalues are: $\lambda= \sum(M),-2\sqrt{6},2\sqrt{6}$

For the deva square the Eigenvectors are:
$\overrightarrow{v_1}= \begin{bmatrix} -\frac{1}{\sqrt{3}} \\ -\frac{1}{\sqrt{3}} \\ -\frac{1}{\sqrt{3}} \\ \end{bmatrix}$;
$\overrightarrow{v_2}= \begin{bmatrix} -\frac{1}{3}-\frac{1}{\sqrt{6}} \\ \frac{2}{3} \\ -\frac{1}{3}+\frac{1}{\sqrt{6}} \\ \end{bmatrix}$;
$\overrightarrow{v_3}= \begin{bmatrix} \frac{1}{3}-\frac{1}{\sqrt{6}} \\ -\frac{2}{3} \\ \frac{1}{3}+\frac{1}{\sqrt{6}}\\ \end{bmatrix}$

Notably for all additive derivatives of the deva squares the eigenvectors take the above forms though the signs of the values might be reversed.

12) For such 3rd order magic squares each row, column, diagonal and off-diagonal can be seen as a triple of numbers that can specify a point in space as its x, y and z coordinates. We have a total of 3 rows + 3 columns + 2 diagonals + 4 off-diagonals= 12 triples. By permuting the order in which each number in the triple is taken we can specify 6 points from each triple. Thus we have a total of $6 \times 12 = 72$ points. Of these $x+y+z=\sum(M)$ for those points derived from the columns, rows and diagonals. Thus, 48 out of the 72 points lie on the plane $x+y+z=\sum(M)$. While those from the off-diagonals lie on 4 different parallel planes on either side of the main plane. From their equations we know
that these planes have a vector of the form $\langle 1,1,1 \rangle$ and will make with the horizontal plane ( $z=0; \langle 0,0,1 \rangle$) the angle: $\arccos\left(\frac{1}{\sqrt{3}}\right)=\arcsin\left(\sqrt{\frac{2}{3}}\right)= \arctan\left(\sqrt{2}\right) \approx 54.7356^o$

Thus, these points define a regular hexagonal bipyramidal frustum, with the 48 points magic constant points on the principal plane forming a nice pattern of 4 concentric hexagons.

Figure 1: the principal plane with 48 points from the deva square

The 12 points from two of the off-diagonals form 2 hexagons congruent to the inner-most hexagon on either side of the principal plane and lie farthest from it. The remaining 12 points from the other two off-diagonals form 2 hexagons on either side of the principal plane and are congruent to the last but 1 of the concentric hexagons counting from inside.

Figure 2: The hexagonal bipyramidal frustum of the deva square

For the deva square the four concentric hexagons have side-lengths of $\sqrt{2}, 2\sqrt{2}, 3\sqrt{2}, 4\sqrt{2}$, with the first and the last being also the sides of the hexagons forming the base and the height of bipyramidal frustum. The pyramidal edge of the frustum is $3\sqrt{5}$ and the remaining off diagonal hexagons cut the pyramidal side into segments in the proportion $1:2$. It may be considered the maṇḍala of the deva-s.

13) Such magic squares can also be converted to the Hindu magic triangles using Narāyaṇa paṇḍita’s simple algorithm. Below is illustrated the 4-fold magic triangle, the pañcaviṃśati-yoni derived from the deva square.

Figure 3: the pañcaviṃśati-yoni

While magic squares might seem to the deracinated modern as a quaint superstition of the Hindu and some other old civilizations like the Chinese and Japanese, they might have some deeper significance. For example there is the deceptively simple looking but yet unsolved mystery: does a order 3 magic square exist where all the elements are perfect squares, i.e. are they squares on sides of a right triangle with integer sides? Here we have looked only at what the Hindus termed nava-koṣṭhaka-viṣaṃagarbha-bhadra-s. But the several other magic figurate numbers described in Hindu tradition might have other interesting mathematics lurking in them. Notably, the great Srinivasa Ramanujan also (re)discovered several saṃagarbha-bhadra-s similar to traditional Hindu versions (like the ones found on Candrātreya temples, including one demolished by the Moslem tyrant Baboor at Gwalior) with interesting properties. Additional also gives the above formula for the general nava-koṣṭhaka-viṣaṃagarbha-bhadra-s

More generally they might be seen as a numerical representative of a conservation law and the structures that emerge naturally once such a law is laid down. One can also use it as an analogical aid for understanding structures emerging under constraints. One could conceive an imaginary model for a protein where the amino acids (standing for numbers in the square) are under interaction constraints with others to attain some total constant stabilizing energy. Under such a scenario the magic square plane as derived above would represent a $\beta$-sheet.

## Marching onward in the American spring but where to?

We hear in the news that the students (and whoever else) at the University of California, Berkeley, are in state of ferment. This is unsurprising in itself given that the it has for long been the center of American student protests and riots. More generally several top American universities are refugia of Marxian ideology, which was made irrelevant in mainstream politics through intense action of the likes of McCarthy and Hoover. This new round has coincided with the election of the new mleccheśa and is centered on clashes between the supporters of the said mleccheśa and the Marxian elements. A connection may also be seen between these upheavals and earlier ones during the reign of the previous kṛṣṇa-mleccheśa in the form of the Occupy movement.

We are no strangers to student rebellions. In our own school we were active in inciting such against preta agents and in college against a tyrannical head of the department. We had also witnessed with interest rebellion that took place among medical students in a military medical school, which was swiftly squashed by the military authorities. We might even concede that there might have been some organic elements in the old Berkeley rebellions relating to the wars the US was fighting far from its shores with little to gain and notable (by American standards) loss of lives. However, rather than being an organic movement the current ferment appears to be one which is underwritten and incited by certain parties that are quite obvious to the discerning. First, while we were a somewhat atypical student we still had to earn a degree via grad school and one thing is certain – you cannot get too far by wasting your time on extracurricular activities like setting fires on the college campus or fighting Nazis. Second, we have visited several American campuses including multiple UCs – one thing is clear – majority of those students can hardly be described as the most needy or oppressed. In fact foreign students, such as us, had much greater pressures to bear due to the uncertainties and threats of the mleccha immigration system and that we were in an alien land with nowhere to really go in the event of failure. Yet, most foreign students in masters and PhD programs lived comfortable lives, often larger than reality – stacking up credit card bills, enjoying a peculiar kind of student life with coethnics and others, sometimes sex, alcohol and other oṣadhi-s as they gradually meandered their way to a PhD. If this was their existence what to say of the American students?

Thus, we posit that in most of these schools these students are not facing some existential crisis or serious want that they need to rebel and riot. They are not facing any real invasion of Nazis or some other great power that is going to obliterate whatever freedoms they currently enjoy. Not that they are too aware that freedom in the Anglosphere is something which is carefully managed too. Yet all in all one can hardly say their lives are intolerable; quite the contrary. In fact what they seem to be primarily demanding is shutting down the free speech of those who do not fit their conception of propriety. As the American psychologist Jonathan Haidt pointed out the student seem to come out of college more fragile and less open to ideas. We would simply state that they are exhibiting the usual convergence with their memetic cousin marūnmāda – the same policing (e.g. ISIS or the hellhole of the Sauds) and parallel “blasphemy” laws typical of the classic Abrahamisms.

The Marxian streak in the American academia has long had such culture. As example one might cite the famous Dick Lewontin. He along with his fellow Marxian authors Leon Kamin and Steven Rose introduce their book titled “Not In Our Genes” thus:
“Over the past decade and half we have watched with concern the rising tide of biological determinist writing, with its increasingly grandiose claims to be able to locate the causes of the inequalities of status, wealth, and power between classes, genders, and races in Western society in a reductionist theory of human nature. Each of us has been engaged for much of this time in research, writing, speaking, teaching and public political activity in opposition to the oppressive forms in which determinist ideology manifests itself. We share a commitment to the prospect of the creation of a more socially just – a socialist – society. And we recognize that a critical science is an integral part of the struggle to create that society, just as we also believe that the social function of much of today’s science is to hinder the creation of that society by acting to preserve the interests of the dominant class, gender, and race. This belief – in the possibility of critical and liberatory science – is why we have each in our separate ways and to varying degrees been involved in the development of what has become know over the 1970s and 1980s, in the United States and Britain, as the radical science movement.”

Then the three authors go on to thank their colleagues: “But we would like particularly to mention: members of the Dialectics of Biology Group and the Campaign Aganist Racism, I.Q. and the Class Society, Martin Barker… Stephen Gould…Richard Levins…Eli Messinger … Peter Sedgwick … Ethel Tobach.” [we have typed in only few names of their list to illustrate some prominent Marxist figures: e.g. biologist Richard Levins who ridiculously mentioned how work on evolution was closely inspired by Marx]

It should be noted that despite this allusion to being under oppression, people like Lewontin lead a luxurious life, which older American academics enjoy while preaching Marxian doctrines as. For many of these people who have not seen true hardship we cannot stop from psychoanalyzing their revolutionary concerns as a craving for higher “moral ground”. At the same time one can also see the secularized Abrahamistic tendency of screaming about being persecuted while being the persecutor himself.

The dominance of such academics epitomized by Dick Lewontin in American academia over the past 50 years or more has set the ground for the student mass to be indoctrinated in ways that make them putty ready for shaping by the bigger players in rebellions that favor their causes. While Lewontin is a case from biology, we have this phenomenon across the academic spectrum with striking parallels. While the casual observer might characterize their leanings as liberal or leftist (not wrong in general terms), the more observant will notice a specific tendency, which is clearest in the most finished versions of these players: a love for something akin to what Aristotle called eristics. In old yavana eristics one took both sides of an issue and alternately argued for the side they took being correct. Thus, the truth or the consequences of the observed data does not matter much for this academic; rather he would simply take a particular stance he perceives as favorable to his ideology and cause, a build a formal-looking framework for it (“theory”), and argue for its correctness. Thankfully, objectivity being a major factor in the sciences has curbed this practice outside of areas of biology that directly intersect with human biodiversity. However, in the so called social sciences it has been pursued untrammeled.

The way such practices have gotten the upper hand in the American academia is also interesting to observe. One area which illustrates this point rather clearly is the endeavor of white indology. White indology may be divided into three three broad sectors: 1) Old white indology of the English-speaking world (Anglospheric indology). 2) Continental European indology with a dominant German school. 3) Neo-American indology. The first two are the older sectors of white indology which declined in the years following the second world war due to the Hindus getting rid of English rule and the catastrophic defeat of the German state in the same war. The third sector was on the ascendant at the same time as the other two declined, matching the rise of the USA as a superpower. While white indology was always dominated by an anti-Hindu streak, arising from the very forces which engendered it, its neo-American manifestation, to start with, had a somewhat positive set of founders like Ingalls and Brown. However, it attracted in droves the same class of individuals as Dick Lewontin and his cohorts during its first years. These then went on to populate white indology in the coming years spreading the same class of doctrines which were at work in the muzzling of sociobiology (e.g. the attack on Ed Wilson). This, negated any positive start it might have had and turned it towards an even more pernicious strain of prati-dharma than the old flavors

So what is the form of these doctrines? As we have pointed out repeatedly on these pages they stem from the old Abrahamisms but masquerade under a secular mask. Their Abrahamistic roots are often missed by the casual observer because the practitioners of this secular strain are often seen attacking the older religiously expressed Abrahamisms. But even here there is a certain distinction: while the second Abrahamism is strongly attacked the proponents are milder on the root Abrahamism and often even synergistic and admiring of the third version. The resonance with the third stems from a certain essentialism (i.e. back to the basics) that the two share. Indeed, the uncompromising theoclasm of the third Abrahamism coupled with its message of “universal brotherhood” resonates very strongly with the secular variant. More generally this secular mutation shares with one or more of the religious versions an urge to: 1) broadcast aggressively a pathway to a utopian culmination; 2) “save” people against their wishes; 3) to claim rewards for doing the broadcasting and saving among unbelievers; 4) theoclastic frenzy; 5) to kill when the offerings are rebuffed or the doctrine is criticized by unbelievers. If they cannot literally kill in the academic environment they would do all within their means to silence the unconverted. 6) to claim oppression (demand safe-spaces) while being the oppressor himself (compare with Mohammedans demanding safe-spaces from Hindus like Pakistan or Bangladesh while exterminating Hindus themselves). Moreover, the process of conversion involves the normative inversion parallel to the Mosaic distinction of the classical Abrahamism. This can be observed most strikingly in the species Mahometanus secularis, converts from to the secular meme from the 3rd Abrahamism [Appendix 1].

Over 50 years of such indoctrination in American and more generally occidental academia has produced whole lineages of such secular practitioners that the unconverted barely exist any more. Importantly, even as Abrahamism sees idolatry and polytheism as inconceivably sinful wrongs (e.g. see statements from the Saud hellhole or their cousins in the Islamic state) this body of academics and students view the existence of the alternative views as similar inconceivable sins that deserve a kind of punishment similar to the polytheist under Abrahamism; likewise with apostates from the system.

Given this background, one may now look at the “March for Science” which recently took place in capital of the USA and other cities. Was it all about science or a ferment stemming from something else?

From first hand experience we can say that the funding and management of science does face several serious problems. This should be cause of concern of scientists since it directly affects what they do. Yet, beyond the lip-service or an important-sounding position article I have hardly seen anything being done to address the problems because those problems simply do not affect those in power who are the sūtradhārin-s in the system. Beyond funding and management there are indeed other real issues too. While the conduct of science is closer to a meritocracy in principle, in practice there are genuine discriminations against real talent and rewards for the less-deserving. There are even more serious issues of fraudulent activities driven by the high monetary stakes and lust for extended-abstract-type tabloid publications. If scientists were to agitate and swing into action for remedying these matters they would certainly have my support. But these are not exactly the things spurring them into the march it seems:
1) When you see early-career scientists (e.g. post-docs), who really need to be putting their head down and racking up the results to find a job in the ultra-competitive Euro-American market taking not just a weekend off for the march but spending whole months on it, it raises a red flag.
2) When you see issues, which should be peripheral to science, like rare atypical sexualities, the 3rd Abrahamism, and women’s issues being a major factor among the marchers one needs to look more closely.
3) As we have remarked a good part of the “high-power” American faculty lead large lives but when they talk of suffering one needs to ask why?
4) Then when you see such people worrying about the environment when they live in a country with one of the most pristine environments as well as being one of the most inveterate user of fossil fuels one almost recognizes the faux concern for the other typical of the Abrahamist (e.g. the aspersion at Hindu cremation in India – why not stop using your gas-guzzling car before worrying about Hindoos burning their corpses).
5) Making noise about climate studies rather putting ones head down to study the actual data along with falsifications that are in the same league as those seen in cancer biology and high-profile molecular biology publications makes one wonder the politics of the climate-change group

In conclusion, the observant cannot miss the point that the occidental academic, while having shed overt Abrahamisms due to clashes with scientific empiricism, have only adopted a secular mutation thereof, often ensuing in some form from the unkempt prophet Marx. With several decades of preaching this religion in academia and converting students by the droves they are now raw material for those wishing to enact more widely the agenda of this religion and by the usual pulls the 3rd Abrahamism. Now, one may point out that this is purely a struggle inside Abrahamism, like between Shia and Sunni or Orthodox and the western flavors of the preta-cult. But just as Sunni and Shia alike can be united in the qatl al-kāfir, i.e. polytheist heathens that we are, so is this secular cult. With a large number of the Hindu cognitive elite in the US and their children, the ABCDs, undergoing rapid conversion at the hands of these “dāyi-s” to be sent back as yenicheri to attack the Hindus, this does pose a serious threat. Moreover, the Indian scientific establishment closely apes the American one. The most important Indian scientific institutes are filled with such converts who hold and propagate the same views as their teachers in Ma’ssaland. As we have said before, scientific success of a nation can come about only when it has a nationalistic urge to succeed. All of this can be subverted by these converts within the system. Hence, it is not merely a intra-Abrahamistic struggle with no consequences to Hindus.

Appendix 1
We had a conversation in an academic context with a secular TSPian from who acknowledges some problems with Mohammedanism and favors its replacement by “attitudes characterized by a more rational scientific outlook”:
TSPian: “But on the Indian side the brāhmaṇ-paṇḍit people hate Mussalmans and Islam. They discard the fact that it teaches universal brotherhood. Would you not be supporting this hate?”

We: “See – hate for the other is something which is characteristic of Abrahamistic religions like Mohammedanism. Why foist it on the brāhmaṇa-s. Do Mussalman-s not hate the kaffr who simply does not want to convert to and considers that religion as mostly being a negative path? With this premise it is easy to see the other side which simply want’s to defend its space as hateful.”

TSPian: “I know some people misunderstand Islam and unpleasant things happen sometimes due to their actions in the name of Islam. That is why I am myself for a totally modern path which recognizes human rights and does not allow hate from either religion.”

## The upper story in a few pictures

We have always held that there is no substitute to the knowledge gained from handling real data. It is always superior to one gained from mere reading without reproducing or self-investigation. Hence, we have endeavored to play with various objects of scientific or mathematical inquiry as part of appreciating them. That said, what we are presenting here is by no means an elaborately worked out presentation that takes into account all the data, the pitfalls, the controls or the tests. It is merely a “superficial” exploration of the data. The data itself is not first hand. It is gleaned from many sources but mainly from the chapter Hominid brain evolution in “A Companion to Paleoanthropology” by PT Schoenemann that an interested reader might study for the real stuff. One might also consult the review by paleonthropologist J Hawks, and the more lay book by B Hood relevant to the topic.

As primates we pride ourselves over our brains and those of our lineage in general. By all measures even among the highly encephalized placental mammals primates are unusual barring the case of the cetacean lineage of artiodactyls. Below we examine the cranial capacity, a proxy for brain size, of 35 primates which are divided into 9 grades:
1) Lemur-like grade strepsirrhine and possibly basal haplorrhine primates: Smilodectes gracilis, Adapis parisiensis, Rooneyia viejaensis, Necrolemur antiquus, Tetonius homunculus.
2) The parapithecids: Parapithecus grangeri close to simiiform ancestry; the propliopithicids, which are believed to be close to the ancestry of ape and monkey (the old world monkeys i.e. Cercopithecoidea): Aegyptopithecus zeuxis
3) The enigmatic Oreopithecus, which shows features of great apes combined with primitive features suggestive of monkeys.
4) Siamang: the lesser ape.
5) Proconsul nyanzae which is close to ancestry of apes
6) great apes: Dryopithecus brancoi, Pongo species, Gorilla, Pan troglodytes.
7) The grade of apes believed to be close to Homo clade than other great apes: Sahelanthropus tchadensis, Ardipithecus ramidus.
8) Australopithecine grade: Australopithecus afarensis, Paranthropus aethiopicus, Paranthropus boisei, Australopithecus africanus, Paranthropus robustus, Australopithecus garhi
9) The Homo grade: Homo neanderthalenis, Homo helmei, Homo sapiens, Homo antecessor, Homo heidelbergensis, Homo erectus, Homo ergaster, Homo rudolfensis, Homo georgicus, Homo habilis, Homo naledi, Homo floresiensis

Of course we are not taking any particular stance of whether the Homo species defined above are genuine phylogenetic divisions or in part merely intraspecific variation.

What this graph along with recent finds like Homo naledi and Homo floresiensis suggests is that whereas the initial increase in brain size was probably ancestral to the great apes the dramatic increase that happened in Homo was likely not ancestral to Homo. Rather Homo probably began with a brain within the general range of other great apes and the australopithecine grade apes which are believed to be closer to Homo than to Pan. Homo then showed a trend of increase on average, though small-brained forms like H.naledi and H.floresiensis seem to have survived for long alongside the large-brained forms.

The next figure plots the cranial capacities of the same 35 primates against their age. The blue dashed line indicates the cumulative mean of mean cranial capacity of these animals starting from 55 million years ago. They are color-coded as in above plot as per the 9 grades discussed above. One immediately notices that for this sample there is a dramatic rise of cumulative mean cerebral capacity starting from 55 Mya to present. The temporal phases corresponding to the origin and radiation of the great apes followed by the explosive expansion in Homo within the last 2 Myrs is also rather clear. To the extreme right the ranges of each of the nine grades that have two or more representatives with the same color code is shown. The distinction between the great apes and the lesser apes and the primitive apes (Proconsul, Oreopithecus(?)) is clear. But within the great apes the non-human extant great apes, the basal lineages of the human clade still mostly have overlapping ranges. However, in Homo not only is the distinction clear but also the range is dramatic (even discounting the recently described enigmatic forms like Homo naledi and Homo floresiensis). This does raise the question of whether H.naledi and H. floresiensis should be included in the genus Homo (of course generic definition is a subjective one, but their phylogenetic positions need more attention).

We had earlier postulated that apes and monkeys parallely evolved larger brains due to inter-specific antagonistic interactions. Chimpanzee and Homo are great hunters of other primates. While the evidence for hunting monkeys or other animals by the gorilla is rather indirect, we do have such evidence for hunting of the lemur-grade primates, the slow lorises, by the orangutan. Thus, we speculate that hunting other primates emerged early among great apes and has. persisted over more than ten million years. The study of the Ngogo chimps showed that the red colobus monkey population fell by ~89% between 1975 to 2007 with the killing by chimps playing a big role in this fall. The colobus monkey tends to hold fort and fight rather than flee when confronted by chimps but with large chimp troops as in Ngogo this goes badly for them. Thus, the primate on primate predation can seriously affect the survival of populations. This appears to have sparked a classic biological arms race in the form of parallel brain size increase in both apes and monkeys. We believe that the final explosive expansion of brain size within Homo was primarily driven by similar arms race scenario with inter-primate conflict playing a major role.

Now what about the details brain-size evolution within Homo? A lot of ink has been expended on this topic and our intention here is not to review all of that. All we do is to present one final plot of the cranial capacities of Neanderthals. We plot below the c.cs of 34 Neanderthals from:
Israel, Syria, Belgium, Germany, Slovakia, Gibraltar, Morocco, Croatia, France, Italy, Iraq and Uzbekistan against the respective ages which span from 40,000 to 130,000 years bp. They are color-coded as per their geographic origin and labeled with the initials of the countries.

One notices that the cumulative median cranial capacity shows a modest increase from 1255 to 1381 cc in this 90000 year window for these specimens, which is in the range for modern humans. However, that said throughout the 60 kyr period from 100-40 kyrs we see a consistent presence of males with c.c >1500 cc across the geographic distribution of Neanderthals: the largest is the impressive 1740 cc Amud 1 cranium from Israel. To the right of the plot we see the range of the modern human c.c. (M.H) and we can see that the Neanderthals had a tendency for larger c.c than us. Equally striking is the illustration of the range of the archaic specimens of Homo sapiens shown the right (Ar.H). This is drawn from specimens ranging from the Idaltu cranium from 160,000 yrs down to the Veyrier 1 cranium from France at 10,000 years. Here too we see a tendency for higher c.c. than us with a Cro Magnon cranium from Europe Grimaldi 4 having a whopping c.c of 1775 cc. The interesting South African Florisbad Skull from ~260,000 ya already shows a c.c of 1400 cc and has often been classified as distinct species, Homo helmei.

This indicates that the trend for c.c growth actually peaked in late Homo with likely parallel increases across distinct lineages such as sapiens, Neanderthals, and Florisbad-like Homo. To us this peaking is a sign of selection being ratcheted up inter-primate conflict. Then why did the c.c decline with the emergence of modern humans. One possibility is that a combination of distinct forces were acting: 1) Homo sapiens triumphed in the inter-specific conflict against other coeval Homo leaving them with only intra-specific conflict with similar IQ individuals. 2) This was accompanied by shift towards ever-increasing group cooperation and group size in Homo sapiens. 3) Large brains come at a steep cost – the head has to pass through the pelvis of the female. So possibly there was also some selection against the big size if it could be offset in some other way (there was also potentially inter-sexual genetic conflict at play). The group cooperation allowed Homo sapiens to potentially bank more on the “hive-brain” with intra-group specialization – a parallel to caste-formation hymenopterans and isopterans. An individual wasp might not have much smarts but as super-organismic assemblage they can show considerable intelligence. A similar rise of groups simulating super-organismic assemblage among Homo might have had its negative effects on brain size especially if there was selection against larger brains from the reproductive tract. Thus, together these forces sometimes acting at the same time sometimes with slight temporal offset allowed the brains to shrink. Our thinking in this direction has some overlap with the domestication hypothesis of Bruce Hood.

## Trigonometric tangles-3: the fractals

This exploration began in days of youth shortly after we learned about complex numbers. It culminated only much later in adulthood when we discovered for ourselves a class of fractal curves related to a celebrated curve discovered by the great mathematicians Bernhard Riemann and Karl Weierstrass. We detail it here covering some very elementary mathematics because retracing the path one has taken often helps when one has to teach the same to a young student of pedestrian quantitative IQ like ourselves.

A unit circle with center at origin ($0+0i$) may be defined in the complex plane by the equation:
$z=\cos(\theta)+i\sin(\theta); \; \theta \in [0,2\pi]$,
which by the fundamental discovery of the great Leonhard Euler becomes:
$z=e^{i\theta}$
From the above it is apparent that equation of this unit circle might be written as:
$z\overline{z} = |z|=1; \; z \in \mathbb{C}$
Now this definition of the unit circle in the complex allows us carry out a variety of interesting mappings. A mapping is an operation which transforms the points on the unit circle $z$ to another set of points $z'$. The simplest of these is the power operation:
$z' \mapsto z^n \; n \in \mathbb{N}$
This mapping simply involves the operation of raising every $z$ to a positive integer $n$ to obtain $z'$. What this operation does is to simply redistribute the points $z$ to other points $z'$ back on the unit circle. However, if we connect every $z$ to its corresponding $z'$ by a segment then the envelope of those segments defines an epicycloid in the unit circle with $n-1$ cusps. Thus, for squaring we get a cardioid, for cubing a nephroid, for power 4 a tricuspid epicycloid and so on.

Figure 1: Epicycloids obtained via the map $z' \mapsto z^n \; n \in \mathbb{N}$ map

It is this relationship to the exponent which gives basis for the form of the core region of the famous fractal known as the Mandelbrot set. Thus, for a Mandelbrot set created by the the map of the form $z'=z^2+c$ we get a cardioid; for $z'=z^3+c$ we get a nephroid; so on.

Figure 2: Mandelbrot set for $z^3$ showing nephroid

Now, you can distort the unit circle by the map of the form:
$z'\mapsto az+b\overline{z}; \; a,b \in \mathbb{C}$
This yields an ellipse with foci at $\pm \sqrt{ab}$ and can be visualized as the projection of the great circles of a sphere on to a plane.

Figure 3: Mapping a unit circle on to ellipses.

Instead of the unit circle with center at origin let us consider a general circle in the complex plane. Its equation is:
$|z-c|=a; \; a \in \mathbb{R}; c \in \mathbb{C}$
Thus the center of the circle is at $c$ and radius is $a$. Now let us consider a circle with $c=1+0i, a=1$ — a unit circle passing through origin. If we deploy the following map on it $z'\mapsto \sqrt{z}$, the square root function being two-valued creates a $1\mapsto 2$ mapping. This results in the bilobed lemniscate discovered by Jakob Bernoulli which crosses over at origin and with foci at $\pm 1$. If we instead deploy the squaring map $z'\mapsto z^2$ on this circle we end up with a cardioid. Instead of the above unit circle consider a vertical line passing through $c=1+0i$. Its equation would be $z=1+iy$. If we similarly apply the square root operator on it gets mapped onto to a double branched curve the rectangular hyperbola. On the other hand application of the squaring operator on this vertical line bends it into a left facing parabola. Thus, the ellipse, hyperbola and parabola can be seen as secondary conics generated from the line and circle.

Figure 4

Now if the circle has $a<1$ then the same square root map creates two disjoint lobes which are the ovals discovered by the astronomer Cassini, whereas the square map creates a limacon. Similarly if $a>1$ the mapping merges the lobes into continuous Cassinian curves and limacons with an internal lobe.

Having seen these very simple maps we now move on to the fractal mappings of the unit circle at origin which use the same basic principle but a mapping function that can generate fractal structure. Upon discovering these we realized that what we arrived at what is a generalized form the Riemann-Weierstrass function. Hence, before we look a those mappings we shall take a brief look at this remarkable function that marked the beginning of the study of fractals. The great Carl Gauss wondered if all continuous functions are differentiable except at a “limited” set of special points (e.g. the cusp of an epicycloid). A few years after his death his brilliant successor Bernhard Riemann discovered a function which is continuous everywhere but is most undifferentiable. He was probably unable to develop this further due to his early death a few years later. Karl Weierstrass presented this function in a more complete form and subsequently Hardy established a partial proof for its undifferentiablity. These self-similar curves can be formulated in multiple different ways of which the simplest is of the form:
$y=\displaystyle \sum_{n=1}^\infty \dfrac{\sin(n^a \pi x)}{(n^a \pi)}; \; a,n \in \mathbb{N}_1$
Here $a$ is the power which above 5 considerably smoothens the curve.

Figure 5

Another formulation which generates a greater variety of these curves is given by:
$y=\displaystyle \sum_{n=0}^\infty a^n \cos(b^n \pi x);\; a \in (0,1), b \in (1,\infty)$

This form recapitulates a range of interesting behavior like the outlines of coastlines, clouds and mountains, and seemingly chaotic fluctuations of values like light output of variable stars, climatic variables and market prices.

Figure 6

Now, our mappings on in complex plane are generated by the below map operating on the unit circle described with center at $0+0i$:

$z' \mapsto \displaystyle \sum_{n=0}^\infty \dfrac{z^{\left(a+bn\right)^c}}{\left(a+bn\right)^c}; \; a,b,c \in \mathbb{R}$
“Good” forms are obtained for relative small $(a,b,c)$. In particular $c \in [1.5,3]$. The fractal forms generated by these mappings appear to have some value in capturing various biological forms. One of the most obvious forms that becomes apparent is the crenulated margin of a leaf (e.g. first curve below). Indeed, we have used this and other related Riemann-Weierstrass function formulations to generate a range of leaf like forms.

Figure 7: various fractal maps of the above form with $c=2$

Another problem for which we found inspiration as we studied these curves was that of packaging DNA in the cell. A bacterium like the laboratory Escherichia coli has a cell of length ~.002 millimeter and 0.00157 mm circumference. However, its genome when fully extended is a circle of circumference ~1.5 mm. So how is that circle of DNA fitted into a cell with much smaller circumference and length? This is achieved by coiling the chromosomal circle into loops and those loops to further loops by the action of topoisomerases. Maps such as the above can provide a means of visualizing such a looping processing.

Figure 8: Further fractal maps with $c=2$ showing intricate looping.

Another activity in which these functions may be put to use is to generate “music”. However, we are not presenting any samples here because we had generated them long ago using a different programming language we no longer use and are not sufficiently motivated to re-write the “musical” code in the language we are currently using for these demonstrations.

Figure 9: Further fractal maps with fractional $c$.