## From Plato to Euler and back

This is primarily meant as an educational handout on some very basic theorems of geometry that one might have studied in school. Some educated adults whom we asked about these had either forgotten them or claimed to have never studied them. Hence, we provide these here for those who might be interested. In the below discussion all angles are in radians unless explicitly specified otherwise.

There are only 5 Platonic solids
The first question is at least as old as the times of Plato: Why are they only 5 Platonic solids? Platonic solids are regular polyhedra, i.e. they have all their faces as the same regular polygon. The simplest regular polygon is an equilateral triangle. Its angle is $\tfrac{\pi}{3}$. In order to fold a solid the sum of the angles of the polygonal faces must be less than $2 \pi$ radians because $2 \pi$ is the sum of the angles about a point on a plane in Euclidean space. Further, to make a solid we need at least 3 polygonal faces meet at a vertex. Hence, with an equilateral triangle we can make solids with 3 or 4 or 5 faces meeting at a vertex, $3 \times \tfrac{\pi}{3}; 4 \times \tfrac{\pi}{3}; 5\times \tfrac{\pi}{3}<2\pi$. These respectively yield the tetrahedron, the octahedron and the icosahedron (Figure 1). With the next regular polygon, a square, we can get 3 squares to meet at a vertex, i.e. $3 \times \tfrac{\pi}{2}$. This yields us a cube. Next, with a regular pentagon we can get three pentagonal faces at vertex, i.e. $3 \times \tfrac{3\pi}{5}$. This gives us the dodecahedron. Thus, we can have only 5 Platonic solids in 3-dimensional space $_\blacksquare$.

In principle, one can join the centers of adjacent faces of regular polyhedron to get another regular polyhedron due to their inherent symmetry (Figure 1). Performing this operation one can also see that the Platonic solids have a duality with another Platonic solid: (cube-octahedron) – (tetrahedron-tetrahedron) – (dodecahedron-icosahedron). In each dual pair there is one of the three formed by equilateral triangle faces. Given that the duality operation on the tetrahedron yields a congruent tetrahedron, you again will have only 5 Platonic solids. It made a such a profound impression on me when I first discovered this for myself as a kid that I could entirely appreciate the profound impact the discovery of these solids (apocryphally by Pythagoras and Theaetetus) had on Plato of yore. It tells you some very basic thing about Euclidean space.

Figure 1. The 5 Platonic solids and their duals.

The sum of the angles of a convex polygon
A polygon is a closed path with $B$ vertices where adjacent vertices are connected by a single edge. Hence, it follows that the polygon has $B$ edges or sides as well (Figure 2). This gives our next question: what is the sum of the angles of a polygon? The simplest polygon, the triangle, has angles summing to $\pi$ radians in Euclidean space. Every other planar convex polygon of $B$ vertices can be constructed from $B-2$ triangles all of which share a common vertex (Figure 2). From Figure 2 it is apparent that the sum of angles of the polygon is equal to the sum of the angles of the constituent triangles. Thus, the sum of the angles of a convex polygon is $(B-2)\pi$ radians. This result was first derived by Proclus.

Figure 2. The sum of the angles of a convex polygon.

The triangulation theorem
Let a convex polygon have $B$ vertices and $I$ internal points (Figure 3). We triangulate it by connecting the internal points to the vertices of the bounding polygon or each other without any edge crossing over such that the whole polygon is dissected into $F$ triangles (Figure 3). We then ask: Is there are relationship between $B$, $I$, $F$. The answer is the triangulation theorem, which states that $F=2I+B-2$.

Figure 3. Triangulation of a polygon.

Proof:
1) The sum of the angles of the bounding polygon (as shown above) is $(B-2)\pi$.
2) The sum of all the angles on a plane sharing common vertex is $2\pi$ radians. Hence, the sum of the angles at all internal points is $2I\pi$.
3) Since the sum of the angles of a triangle is $\pi$, the sum of the angles of all the $F$ constituent triangles of the triangulation of the polygon is $F\pi$. From Figure 3 it can also be seen that it is $(B-2)\pi+2I\pi$.
4) $F\pi=(B-2)\pi+2I\pi; \; \therefore F=2I+B-2\;\; _\blacksquare$

Euler’s theorem

Figure 4. Two polyhedra and the polygonal tilings derived from them.

The above result regarding the triangulation of polygons in 2D space Euclidean space and Plato’s discovery that only 5 regular polyhedra exist in 3D space are both related to the sum of all the angles on plane about a vertex being $2\pi$. This in turn leads to the celebrated theorem of Euler regarding polyhedra, which while deceptively simple had to wait for 2 millennia since Plato’s days. Consider the two polyhedra in Figure 4. The first is a pentagonal pyramid while the second is a hexagonal prism. Let $F$ be the number of faces, $V$ the number of vertices and $E$ the number of edges of the polyhedron. For the pyramid in question we have $F=6; V=6; E=10$. Thus, $6+6=10+2$. For the prism in question we have $F=8; V=12; E=18\; \therefore 8+12=18+2$. You can try this out for other polyhedra like those in Figure 1. We can as the question: Is there a general relationship here? We observe that always: $F+V=E+2$. This is Euler’s theorem and and we prove it below.

Proof:
1) We first reduce the 3D polyhedron to a 2D graph (Figure 4). The way we do this is by choosing one face of the polyhedron as the bounding polygon. We then flatten all other faces inside that bounding polygon while still maintaining the topology of the face. This means that a triangular face remains a triangle , a quadrilateral remains a quadrilateral, and a $n$-gon remains a $n$-gon in the flattened 2D graph. Thus, the 2D graph is a polygonal tiling of the selected face polygon of the starting polyhedron. This is easy to conceive for pyramids and prisms (Figure 4). It is more complicated for some of the Platonic polyhedra (Figure 5).

Figure 5. The polygonal tilings derived from the Platonic polyhedra.

Thus, we observe that the 2D graph retains all edges and vertices of the starting polyhedron. However, since it is constructed by flattening all other faces onto one face we lose that face and the resulting number of tiling polygons is 1 less than the number of polyhedral faces. Thus, the 3D equation $F+V=E+2$ becomes $F+V=E+1$ for a 2D polygonal tilings. Proving the latter in 2D effectively results in proving the former.

2) We define the number $P=F+V-E$ for a given polygon tiling. Our objective is to prove that always $P=1$. We next triangulate all non-triangular polygons in a given polygon tiling into triangles. Let us define $T=F_t+V_t-E_t$ as the number we get from the faces, vertices and edges after triangulation. From Figure 6 it is apparent that for each new edge we add during triangulation we get a new face. Thus, $P=T$. Hence, proving $T=1$ will prove $P=1$.

Figure 6. Conservation of $F+V-E$ upon polygon triangulation.

3) Now that we have triangulated the polygonal tiling, we can apply the triangulation theorem on the resulting graph to show $T=1$. The faces $F$ in the triangulation theorem is the same as the $F_t$ in our triangulated graph; hence we can write $F_t=2I+B-2$. Further, the number of vertices $V_t$ of the triangulated graph is the sum of the number of boundary vertices of the bounding polygon and the internal points. Thus, we get $V_t=I+B$. Let $s$ be the sum of the number of sides of all triangles in the triangulation redundantly counting all shared edges for each triangle. Since there are $F_t$ triangles we have $s=3F_t$. Now $s$ can be expressed in another way. $B$ will be the number of all the sides of triangles making up the boundary polygon. The $E_t-B$ will be the remaining edges. Now they will fall on the side of 2 triangles each (e.g. see Figure 6) so to make up $s$ we have count them twice. Thus, we can write $s=B+2(E_t-B)$.

Thus, we get:
$B+2(E_t-B)=3F_t; \; \therefore B=2E_t-3F_t$
From the triangulation theorem and $I=V_t-B$ we can write: $F_t=2V_t-2B+B-2; \therefore F_t=2V_t-B-2$
Plugging the above value of $B$ into this we get: $F_t=2V_t-2E_t+3F_t-2; \; \therefore F_t+V_t-E_t=T=1$
Since $P=T$, it follows that $F+V=E+1$ for a polygonal tiling. From this it follows that:
$F+V=E+2 \;\; _\blacksquare$

Back to the Platonic solids
We may next ask: Given Euler’s theorem can we prove that there are only 5 Platonic solids? To answer this we need to prove two further relationships first. For this we take into account that a Platonic polyhedron has regular polygons as its faces and that each vertex must necessarily belong to the same number of polygonal faces. Let $n$ be the number of edges of the regular polygonal face of a Platonic solid. Let $m$ be the number of edges of meeting at any given vertex of a Platonic solid. Let $s$ be the sum of the number of sides of all polygonal faces of a Platonic solid counting redundantly. Given that we have $F$ faces in the solid, we get $s=nF$. Given that we have $V$ vertices with $m$ edges meeting at each of them we get $s=mV$. Finally, we note that each edge of the polyhedron simultaneously belongs to 2 of its polygonal faces; hence, we get $s=2E$. From this we get the relationship for any Platonic solid: $2E=nF=mV$. For example, in an octahedron we have $E=12, n=3, m=4$; thus, $2\times 12=3 \times 8 = 4 \times 6=24$.

From the above we have for a Platonic solid: $F=\tfrac{2E}{n}, V=\tfrac{2E}{m}$

Next, we plug the above into Euler’s formula $F+V=E+2$ to get:
$\tfrac{2E}{n}+\tfrac{2E}{m}-E=2\\[7pt] \tfrac{2}{n}+\tfrac{2}{m}-1=\tfrac{2}{E}\\[7pt] \therefore \tfrac{2}{n}+\tfrac{2}{m}-1>0\\[7pt] \therefore \tfrac{2}{n}+\tfrac{2}{m}>1$

By multiplying both sides by $nm$ we get the Eulerian inequality for a Platonic solid: $2m+2n>nm$

We can now use this to prove that there can be only 5 Platonic solids. For this we should first keep in mind that to have a solid at least 3 faces should meet at a vertex; hence, minimally $m=3$. We then begin with a triangular face $n=3$ and plug it into the Eulerian inequality:

$2m+6>3m;\; \therefore m<6$
Thus, for a triangle, $m=3, 4, 5$: these yield the tetrahedron, octahedron and icosahedron.

Next we take a square face $n=4$
$2m+8>4m;\; \therefore m<4$
Thus, we can only have $m=3$ for a square face and we get the cube.

Next we take a pentagonal face $n=5$
$2m+10>5m;\; \therefore m<\tfrac{10}{3}$
Thus, we can only have $m=3$ for a pentagonal face and we get the dodecahedron.

Next we take a hexagonal face $n=6$
$2m+12>6m;\; \therefore m<3$
Thus, we cannot get any polyhedron because $m<3$. The same applies for any $n>6$. Thus there can only be 5 Platonic solids $_\blacksquare$

Tailpiece
While we can have infinite regular polygons, only the first 3 of them yield a total of just 5 Platonic solids. This is a strong constraint in 3D space; thus, one would expect these Platonic ideals to occur as reflections throughout nature. Indeed, that is the case and we may note the following:
1) The 4 bonds formed by the $sp^3$ hybridized orbitals of carbon define a tetrahedron. Similarly, phosphates $PO_4^{3-}$, such as those forming the backbone of DNA and RNA also assume a tetrahedral geometry. $Fe_3O_4$ assumes tetrahedral crystals. Thus, the tetrahedron is a fundamental structure of nature and life.
2) The cube is a ubiquitous habit of crystals: $NaCl$ is a good example of such.
3) The octahedron is the habit of diamond and alum $KAl(SO_4)_2 \cdot 12H_2O$.
4) The dodecahedron is the habit of the quasi-crystal of the holmium–magnesium–zinc alloy. The RNA of certain nodaviruses is packed in a dodecahedral form within their capsids.
5) The icosahedron is famously the form of the capsids of numerous viruses. It is also seen in some non-viral protein assemblies like those which form the propanediol utilizing and ethanolamine utilizing bacterial microcompartments.

In the human world, a deliberately made cube used as a gaming die and a fixed weight was seen among the Harappan people of bronze age India. Harappans also made regular tetrahedral weights, such as those found in Mohenjo daro. Some believe that the occurrence of pyrites which are found in certain places in Italy and naturally assume the forms of cubes, dodecahedra and icosahedra might have inspired the yavana-s to think about these solids. Among the yavana-s, we are informed by Proclus that Pythagoras was the first to discover the Platonic solids. However, there is no evidence that he actually knew all 5 of them and that there were only 5. Plato mentions them in his Timaeus and by then it was known that there are 5 of them. It is believed that Plato obtained that knowledge from Theaetetus, his mathematical interlocutor, who is believed to have first recognized all 5 of them. However, the evidence for this comes from an apocryphal commentary on Euclid. Plato thought that the tetrahedron, octahedron, cube and icosahedron corresponded to the 4 “elements” of Greek tradition: fire, air, earth and water. He then added that a god used the dodecahedron “for embroidering the constellations on the whole heaven.” This is perhaps an allusion to the 12 Zodiacal constellations of the Greeks. It is also held that Theaetetus’ explorations inspired the Elements, where Euclid presumably following his successors gives these 5 Platonic polyhedra and indicates that there are only 5 of them. Material evidence for their recognition in the Classical world comes from the discovery of the icosahedral dice from Ptolemaic Egypt. Further, the Romans made metal dodecahedra and icosahedra (mostly the former), whose function remains unknown to date as far as we know.

In Indian mathematics we are unaware of a specific mention of Platonic solids as a group. However, we have evidence for a “ghost-lineage” of Hindu mathematical knowledge regarding Platonic solids in the form of beads shaped as those solids from at least the Gupta age. There is a persistent belief among white indologists and their imitators that Hindu mathematical tradition is a poor derivative of the Mesopotamian and Greek traditions. However, there is absolutely no evidence for transmission of Greek texts with knowledge of Platonic solids to Hindus prior to the 1700s of the common era. Thus, there is no evidence that the Indian production of Platonic solids was inspired by the Greek textual tradition. One may point out that their emergence in Indian art is approximately contemporaneous with their emergence in material depictions in the Classical world. However, the Indian versions are not used as dice as in Ptolemaic Egypt or as made like or of the large size of their Roman counterparts. They are part of a distinctly Indian use in jewelry. Indeed, in this regard one may point out that already in the Sāmaññaphala-sutta-84 the tāthāgata mentions such a well-polished octahedral bead. One may also note that some tāthāgata atomic theories in India saw substances as being formed from their constituent molecules by an octahedral packing. It is also notable that this style of polyhedral beads was transmitted to Myanmar, Thailand and Vietnam along with early gold-working traditions and at least dodecahedral and octahedral beads have been found in those regions.

Posted in Scientific ramblings |

## Śiva-gaṇa-s and Andhakāsura-vadha in the Vāmana-purāṇa

On Twitter, one of our acquaintances going by the name @GhorAngirasa had a discussion on the significance of the number 66 in śaiva tradition. That reminded us of an unfinished article where we had noted this number in the context of a śaiva paurāṇika narrative. This lengthy discursive article looks at some additional features of the ancient tale of the killing of Andhaka by Rudra beyond what we had discussed earlier in a note on the evolution of the “fertilizing sweat” motif. Accordingly, we took this piece out of the shelf and worked it up so that it could be minimally presentable for publication. It provided as a pdf file because of the length and the number of illustrations in it.

Śiva-gaṇa-s and Andhakāsura-vadha in the Vāmana-purāṇa

Posted in Heathen thought, History |

## Packing constants for polygonal fractal maps

Among the very first programs which we wrote in our childhood was one to generate the famous Sierpinski triangle as an attractor using the “Chaos Game” algorithm of Barnsley. A couple of years later we returned to it generalize it as a map in the complex plane. Consider the polynomial equation,

$z^m+1=0$, where integer $m=3,4,5...$

The roots of this equation, $z_j: z_1, z_2... z_m$, define the vertices of a $m$-sided polygon in the complex plan. For example, if $m=3$, we get an equilateral triangle defined by the roots $z_1= \tfrac{1}{2}+\tfrac{\sqrt{3}i}{2}, z_2= -1, z_3= \tfrac{1}{2}-\tfrac{\sqrt{3}i}{2}$.

With this in place the Chaos Game map is defined for a given $m$ as:

$z_{n+1}=r(z_n+z_j)$,

where $z_j$ is one of the $m$ roots chosen randomly with equal probability as the others in each iteration of the map and $0. If $r=1$ for any $m$ we get a random-walk structure (Figure 1).

Figure 1

For other $m, r$ we get chaotic maps and for particular values of $m, r$ we get attractors with a fractal structure. Thus, the Sierpinski triangle is obtained with $m=3, r=\tfrac{1}{2}$. This fills the triangle defined by $z_1, z_2, z_3$

Figure 2

For $m=4, r=\tfrac{1}{1+\sqrt{2}}$ we get the fractal street block (Figure 3).

Figure 3

We currently revisited this map because of a curious problem that emerges when we continue this operation as below for further polygons. For $m=5, r=1-\tfrac{1}{\phi}$ (where $\phi=\tfrac{1+\sqrt{5}}{2}$, the Golden Ratio) we get the fractal pentagons surrounding the interior penta-flake (Figure 4).

Figure 4

For $m=6, r=\tfrac{1}{3}$ we get the fractal hexagons surrounding the interior Koch’s snowflake (Figure 5).

Figure 5

For $m=7, r=\tfrac{1}{S}$ (where $S=2+2\cos\left(\tfrac{2\pi}{7}\right)$, the Silver constant. The Silver constant is an algebraic number which is the real root of the cubic $x^3-5x^2+6x-1$) we get the fractal heptagon necklace surrounding the interior hepta-flake (Figure 6).

Figure 6

For $m=8, r=\tfrac{1}{2+\sqrt{2}}$ we get the fractal octagon necklace (Figure 7).

Figure 7

For $m=9, r=\tfrac{1}{2+2\cos\left(\pi/9\right)}$ we get the fractal nonagon necklace (Figure 8).

Figure 8

For $m=10, r=\tfrac{1}{1+2\phi}$ we get the fractal decagon necklace (Figure 9).

Figure 9

For $m=11, r\approx 0.2209$ we get the fractal hendecagon necklace (Figure 10).

Figure 10

For $m=12, r=\tfrac{1}{3+\sqrt{3}}$ we get the fractal dodecagon necklace (Figure 11).

Figure 11

Given that these attractors are fractal, they have an infinite perimeter but occupy a finite area. Thus, one can define a common feature for the above attractors namely “tangency” of the fractal elements, i.e., each polygonal unit is distinct from its neighbor with the same scale-factor but at same time makes a contact with it like a tangent. While for the triangle and the square this definition is a bit murky, it is clearly visible from the pentagon onward. It can be contrasted with other fractal attractors obtained by this method where the elements overlap or are shared. For instance for $m=12, r=\tfrac{1}{2+2\cos\left(\pi/6\right)}$, we have the expected dodecad structure but two dodecad sub-elements are shared by each of the adjacent elements (Figure 12).

Figure 12

In the above examples (Figure 2-11) we have determined for each polygon the value of $r$ which results in a tangent attractor:

$m=3, r=\tfrac{1}{2}; m=4, r=\tfrac{1}{1+\sqrt{2}}; m=5, r=1-\tfrac{1}{\phi}; m=6, r=\tfrac{1}{3};\\ m=7, r=\tfrac{1}{S}; m=8, r=\tfrac{1}{2+\sqrt{2}}; m=9, r=\tfrac{1}{2+2\cos\left(\pi/9\right)}; \\ m=10, r=\tfrac{1}{1+2\phi}, ; m=11, r\approx 0.2209; m=12, r=\tfrac{1}{3+\sqrt{3}}$

For $m=5..8$ it is the reciprocal of $2+2\cos\left(\tfrac{2\pi}{m}\right)=4\cos^2\left(\tfrac{\pi}{m}\right)$, which are known as Beraha constants ( $B_n$) that appear in graph-coloring theory. However, for $m=9..12$ this principle clearly breaks down: e.g. for $m=12$, we see that $\tfrac{1}{B_{12}}$ yields the overlapping attractor (Figure 12). We have been able to obtain closed forms for the $r$ that yield tangent attractors for all $m$ except 11, where we report an empirically determined approximate value. What is the closed form expression for it? This leads to the question of whether there is a general formula to obtain our $r$ that results in tangency? To our knowledge these have not been answered.

Posted in art, Scientific ramblings |

## Cows, horses, sheep and goats

It goes without saying that humans are what they are today because of cows, horses, sheep and goats. Hindu civilization in particular is in the very least the product for the first two, while remaining two contributed to it from “behind the scenes”. The major civilizational transitions in human history were built atop the bones of these four animals: the Neolithic developments in West Asia, the urban Nilotic civilization in Egypt, the rise of the Indo-Europeans on the Eurasian steppes, the rise of the Turco-Mongolic peoples on the same steppes but at their eastern end, and the expansions of the African pastoralists. We were curious if the present distribution of these animals has any notable features or relationship to their history. Accordingly, we used the UN Food and Agriculture Organization data on these animals to visualize their distribution across different countries and continents. Table 1 shows their total population (here the term “cows” stands for what is provided in the FAOSTAT database as cattle).

Table 1. The latest UN FAOSTAT population data
$\begin{tabular}{lr} \hline Animal & Population \\ \hline Cows & 1491687240 \\ Sheep & 1202430935 \\ Goats & 1034406504 \\ Horses & 60566601 \\ \hline \end{tabular}$

We first plot the populations of each pair of these four animals for all countries that possess both of them using the $\log_{10}$ scale (Figure 1) and tabulate the correlations between them in Table 2.

Figure 1

Table 2. The Pearson correlations between life stock on the $\log_{10}$ scale
$\begin{tabular}{rrrrr} \hline & Cows & Horses & Goats & Sheep \\ \hline Cows & 1.00 & & & \\ Horses & 0.85 & 1.00 & & \\ Goats & 0.81 & 0.66 & 1.00 & \\ Sheep & 0.80 & 0.72 & 0.83 & 1.00 \\ \hline \end{tabular}$

The populations of the horses and cows are the most correlated pairs. The sheep and goats are the next most correlated. Thus, animals of comparable size tend to be more correlated in their populations. The horse-cow pair is also the primary pair of the old Indo-European pastoralism. It is possible there is a historical echo of the Indo-European conquests that made this pair a widely adopted unit.

Figure 2

We then looked at the distributions of the ratios for these six pairs (Figure 2). In each plot we mark the following: the mean(red), India (blue), Mongolia (green), Kazakhstan (gray), Kyrgyzstan (cyan). We can see that the plot for the cow:horse plot is the least skewed and is approximately normal. India presents a large country that is heavily skewed in the favor of the cow over the horse. The horse evidently did not do very well over larger swaths of India unlike the cow. On one hand this had its consequences for the Indians in course of their conflict with horse-borne invaders. On the other it made India the land of cows. In contrast, Mongolia, Kazakhstan and Kyrgyzstan show the reverse trend, being horse-heavy pastoralist nations. Kazakhstan was one of the first sites of horse-domestication in horse-centric pastoralist Botai culture. This form of pastoralism persisted on the eastern steppes among the Altaic type of peoples long after the Botai people became extinct after colliding with our Indo-Iranian ancestors . Mongolia still retains this state in its most drastic form.

Mongolia is also unusual in favoring both goats and sheep over cows. Notably it also has a comparable number of sheep and goats. The sheep:goat ratio shows multimodality with some countries investing heavily in favor of sheep over goats. This is where the two steppe nations Kazakhstan, Kyrgyzstan differ from Mongolia. India in contrast weighted in favor of goats over sheep, again reflecting the environmental conditions favoring goats. Further, goat pastoralism might have come early to India along with the Iranian farmers who played a major role in the foundation of the Harappan civilization. Nevertheless, sheep pastoralism eventually spread deep into peninsular India with the use of sheep milk and a memory of it is preserved in the origin myths of the Reḍḍi-s of Andhra. The goat:horse ratio is also notably multimodal suggesting that originally goats and horses defined very different pastoralist niches. The effects of this ancient niche distinct appears to persist to this date.

Figure 3. India (blue), Mongolia (green), world (black)

The above are based on the latest population statistics but one can also ask about their dynamics. In this regard the FAOSTAT database has data for many countries and the world for about 57 years. One simple way to look at population change in this period is fraction of population change with respect to the start and end of this period. This can be assessed using the formula:

$f_{pc}=\dfrac{2(p_f-p_i)}{p_f+p_i}$

Here $f_{pc}$ is the fraction of population change, $p_i$ the initial population and $p_f$ the final population. The distribution of $f_{pc}$ for the four animals is plotted in Figure 3. Except for horses which have shown a slight decline, the populations of the other animals have shown growth world-wide. One sees that, except for sheep, the $f_{pc}$ for India is lower than for the world for the remaining animals. In contrast, for Mongolia it is higher than for the world define a truly pastoralist nation. The Indian horse populations have shown a notable decline in this period.

Figure 4. $y$-axis in $\log_{10}$ scale.

This measure only gives a coarse view of the population change. Hence, we looked at the absolute population change for 3 nations (Mongolia, India, USA) in Figure 4 over a 57 year period for which data exists. The dynamics of the precipitous decline of the horse in India is apparent. As noted above, much of India is not suited for the horse and in the absence of the old military and royal-display derived pressure to keep the horse numbers high it has mostly had a free fall. USA shows almost the reverse trend with respect horses and cows. We are not sure if this trend for cows relates to the decline in beef consumption among the Americans. The India cattle situation two booms followed by busts. The turning points for these busts seem to correspond to El Niño-related droughts and it needs to be see if they were indeed the triggering factors for the declines in Indian cattle. In Mongolia there seem to be generally similar trends for both cattle and horses. The marked rise in productivity after the fall of the Soviet Empire suggests the release from Soviet collectivism allowed the Mongols to recover their traditional pastoralist lifestyle. The Mongolian situation also shows the strongest evidence for climate effects, given that after the Soviet collapse the Mongolian cattle and horse populations have shown similar busts and booms. These seem to correspond to the aftermath of the severe Mongolian winters known as the jud-s that take a heavy toll on the animals and the pastoralists. However, the warming in northern latitudes might be allowing a rapid bounce back from the juds.

Posted in History, Scientific ramblings |

## Fraudulent science by Indians: some really bad news

Around 2011 we were approached by a researcher of Indian origin for a collaboration in biochemistry regarding a family of proteins whose biochemical functions we had uncovered. After more than an year of dealing with him, it became clear that his research practices were questionable. Based on information from a whistle-blower we then reached the conclusion that he was involved in outright scientific fraud. This prompted us to investigate his work a little more and we found at least 15 published papers of his with fraudulent data. This brought to our mind one of the earliest cases of fraud in molecular biology by an Indian postdoc from Mumbai working with Jim Watson. A broader investigation revealed that the person whom we were dealing with was merely one of at least five researchers of Indian background all with professorial positions in the United States of America and India who were involved in a very similar pattern of fraud. This concerned us for it brings a bad name to researchers of Indian origin. However, simultaneously we also observed several examples of similar fraud by people of European, Jewish, Chinese and Japanese ethnicity. Indeed, very recently we had another case to deal with involving comparable, painful fraud by Chinese collaborators. Hence, we thought this is a universal problem with no special predilection for such fraud among Indians.

However, as the days went by since our original encounter with fraud, we seemed to accumulate more and more cases of Indians engaging in such fraud. Recently, we uncovered yet another case of fraud involving a family of proteins whose evolution and biochemistry we had helped characterize in the first decade of the 2000s. This again involved an Indian lead author giving us the gnawing doubt that things were not right among our people. This prompted a colleague and me to look at federal registry of scientific misconduct issued by The Office of Research Integrity (ORI) of the Department of Health and Human Services (DHHS), USA to see if some of the cases we had detected were in it. This register records cases in which fraudulent research performed using public money in form of grants given by the DHHS were reported to the ORI for examination. While the federal register released by ORI does not cover all cases of fraud performed using public funds, it has at least 60 confirmed cases along with the details of the case and the punishment recommended by ORI. These 60 cases were in period from 2011 to 2019 CE. The majority of them were biochemistry/molecular biology fraud of the kind we had encountered in our starting investigation: 1) manipulation of gel image; 2) creation of fake images of biological material; 3) Some cases of simple plagirism; 4) Out right creation of fake numerical data.

We realized that these 60 cases could provide a means of examining if Indians were particularly prone to fraud or not. Notably, the cases released in the register allowed the names to be classified quite unambiguously into 6 ethnic groups (Table 1). Among West Asians there are representatives from each of the 3 Abrahamistic religions of the region.

Table 1
$\begin{tabular}{lrrrr} \hline Origin & Count & Percentage & Cheaters/million & Cheaters/1000 \\ \hline European & 27 & 45.00 & 0.14 & 0.67 \\ Indian & 20 & 33.30 & 6.29 & 4.72 \\ Chinese & 7 & 11.70 & 1.85 & 0.57 \\ West Asian & 4 & 6.70 & 0.38 & 2.32 \\ Korean & 1 & 1.70 & 0.59 & 0.25 \\ Japanese & 1 & 1.70 & 0.77 & 0.33 \\ \hline \end{tabular}$

Indians constitute $33\%$ of the fraud cases in the federal register. This itself is quite striking because it is rather clear that they do not make up that high a fraction of the biomedical research workforce, which is being probed here. Now, the simplest normalization for the counts is by the representation of these ethnic groups in the USA. This population data can be assembled quite easily by an internet search for the period under consideration. Normalization by this population share is shown in column 4 of table 1 as the number of cheaters per million of the population. By this reckoning and Indian would be nearly 46 times as likely to be involved in fraud as a person of European descent.

One could object that the US population share of the ethnic groups is not a valid normalization for it does not accurately reflect their representation in the biomedical work force. While the latter part of the statement might be true it is amply clear to anyone in the business that Indians do not constitute a fraction greater than Chinese in this workforce, making this objection quite facile. Yet, one would want to perform a more objective normalization based on the proportions of these groups in the workforce. This data is much harder to get in a clean form. However, an article published in the Nature magazine “The new face of US science” by Misty L. Heggeness, Kearney T. W. Gunsalus, José Pacas and Gary McDowell (03 January 2017) allows us to get approximate figures needed for such a normalization. This article informs us that in 2014 (which is in the time range of the data under consideration and also given that a typical graduate student/postdoctoral career in about 4-5 years) that there were about 69000 biomedical researchers in the US. Of these about 40020 were of European descent (excluding Hispanics) and 23500 with ancestry in various Asian countries. That primarily includes Indians, Chinese, Koreans and Japanese. To obtain the breakdown we can use two methods: 1) Sample a random set of 100 biomedical research publications and see what proportion of these Asians are found in them; 2) Use the proportions of graduate students from these nations in American biomedical research programs to get a sense of the breakdown. Based on this we can estimate the approximate maximum number of biomedical researchers of these ethnicities to be: 4236 Indians, 12236 Chinese, 3928 Koreans and 3059 Japanese. For West Asians it is harder to get an estimate but based on the biggest contributing group of these the Israelis from a publication in the Jewish newspaper Haaretz we an infer this number to be $\approx 1725$. This allows a more specific normalization, which is shown in column 5 of Table 1 as the number of fraudsters per 1000 researchers.

The Indians continue to remain the most prone to fraud even after this more realistic normalization. If the 60 fraudsters were evenly distributed among the 69000 biomedical researchers then the probability of finding a fraudster at random would be 0.00087. Given this and their numbers in the biomedical workforce, the probability that at least 20 Indians are frauds in the federal register by chance alone is $4.44 \times 10^{-10}$ or they are $\sim 8$ times more likely than other populations to commit biomedical research fraud. From the names of the Indian fraudsters in the federal registry we can infer that in all likelihood all of them come from the “forward castes”. Up to $1/3$ of them are likely of brāhmaṇa descent. This means they are drawn from the Indian elite. As can be seen in Table 2 they are drawn from all over India with a particular over-representation of individuals from South Indian states.

Table 2
$\begin{tabular}{lr} \hline Region & Count \\ \hline Tamil Nad/Kerala & 7 \\ Telugu states & 4 \\ Bengal & 2\\ Sindh/Panjab & 2 \\ Hindi belt'' & 3 \\ Maharashtra & 1 \\ Karnataka & 1 \\ \hline \end{tabular}$

Further, since entry into the US biomedical work force typically involves an IQ test (administered either for admission to an American graduate school or to an Indian institute), we are dealing with people most likely with IQ $\ge 123$. Thus, what we are seeing is not per say a problem of cognitive capacity but a problem of “ethics” or “corruption” in the Indian cognitive elite.

One could point out that there is some bias against Hindus in the US academia; hence, they might be specially picking on the Indians as opposed to the European origin majority or other foreigners, though all commit fraud to a similar degree. Undoubtedly there is bias — powerful fraudsters of European or Jewish descent are more often “rehabilitated” or overlooked than those of Indian descent. Nevertheless, that is unlikely to have been the primary cause for at least the cases in the federal register. Several of the Indian cases in the register were considered “golden boys” by the American institutions or had vanity articles about their fraudulent research in American outlets. Thus, it does not appear that at least in these cases they were being specially targeted. Thus, we posit that there is a real problem.

A part of this problem is a general one. The Euro-American biomedical research (which sets the trend in most of the world except to a degree in Japan) is beset with several serious problems:
* There is very little attention paid in biology education to the theoretical foundations of the science. We would go as far as to say that less than 50 $\%$ of the practitioners in biomedical research have good grasp of the foundations of biology. To give an analogy of how bad this is, imagine more than half the physicists and real engineers plying their trade without knowing classical mechanics in any serious sense (e.g. having not much of a clue of how to set up a Hamiltonian or a Lagrangian of a system). As a result poor hypotheses abound, which in turn spawn a glut of bad ideas.

* The Euro-American scientific system has an unhealthy model of competition and the fetish of peer-review, which favor both an urge to cheat to get an article published as well as rich dividends for nuanced plagiarism -i.e. plagiarism of ideas without citation rather than outright copying of text. To put it bluntly we have seen some form a plagiarism of our work with total impunity almost every other month in the past few years. Plagiarism also contributes to confirmation bias and fake reproduction of bad ideas. A part of this competition is fostered by big labs in several Euro-American institutions, where powerful principal investigators run the show like industrial sweat shops. These environments also do not allow for proper oversight — I am aware of cases where the graduate student or postdoc did not see the PI for more than a month at a stretch. Further, most interaction was limited to sanitized presentations in lab meeting rather than direct oversight by the PI at the bench.

* The magazine culture: Euro-American biomedical research assessement and funding agencies place enormous emphasis on publication in the two famous British and American science tabloids or the journal Cell, their many offspring and upstarts like eLife, PLOS Biology etc. In some of these venues, especially the tabloids and Cell articles are subject to insane review processes with time lags of 6-12 months from submission to acceptance for publication. These venues are thus high stakes venues that increase the urge for plagiarism and chicanery.

These are general causes that affect both Indians and others. However, as the scientific system in India increasingly emulates these practices in some form, it increases the incentive for Indians in India to commit fraud rather than do good science. Now let us consider some factors that might predispose Indian origin researchers to commit fraud more often than others:
* Some of the fraudsters who are in the federal register come from labs run but Indian origin PIs in American institutions. Their labs are reported as having an unhealthy environment — high pressure to produce results at short notice without adequate mentoring or oversight. We have evidence that this was the case even in the fraud cases which are not in the register. As noted above this is a major recipe for malpractice. Thus, it appears that there is some tendency for Indian origin PIs to be less than professional in managing their labs.

* Indians face major immigration constraints in the US. This can be used as an anvil both by American and Indian origin PIs to pressurize their students, who typically have no other avenues for escape or alternative employment due to sword of deportation hanging above their necks. Hence, science takes a back seat to survival and the incentives are slanted towards getting ahead by means of malpractice.

* Training in Indian schools and colleges does not emphasize aspects of honest scientific practice. Students often manipulate laboratory experiments to get results that their instructor expects. Little training is given in the statistics of variation and experimental error in school and college. For example, in my first semester in college I vividly remember the instructor conducting physics lab demanding that we exactly get $g=9.8\tfrac{m}{s^2}$ in single trial experiment!

* Importantly, most students in the science stream in India have no real interest in science per say but merely see it as a means for obtaining a seat in engineering or medical school. Those who do not make it typically enter the sciences and gradually drift their way through a B.Sc. and then a M.Sc. to finally reach a graduate program. Most of them are not from the cream of the educational system and are often not suited for cutting-edge science. Thus, when they make it to graduate school in the US they come ill-equipped for science and when subject to pressure might have some incentive to “game the system” just as they gamed the exams in India to get ahead.

* After the near-death encounters with Islam and Christianity, Hindu civilization is a shadow of its former self. There is serious decline of the internal system known as dharma. The decline of dharma shows up in the form of loss of discernment regarding the pursuit of knowledge and the pursuit of gain. In the past when a person committed himself to the pursuit of knowledge, there was a strong demand on him to observe certain ethics. These were enforced by the gate-keepers of the system and quacks would be punished. Indeed, father Manu the law-giver has a long list of criminals engaging in fraud like palmistry or future-prognostication, quacks, scams, and fellows claiming qualifications which they do not possess. He recommends that such be caught by the Rājan using investigators sent to keep an eye on such and if their crimes are proven to fine them. However, multiple repeat offenders could be put to death. Similar the legal tradition of Yājñavalkya has various provisions for the punishment of the quack physician.

In conclusion, we do not find any pleasure in presenting this. It personally only affects us in a negative way given that it has brought a bad name for Indian researchers, which is not going to go away in the near future. That is why I tell people that it is better to be a small man doing some low key but real science rather than professor Big who appears in the newspapers but does fake science. The approach of fake it till you make needs to be adopted with greatest care and does not apply to every aspect of human activity. These observations might have much deeper, unpleasant implications. It is relatively clear that the Hindus have not performed too well for being an old nation with clearly visible past achievements after they saw off the English tyrants more than 70 years ago. There are many reasons offered like the effects of the medieval Mohammedan incubus, the English tyranny, and Gandhi and Nehru. The former factors certainly have had their effect. But the past 70 odd years since independence have seen no major revival of scientific pursuit. There have been no major successors to many of the exemplary solitary Hindu researchers from the pre-Independence or early post-Independence era. We suspect that one cause for this is a terrible culture of knowledge generation among Indians in recent times. An offshoot of this is this tendency of dishonesty that we are seeing among Indian biomedical researchers. It shows in terms of tangible technology too: despite having an big need for the aeronautical engine or a proper assault rifle, Indians have had considerable difficulty in successfully mastering these technologies. Facets of this are also seen in other areas of Indian creative expression, e.g. journalism and cinema. I do not watch cinema but I am reliably informed by someone who does that there is some tendency for plagiarism from the occident. As with science, this tendency in journalism along with unthinking adoption of occidental memes are damaging for the nation. In journalism the idea is merely to produce uncritical stuff that the pay-master (e.g. the mleccha) likes. Hence, we feel that this data should be presented so that our people make take deep look at their problems and consider their science policy implementations accordingly.

Big man’s story
The fake scientist

## Leaves from the scrapbook-3

As described here these entries are from the scrapbook of Somakhya.

Entry 4; Āgantu, year Anala of the first cycle: I was at station to board the train to Kshayadrajanagara. My extended clansman’s wife got the young lady who I had accompany. She seemed like a quiet and shy person who said nothing beyond her name which was Shallaki. While remarkably unusual like that of the caturbhaginī, it still sounded vaguely familiar though I could not place it. Since I am no conversationalist myself I did not say anything beyond my name and just helped her a bit with her luggage. She seemed to be in the age range as the two younger caturbhaginī but something again struck me as unusual beyond her name: she had some lakṣaṇa-s similar to them unlike no other women, which are only visible to the insiders of the gaṇacakra. I again quietly remarked to myself that it was a strange thing. While some ruffianly dāḍhīvāle got into the train with us, thankfully, they passed ahead and the two other travelers in the compartment seemed innocuous. One was a soldier and the other was a pious vaiṣṇava woman. As the train got moving I spent my time looking out of the window as the vast expanses of Bhārata had always fascinated me. Occasionally an interesting temple and on other occasions a cemetery would pass by. When the latter would come in sight I would quietly utter the first verse of the Nīlarudra. Sometimes a strange-looking granite mountain would pass by. By examining its rock closely, I realized that these were the Archean granites — some of the oldest rocks on the Earth. As the train coursed ahead I caught sight of some sedimentary rocks likely from the Proterozoic and remarked to myself that someday I should perhaps journey there to see if I could collect some of that rock.

The reddish dusk soon gave way to the inky darkness of night and I could catch sight of the great star $\alpha$ Cygni giving me a good indication of the precise direction in which we were traveling. In clear air away from the cities I could catch sight of the Milky Way sprawling in the midst of the summer triangle. My fellow-traveler Shallaki then opened a large packet my extended clanswoman had provided her and asked if I might want to have dinner. I acquiesced and she quietly gave me a share while telling me that I could take more if I wished. She also courteously offered the food to the other two passengers who had, however, purchased their own fare. I received a message from Lootika who was also journeying back home to take the same exam but at a different center. Like me she too was unhappy about the break it was causing to the flow of our work. She also had some other sense of unease, which she stated arose from an encounter with a bhūta that Vrishchika had told her about. But the exchange grew more lively when she said that she had collected several specimens of microthelyphonid whip-scorpions in course of her field work that had been totally ignored since the English naturalists had found them in the days when their tyrants were lording it over our land. We exchanged messages regarding the peculiar bristles on their second limb wondered what their function might be. Thinking about this I seem to have fallen asleep.

The next morning when I arose the rest of the passengers in the compartment were already awake. The elderly vaiṣṇava woman was reading some devotional material and asked me if a certain station had passed since I was the only one who kept gazing out of the window. I replied in the negative. She was happy as she said that she wanted to get some surasā and uttered it with a Dravidian accent. I was a bit puzzled and did not get what she said. Shallaki pointed to me that she meant surasā as in tulasī. A little while later the train stopped for at that station and I darted out and got bunches of basil for the vaiṣṇava woman. Since she said she was proceeding to a shrine of Nṛsiṃha where she wanted to offer them I got some extra bunches so that she might offer some on our behalf too. She was very thankful that I had obtained the basil given that she might have not been fast enough with her aging limbs to get back into the train. She insisted on getting me and Shallaki dośaka-s in the next stop despite our strong protestations. Invoking Rudra that his darts might not harm me in the form of food I consumed the dośaka in order not offend the elderly woman.

When Shallaki told me that the lady meant tulasī by surasā, I made a remark that it gives some hint about the obscure etymology of the plant’s name. Till that point in the journey Shallaki and I had exchanged just a couple of sentences but her eyes suddenly lit up at that comment of mine and we had a interesting discussion on the etymology of the Indian words for the basil and wandered off into a discursive chat on the substrate in Indo-Aryan. It was in course of that conversation that I learned that she was interested in the evolution of languages and that it was the object of her study. The conversation also convinced me that the crossing of our paths had some deeper significance but I still did not known what it was.

Entry 5; Cakram, year Anala of the first cycle: Finally, we arrived at Kshayadrajanagara and upon getting off the train I saw Indrasena and his brother Pinakasena who had come to pick me up. I introduced them to Shallaki. The brothers stole a quick glance at each other with utter and unbelievable surprise but did not say anything. Suddenly, I realized that a prophesy from a few years back by might be playing out. We boarded the bus to go to Indrasena’s home. While my mother had wanted that I stay with my uncle during this visit, I instead had made up my mind to stay with Indrasena’s family. My cousin Saumanasa was also writing the same exam and I thought I could meet her briefly after the exam and convey my familial sentiments.

As the bus labored through the crowded roads of Kshayadrajanagara we saw many a dirty sight — a consequence of a roguish political party Congress-S which ruled the state and paid little heed to cleanliness. Coming from a small town Shallaki was more startled by the sights than the rest of us. Suddenly the bus came to a standstill and showed no signs of even inching forward. Despite craning my neck, I could not see much but Pinakasena clarified that the city was being visited by an important CEO of a mleccha multinational, a person of Indian origin, Pachchaisundari by name. Her convoy was passing by and had held up all the traffic. She was recently in the news for the supreme court had green-lighted her plan of online-social credit which she had established in collaboration with another CEO Lundberg. It works thus: If you say publicly made a statement on social media like ”Mohammad was perhaps the greatest man who ever walked on the earth” or ”Mahmud Ghaznavi employed Hindus and Moslems alike based on their merit and service credentials” then you got a positive credit. On the other hand if you made a statement like ”Baboor demolished a famous temple of the Hindus at Ayodhya” then you got a negative credit. Building negative credit could eventually lead to such a score that your email on Pachchaisundari’s platform could be locked up for a week or you could not post on social media and so on. But if you built positive credit you could cash it for discounts on online purchases, subsidized tickets for online movies and serials and the like. It was being touted as a great tool to aid the building of secular democracies although it was a model pioneered first the neo-Han empire of Xi in China.

After nearly a half hour wait, we finally got moving and reached Indrasena’s house. We let Pinakasena to lead Shallaki to the apartment she was to stay in with two other women. After I had refreshed myself with a bath, I went to Indrasena’s room and he mentioned the prognosis of the Vīrabhadra-nartaka. I nodded and said that everything was falling in place indeed and confirmed to him that till the moment I saw him and his brother I did not precisely realize what was playing out. Indrasena asked me if had heard anything from Lootika and before I could answer told me of Vrishchika’s encounter with the bhūta. I remarked that Lootika was rather worried of the same and that the alignments of the prophesies were rather striking and even quite unexpected to me.

Entry 6; Luki, year Anala of the first cycle: After the exam was over I briefly dropped by at my uncle’s workplace to wish him and then went to his house to meet my cousins and aunt. Then I ambled back to Indrasena’s home for the evening. Pinakasena had sought the permission of his parents to have Shallaki over over for dinner. Thus, when I came in the three of them were lost in a discussion on certain intricacies of the upāsanā of Guhyakālī. The prophesy was now confirmed in my mind beyond any smidgen of doubt. Leaving the other two to continue their discussion, for it was after all their kula, Indra and I went over to his room. He asked me a bit about the exam since he was to write the same the following year. I then offered worship to his idols and pictures of Vaiśravaṇa. He told me of a strange dream he had witnessed the prior night — he seemed think there was something to it. He had dreamt of a man being killed by a centrifuge rotor exploding out of its spin-drive and striking him. The four of us then went up to the terrace to ply the planchette as we all felt a strong premonition of encountering at least one bhūta. Surely enough, we had steady stream of phantoms animating our device one after the other. The first bhūta startled Indrasena greatly. It was a ghost of a young man who said that he had been killed in a centrifuge blast at Turushkarajapura. Indrasena wanted to engage him a bit more but I sensed danger in letting him hang around. So I signaled to Indra to dismiss him right away. Before leaving he tried to seize Shallaki but she repulsed him with a mantra that neither I nor Indra knew but was apparently known to Pinakasena. I knew, however, that it was not the last that our gang was going to see of this phantom.

Posted in Life | Tagged , ,

## Mongolica: The Tangut empire

In the early 1100s of the CE Rtsa-mi lotsawa Sangs-rgyas grags-pa was in Nālandā, India, to study and transmit the latest that the tāntrika strain of Bauddha-mata had to offer. Within a century both his world and that of his Indian hosts was to come crashing down. It was almost as if the prophetic section of the nāstika Kālacakra-tantram that he was there to study and take to his lands would play out unerringly as the yuga-cakra turned with the irruption of the demons of makkhaviṣaya and the mlecchendra-duṣṭa-s. Sometime close to the beginning of the 1200s a marauding band of Mohammedans led by the Moslem Turk Ghazi Ikhtiyar al-Din Muhammad Bakhtiyar Khalji razed to ground this famed mahāvihāra of Nālandā. Rtsa-mi lotsawa Sangs-rgyas grags-pa was a Tangut from their kingdom known as Xi Xia in inner Asia. In 1205 CE the Tanguts supported the fleeing rivals of Chingiz Khan and were subject to a swift punitive assault by the Mongols from the western corridor. The Mongols followed it up by further invasions in the Tangut territory in 1207-8, 1209-10 and 1215 CE, in each case punishing them, seizing bits of their kingdom and forcing them to vassalage. In 1217 CE the great Khan sent an emissary asking the Tangut kingdom to join the Mongols in the great assault on the Mohammedans of Khwarazm. The Tangut king arrogantly responded: “If you have no strength [to fight on your own] there is no reason to be a Khan.” The Tanguts were pay dearly for this: they became the target of the final campaign of Chingiz Khan. By 1227 CE the Mongols had erased the Tangut kingdom off the face of the earth.

What was the origin of this Tangut Kingdom and what was its story? This remained somewhat mysterious given that it had been annihilated by the Mongols. However, their language survived that event and continued to be used by the remaining Tanguts until it was finally exterminated by the Ching. They would have remained a mystery had they not produced an enormous volume of written literature, a library of which was discovered by the Russian explorer PK Kozlov in 1908 in the Tangut fort of Qara Khoto. This library has formed the bed rock of Tangutology, which has since then has had a small but vigorous set of practitioners. It is rather unfortunate that there has been no Indian interest or participation in Tangutology especially given the religious links they had to India and the importance they attached to knowledge of the Sanskrit language and production of translations from it. Further, after the destruction of their kingdom the surviving members of the ruling clan became notable in the Indian state of Sikkim. The Tangut language was itself written in a very complex system where basic logograms are successively put together to make a new array of logograms that represent further new words. They superficially resemble Chinese but are mostly unrelated to it. This complex Tangut script was almost designed like a secret code so that none of their competitors and neighbors like the Tibetans, the Uighurs, the Khitan or the Han could read their documents. It was also designed with the view that its structure encoded a certain innate “mantraic power” — ultimately a Hindu concept acquired from the bauddha tradition (It is conceivable that the mysterious Khitan script also was designed with such ideas in mind). This script was in place by at least 1036 CE. The Tangut language written in this script found in the books discovered by Kozlov was deciphered by the great Russian Japanologist NA Nevsky, who was murdered by Stalin’s henchmen before he could publish it. This decipherment has formed the foundation of the Tangut studies that followed and suggest that it was member of the Tibeto-Burman family of languages. These texts give us a picture of the heydays of a rather interesting state, the Tangut Kingdom.

A Tangut document showing their script and artwork

From the Chinese evidence we observe that their predecessors or at least that of the ruling Ngvemi clan of the Tanguts might have originally had links to the Tabgach (Chinese: Tuoba) people with Altaic affinities: they were either early Turks or Mongols or a distinct extinct branch of the Altaic family. Consistent with this, the Mohammedan Qarakhanid Turk, Mahmud Kashghari, who wrote a Turkic-Arabic lexicon mentions the contemporary Tanguts as being one of the more sedentary of the 20 tribes of Turks that he lists. A Chinese Song record also claims that they were a clan of Turks. A memory of the pastoralist origin of the ruling clan, consistent with Altaic roots, appears to have been preserved in the poem of one of their ancestors that reflects the simple concerns of such herders:

They fixed the livestock enclosure, a wolf cannot get it,
They dug a well in the thicket, the livestock will not suffer from thirst.

If the courageous and wise do not sit [there], the meeting will not be successful,
If there is no bull with high horns in the herd, the herd is empty.

If you cannot ride the rounds on a horse, it is no good for riding.
If the livestock are beyond count, the owner deals only with livestock.

If you know the saying poorly, you will not be able to have a conversation,
If you have a few horses and yaks, you will not eat your fill.

There are no better close ones than [one’s] father and mother,
there is no meat tastier than the meat on the bones.

He who has livestock is not rich,
He who has a [good] mind is rich.

-translated from the original Tangut by the famous Tangutologist Kychanov.

In their own mysterious verse on their ancestry, they claim that their ruling Ngvemi clan stemmed from their ancestral mother ‘A-mbah (the etymology of this name is unknown to me but of interest!):

Our mother ‘A-mbah, source of families and clans,
silver-wombed and golden breasted,
the valiant tribe did not die out and
carries the name of Ngvemi.

-translated by Kychanov

In this regard it is notable that the Mongol sources in China mention a Tangut ethnonym as Yü-mi, which is believed to be derived from the name of trans-Himalayan goddess Umai who is mentioned as consort of Möngke Tengri in Turko-Mongol sources. In the Indo-Aryan world this goddess was incorporated as an ectype of Rudrāṇī i.e. Umā. It is hence conceivable that her other name Ambā has a relationship to the ancestral mother of the Tanguts mentioned in the above verse.

However, in their own writings the Tangut also mention that they as a people, the Tibetans and the Han Chinese have a common origin: was this some kind of early recognition of the monophyly of their Tibeto-Burman languages? In their earliest days going back to the 300-400 CE they appear to have had marriage relationships with the para-Mongolic Tuyuhun (Tuguhun) Khaganate and played a role in the formation of this state in between the Tibetans and the Chinese. It was in this period they appear to have associated with the speakers of the Tibeto-Burman language that became the dominant language of the Tanguts. Other Tabgach people appear to have sinicized early and adopted Chinese in the more eastern regions. As the Tibetans started expanding their empire in the 600s and eating into the Tuyuhun Khaganate and marching towards Tang China the proto-Tangut fled North-East and submitted to the Tang empire in return for protection of their identity against the Tibetan advance. Thus, having survived the Tibetan assault and the proto-Tangut developed a close client relationship with the Tang Chinese empire. However, this did not help them when they came under attack again, this time from a more determined foe — the Blue Turks (Gök Türks) of Mongolia. Blue Turk ruler Qapaghan Khagan had charted out a comprehensive plan to attack the Tang empire and punish the Chinese severely. He first aimed to capture the Chinese towns of Paoting and Chengting and for this decided to perform a flank-clearing operation by neutralizing those who might come to the aid to the Tang. Thus, in 700 CE he sent his 17 year old nephew the rising star of the Turks and their future Khagan, Tengrida Bilge, against the Tangut. This is the first time we hear of them under that name. The results of the devastating Turkic attack on them is summarized in the Orkhon inscriptions thus: “When I was 17 years old I went on a campaign against the Tangut. I put the Tangut to rout; there I took their wives and children, horses and possession” [Translation: Talat Tekin from the Bilge Khagan inscription in Mongolia].

Having barely survived this assault they got a chance to enhance their profile when the Blue Turks and the Uighurs were gone or had declined and Tang empire was wracked by natural calamities in the late 800s. The Tang excesses in their wake triggered the rebellion of farmers under Wang Xianzhi. At the same time a Chinese smuggler Huang Chao was taking advantage of salt shortages to run a salt trafficking operation. Having amassed a force of fighters he made an initial alliance with Wang Xianzhi and then broke up with him to lead his own rebellion, which in many ways paralleled that of the Chinese brother of Jesus Christ closer to our times. At the height of his rebellion Huang Chao took Guangzhou and then the Tang capital of Chang’an. The Tang emperor fleeing from these attacks called upon the proto-Tangut lord Tuoba Ssu-Kung (880-884 CE) to come to his aid. The Tangut crushed the Huang Chao rebellion in the west between and were rewarded the regions of Inner Mongolia by the Tang and title of “Prince Pacifier of the West”. The Tangut lords then welded together a multi-ethnic state which other than themselves contained Chinese, Tibetans and the remnants of the Tuyuhun Khanganate (Some of whom later joined Chingiz Khan; e.g. the lord Alaqush-digit-quri).

Subsequently, with the decline of the competing Central Asian powers operating in the region the Tangut were able to solidify their power. In the 900s when the Tang collapsed the Tangut established themselves as an independent state. While the Song tried to reconquer the old Chinese territory in 960 CE they could not subjugate the Tangut who continued to grow in power. In northern China the Khitan of para-Mongolic origin had established the Liao dynasty, which was also poised for potential conflict with the Tangut. However, negotiating their hold through these times they consolidated their kingdom as the Xi Xia, which was what Chingiz Khan eventually destroyed. It came to include what is today the Western and Central part of China’s Gansu province, Northern Ordos in Inner Mongolia and Ningxia Hui.

By around 1000 CE it appears that the Tangut queen Lady Wang had become a practicing bauddha with special attachment to the deity Mañjuśrī. At her death in 1007 CE the Tangut requested that the Song allow offerings to be sent to Mañjuśrī at his holy mountain Wutai in the name of the late queen. The Song allowed these offerings to proceed but had reasons to be vary of the Tangut. The Tangut prince Li Yuanhao, who also had an affinity to Mañjuśrī, was a noted warrior and began a series of conquests of Chinese and Uighur lands starting in the 1020s. He took Hexi and then in 1028 advanced against the Uighurs. The Tanguts defeated them after a bloody battle which is recorded by Mahmud Kasghari with some glee because that relieved the Uighur pressure on the Qarakhanid Mohammedans. Thus, the Tanguts seized Ganzhou and moving on in 1031-31 Li Yuanhao captured Liangzhou and finally took Dunhuang. This brought him in possession of various bauddha domains strengthening his association with the religion. Sometime after 1035 CE an Indian śramaṇa, evidently fleeing from the assault of the Mohammedans led by Mahmud and Masud of Ghazni on Gandhara and the Panjab, reached the Tangut kingdom. He presented Li Yuanhao with what he claimed were 150 relics from the cremation of the Buddha. These were supposed to include an ungual and a cranial fragment (likely from Gandhara). Filled with piety Yuanhao had an elaborate coffin made for them with a stone inner chamber in which they were placed in a silk wrap with gems. This in turn was placed in an iron box which was then placed inside a golden casket which in turn was placed in a silver outer coffin. The coffin was then placed above an underground spring. On top of the coffin he had an image of Kubera installed and the yakṣarāṭ was invoked to protect the king and state and provide overflowing granaries. In this period Yuanhao (1035 CE) also bartered several horses that they were good at breeding as pastoralists for bauddha texts in Chinese translation from the Song. He obtained another batch from the Khitan for a similar barter of horses. He then commissioned translation of all of these into the Tangut language by a panel of scholars — this illustrates the esteem in which he held the direct textual study of bauddha knowledge.

Thereafter, around 1038 CE Yuanhao declared himself to be cakravartin, superior to all the Han emperors of the past, and his kingdom to be a dharmarājya. Despite claims of being an embodiment of humaneness as per the bauddha-mata Yuanhao’s coronation rituals as cakravartin had elements that were clearly aimed at a display of military power. It remains unclear where these elements of the rituals came from — the pre-bauddha Tangut religion or from some Altaic adaptation of the bauddha-mata. We hear that:

“[Yuanhao] together with his braves smeared blood on their lips and took an oath first to attack Fuyan, desiring to enter [Song territory] simultaneously from three routes out of Dejing, Saimen fort, and Chicheng. Then [he] built an altar, received appointment, and assumed the imperial position (ji huangdi wei). At that time [he] was thirty sui.” – translation from noted Tangutologist Ruth Dunnell.

Hence, when in 1038 CE when Li Yuanhao, now king of the Tanguts, wanted to send offerings to Wutai the Song blocked them claiming that he was using it as ploy to spy on the defenses of Shanxi, which they felt he might raid next. Enraged Li Yuanhao commissioned the construction of this own mountain-shrine of Mañjuśrī for worship according the tantra known as the Mañjuśrīya-mūlakalpa. This was set up in the Helan mountains west of modern Yinchuan oriented in the holy Northeast axis of Mañjuśrī with respect to India.

By this time the Tangut state came into its own as a well-oiled military machine. Their law stipulated that every able-bodied man from age 15 to 70 was to enlist to serve in the army. Failure to enlist was punished right away with death. Every boy at age 15 was subject to a special medical checkup by state doctors and if found fit was registered in the army. Half were assigned to the reserves and half were placed in the current fighting register. Based on their abilities the men were either placed in the fighting force or in service and logistics or in military engineering. The latter included military manufacture, which produced iron and fashioned cavalry swords (= talvār-s) on a large scale that were reported as being among the best in the world by the contemporary Chinese. The recruits were classified based on the household income: the men from lower income families had to join the army with either one horse or one camel; those with higher income had to join with two horses or two camels. The main fighting force of the Tanguts was a formidable cavalry with each unit placed under a unit commander who was to be protected by his men to death. The best men were drawn into elite forces units known as the “Iron hawks” who conducted special operations and deployed special weapons. They also had a special weapons camel-division, which comprised of mobile stone-throwing machines with a rotating base mounted on camel backs which allowed all-round bombardment. On the theoretical front the Tangut made a close study of the military theories of other civilizations. They made several translations of the Chinese military works like the “The General’s Garden” and the “Art of War”, both of which they appear to have adapted to their needs. The Chinese record the terror of the movement of the Tangut army which they mention as “moving like a tornado this way and that…” This martial background of the Tanguts provides a measure of the military achievement of Chingiz Khan and his men when they smashed the Tangut armies one after another.

With a strong military system in place in Li Yuanhao eyed further hostilities with this neighbors even as he had vowed during his coronation as cakravartin. In 1038 he faced the possibility of enmity the with the Khitans as his wife who was a Khitan died under mysterious circumstances and the Khitans suspected that he might have eliminated her. Having for the time being pacified the Khitan ambassador sent to his court Li Yuanhao marched against the Qingtang Tibetan state and crushed them. Then in 1039 he opened the big war with the Han Chinese of the Song empire. This war lasted 5 years and towards its end the Khitan also joined it against the Tanguts. However, the cakravartin proved his name and emerged victorious on both fronts solidifying the Tangut state and perhaps now justifying the the term empire applied to it due to his conquests. In 1046 Yuanhao faced a second Khitan invasion which ended inconclusively on the battlefield but the Tangut ended up retaining their territory. A couple of years later Yuanhao and his eldest son with whose mother he had a conflict had an armed duel and he was killed as a result.

In the interim period where a tribal council was ruling them the Tanguts faced a defeat at the hands of the Khitan (1051) and a few years later they negotiated peace with the Khitans who, however, refused to have any marriage with them from then on. In this period they were ruled by a queen who was assassinated in 1056 CE. In 1067 for the first time in a while the Song were able to retaliate for the successive defeats they had been facing at the hands of the Tangut and defeat them even as their cakravartin Liangzuo, bastard son of Yuanhao born of incest died. The Han Chinese followed this up with successive victories against the Tanguts and in 1081 taking advantage of their internal strife launched an invasion of their empire. Finally, in 1084 after heavy losses the Tanguts repulsed the Chinese. They faced a Tibetan invasion shortly thereafter but managed to negotiate a settlement without territorial loss. In 1096 the Song attacked the Tanguts again in attempt to destroy them but they repeatedly asked the Khitan for help and finally in 1099 managed to repel the Chinese attack albeit with loss of some territory. The Chinese then took advantage of the turmoil in Tibet and tried to invade it. The Tangut quickly formed an alliance with the Tibetans against the Song and established a marriage alliance with the Kokonor Tibetan ruler to face the Chinese in 1102. They then negotiated with the Khitan to aid them to win back their territory lost to the Song. They appear to have partially succeeded in this with Khitan help by 1106. In 1114 the Chinese resumed their war with the Tanguts and a few years later they lost their Khitan allies, whose empire in China was dramatically overthrown by the Tungusic Jurchen invasion (proto-Manchu). Taking advantage of the chaos that followed the Tanguts led by their emperor Li Qianshun recovered all their lost territory and also seized Qingtang from the Tibetans.

This was followed by a phase of sinophilia among the Tanguts where they established the cult of Confucius and appointed a Chinese commander Ren Dejing who surrendered to them as a high official. With his internal Chinese faction he staged a coup to nearly precipitate a civil war after they threatened to take over the state. He and his Chinese faction were all killed in 1170 and the Tangut empire’s unity was restored. At that point with their army at the peak of its size and performance it looked as thought the Tanguts were unconquerable. Indeed, in 1193 the Jurchen tried to invade them at multiple points along the border but were forced to retreat. But in 1205 CE they faced a challenge of the kind they had never faced in their whole existence: Temüjin, who was soon to be Chingiz Khan. Those who they thought might be a weaker version of their para-Mongolic cousins the Khitan proved to be something else — within 5 years from first defeat at the hands of the Mongols the mighty Tangut empire was reduced to vassalage followed by its total destruction.