## Two squares that sum to a cube

Introduction
This note records an exploration that began in our youth with the simple arithmetic question: Sum of the squares of which pair integers yields a perfect cube? Some obvious cases immediately come to mind: $2^2+2^2=2^3; 5^2+10^2=5^3$. In both these cases we can see that the addition of a square the to the square of the first number yields its cube. Hence, we can ask another related question: Which are the integers whose squares when added to another perfect square yield their cubes? In general terms the answers to these questions are the non-trivial (i.e. where $x, y, z \ne 0$) solutions of the indeterminate equation:

$x^2+y^2=z^3 \;\;\; (\S 1)$

Figure 1 shows the first few solutions lying on the surface defined by $\S 1$. Due to the 8-fold symmetry we restrict ourselves to positive solutions such that $x \le y$ for a given $z$.

Figure 1

Since $z$ is explicitly defined by $x, y$ we can study the solutions of $\S 1$ by plotting $x, y$ on the $x-y$ plane (Figure 2). At first sight, a mathematically naive person might perceive only a limited order in this plot of the solutions. However, as we shall see below, a closer examination and some algebra reveals a deep structure.

Figure 2. Some of the more obvious structure is indicated in different colors and is discussed below in detail.

Solution circles, 2 square numbers and Brahmagupta’s identity
Some simple algebra can provide an understanding of the structure of the solutions of $\S 1$. From $\S 1$ it is obvious that all solutions would be lattice points $(x,y)$ on circles of radius $z^{3/2}$. Based on this, we can determine a set of $x, y, z$ that are to solutions of $\S 1$ thus: let $z$ be an integer that is the sum of two perfect squares, $z=m^2+n^2$. Then,
$\left(m^2+n^2\right)^3= m^6 + 3m^4n^2 + 3m^2n^4 + n^6\\[6pt] =(m^6+2m^4n^2+m^2n^4)+(n^6+2m^2n^4+m^4n^2)\\[6pt] =m^2(m^4+2m^2n^2+n^4)+n^2(n^4+2m^2n^2+m^4)\\[6pt] =(m(m^2+n^2))^2+(n(m^2+n^2))^2 \; \; \; (\S 2)$

Thus, if $z=m^2+n^2$ (i.e. it is a 2 square number) $\S 2$ gives us an expression for $z^3$ as the sum of 2 squares. Hence, the valid solutions to $\S 1$ will be all
$x=m^3+n^2m\\[6pt] y=n^3+m^2n\\[6pt] z=m^2+n^2$

For example, if $z=5 = 1^2+2^2$ we have $m=1; \; n=2$. Thus, $x=1^3+2^2\times 1=5$ and $y=2^3+1^2\times 2= 10$. Hence, $z$ which belong to the 2 square sequence, i.e. numbers that can be non-trivially (i.e. one of the terms is not 0) expressed as the sum of 2 perfect squares, are valid solutions: 2, 5, 8, 10, 13, 17, 18, 20, 25, 26… The corresponding $x, y$ can be derived as above. One also notices from the above expressions that $m^2+n^2=z$ is a common factor for $x, y, z$ in all such solutions to $\S 1$.

While this method yields $x=5, \; y=10, \; z=5$, we notice that the same $z=5$ also yields a second pair of $x=2, y=11$. Now how do we account for these additional pairs and for what values of $z$ they arise?

The most obvious set of these correspond to $z$ being equal to special numbers such as 5, 13, 17, 29… One notices that these are both 2 square numbers and primes of the form $4n+1$. Fermat first noticed that such primes are always 2 square numbers. This attracted the attention of some of the greatest mathematicians like Euler, Lagrange and Gauss, who proved it to be so by using different methods. What is important for us is that such primes are also the karṇā-s (hypotenuses) of primitive bhujā-koṭi-karṇā triples. Thus, we have $z=m^2+n^2$ from the fact that they are 2 square numbers and $z^2=a^2+b^2$ from the fact that they are bh-k-k hypotenuses. Hence, we can write,
$z^3=z \cdot z^2 =(m^2+n^2)(a^2+b^2)\\[6pt] =a^2m^2 + a^2n^2 + b^2m^2 + b^2n^2\\[6pt] =(a^2m^2 + b^2n^2 +2am \cdot bn) + (a^2n^2 + b^2m^2 - 2am \cdot bn))\\[6pt] =(an-bm)^2+(am+bn)^2 \; \; \; (\S 3)$

Alternatively,
$(a^2m^2 + b^2n^2 -2am \cdot bn) + (a^2n^2 + b^2m^2 + 2am \cdot bn)\\[6pt] (am-bn)^2 + (an+bm)^2 \; \; \; (\S 4)$

$\S 3$ and $\S 4$ give us the famous identities of Brahmagupta from which we get two expressions for $z^3$ as the sum of a pair of perfect squares when $z$ is a $4n+1$ prime. From $\S 3$ we get:
$x =|bm-an| \\[6pt] y = am+bn\\[6pt] z = m^2+n^2=\sqrt{a^2+b^2}$

From $\S 4$ we get:
$x= |am-bn|\\[6pt] y= an+bm\\[6pt] z=m^2+n^2=\sqrt{a^2+b^2}$

For example, if we take $z=5$, we have $m=1, \; n= 2, \; a=3, \; b=4$. From first set of expressions we get $x=|4-6|=2, \; y=3+8=11$. From the second we get $x=|3-8|=5; \; y= 6+4= 10$. The second pair $x, y$ is equivalent to what we can get via $\S 2$. Thus, when $z$ is a 2 square prime its cube can be expressed as the sum of distinct two pairs of squares.

Now, one can get a further set of $z$ corresponding to solutions with two pairs of $x, y$ when say $z=2p$ where $p$ is a $4n+1$ prime. Here $z^3=2p \cdot (2p)^2$. From above we can write,

$p=m^2+n^2\\[6pt] \therefore 2p= (m+n)^2+(m-n)^2$

Further,
$p^2=a^2+b^2\\[6pt] \therefore (2p)^2= (2a)^2+(2b)^2\\[6pt] \therefore z^3= ((m+n)^2+(m-n)^2)((2a)^2+(2b)^2)$

Thus, from Brahmagupta’s identity we have:
$x = 2a(m+n)-|2b(m-n)|\\[6pt] y = 2b(m+n)+|2a(m-n)|$

Alternatively,
$x=2a(m+n)+|2b(m-n)|\\[6pt] y=2b(m+n)-|2a(m-n)|$

For example, if $z=26= 2 \times 13$ then $m=2, \; n=3, \; a=5, \; b=12$. Thus, $x=74, \; y=110$ or $x=26, \; y=130$. Similarly, one can derive formulae for $z=4p, 8p...$

When $z=p^2 =a^2+b^2$ where $p=m^2+n^2$ is a $4n+1$ (2 square) prime we can show via repeated application of Brahmagupta’s identity that there 3 possible $x,y$ pairs whose squares can compose $z^3$. We can derive the below formulae for them:
$x=a^3+b^2a\\[6pt] y=b^3+a^2b$

Or,
$x=p(a^2-b^2)\\[6pt] y=2pab$

Or,
$x= |m (b^2 n + 2 a b m - a^2 n ) - n | a^2m + 2 ab n - b^2 mb| | \\[6pt] y= (a m - a n + b m + b n) (a m + a n - b m + b n)$

As an example, by applying the above formulae we can see that for $z=25 = 5^2$ its cube, $25^3$, can be split up into 3 distinct pairs of squares: $35^2+120^2 = 44^2 + 117^2 = 75^2 +100^2$.

The nested combinatorial application of Brahmagupta’s formula can thus result in increasingly complex formations with multiple alternative partitions of a perfect cube into perfect squares for different multiples of the 2 square primes. Thus if $z=50=2 \times p^2$ we get 4 distinct pairs of $x, y$; for $z=125= 5 \times 5^2$ we get 5 distinct pairs of $x, y$.

When a number is product of two distinct $4n+1$ primes then its cube can be partitioned into 8 distinct pairs perfect squares. For instance,
$65^3= 7^2 + 524^2 = 65^2+ 520^2 = 140^2 + 505^2 = 191^2 + 488^2= 208^2 + 481^2 = 260^2+ 455^2= 320^2+ 415^2 = 364^2+ 377^2$
The same applies to the multiples of these numbers by 2, 4… When a number is a product of a $4n+1$ prime and the square of a distinct $4n+1$ prime then the cube of that number can be partitioned into 14 different pairs of perfect squares. Thus, $325=5^2 \times 13$ is the first number whose cube can be thus partitioned. The next is $425 = 5^2 \times 17$. Figure 3 shows some circles with multiple pairs of $x, y$ whose squares sum to the same $z^3$.

Figure 3

Ramanujan and Landau had independently discovered that the number of positive integers, $N(x) \le x$, that can be written as a sum of 2 squares including 0 defines a constant:

$\displaystyle K_{RL} = \lim_{x \to \infty} N(x)\dfrac{\sqrt{\log(x)}}{x} = 0.7642236535...$

Based on this, we can also look at how many unique $z \le x$, i.e. $N(z)$ by defining:

$K=N(z)\dfrac{\sqrt{\log(x)}}{x}$

We empirically observe that this $K \approx 0.8071$ (Figure 4). The $K_{RL}$ has a closed form which has deep connections to the Riemann $\zeta(x)$ and the Dirichlet $\beta(x)$. However, we are not aware of a closed form for the constant $K$ in our case or even its exact value as $x \to \infty$. This $K$ seems to reach a fairly stable value around that reported above but $K_{RL}$ converges very slowly.

Figure 4

Solution families on curves
In addition to the arrangement of solutions as lattice points on circle of radius $z^{3/2}$, there are also other patterns that become apparent from a closer look at the solutions in the $x-y$ plane (Figure 2). The most obvious is the set of points lying on the $y=x$ line at the right diagonal boundary of the plot. These are defined by a family of the form:

$x= y =2n^3; \; z= 2n^2$

This family defines the sequence of integers twice whose square equals a perfect cube: 2, 16, 54, 128, 250, 432, 686, 1024 … (blue line in Figure 3).

Then we see pairs of regularly positioned points that eventually lie closer to the right diagonal boundary (emphasized in red and blue in Figure 3). These are families that lie on one of two parallel curves. The first is defined by the parametric equations:

$x = 2t^3 + 6t^2 + 3t - 2 \\[6pt] y = 2t^3 + 12t^2 + 21t + 11$

This curve has a lobe (Figure 5) and generally resembles the shape of the curve of von Tschirnhaus (see below). The second curve takes the form:

$x=2t^3+6t^2+9t\\[6pt] y=2t^3+12t^2+27t+27$

This curve has no lobe and lies inside the divergent arms of the first one (Figure 5).

Figure 5

A further family lies on the curve defined by the below parametric equations (emphasized in red in Figure 3):

$x=3t^2-1\\[6pt] y=t^3-3t$

This curve has a single lobe and two divergent arms and is a version of the eponymous curve discovered by Ehrenfried Walther von Tschirnhaus, the famous German polymath, who among other things reinvented porcelain in the Occident. It crosses over the above two curves and proceeds closer to the vertical left boundary of the $x-y$ plot.

Finally, we a family of families lying on the family of curves (shown in blue in Figures 3, 5) defined by the parametric equations:

$x=kt^2 + k^3\\[6pt] y= t^3 + k^2t$,
where $k=1, 2, 3 ...$

The first member of each of the new families on this family of curves starts from where the curve intersect the $y=x$ line. Thus, each starts with the points $x=y=2k^3$ (Figure 3). The first of this gives yields the answer to the second question posed in the introduction. When the square of $x=n^2+1$ is added to the square of $y=n^3+n$ we get the cube of $z=n^2+1$, i.e. $x=z$. It remains unknown to us if there are any further families beyond these.

Posted in Scientific ramblings |

## Difference of consecutive cubes, conics and a Japanese temple tablet

Introduction
In our part of the world, someone with even a nominal knowledge of mathematics might be aware of the taxicab number made famous by the conversation of Ramanujan and Hardy: the smallest number that can be expressed as the sum of two distinct pairs positive cubes: $1729=1^3 + 12^3 = 9^3 + 10^3$. This number is just one of a family of such taxicab numbers with deep connections to other objects in the mathematical world that were discovered by Ramanujan long before anyone knew of their significance. There are several other interesting questions, which, in a similar vein, relate to the sum and difference of cubes. From Fermat’s last theorem we know that the indeterminate equation $x^3+y^3=z^3$ cannot have any non-trivial integer solutions (non-trivial being where none of the solutions are 0). However, this still leaves open other possibilities with sums of 3 cubes and differences of the cubes and the like. For example, one popular and widely investigated one asks which integers can be expressed as a sum of any three cubes. When we learned that Fermat’s last theorem precludes integer solutions for $x^3+y^3=z^3$ in our early youth, we wondered if there are non-trivial positive integer solutions indeterminate equation: $x^3+y^3+z^3=w^3$. Soon we became interested and investigated one specific version of it of the form:

$x^3+y^3=w^3-z^3$, where $w=z+1$; thus, we have $x^3+y^3=3z^2+3z+1 \;\;\; (\S 1)$

This led us to discovering and proving for ourselves a simple arithmetic theorem concerning such cubes. Later, we read to our amazement that a related problem had been considered on a Japanese temple tablet in a beautiful “mystical”-sounding verse and solved in a commentary on it. This inspired us to look at the problem again and we discovered further interesting links between conics and these cubes. We detail these explorations and the Japanese temple tablet below.

The difference of consecutive cubes and an arithmetic theorem
We first asked when does the difference of consecutive cubes result in a sum of two cubes. This idea came to us as a parallel to the bhujā-koṭi-karṇa triples. While we have 4 cubes in $(\S 1)$, given that two are consecutive, we only have a triple of distinct positive integers $x, y, z$ bearing the relationship:

$z = \dfrac{\sqrt{12x^3 + 12y^3 - 3} -3}{6} \; \; \; (\S 2)$

Thus, we have to search for the all the cases where $(\S 2)$ evaluates to an integer given dyads of positive integers $x, y$. If we order the resultant triples such that $x < y < z$ we get a unique set of triples, e.g. 1, 6, 8: $1^3+6^3=9^3-8^3$. Figure 1 shows a plot of $x, y, z$ for all $x, y<20000$, a total of 1173 triples.

Figure 1

The plot initially reminds one of a first time observer of the sky with a great mass of stellar points with some vague patterns among them. Given that $z$ is dependent on $x, y$ and the points lie on a single curved surface in 3D, we can reduce dimensions for simplicity and plot just $x, y$ (Figure 2).

Figure 2.

As we have ordered our triples $x, y$ all points obviously lie above the $y=x$ line. We then notice the first clear pattern. There are set of regularly arranged points (in red in Figure 2) that mark the left margin of the plot. These points lead us to the following theorem:

The cube of every positive integer is equal to the difference of the cubes of two consecutive positive integers minus the cube a third positive integer.

To prove this let us consider those regularly arranged points. The first few of them are tabulated below:

x y
1 6
2 17
3 34
4 57
5 86
6 121

We observe that the $x$ values of these points include each positive integer in order. The corresponding $y$ grows rapidly and follows a peculiar pattern. This pattern is represented by the numbers lying on the 5th spoke hexagonal spiral (Figure 3) where 6 spokes separated by the rotation angle of $\tfrac{\pi}{3}$ radians pass through the origin.

Figure 3.

The numbers lying on the 5th spoke of this spiral can be represented by the formula: $3n^2+2n+1$. Thus, we have $x=1, 2, 3...n$ and the corresponding $y=3n^2+2n+1$. Hence,

$x^3+y^3=n^3+(3n^2+2n+1)^3= 3(3n^3 + 3n^2 + 2n)^2 + 3 (3n^3 + 3n^2 + 2n) + 1 \;\;\; (\S 3)$

If we write $z=3n^3 + 3n^2 + 2n$, the right hand side of $(\S 3)$ becomes $(z+1)^3-z^3$

$\therefore x^3=(z+1)^3-z^3-y^3$, where $x=n;\; y=3n^2+2n+1;\; z=3n^3 + 3n^2 + 2n \;\;\; _{...\blacksquare}$

Thus, this theorem illustrates one deep connection between the cubes of numbers the hexagonal number spiral.

The Japanese temple tablet
The Samurai intellectual Shiraishi Chochu recorded a problem inscribed in tablet hung at a temple in the 1800s by the poorly known mathematician Gokai Ampon:

“There are three integral numbers, heaven, earth, and man, which being cubed and added together give a result of which the cube root has no decimal part. Required to find the numbers.” – translation from Smith and Mikami

In essence, Gokai Ampon wants us to find integer solutions to indeterminate equation $x^3+y^3+z^3=w^3$, which is the same question that had originally prompted our quest. His solutions recorded by Shiraishi Chochu are a particular class of solutions to $x^3+y^3=3z^2+3z+1$, i.e. the sum of the cubes of two positive integer being the difference of the cubes of consecutive positive integers.

If we look at Figure 2, we find that among the mass of points with apparently no discernible order there are a group of regularly arranged points coming in pairs with the same $x$ value and lying on a slim parabola (colored purple). It was this group of points that caught Gokai Ampon’s attention. The first few of the pairs are listed below:

x y1 y2
3 4 10
12 19 31
27 46 64
48 85 109
75 136 166
108 199 235

We observe that for these points $x$ takes the form $3n^2; \; n=1, 2, 3...$. For $y_1$ we get a fit with $6n^2 -3n +1$ and for $y_2$ with $6n^2 +3n +1$. Thus, with $y_1$ we have:

$x^3+y_1^3=(3n^2)^3+(6n^2 -3n +1)^3 = 3 \cdot 3^2(2 n^2-3 n^3 - n)^2 + 3 \cdot 3 (2 n^2-3 n^3 - n) + 1 \;\;\; (\S 4)$

By writing $z_1=3(2 n^2-3 n^3 - n)$ and plugging it in $(\S 4)$ we get $x^3+y_1^3+z_1^3= (z_1+1)^3$, which is a family of valid solutions to the Japanese problem. Given that we are only considering positive integers, the final parameterization will be $z_1=3(3 n^3- 2 n^2+ n)-1$. Similarly, with $y_2$ we get:

$x^3+y_2^3=(3n^2)^3+(6n^2 +3n +1)^3 = 3 \cdot 3^2 (3 n^3 + 2 n^2 + n)^2 + 3 \cdot 3 (3 n^3 + 2 n^2 + n) + 1 \;\;\; (\S 5)$

By plugging $z_2= 3 (3 n^3 + 2 n^2 + n)$ in $(\S 5)$ we get $x^3+y_1^3+z_2^3= (z_2+1)^3$, the second valid family of valid solutions to the Japanese problem.

One can see that these two solutions are lattice points on a parabola whose parametric equation is $(x=3t^2, y=6t^2 \pm 3t +1)$. Thus, both our solution which covers the cubes of every positive integer and the Japanese solutions are 2 distinct parameterized families corresponding to parabolas in the $(x,y)$ plane.

The elliptical families
When we learned of the Japanese solutions, we wondered if there might be any other families of solutions hidden within the apparent disorder of the total set of all solutions. Returning to the $x-y$ plot, we noted that several points lie on arcs of increasing size (shown as orange and pink points in Figure 2). Examining these, we discovered that they are integer points lying on pairs of related ellipses of eccentricity $\sqrt{\tfrac{2}{3}}$ that have equations of the form:

$\begin{cases} x^2-xy+y^2-ax-by+c=0 \\[6pt] x^2-xy+y^2-bx-ay+c=0 \end{cases} \; \; \; (\S 6)$

Where the 3 parameters $a, b, c$ are defined by:

$c=27u^4+9u^2+1\\[6pt] a=c+(9u^2+1) = 27u^4 + 18u^2 + 2\\[6pt] b=c-(9u^2+2)= 27u^4 - 1$

The lattice points of $(\S 6)$ are solutions to $(\S 1)$ and emerge at special values of $u$. First few values of $u$ and the number of solutions they yield on the 2 corresponding ellipses are shown in Figure 4 and Table 1.

Figure 4

Table 1

The ellipses first corresponding to the few values of $u$ that yield solutions numbering $\ge 2$ are show in Figure 5. Not all values of $u$ are equally rich in terms of solutions for $(\S 1)$. The integers (1, 2, 3…), thirds i.e. $n \pm \tfrac{1}{3}$ consistently yield solutions but the integers tend to be clearly richer than the thirds (Figure 4). Other than integers, certain irrational values of the form $u=\sqrt{\tfrac{6n+1}{6}}$ are particularly rich in solutions. Other than those, $u=\sqrt{12}$ and certain values of the form $u=\sqrt{\tfrac{6n+1}{18}}, u= \sqrt{\tfrac{6n+2}{18}}$ are also rich in solutions. Thus, $12a, 12b, 12c$ are integers for all $u$, yielding solutions lying on ellipses. However, it remains unclear if there is a general rule to determine which of the quadratic surds that take the above forms will be $u$ that yield solutions for $(\S 1)$. In any case the given that integer $u$ yield solutions for $(\S 1)$ that lie on defined ellipses the family of such elliptical solutions is infinite.

Figure 5

Finally, we may note one special feature of the family of elliptical solutions that are specifically associated with the integer values of $u$. Given that the solutions $x, y, z$ define a simple curved surface (Figure 1), i.e. the surface does not show any folding, there can utmost be 2 pairs of $x, y$ that yield the same $z$. There are 29 $z$ which can be derived from 2 distinct pairs of $x, y< 20000$. For example: $9^3+ 58^3= 22^3 + 57^3=256^3-255^3$. Thus, (9, 58) and (22, 57) form a pair that yield the same $z=255$. The majority of such pairs of solutions with the same $z$ lie on the ellipses arising from integer $u$. Further, each such ellipse contains a pair whose whose $x$ values are respectively defined by $x_1 =9u^3; x_2= 9u^3+9u^2+3u+1$. The corresponding $y$ values can be obtained by plugging $x_1$ into the second ellipse and $x_2$ into the first ellipse in $(\S 6$). Thus we get $y_1= 27n^4 + 18n^3 + 9n^2 + 3n + 1; y_2=27n^4 + 18n^3 + 9n^2 + 3n$; thus, for these cases $y_1=y_2+1$. These pairs are shown below for $u=1, 2, 3, 4$.

Thus, these paired values define an infinite family by themselves lying on in the curves defined by the below parametric equations (Figure 5, blue curve):

$\begin{cases} x= 9t^3\\[6pt] y= 27t^4 + 18t^3 + 9t^2 + 3t + 1 \end{cases}\\[10pt] \begin{cases} x=9t^3+9t^2+3t+1\\[6pt] y=27t^4 + 18t^3 + 9t^2 + 3t \end{cases}$

There are a minority of pairs which lie outside of the ellipses. We do not know as yet if they define any other families of solutions. More generally, it is also not clear if there are any other families of solutions beyond the above parabolic and elliptical families.

Sum of cubes of 2 positive integers that equal the difference of cubes successive same-parity positive integers
In this final section we shall briefly consider a related indeterminate equation:

$x^3+y^3=(z+2)^3-z^3=6z^2+12z+8 \; \; \; (\S 7)$

One can see right away that some of the regular families of solutions of $(\S 7)$ are related to those of $(\S 1)$. The first relates to the above theorem regarding the cubes of every positive integer. In this case the equivalent is:

The cube of every positive even number is equal to the difference of the cubes of two consecutive even numbers minus the cube of another even number.

These correspond to the solutions to $(\S 7)$ of the form $(x=2n, y= 6n^2 + 4n + 2$ (Figure 6, violet curve). Further, the equivalents of the solutions to Gokai Ampon’s points in this case lie on the parabola defined by the parametric equation: $(x=6t^2, 12t^2 - 6t + 2)$ (Figure 6, purple curve). However, the solutions to $(\S 7)$ feature a unique parabolic family of solutions with no equivalent among the solutions of $(\S 1)$. These lie on the parabola defined by the parametric equation: $(3t^2 - t + 1, 3t^2 + t + 1)$ (Figure 6, orange curve). These correspond to $x, y$ such as:
(3, 5); (11, 15) (25, 31) (45, 53)… Thus, $x=3n^2 - n + 1$ and $y=3n^2 + n + 1$ provide another link to the hexagonal number spiral as they correspond to numbers that respectively lie on its 2nd and 4th spoke (Figure 3). With this in hand, we can show that for these $x, y$ give rise to $z=3 n^3 + 2 n - 1$, which defines the sequence: 4, 27, 86, 199, 384, 659, 1042…

As with the solutions to $(\S 1)$, here too we have the equivalent elliptical families corresponding to the integer lattice points on ellipses of eccentricity $\sqrt{\tfrac{2}{3}}$ that have equations of the form:

$x^2-xy+y^2-ax-by+c=0 \\[6pt] x^2-xy+y^2-bx-ay+c=0$

Here the 3 parameters $a, b, c$ are defined by:

$a=54u^4 + 36u^2 + 4\\[6pt] b=54u^4 - 2\\[6pt] c=108u^4 + 36u^2 + 4$

However, the $u$ which yield elliptical solutions for $(\S 7)$ are the same as those that yield solutions for $(\S 1)$ and there is an equivalence in the corresponding solutions. Figure 6 shows a few elliptical solutions $\left(u=2, \sqrt{\tfrac{37}{3}}, \tfrac{\sqrt{73}}{3}, 3\right)$.

In conclusion, this exploration reveals connections between a certain class of cubic indeterminate equations and families of solutions defined by particular parabolas and ellipses. It is not known to us if any one previously studied these elliptical families or reported any other families beyond those considered here.

Figure 6

Posted in Scientific ramblings |

## The Mātrā-meru and convergence to a triangle

What is presented below will be elementary for someone with even just the mastery of secondary school mathematics. Nevertheless, even simple stuff might present points of interest to people who see beauty in such things. Consider the following question:

Given the first 2 terms $0 \le x_1, x_2 \le 1$, what will be the behavior of the sequence defined by the recursive relationship:

$x_{n+1}=x_n\sqrt{1-x_{n-1}^2}+x_{n-1}\sqrt{1-x_n^2}$

Answer: It will converge to a cycle of length 3, where $x_n, x_{n+1}, x_{n+2}$ will be the sines of the 3 angles of a right or an acute triangle. Further, let $M$ be the well-known Mātrā-meru sequence: 1, 1, 2, 3, 5, 8, 13, 21…, then:

$x_n=\sin\left(M[n]\arcsin\left(x_1\right)+M[n+1] \arcsin\left(x_2\right) \right)$;
$x_{n+1}=\sin\left(M[n+1]\arcsin\left(x_1\right)+M[n+2] \arcsin\left(x_2\right) \right)$;
$x_{n+2}=\sin\left(\pi-M[n+2]\arcsin\left(x_1\right)+M[n+3] \arcsin\left(x_2\right) \right)$

Where $M[n+1]\arcsin\left(x_1\right)+M[n+2] \arcsin\left(x_2\right)$ is the largest such angle that is $\le \tfrac{\pi}{2}$

This can be easily proved thus:

1) Since $0 \le x_1, x_2 \le 1$, we can write $x_1=\sin(A), x_2=\sin(B)$.

2) Thus, given the recursive relationship the next term becomes,
$x_3=\sin(A)\cos(B)+\sin(B)\cos(A)=\sin(A+B)$

3) Continuing this way, we can write,
$x_4=\sin(A+2B); \; x_5=\sin(2A+3B); \; x_6= \sin(3A+5B)$.

We notice the multiplicands of $A, B$ are the successive terms of the Mātrā-meru sequence. Thus, $x_n=\sin\left(M[k]A+M[k+1] B \right)$.

4) This will continue till $M[k]A+M[k+1]B$ comes closest to $\tfrac{\pi}{2}$. Then the next term $M[k+1]A+M[k+2]B \ge \tfrac{\pi}{2}$. But due the symmetry of the sine function,

$\sin(M[k+1]A+M[k+2]B)=\sin(\pi-M[k+1]A+M[k+2]B)$.

Since, $M[k+1]A+M[k+2]B=M[k-1]A+M[k]B+M[k]A+M[k+1]B$,

we get $M[k-1]A+M[k]B+M[k]A+M[k+1]B+\pi-M[k+1]A+M[k+2]B = \pi$.

Thus, at this stage the three successive terms $x_n, x_{n+1}, x_{n+2}$ are sines of the 3 angles of an acute or right triangle and they will settle into a cycle of those 3 values $_{...\blacksquare}$

Hence, the above recursive relationship results in any pair of $x_1, x_2$ converging to the 3 sines of an acute or right triangle. As a corollary if you start with $x_1, x_2$ which are already sines of an acute or right triangle then you stay on that triangle. Let us consider some special cases below (Figure 1).

Figure 1. The angles are given in degrees for ease of representation

When $x_1=x_2=\tfrac{\sqrt{3}}{2}$ then the values are on an equilateral triangle and the iterates remain fixed on that triangle (Figure 1, panel 1).

When $x_1=\tfrac{3}{4}; \; x_2=\tfrac{4}{5}$ then the values are on the 3-4-5 right triangle and the iterates remain fixed on that triangle (Figure 1, panel 2).

When $x_1=\tfrac{1}{\sqrt{2}}; \; x_2=\tfrac{1}{\sqrt{2}}$ then the values are on the half-square right triangle and the iterates remain fixed on that triangle (Figure 1, panel 3).

When $x_1=x_2=\tfrac{1}{2}$ then they are not an acute or right triangle. However, within one iteration the iterates converge to the sines of a right triangle, namely the $30^\circ-60^\circ-90^\circ \; \triangle$ (Figure 1, panel 4).

It is easy to see that if $\tfrac{1}{\sqrt{2}} \ge x_1=x_2 \le 1$ then they are on an isosceles triangle and remain on that. However, if $\arcsin(x_1)+\arcsin(x_2) < \tfrac{\pi}{2}$ then can we converge to an isosceles acute triangle? This happens in special cases which can be determined by solving an equation. In order to do so we shall take $x_2=x; \; x_1=k-x; \; k< \sqrt{2}$. From the above proof the successive angles corresponding to the iterates of $x_n$ are:

$\arcsin(k-x); \; \arcsin(x); \arcsin(k-x) + \arcsin(x); \; \arcsin(k-x) + 2\arcsin(x); \; 2\arcsin(k-x) + 3\arcsin(x); \; 3\arcsin(k-x) + 5\arcsin(x)...$

Thus, we have to look for real solutions of the equations such as:

$\arcsin(k-x)=\pi - (\arcsin(k-x)+\arcsin(x))$
$\arcsin(x) = \pi -(\arcsin(k-x) + \arcsin(x)$
$\arcsin(x) = \pi -(\arcsin(k-x) + 2\arcsin(x)) ...$

Let consider the example of $k= 1$: with either of the first two equations we get degenerate triangles (e.g. $x_1=0, x_2=1, x_3=1$). However, if we instead take $x_1=\tfrac{1}{m}, x_2=\tfrac{m-1}{m}$ for some large $m$ we get near-isosceles triangles (Figure 1, panel 5).

The one equation with a real solution for $k=1$, which gives a unique isosceles triangle, is seen when:

$\arcsin(1-x) +3 \arcsin(x) = \pi, \; x \approx 0.83756543528332$

This $x$ is the greatest root $(r_1)$ of the cubic equation $4x^3-4x+1=0$. Thus, $x_1=1-r_1, x_2= r_1$ yields an isosceles triangle with its equal angles $\arcsin(r_1) \approx 56.88^\circ$ (Figure 1, panel 6).

Next we shall consider the evolution of certain special sequences of triangles. The first is where $x_1, x_2$ are constituted by successive terms of the Mātrā-meru sequence (Figure 2).

Figure 2.

Here, the triangles start with the $30^\circ-60^\circ-90^\circ \; \triangle$ and converge to a unique scalene triangle with angles $\arcsin\left(\tfrac{1}{\phi}\right) - \arcsin\left(\tfrac{3\phi-1}{2\phi+1}\right) - \arcsin\left(\tfrac{2\sqrt{\phi}}{\phi+1}\right) \approx 38.17^\circ - 65.48^\circ - 76.35^\circ$, where $\phi= \tfrac{1+\sqrt{5}}{2}$ is the Golden ratio.

The last panel here shows an interesting numeric coincidence. If you start with $x_1=r_1, x_2=1-r_1$ (see above for $r_1$) you converge to a triangle close to that emerging from the Mātrā-meru sequence. Is there more to this than the coincidence of values?

Finally, let us consider 2 other special triangles that emerge as convergents for 2 related types of operations based on the Mātrā-meru sequence (Figure 3).

Figure 3.

The first 2 rows (in light green) show triangles emerging from $x_1=\tfrac{1}{M[k]}, x_2=\tfrac{1}{M[k+1]}$, where $k=2, 3, 4...$. Here again, we start with a $30^\circ-60^\circ-90^\circ \; \triangle$ and converge to a triangle of the form $\approx 70.82^\circ - 65.41^\circ - 43.77^\circ$.

The second 2 rows (in yellow) show triangles emerging from $x_1=\tfrac{1}{M[k]}, x_2=\tfrac{1}{M[k]}$, where $k=3, 4, 5...$. These converge to a triangle of the form $\approx 87.29^\circ - 57.30^\circ - 35.41^\circ$.

Posted in Scientific ramblings |

## The Aśvin-s and Rudra

The twin Aśvin-s and Rudra are both Indo-Aryan reflexes of two deity-classes which can be reconstructed as likely being present in the Proto-Indo-European religion. Both are likely to have even deeper roots going back to even earlier religious traditions across a wide swath of humanity. Indeed, the divine twins feature even outside the IE religions. At the face of it it is not obvious if these two deity classes show any special links. For example, with Rudra as a focus we can use the Ṛgveda to sample his association with other deities. One proxy for association can be how often Rudra is mentioned with another deity in the same pāda or hemistich of a ṛk. Below is a table showing his co-occurrences by this metric.

Table 1. The association of various Devatā-s with Rudra in the RV

Devatā #
Marut-s 21
Vasu-s 18
Āditya-s 13
Agni 10
Ṛbhu-s 7
Indra 6
Aśvin-s 6
Soma 6
Varuṇa 5
Mitra 5
Bhaga 5
Puṣaṇ 4
Viṣṇu 3
Bṛhaspati 2
Sarasvatī 2
Tvaṣṭṛ 2
Aryaman 1
Savitṛ 1
Vāyu 1

Not surprisingly, he is most closely associated with his sons the Marut-s who are often referred to as the plurality of Rudra-s. The other close associations come from the fact that the Rudra-s are mentioned as a group of gods along side to two other big groups of gods: the Vasu-s and the Āditya-s. He further shows notable associations with one of the archetypal deities of the Vasu-group, Agni with who he shares a duality. Beyond that his mentions along with the Aśvin-s is no more frequent than with Soma, Indra or the Ṛbhu-s (the divine craftsmen whose masters are said to be the Rudra-s: RV 8.7.12). That said a closer look reveals a deeper link between Rudra and the Aśvin-s.

uta tyā me raudrāv arcimantā
mandū hitaprayasā vikṣu yajyū || RV 10.61.15

Also these two sons of Rudra, the Nāsatya-s [worshiped] with ṛk-s, are to be welcomed and made offerings by me, O Indra. The two who are liberal to him, who as Manu [had done, invites them] to the woven grass (the twisted barhiṣ in the vedi), the delightful [twins] for whom pleasing offerings are made, the twins who seek the ritual among the people.

Here the ṛṣi Kakṣīvān calls the Aśvin-s the sons of Rudra in a manner similar to the Marut-s. Further, this sūkta mentions our ancient ancestor Cyavāna as the one who had measured out the vedi for the for the Aśvin-s. Cyavāna is mentioned in tradition as the one who instituted the rite where soma is offered to the Aśvin-s. Indeed, in the above ṛk it appears that Kakṣīvān seeks Indra permission for the same as there are indications from later narratives that Indra was not entirely on board with that. In any case, what is important for our current discussion is the relatively unambiguous link of the Aśvin-s and Rudra in this ṛk.

This is not the only instance the Aśvin-s are called Rudra-s or the son-s or Rudra. Indeed, we encounter another such example elsewhere in the RV:

tāv id doṣā tā uṣasi śubhas
patī tā yāman rudravartanī |
mā no martāya ripave vājinī-
vasū paro rudrāv ati khyatam || RV 8.22.14

Just these two in the evening, these two auspicious lords at dawn, the two who follow the tracks of Rudra in the course. Don’t look over and beyond us to a roguish mortal, O Rudra-s with booty-bearing mares.

In this ṛk of Sobhari Kāṇva, they are not just called Rudra-s but also specific described as Rudravartanī. This word is of considerable interest in regard to this connection between these deities. It belongs to class of compounds of the form “x+vartani” that are found throughout the RV and used for different deities. The word vartani means wheel (typically of the chariot) or track of the chariot wheel. Thus, x+vartani compounds are usually interpreted as bahuvrīhi-s. Below we list all the instances of such compounds in the RV along with the gods they denote and the number of occurrences of each case:

• Rudravartani
• Aśvin-s, 4
• Hiraṇyavartani
• Aśvin-s, 6
• Sarasvatī, 1
• river goddess, 1
• Raghuvartani
• Aśvin-s, 1
• Soma, 1
• Ghṛtavartani
• Aśvin-s, 1
• Vṛjinvartani
• Agni, 1
• Kṛṣṇavartani
• Agni, 1
• Gāyatravartani
• Indrāgnī, 1
• Dvivartani
• Agni, 1

It is immediately apparent that this class of compounds are special descriptors of the Aśvin-s for 12 of the 19 occurrences of them are used for the Aśvin-s. This is likely a special allusion related to the oft-mentioned speeding tricycle (tri-cakra) chariot of these gods. However, it should be noted that such compounds, while most frequently used for the Aśvin-s are not limited to them; e.g. Hiraṇya-vartani is use for at least 2-3 distinct deities (river goddess Sindhu could be a cryptic allusion to Sarasvatī). Some of the usages, such as Hiraṇya-vartani or Raghu-vartani can be simply interpreted as the ones with the golden wheels or one with swift wheels and appear to apply to other deities than the Aśvin-s. Indeed, a related term “Hiraṇya-cakra” is used for the actual Rudra-s, i.e. Marut-s (e.g. RV 1.88.5) or for that matter Hiraṇya-ratha used for Indra (e.g. RV 1.30.16). Further, some might be even typical of other deities: e.g. Kṛṣṇa-vartani (with black tracks: alluding to the smoke) and Vṛjina-vartani (with curving tracks, alluding to the flames) are apt for Agni for whom they are used. The form dvi-vartana used Agni is simply indicative of two tracks left behind by the two chariot wheels.

However, of all these Rudra-vartani is specifically used only for the Aśvin-s on multiple occasions and by multiple composers. This suggests that this term has a special connection with the Aśvin-s. Drawing the cue from the more frequent Hiraṇya-vartani, also used for the Aśvin-s, Rudra-vartani has been tradition interpreted as either Rudra= ruddy or Rudra = fierce. Thus, the compound is understood as the Aśvin-s with ruddy tracks, or those with fierce tracks — perhaps as an allusion to their speeding chariot that is frequently seen in the RV, or as those who go along terrifying paths. Entirely, independently of our investigations, we learned that this line of reasoning was first explored in detail by the great patriot Aurobindo Ghose. But the key is the observation that this term is specific to the Aśvin-s. Rudra is not used in the sense of “ruddy” elsewhere in the corpus. “Fierce or terrible tracks” would have implied the form raudra-vartani, which we do not ever encounter in this corpus. Further, ruddy horses or chariots are described by terms like aruṇayugbhir aśvaiḥ (RV 6.65.2) and fierce chariots by terms like tveṣa-ratha for the Marut-s (RV 5.61.13; also perhaps a personal name of a Mitanni ruler among the Indo-Aryans of West Asia). Indeed, the ferocity of the Marut-s’ chariots with ruddy horses are alluded to elsewhere too (e.g. RV 1.88.2) but the term Rudravartani is never applied to them even if it might be natural in this sense.

Thus, taken together with the instances where the Aśvin-s are called Rudra-s or the sons of Rudra (see above and also below) we conclude that Rudra-vartanī specifically indicates the association of the twins with Rudra and means “the two who follow the track of Rudra”. After we reached this conclusion, a search revealed that such a translation had been independently conceived by the German indologist Hermann Oldenberg. Notably, this link to Rudra is further strengthened by another instance where they are called both Rudra-s and Hiraṇyavartanī:

ā no ratnāni bibhratāv
aśvinā gacchataṃ yuvam |
rudrā hiraṇyavartanī
juṣāṇā vājinīvasū
mādhvī mama śrutaṃ havam || RV 5.75.3
Bearing treasures to us, Aśvins, come here, you two, O Rudra-s with golden wheels, with booty-bearing mares, being pleased, the holders of the honey-lore, hear my invocation.

We believe there are many dimensions to this connection:
1) The ancient name of the Aśvin-s is Divo Napatā. The twin sons of Dyaus. This is an equivalent of the name of their Greek cognates the Dioskouroi (the national deities of the Spartans), meaning the youths of Zeus (the cognate of Dyaus; Skt Divaḥ Kumārau) or their Lithuanian cognates Dievo Suneliai (Sons of Dieva = Dyaus). Now Dyaus on occasion is identified with Asura Rudra in the RV:

tvam agne rudro asuro maho divaḥ (RV 2.1.6)
You, O Agni, are Rudra the Asura of heaven (Dyaus)

yathā rudrasya sūnavo divo
vaśanty asurasya vedhasaḥ | (RV 8.20.17)
It shall be [just] as they wish, the sons of Rudra, the Asura of heaven (Dyaus) are the wise ones.

And like in this case too:
Indra, this is for you and that heaven (Dyaus), for that self-glorious Rudra.

The above indicate that there was an early Vedic tradition that identified Rudra with Dyaus, in which sense he was also seen as the father of world by the Bharadvāja-s (RV 6.49.10). This, together with the appellation Divo Napātā for the Aśvin-s, hints a parallel Vedic tradition which saw them as deities in the Rudra-class associated with the leader of that class Rudra, the great Asura of Dyaus. Their “Raudra nature” is clearly brought out in the ṛk RV 10.93.7: uta no rudrā cin mṛḻatām aśvinā : Also, though being Rudra-s, may the Aśvin-s be merciful. This plea for mercy to them is comparable to that typically made to Rudra or the Marut-s. Consistent with this, they share their medical prowess with Rudra (as physicians of the gods) and even more tellingly also their knowledge of poisons with Rudra (RV 1.117.16: where they either kill the brood of Viṣvāc with poison or destroy the poison associated with the brood of Viṣvāc). Thus, across the RV, composers from different clans occasionally saw the Aśvin-s as Rudra-s or Rudra’s sons and allude to their Rudrian properties.

2) One of the notable aspects differentiating the early Atharvaṇ tradition from the RV tradition with regard to Rudra is the use of the twin appellation Bhavā-Śarvā for the deity in the former. These names of Rudra also persist in the ādhvaryava tradition preserved in the Yajurveda-s but the twinning is less prominent relative the AV tradition. Notably, in the celebrated Mṛgāreṣṭi ritual the AV tradition features a sūkta to the twin Bhavā-Śarvā whereas in its place the KYV tradition has ṛk-s to the Aśvin-s. Notably, in the incantations for the Śūlagava ox sacrifice laid out in the Śāṅkhāyana-śrautasūtra (4.20.1-2), Bhavā-Śarvā are called the sons of Mahādeva.

tasya te dhanur hṛdayam mana iṣavaś cakṣur visargas tam tvā tathā veda namas te astu somas tvā avatu mā mā hiṃsīḥ | yāv araṇye patayantau vṛkau jambavantau iva | mahādevasya putrābhyām bhava-śarvābhyām namaḥ ||

The heart is your bow, the mind is your arrow, the eye is your shooting. Thus we know you. Obeisance to you. May Soma protect him and may you never ever harm me. The two who roam around in the forest like wolves with jaws wide open; obeisance to the two sons of Mahādeva, Bhava and Śarva.

This points to two parallel streams within the early Vedic tradition which featured Rudra in singular form (apart from the plurality of the Rudra-class) as seen in the RV or in twin form of Bhavā-Śarvā as seen in the AV and the Śāṅkhāyana-śrautasūtra (as the twin sons of Rudra). This suggests that the Rudra-class had an ancient intrinsic twin nature shared with the Aśvin-s which lingers in the Bhavā-Śarvā dyad. Notably, even in Greek tradition the deity of the Rudra-class, Apollo is born with twin (albeit female), Artemis. While the RV Aśvin-s are identical twins in character, sometimes, in the IE world one sees some differentiation of the the twins with one of them associated with healing and animal-husbandry and the other with warfare. For example, among the Yavana-s, one of the pair, Castor is a horse-trainer while the other one Polydeuces is a boxer. A similar differentiation is perhaps reflected in the twin Rudra-s, with the name Bhava indicating welfare and health, and the and Śarva meaning and archer and indicating the warrior nature of the second twin. Thus, it is likely that Bhava and Śarva were part of the parallel Vedic tradition where they played the role of the Aśvin-s as twin Rudra-s.

3) A later reflex of this twinning in the Rudra-class appears to have emerged via developments in the form of the Kaumāra tradition emerging in the Indo-Iranian borderlands. There we see the dual form of the god Kumāra as Skanda-Viśākha. This dyad is earliest attested in the Atharvavedīya-Skanda-yāga from the AV-pariśiṣṭha-s and has a prolonged presence in the Hindu tradition. The grammarian Patañjali mentions metal images of Rudra along with the Skanda-Viśākha dyad in the Mauryan age (~322-185 BCE). The persistence of this tradition is illustrated by a much later attestation of this dyad, evidently drawn from a now lost early tradition, seen in the Kālikā-purāṇa with a predominantly East Indian locus:

dahano ‘pi tathā kāle prāpte gaṅgodare svayam |
retaḥ saṅkrāmayāmāsa śāṃbhavaṃ svarṇa-sannibham ||
Agni himself, in due course of time, transmitted the semen of Rudra, which shone like gold, to the womb of Gaṅgā.

sā tena retasā devī sarva-lakṣaṇa-saṃyutaṃ |
pūrṇa-kāle ‘tha suṣuve putra-yugmaṃ manoharam ||
Then by that semen the goddess, upon completion of pregnancy, gave birth to charming twin sons endowed with all [good] features.

ekaḥ skando viśākhākhyo dvitīyaś cāru-rūpa-dhṛk |
śakti-dvaya-dharau dvau tau tejaḥ kānti-vivardhitau|| KP 46.82-84
The first was Skanda and the second bearing a beautiful form was known as Viśākha. The two held a spear each and two shone with their radiance.

Thus, we see Rudra siring the twin Kaumāra deities Skanda and Viśākha who are described as bearing spears much like the depiction of the spear-bearing Dioscuri in the yavana tradition. Interestingly, their beauty is specifically described much like that of the Aśvin-s in Vedic tradition. Further, the Kālikā-purāṇa recommends the worship of this dual Kaumāra form for the ṣaṣṭhī night:

rātrau skanda-viśākhasya kṛtvā piṣṭa-putrikām |
pujayec chatrunāśāya durgāyāḥ prīyate tathā || KP 60.50
In the night having made images of Skanda and Viśākha from flour one should worship them for the destruction of enemies and for pleasing Durgā.

In archaeological terms we find depictions of the Skanda-Viśākha dyad on Kuṣāṇa coins and also Kuṣāṇa age images from a lost Kaumāra shrine from the holy city Mathura (now housed in the collections of the Mathura museum). What is notable about their numismatic appearances is their resemblance to the twin gods on Greek Dioscuri coins. Thus, like Bhava-Śarva in a parallel Vedic tradition, Skanda-Viśākha are likely developments of the ancient Raudra twins in a Para-Vedic tradition that then entered the Indo-Aryan mainstream.

4) Finally, it is plausible that the term Rudravartanī and the association of the Aśvin-s with Rudra have an astronomical significance. In the classic nakṣatra system that developed by the time of the AV and the YV the Aśvin-s are associated with the constellation of Aśvayujau which corresponds to part of Aries. However, the obvious constellation that resembles the divine twins is Gemini, which was recognized as the Dioscuri in the Greek tradition. While it was termed the constellation of Aditi, the ārya-s too recognized the dual nature of the asterism Punarvasu made up two stars Castor and Pollux — it is occasionally used in dual like: punarvasū nakṣatram aditir devatā ||. Early on we see the recognition of the twin nature in the statement that Aditi is two-headed in the Yajurveda (Taittirīya Saṃhitā in 1.2.4; Śatapatha Brāhmaṇa 3.2.4.16). Ironically, Pāṇini reinforces the dual nature with the sūtra: chandasi punarvasvor ekavacanam || PAA 1.2.61. In the Veda Punarvasu might be [optionally take a] singular declension. The grammarians clarify that this is limited to the Veda while in common speech it is always dual indicating its twin nature. The only direct allusion to the asterism in RV (along with Revati) appears to be in RV 10.19.1 (by our ancestor Cyavāna), which associates it with Agni and Soma in a cryptic hymn whose actual meaning has been hard to discern. However, a potential connection is seen in RV 10.39.11 where the Aśvin-s are called Rudravartanī and explicitly linked Aditi — pairing that is otherwise rather unusual:

na taṃ rājānāv adite kutaś cana
nāṃho aśnoti duritaṃ nakir bhayam |
yam aśvinā suhavā rudravartanī
purorathaṃ kṛṇuthaḥ patnyā saha || RV 10.39.11
From nowhere troubles nor evil nor fear reach him, along with his wife, for whom you two kings and Aditi prepare a chariot to be in the forefront. O Aśvins, you who are easy to invoke, follow the track of Rudra.

We take this to mean that the Aśvin-s here are associated with Aditi and thereby the asterism of Punarvasu. Now, Punarvasu rises after the constellation of Rudra, i.e. Ārdra (corresponding to Sirius and the proximal bright stars of Canis Major). Thus, we can see the twins literally following the track of Rudra in the sky.

Posted in Heathen thought, History |

## The roots of Vaiṣṇavam: a view from the numerology of Vedic texts

While glorified with a 1000 names in the famous stotra of the early Sātvata tradition of the Mahābharata, in the texts of an even earlier period the god primarily went under that name Viṣṇu. Indeed, even the litany of the 1000 names begins with viśvaṃ viśṇur vaśatkāraḥ signaling the connection to that ancient layer of tradition, with viśvam i.e, “everything” being an etymological elaboration of his name Viṣṇu, “the all-pervader”. Further, vaṣaṭkāra indicates his link to the Vedic ritual (the vaṣaṭ offering to Viṣṇu is already specifically mentioned twice in the Ṛgveda) wherein we can trace the meteoric rise of this god as the head-deity of one of the great sectarian streams of the āstika-s.

His later rise notwithstanding, it should be borne in mind that Viṣṇu is a prominent deity of early Indo-European vintage. Mentioned 113 times in Ṛgveda itself (Table 1), one can already sense his prominence in the pantheon. His cognate in the Germanic world Víðarr provides evidence of his importance in the early undivided IE days. In the Gylfaginning of the Icelandic prose Edda, Snorri Sturluson, who still had links to the old Germanic heathenism, described Víðarr thus:
“Víðarr is the name of the silent asa. He has a very thick shoe, and he is the strongest next after Thor. From him the gods have much help in all hard tasks.”
Here, his being “strongest next to Thor” is mirrored in the ārya tradition where Viṣṇu is nearly equal/equal to his elder brother/friend Indra. Víðarr “helped the gods in the hard tasks”, a specific feature shared with Viṣṇu in the ārya tradition. Finally, his thick shoe is related to the stride Víðarr takes in the final battle of the gods, Ragnarok, where he tramples the nether jaw of the demonic Fenris-wolf and opens its mouth wide apart to slay it. In the ārya tradition, likewise Viṣṇu’s three demon-conquering wide strides are well-known and this gives him his alternative name in the Veda, Urugāya, “the wide-goer”. This is related to again related to the etymology of Víðarr = “wider” and the very cognate of Víðarr, vitara, appears in the śruti in the context of Viṣṇu: athābravīd vṛtram indro haniṣyan sakhe viṣṇo vitaraṃ vi kramasva || : Then Indra spoke as he prepared to slay Vṛtra: “friend Viṣṇu stride widely.” The strides of Viṣṇu are also said to make space by widening the universe for Indra to swing his vajra to slay Vṛtra. This is parallel to Víðarr holding the jaws of the Fenris-wolf wide apart.

Both Viṣṇu and Víðarr are mentioned as possessing a special world/realm. In the case of Viṣṇu, who is called the “cowherd” or the cattle-protector, it is wide pasture in a mountainous realm. Víðarr’s is a case it is mentioned a thick with grass. Finally, we may note that Víðarr is one of the deities to survive the Ragnarok and usher in the new Germanic “satyayuga”. Among the ārya-s, as the “time-god”, he is seen as again surviving the yuga-s in the later Viṣṇu-centric traditions.

Figure 1. Viṣṇu (an image of seal impression of seal from Gandhāra; Northwestern India showing a Hunnic or Iranic lord worshiping the god) and Celtic Taranis (an ancient Gaulish bronze found in 1774 CE at Le Chatelet, France)

In the Celtic world, the chief Gaulish deity of the Indra-class, Taranis, appears to have absorbed elements of his partner, the deity of the Viṣṇu-class. We go somewhat out on a limb to suggest that the late iconography of Taranis, before the end of the Gaulish religion, was actually influenced by that of Viṣṇu carried either directly by Indian or Indianized Iranic travelers to the West or from them via intermediaries to the Gauls (Figure 1). Examples of this influence are seen in the wheel-wielding images of Taranis and his depiction on the famous Gundestrup cauldron. In the case of the latter, the Indian influence is clinched by the elephants associated with the goddess (probably Rhiannon) who is depicted similar to Lakṣmī the wife of Viṣṇu. We postulate that this influence was because they could see obvious parallels between their deities and the Indian counterparts, which in turn was a consequence of Taranis absorbing elements of the original Viṣṇu-class deity.

Closer to the Indo-Aryan realm, our Iranic cousins have deity called Rashnu Razishta, the heavenly judge, who superficially plays a Citragupta-like role. His name and sudden appearance without parallels elsewhere in the IE world suggests that he is none other than a cognate of the Indo-Aryan Viṣṇu. His name probably underwent a folk-linguistic mutation from ‘vi’ to ‘ra’ for the Iranians probably (wrongly) interpreted the ‘vi’ as a prefix with a negative connotation and “corrected” it to ‘ra’ the proper one (A suggestion also made by Puhvel/Dumezil though, in my opinion wrongly, opposed by some Indo-Europeanists). His popular worship among the Iranians is hinted by the occurrence of his name as part of personal names found in the Persepolis Fortification Tablets from the time of the Achaemenid emperor Darius-I: Rashnudāta and Rashnuka and the Parthian name Rashnumithra. Rashnu’s link to Viṣṇu is indicated by his accompanying Mithra to his right on his chariot when he rides forth for battle. In the Zoroastrian strain of the Iranic religion, Mithra with his vazra has taken the place of Indra and Rashnu retains the role of Viṣṇu now as Mithra’s assistant instead (Note that even RV Viṣṇu is specifically linked to Mitra, e.g. in RV 1.156.1 and RV 8.52.3). Notably, Yasht 12 to Rashnu (which has been clearly redacted to interpolate Ahura Mazdā’s name for Rashnu in the initial manthra-s) shows that he held an important role in the ritual and he is described as tall, forceful deity praised in superlative terms indicating the importance he once held in Iranian realm:

rashnvô ashâum rashnvô razishta rashnvô spênishta rashnvô vaêdhishta rashnvô vidhcôishta rashnvô parakavistema rashnvô dûraêdareshtema rashnvô arethamat bairishta rashnvô tâyûm nijakhnishta…

O holy Rashnu! O most-true Rashnu! most-beneficent Rashnu! most-knowing Rashnu! most- discerning Rashnu! most-fore-knowing Rashnu! most far-seeing Rashnu! Rashnu, the best doer of justice! Rashnu, the best smiter of thieves (tâyu = Skt stāyu)…

Keeping with his far-seeing nature, the yasht is unusual in describing the Iranic karshvare-s (world regions) and heavenly constellations as the regions seen by Rashnu as he flies along. This peculiar feature of the yasht, seen with no other Iranic deity, points to two parallels with Viṣṇu: (1) The listing of the realms is suggestive of Viṣṇu pervading them or striding over them. (2) Among the places Rashnu is invoked there are 3 stations mentioned in succession the quarters of the earth, boundary of the earth and all over the earth. Apart from reminding one of the three strides, it also reminds one of the ṛk where Viṣṇu is described as fixing the boundary of the earth by the pegs at the quarters (RV 7.100.3).

In conclusion, we see that the Viṣṇu-class of deities, despite their spotty occurrence in what has come down to us of the IE traditions, can be traced to a prominent proto-deity in the early IE religion. Here, by early we mean at least the time when the western thrust of the IE people from their Yamnaya homeland took place — the group from which the Germanic, Celtic and Indo-Iranian branches ultimately derive. This conclusion is of importance in understanding the rise of Viṣṇu in the Indian tradition. We shall explore the early stages of that by looking at the counts of the occurrence of the name Viṣṇu in several Vedic texts (Table 1). These texts span the entire range of Vedic tradition from the earliest RV to the Ṛgvidhāna which clings to the very edge of the late Vedic literary activity. In between lie the Saṃhitā-s of the 3 other Veda-s, the khila of the RV, the Brāhmaṇa-s, the Śrauta-sūtra-s and the Gṛhya-sūtra-s. Across these texts the most common name of the deity under consideration is Viṣṇu. He is known by other names like Urugāya, Śipiviṣṭa and the epithet evaya in the core Vedic period. In the latest Vedic texts we encounter the name Nārāyaṇa (e.g. Taittirīya-āraṇyaka), which became prominent in the epic period. However, all these names are generally rare making the count for Viṣṇu a good proxy for estimating the extent of his mentions.

Table 1. The occurrence of Viṣṇu in Vedic texts

Text Veda #
Ṛgveda RV 113
Ṛgveda-khilāni RV 15
Kauṣītaki-brāhmaṇa RV 14
Aitareya-Brahmana RV 24
Āśvalāyana-śrautasūtra RV 39
Śāṅkhāyana-śrautasūtra RV 48
Āśvalāyana-gṛhyasūtra RV 1
Śāṅkhāyana-gṛhyasūtra RV 10
Ṛgvidhāna RV 14
Atharvaveda (vulgate) AV 69
Atharvaveda (Paippalāda) AV 71
Gopatha-brāhmaṇa AV 22
Vaitāna-sūtra AV 14
Kauśika-sūtra AV 11
Maitrāyaṇī-saṃhitā KYV 233
Kāṭha-Saṃhitā KYV 197
Taittirīya-Saṃhitā KYV 175
Taittirīya-Brāhmaṇa KYV 79
Baudhāyana-Śrautasūtra KYV 165
Āpastamba-Śrautasūtra KYV 101
Hiraṇyakeśi/Satyāṣāḍha-Śrautasūtra KYV 88
Mānava-Śrautasūtra YYV 84
Vārāha-gṛhyasūtra KYV 5
Mānava-gṛhyasūtra KYV 7
Vaikhānasa-gṛhyasūtra KYV 23
Hiraṇyakeśi-gṛhyasūtra KYV 7
Laugākṣī-gṛhyasūtra KYV 2
Vājasaneyi-Saṃhitā (Mādhyandina) SYV 76
Śatapatha-Brāhmaṇa (Mādhyandina) SYV 223
Kāṭyāyana-Śrautasūtra SYV 40
Sāmaveda Kauthuma-saṃhitā SV 22
Pañcaviṃśa-brāhmaṇa SV 28
Mantra-brāhmaṇa SV 18
Khādira-gṛhyasūtra SV 4
Jaimini-gṛhyasūtra SV 12
Kauthuma-gṛhyasūtra SV 4

First, let us get some caveats regarding this table out of the way. These texts are of very different sizes; thus, someone could claim that the counts could simply reflect the size differences. Hence, one would wish to normalize it by text size. But what unit do you use for normalization? The word would be the ideal unit but it is difficult to obtain for all these texts because of samasta-pada-s typical of Sanskrit not being separated in each case. An alternative is the size of the file in bits. However, this depends on the encoding/format used and we do not have all the texts in a common encoding/format. Next, we have ascii or utf-8/16 encoding files for many of them but not all. The rest were counted using the html files on the TITUS system. We cannot be sure of the completeness of all the texts in TITUS. This said, we can still use the absolute counts reasonable well by comparing “apples with apples”. The Saṃhitā-s may be approximately seen to be of the same order of magnitude. The Brāhmaṇa can be again approximately compared, and the equivalent classes of kalpa texts can be similarly compared.

With these caveats in place, one thing that stands out is the extraordinary prominence of Viṣṇu in the Vedic tradition represented by the Yajurveda. This is clearly in contrast with the other Vedic tradition, namely that of the Samaveda. Tellingly, the one Samavedic text that is enriched in Viṣṇu is the Mantra-brāhmaṇa, which is primarily a collection of yajuṣ-es within the Samavedic tradition. This short collection of yajuṣ-es has the same order of magnitude of mentions of Viṣṇu as the SV Kauthuma-saṃhitā and Pañcaviṃśa-brāhmaṇa, both of which are several times the size of the Mantra-brāhmaṇa.

The divisions in the Vedic tradition correspond to the roles of the ritualists: the hautra tradition of the RV practitioners, the ādhvaryava tradition of the YV practitioners and the chāndoga tradition of the SV practitioners. The AV practitioners are associated with the role of the Brahman but in reality have their own parallel śrauta tradition. We see that in the ādhvaryava tradition the prominence of Viṣṇu is across the three sections, i.e. the Saṃhitā, the Brāhmaṇa and Śrauta-sūtra-s that roughly corresponding to temporal layers within the tradition. Further, in the hautra tradition, the Śrauta-sūtra-s have more mentions of Viṣṇu than their Brāhmaṇa-s, which are larger than the sūtra-s in size. Again, notably, with one exception (we will come to that in the end) the Gṛhya-sūtra-s have relatively low occurrences of Viṣṇu across the board relative to the corresponding Śrauta-sūtra-s.

Now, in the śrauta practice the adhvaryu plays the most important role in terms of the physical actions of the ritual (of course accompanied by yajuṣ incantations). He also issues calls to the hotṛ and udgātṛ to play their parts. The hotṛ in contrast primarily plays the role of reciting the Sāmidhenī-s and the śastra-s and the like, while the udgātṛ’s main role is the singing of the stotra-s during the Soma ritual. While the YV and SV texts as we have them today clearly postdate and presuppose the core RV, we know from the internal evidence of the RV that there were already YV and SV practitioners alongside the composers of the RV. E.g.: udgāteva śakune sāma gāyasi : You O bird sing a sāman like the udgātṛ. tvam adhvaryur uta hotāsi pūrvyaḥ praśāstā potā januṣā purohitaḥ | : You (Agni) are the adhvaryu, the primal hotṛ, the praśastṛ (an assistant of the hotṛ who is also known as the upavaktṛ or the maitrāvaruṇa who plays a special role in recitations to those two gods), the potṛ (the assistant of the Brahman who specializes in prāyaścitta-s) and from birth the purohita. Further, the parallel in the Zoroastrian tradition suggest that at least the hotṛ, the adhvaryu and some version of the brahman go back to the proto-Indo-Iranian period.

While these ṛtvik-s function in unison in the śrauta rituals their mantra collections and activities suggest that they originally represented alternative ritual traditions within the early Indo-European fold, which were brought together under a single framework by at least the proto-Indo-Iranian period. While under a common framework they clearly maintained their own parallel traditions, techniques of composition and ritual principles. When we take this in to account, along with the evidence for the presence of a prominent Viṣṇu-class deity in the early Indo-European religion, we can account for the development of Vaiṣṇava tendencies in the Veda: we posit that it was not that Viṣṇu rose to prominence in the middle Vedic period (as opposed to the early Vedic core RV with the Aindrāgna and old Āditya dominance) from nothing but was always a special deity in the ādhvaryava tradition of the Indo-Aryans. What happened was the rise to dominance of the ādhvaryava tradition in the Indo-Aryan śrauta ritual. Conversely, it is conceivable that on the Iranian side the hautraka tradition dominated, at least in early Zoroastrianism where Zarathustra calls himself the cognate zaotar.

These observations can be summed up by one statement in the ādhvaryava tradition: yajño vai viṣṇuḥ |: The ritual is verily Viṣṇu. Thus, by identifying Viṣṇu with the ritual he is identified with the core activities of the ādhvaryava tradition. Hence, this is consistent with this frequent mention within this tradition. This proposal is further strengthened by the two special devatā-dvandva-s that are common though not unique to the YV tradition but not found in the RV Saṃhitā, which is otherwise rich in devatā-dvandva-s. The main dvandva featuring Viṣṇu in the RV is Indrāviṣṇū, where he is linked to his usual partner Indra. The devatā-dvandva Agnāviṣṇū, found in the AV, RV-brāhmaṇa and frequently in the YV tradition, marks the special position of Viṣṇu not only as the last of the deities to receive the sacrifice but also tacitly or not so tacitly indicates his supremacy by placing him at the end of the pantheon (Aitaryeya-brāhmaṇa: agnir vai devānām avamo viṣṇuḥ paramas tadantareṇa sarvā anyā devatā | : Agni is the lowest of the gods, Viṣṇu is the foremost, all the other deities lie in between. While primarily positional it also hints the primacy of Viṣṇu). The second is the dvandva Viṣṇū-varuṇā seen, for example, in the Taittirīya and the Aitaryeya-brāhmaṇa traditions. This dvandva has a special role that is typical of the ādhvaryava tradition, proper completion of yajña, the invocation of Viṣṇu protects from the badly done yajña while the invocation of Varuṇa protects the well-done one and between the two all is taken care off viṣṇur vai yajṅasya duriṣṭam pāti varuṇaḥ sviṣṭaṃ | tayor ubhayor eva śāntyai ||).

Thus, the unusual situation of dvandva-s not found in the RV but shared by the RV brāhmaṇa-s and YV tradition, and the prominence of Viṣṇu in the “little YV” within the SV, i.e., the Mantra-brāhmaṇa, suggests that these are late entrants into the SV and RV tradition from the YV tradition. Indeed, the even greater prominence of Viṣṇu in the RV Śrauta-sūtra-s as opposed to the Brāhmaṇa-s again indicates the ādhvaryava dominance now entering the hautra territory. Thus, it suggests that the rise of Viṣṇu was essential a feature of adhvaryava dominance in the śrauta ritual. This in contrast to the protogonic Prajāpati, who while emerging late from the para-Vedic periphery, uniformly affected all the Vedic traditions. The Prājāpatya-s competed with Viṣṇu for the two figures of the primordial turtle Kaśyapa and the primordial boar Varāha. While the former was originally associated with Indra (from the RV itself), the latter is hinted to be associated with Viṣṇu in the early AV tradition recorded in the Paippalāda-saṃhitā. However, the Prājāpatya-s laid a strong claim to both before Viṣṇu eventually won and claimed both the figures as his avatāra-s in the Post-Vedic period.

Finally, we saw that the Gṛhya-sūtra-s have a low frequency of mention of Viṣṇu with the exception being that of the Taittirīya-affiliated Vaikhānasa tradition. The Gṛhya-sūtra-s for most part generally record an archaic core of household rites of passage quite different from the śrauta rituals. Thus, the absence of a special role of Viṣṇu beyond specific contexts like the fertility ritual for preparing the womb for the embryo is not unexpected. Now, one of the two earliest sectarian Vaiṣṇava traditions arose among the Vaikhānasa-s. Thus, what we are seeing in the case of the Vaikhānasa-gṛhyasūtra is the emergence of this tradition which went on to become a still extant system of the combined iconic and fire worship of Viṣṇu. The second sectarian tradition centered on Viṣṇu, the Pañcarātra, explicitly identifies itself with the Śukla-yajurveda. Thus, we may say that rise of Vaiṣṇavam itself is an internal development within the ādhvaryava tradition, with the two early schools emerging from each of the main Yajurvedic divisions. Once these had emerged they influenced the latest of the Vedic text across the traditions. We see this in the case of the Ṛgvidhāna, which for a relatively short text has several mentions of Viṣṇu in the context of what appears to be an early Pāñcarātrika section.

To conclude we may ask what about the Sātvata tradition that affiliates itself to the Pāñcarātrika tradition? We have evidence that Viṣṇu’s primary manifestation in that system the Vāsudeva along with the 3 other vyūha deities and the goddess Ekānaṃśā probably have para-Vedic roots in the Indo-Iranian borderlands. Likewise, with the watery Nārāyaṇa or his humanized dyadic form of Nara-Nārāyaṇa. But these traditions always saw itself as a part of Vaiṣṇava system. Hence, we posit that they are part of greater “Vaiṣṇava” tradition of old IE provenance that in addition to influencing the ādhvaryava tradition also had “lower” para-Vedic registers such as these. Nevertheless, this tradition’s links to the old Viṣṇu are hinted by certain specific features, for example: (1) The Vāsudeva in his human manifestation (Kṛṣṇa Devakīputra) consorts with a large number of women. Viṣṇu consorting with a bevy of goddesses is already mentioned in RV 3.54.14. This also relates to his early fertility role as the protector of the sperm of males. (2) Already in the RV Viṣṇu is repeatedly mentioned as being the cowherd or the cow-protector (RV 1.22.18; RV 3.55.10). Thus, this aspect of the Devakīputra of the Sātvata religion are likely merely a humanization of an old trait of the deity.

Posted in Heathen thought, History |

## Matters of religion-3

Lootika seeing Somakhya contemplating something remarked: “That great clash of men is upon us, where, as the śruti says, our flag will be mingled with that of the enemy in close combat.”
Somakhya: “It’s almost as if your mind is reflecting mine!”
Lootika: “Indeed, though born V1s our life is like that of V2s in a sense like as this is the downward turn of the yuga-cakra. Hence, may Vajrabāhu sink the dasyu to his left and bring light to the ārya on the right.”
Somakhya: “The founders of my clan or your Śaradvān clansman never shied from yuddha. It is with that in mind O Gautamī we have to invoke the fierce spells of our ancestors as enjoined by father Manu and as Harihara and Bukka had done in the southern country before embarking on the destruction of the makkha-rākṣasa-s. Before we make those oblations we have protect and purify ourselves with the śāntyudaka as was done by the young brahman Vicārin, the knower of the Atharvāṅgirasa-śruti.”

Early in the day Lootika placed the leaves of the secret plants known as citi, tumburu, śigru, devadāru, pīlu, kapu, viparvā, rodākā, vṛkkavatī, nāḍā and nirdahantī in the śāntyudaka pitcher. As she did so Somakhya recited the following incantation of his ancient ancestors:

sahasva yātudhānān
sahasva yātudhānyaḥ |
sahasva sarvā rakṣāṃsi

Conquer the the yātudhāna-s, conquer the yātudhānī-s, conquer all the rakṣa-s. You are conquering, O herb!

L: “O Ārya, tradition holds that a subset of these plants belong to the Atharvan-s or the Bhṛgu-s and the other subset to the Āṅgirasa-s. How do we place them in each of these categories.”
S: “That is indeed so Varārohā. Those from citi to pīlu are those of my ancestors while those from kapu to nirdahantī belong to yours. Such is the lore passed down by the knowers of the Atharvaṇa-śruti.”

Then, as Lootika poured water from her kamaṇḍalu into the pitcher Somakhya standing up held his palms in the añjalika-mudrā and recited the following incantations of his ancient ancestors:

viśvād riprān muñcata sindhavo no
yāny enāṃsi cakṛmā tanūbhiḥ |
indra-praśiṣṭā varuṇa prasūtā
ā siṅcatāpo madhva ā samudre ||

O rivers, free us from all the impurity and from whatever sins we have committed with our bodies. Directed by Indra, impelled by Varuṇa, O waters pour honey into the sea.

yūyaṃ mitrasya varuṇasya yonir
yuṣmān devīr deva ā kṣiyantīndur
yūyaṃ jinvata brahma-kṣatram āpaḥ ||

You are the seat of Mitra and Varuṇa, you are the sweetest cows of Soma. In you, O goddesses, dwells the godly Moon. You, O waters, must impel the the brahma and the kṣatra.

punānā āpo bahudhā sravanti
imāṃś ca lokān pradiśaś ca sarvāḥ |
muṅcantu mṛtyor nirṛter upasthāt ||

The waters, becoming pure, stream in many directions, throughout these worlds and throughout all the quarters of space. Let them purify us from evil, from disgrace; let them release us from death, from the lap of the goddess Nirṛti.

Then he muttered the Sāvitrī and recited the Triṣaptīya hymn.

L: “Should the goddesses of the waters be correctly understood as Apsaras-es with Urvaśī, Sahajanyā, Ghṛtācī and Menakā at their head? Their being called upon to called upon to impel the brahma and the kṣatra, who lead the society of the ārya-s, is similar to their being invoked to protect the brahma-kṣatra in the Yajuṣ incantations known as the Rāṣṭrabhṛt-s.”
S: “One may understand it that way from the etymology of Apsaras-es, which is a extension of the more basic form āpaḥ. Indeed, the Rāṣṭrabhṛt-s further the equation with the Apsaras-es. It has also been explicitly indicated by my ancient clansman Kavi, the son of Bhṛgu:

samudriyā apsaraso manīṣiṇam
āsīnā antar abhi somam akṣaran |

The marine Apsaras-es, sitting within, have streamed toward Soma of inspired mantra-thought.

Here the word samudriyā indicates their belonging to the watery realm. Further, the sense in which they are used in the ṛk of Kavi relates to them being termed “somasya dhenavo madhiṣṭhāḥ” in the mantra-s we just deployed. This identity of the Apsaras-es as the goddesses of the waters is reinforced in the famous ritual of the apsaras-es and the gandharva of the ocean, which we shall perform at a different time. There we shall encounter another dangerous class of 80 Apsaras-es who are known as the Uluṅgulukā-s, coming in groups of 31, 4, 10, 10, 25, associated with the asura Uluṅgula. We shall see in a later rite how to ward off danger from them by invoking Indra and obtain benefits instead.”

L: “Ah, I recall that while studying that rite I saw parallels between the Atharvaṇic divisions of the Apsaras-es and the Greek division of the water-nymphs into Naiads, Nereids, Oceanids and the like. This brings up another point for discussion. The waters are described as streaming through the worlds. The earthly realm is easily understood: the oceans and rivers and the like. The atmospheric realm too is obvious. But what about the heavenly realm? Even the moon is said to be contained in them. This seems to be reiterated in the Taittirīya-śruti where waters are described thus in the context of ‘watery’ nakṣatra of Pūrvāṣāḍhā:

yā divyā āpaḥ payasā saṃbabhūvuḥ |
yā antarikṣa uta pārthivīryāḥ |

Here, the enumeration of the waters begins with the divine waters followed by the more familiar ones. This suggests that our ancestors saw the ‘waters’ not just in the earthly regions but extending even to the heavenly realm. What might this mean?”

S: “That used seem puzzling to me too. But the clue comes from that very ṛk regarding Pūrvāṣāḍhā that you cited. I slips out of my mouth involuntarily whenever I see the said constellation on a clear night with Milky Way. It is telling that its states: ‘yā divyā āpaḥ payasā saṃbabhūvuḥ |‘ Which heavenly waters had their being by milk. We hold that the milk mentioned here is none other than the Milky Way — an analogy which is rather clear to anyone who has beheld the core of our galaxy on a good night. Hence, it is possible that the heavenly waters were an allusion to the Milky Way. This might also apply to the term divyaṃ nabhas to which we offer the barhiṣ at the end of the ritual.”

L: “The idea of divyā āpaḥ is also philosophically satisfying in a current sense. After all where did water on the earth come from? Most likely from the asteroids, the comets, and the objects of the dwarf planet belt. Thus, we may say all the earthly waters are indeed from the divyā āpaḥ.”

Later that evening knotting the yoktra girdle around Lootika’s waist Somakhya indicated to her to cover the mouth of the pitcher with her palms. Then he uttered the incantations:

śaṃ no mitraḥ śaṃ varuṇaḥ
śaṃ viṣṇuḥ śaṃ prajāpatiḥ |
śaṃ na indro bṛhaspatiḥ
śaṃ no bhavatv aryamā ||

Weal to us Mitra, weal to us Varuṇa, weal to us Viṣṇu, weal to us Prajāpati, weal to us Indra, weal to us Bṛhaspati. May Aryaman be beneficent to us.

śaṃ no grahāś cāndramasāḥ
śam ādityaś ca rāhuṇā |
śaṃ no mṛtyur dhūmaketuḥ
śaṃ rudrās tigmatejasaḥ ||

Weal to us the planets and the Moon, weal to us the Sun and the eclipses, weal to us death and the comets, weal the Rudra-s of sharp luster.

śaṃ rudrāḥ śaṃ vasavaḥ
śam ādityāḥ śam agnayaḥ |
śaṃ no maharṣayo devāḥ
śaṃ devyāḥ śaṃ bṛhaspatiḥ ||

Weal the Rudra-s, weal the Vasu-s, weal the Āditya-s, weal the Agni-s, weal to us the great seers and gods, weal to us the goddesses and Bṛhaspati.

Then Somakhya steeped a fistful of darbha in the water and uttered the following incantation:

pṛthivī śāntir antarikṣaṃ śāntir dyauḥ śāntir
āpaḥ śāntir oṣadhayaḥ śāntir vanaspatayaḥ śāntir
viśve me devāḥ śāntiḥ sarve me devāḥ śāntiḥ
śāntiḥ śāntiḥ śāntibhiḥ |
tac chivaṃ sarvam eva śam astu naḥ ||

The Earth peaceful, the atmosphere peaceful, the heaven peaceful, the waters peaceful, the plants peaceful, the trees peaceful, all gods my peace, the collection of gods my peace, peace be peaceful. By peace, I render all that is terrible here, all that is cruel here, all that is sinful here, peaceful and auspicious. Let all this be weal to us.

Then he sprinkled the water on himself, her and around the vedi with the below incantations:

agnir āyuṣmān sa vanaspatibhir āyuṣmān |
sa māyuṣmān āyusmantaṃ kṛṇotu ||

Agni is full of life: with the trees he is full of life. Full of life let him make me full of life.

vāyur ayuṣmān so ‘ntarikṣeṇāyuṣmān |
sa māyuṣmān āyusmantaṃ kṛṇotu ||

Vāyu is full of life: with the atmosphere he is full of life. Full of life let him make me full of life.

indra āyuṣmān sa viryeṇāyuṣmān |
sa māyuṣmān āyuṣmantaṃ kṛṇotu ||

Indra is full of life: with manliness he is full of life. Full of life let him make me full of life.

devā āyuṣmantas te ‘mṛtenāyuṣmantaḥ |
te māyuṣmanta āyuṣmantaṃ kṛṇvantu ||

The gods are full of life: with ambrosia they are full of life. Full of life let them make me full of life.

They seated themselves on the reddish brown bull skin to commence the ritual. Somakhya then placed a fire-stick of the vikaṅkata plum in the vedī dipped in ghee: “Gautamī, it has been stated in the śruti: vajro vai vikaṅkataḥ ।; hence, for such a ritual we use this samidh. It is an ancient medicinal plant, perhaps used an Indian substitute for the willow.” He then muttered the vāravantīya incantation to the god Agni of his ancient clansman who had transferred to the clan of the Vaiśvāmitra-s:

aśvaṃ na tvā vāravantaṃ

This is to praise you, Agni, with salutations, the one like tail of the horse | the one of excellent things, presiding like the emperor over the ritual.

The great god Vaiśvānara, who takes the oblations to the deva-s, blazed forth illuminating their fire-room with a purple hue in the dim twilight even as the last rays of the sun withdrew from the sky. Lootika looking at the ever-moving flames was deeply moved by this thought: It is indeed for this reason our ancient ārya ancestors called him vāravant — his sinuous movements are verily like the prancing tail of a horse dashing across the grassy steppe. His changing from form to form, like the vibhakti-s of the substantives, he truly embodies the ever-moving aspect of the ṛta. Hence, my ancestor Nodhas had stated thus when he performed a sacrifice for the king Śātavaneya:

mūrdhā divo nābhir agniḥ pṛthivyā
taṃ tvā devāso ‘janayanta devaṃ
vaiśvānara jyotir id āryāya ||

The head of heaven and the navel of the earth is Agni. Then he became the spoked wheel of the world-hemispheres. The gods generated you, that god Vaiśvānara, verily as light for the Ārya-s.

Indeed, that wheel in the midst of the world-hemispheres indicates his ever-moving moving form as the celestial cycles. She snapped out her reveries as she had to fill the sruk and hand it over to Somakhya to pour out the great attacking oblations.

dipsato yaś ca dipsati |
vaiśvānarasya daṃṣṭrayor
agner api dadhāmi tam || + vau3ṣaṭ

Him who, though unharmed, would harm us, and him who, harmed, would harm us,
I verily place between the two fangs of Agni Vaiśvānara.

Then Somakhya made the sruva offering to the foe-killing Vāyu:

sagarāya śatruhaṇe svāhā | śarṇilāya śatruhaṇe svāhā | samudrāya śatruhaṇe svāhā |
sāndhasāya śatruhaṇe svāhā | iṣirāya śatruhaṇe svāhā | avasyave śatruhaṇe svāhā |
vāyave śatruhaṇe svāhā | vātāya śatruhaṇe svāhā | mātariśvane śatruhaṇe svāhā | pavamānāya śatruhaṇe svāhā ||

L: “These 10, sometimes obscure, names of Vāyu bring to mind some yajuṣ oblations that I have heard my father making from the Taittirīya-śruti.”
S: “There are two sets of such oblations among the Taittirīyaka-s. The first of those are related to the ‘offering of the seven winds’ or the Vāta-nāmāni:

samudrāya tvā vātāya svāhā | salilāya tvā vātāya svāhā | anādhṛṣyāya tvā vātāya svāhā | apratidhṛṣyāya tvā vātāya svāhā | avasyave tvā vātāya svāhā | duvasvate tvā vātāya svāhā | śimidvate tvā vātāya svāhā ||

The second of those are rather remarkable in associating Vāyu with the the Marut-s, a development which completely enveloped the latter in the epic period, as in the Rāmāyaṇa:

samudro ‘si nabhasvān ārdradānuḥ śambhūr mayobhūr abhi mā vāhi svāhā |
māruto ‘si marutāṃ ganaḥ śambhūr mayobhūr abhi mā vāhi svāhā |
avasyur asi duvasvāñ śambhūr mayobhūr abhi mā vāhi svāhā ||”

L: “It is notable that while these are pauṣṭika, our AV prayoga is abhicārika. This is rather reminiscent of the Iranian tradition where a long series of Vātanāmāni are prescribed to be deployed in the sacrifice to be performed when an Iranian is under attack or during war. However, what is consistent across such Vāyu invocations in our vaidika traditions is the association of Vāyu with moisture and the waters: salila, sagara, samudra, śarṇila and ārdra-dānuḥ(!).”
S: “In fact that watery connection is seen in the very same Iranian collection of incantations you alluded to, suggesting that this is an ancient feature from the Indo-Iranian period referring to Vāyu as the bearer of moisture and the stirrer of water-bodies. The Avestan incantations states:

ýazâi apem ca bakhem ca
ýazâi âxshtîm hãm-vaiñtîm ca
suyãm ca kataremcit
tem vayêmcit ýazamaide
tem vayêmcit zbayamahi

I will sacrifice to the waters and to him who divides them. I will sacrifice to the peaceful one, whose breath is beneficent, and to weal, both of them. To this Vayu do we sacrifice, this Vayu do we invoke.”

Next they prepared the sruk for offering to Indra:

indrasya bāhū sthavirau vṛṣāṇau
citrā imā vṛṣabhau pārayiṣṇū |
tau yokṣe prathamo yoga āgate
yābhyāṃ jitam asurāṇāṃ svar yat ||+ vau3ṣaṭ

Indra’s two arms, firm and manly, these two are wondrous as stud bulls. The [time of] yoking has arrived, I first yoke these two, by which the heaven of the Asura-s was conquered.

S: “This is the famous Atharvanic apratiratha-ṛk. The arms of Indra are compared to fierce bulls, something our ancestors would seen in a rather unadulterated form on the steppes from even before the age of the śruti. But do you suspect a wordplay here?”
L: “Seems so. While the reference is to the two bulls, the time of yoking I believe actually refers to horses being yoked to the ratha, that key war-conveyance of the ārya-s. The time of yoking is verily that of the warrior yoking the horses for war. But what the mantra conveys is that even before the horses we first yoke the Vajra- and the arrow-bearing arms of Maghavan that bring victory.”
S: “Good.”

They prepared the sruk for the next oblation with the below incantation:
atisṛtyātisarā
indrasyaujasā hata |
aviṃ vṛka iva mathnīta
tato vo jīvan mā moci
prāṇam asyāpi nahyata || + vau3ṣaṭ

Having overtaken him the spells slay [him] with the might of Indra, snatch him like a wolf a sheep. Then let him not get away from you alive, indeed block his breath.

L: “The atisara which is dispatched reminds one of the missile dispatched in the hair-raising combat between Arjuna and Karṇa in the national epic, which was much like that of Śatamanyu and Vṛtra. There when Karṇa deployed the missile of Rāma Bhārgava it is said to be of Atharvaṇic nature:

rāmād upāttena mahāmahimnā
ātharvaṇena+arivināśanena |
pārthaṃ ca bāṇair niśitair nijaghne ||

With the [missile] of great might obtained from Rāma, the Atharvaṇic foe-destroyer, [Karṇa] smashed that fiery weapon of Arjuna and he pierced Pārtha with sharp arrows.

Then again when finally Arjuna killed Karṇa with a gigantic missile it is said:

kṛtyām atharvāṅgirasīm ivogrāṃ
dīptām asahyāṃ yudhi mṛtyunāpi |
bruvan kirīṭī tam atiprahṛṣṭo
ayaṃ śaro me vijayāvaho ‘stu ||

Having fired the blazing weapon fierce as an Atharvāṅgirasa Kṛtyā incapable of being endured by Death herself in battle, the crowned one (Arjuna) said with great joy: “Let this arrow be victory-bearing to me.”

S: “For a V2 in battle the atisara might indeed be his actual weapon, while a V1 performing such a prayoga might visualize it as striking his foe like a wolf seizing a sheep. In the national epic there is another such account of the deployment of a śakti made by Tvaṣṭṛ for Rudra by Yudhiṣṭhira which was be worshiped with such mantra-s. Let us read it out in full for it is most inspiring:

tatas tu śaktiṃ rucirogra-daṇḍāṃ
cikṣepa vegāt subhṛśaṃ mahātmā

The great soul, the foremost of the Kuru-s [Yudhiṣṭhira] then hurled with great force at the king of the Madra-s that blazing śakti with a well-made and fierce handle blazing [like] with the shine of gems and corals.

dīptām athaināṃ mahatā balena
savisphuliṅgāṃ sahasā patantīm |
praikṣanta sarve kuravaḥ sametā
yathā yugānte mahatīm ivolkām ||

All the Kuru-s together saw that blazing dart emitting sparks as it powerfully flew hurled with great might like a meteorite [falling] at the end of the yuga.

tāṃ kālarātrīm iva pāśa-hastāṃ
yamasya dhātrīm iva cograrūpām |
sabrahma-daṇḍa-pratimām amoghāṃ
sasarja yatto yudhi dharmarājaḥ ||

Guiding it, the Dharmarāja hurled that [missile], which was of dreadful form like the goddess of the night of dissolution holding a noose, the midwife who birthed Yama, and infallible as the rod of Brahman.

gandha-srag argyāsana-pānabhojanair
abhyarcitāṃ pāṇḍusutaiḥ prayatnāt |
saṃvartakāgni-pratimāṃ jvalantīṃ
kṛtyām atharvāṅgirasīm ivogrām ||

With much effort the sons of Pandu had worshiped [that missile] with perfumes and garlands, water pourings, a seat, drinks and food. It blazed like the fire of destruction and was fierce as an Atharvāṅgirasa Kṛtyā.

īśāna-hetoḥ pratinirmitāṃ tāṃ
tvaṣṭrā ripūṇām asu-deha-bhakṣām |
bhūmy-antarikṣādi-jalāśayāni
prasahya bhūtāni nihantum īśām ||

For the sake of Īśāna this missile was engineered by Tvaṣṭṛ; it consumes the breath and bodies of foes. By its force it can violently destroy the Earth, the atmosphere, water-bodies and living beings.

mantraiś ca ghorair abhimantrayitvā |
sasarja mārgeṇa ca tāṃ pareṇa

Thus, with a forceful effort, having inspired it with fierce mantra-s, [Yudhiṣṭhira] released it with utmost momentum along the best trajectory for the destruction of the lord of the Madra-s.

hato ‘sy asāv ity abhigarjamāno
rudro ‘ndhakāyāntakaraṃ yatheṣum |
prasārya bāhuṃ sudṛḍhaṃ supāṇiṃ
krodhena nṛtyann iva dharmarājaḥ ||

“There, you are killed”, roaring thus the Dharmarāja [hurled the missile], having stretched his arm with a firm good hand, as if dancing in wrath, even as Rudra had shot his arrow for bringing the end of Andhaka.

As you would have noted, the weapon is said to be like the Atharvāṅgirasa Kṛtyā and specifically it was inspired by mantra-s. Thus, indeed a V2 might inspire his weapon as an atisara before its deployment.”

L: “But what about the deployment of the Kṛtyā-prayoga related to the famous sūkta starting with yāṃ kalpayanti…?”
S: “That is used primarily in the deflecting mode. But this is primarily in the attacking mode in the vaidika tradition.”

Then they continued with the next oblation:

athainam indra vṛtrahann
ugro marmaṇi vidhya |
atraivainam abhi tiṣṭha
śakra medy ahaṃ tava |
anu tvendrā rabhāmahe
syāma sumatau tava || + vau3ṣaṭ

Now here O fierce Indra, the Vṛtra-slayer, pierce him in his marman. Trample him right here. O Śakra, I am your votary. We take hold of you, O Indra. May we be in your favor.

L: “It is indeed clear from this mantra that the ārya chooses Indra, whereas the śatru is anindra as has been clearly stated in the śruti.”
S: “Next comes the oblation to delude the mind of the foe.”

veda devo bṛhaspatiḥ |
tat satyam cittamohanam || + vau3ṣaṭ

That which King Varuṇa knows, which the god Bṛhaspati knows, which Indra the Vṛtra-slayer knows, is the truth which bewilders the mind [of the foe].

śarveṇa nīlaśikhaṇḍena
bhavena marutāṃ pitrā |
virūpākṣeṇa babhrūṇā
vācaṃ vadiṣyato hataḥ || + vau3ṣaṭ

By Śarva, but the blue-crested one, by Bhava, by the father of the Marut-s, by the odd-eyed one and by the brown one the voice of he who verbally [attacks] us is smitten.

śarva nīlaśikhaṇḍa
vīra karmaṇi-karmaṇi |
imām asya prāśaṃ jahi
yenedaṃ vibhajāmahe || + vau3ṣaṭ

O Śarva, the blue-crested one, O hero, in every rite smite the food of this one with whom we portion this out.

L: This triad strikes me as being one of the vaidika-prayoga-s comparable to the tāntrika- prayoga-s like that of Bagalāmulkhī or Varāhī. Bagalā is used identically for the vāca-stambhana-prakriyā while the sow-headed Saṃketayoginī may be deployed for mohanam like the first ṛk. The last mantra appears a little obscure to me.
S: “One who is has ubhayavīrya in both the tāntrika and the vaidika prayoga-s is truly an accomplished mantravādin, like that one from the Kaliṅga country. Each path has its own rigor and strictures; hence, in the least it is good to be educated in both. But for a V1 the priority is the vaidika rite. As for the last mantra, I believe that it indicates a bhrātṛvya with whom we are competing for space or glory.”

Finally, they made the mysterious oblation to Rudra and his son.

tvaṃ devānām asi rudra śreṣṭha
tavas tavasām ugrabāho |
hṛṇīyasā manasā modamāna
ā babhūvitha rudrasya sūno || + vau3ṣaṭ

You are the chief of the gods O Rudra, the mightiest of the mighty, O one of formidable arms. With fury, delighting in your mind, you had come into being, O son of Rudra.

L: “That is quite remarkable! The son of Rudra clearly in singular here! It seems to indicate the emergence of a proto-Skanda. In my readings I have only encountered rudrasya sunuḥ in singular in that glorious sūkta to the Marut-s of my ancestor Nodhas.”
S: “That sūkta of Nodhas (RV 1.64) is truly an awe-inspiring one to the Marut-s who are entirely spoken of in plural as is usual. But then as you say we mysteriously get this singular occurrence of the singular form in it (RV 1.64.12):

ghṛṣum pāvakaṃ vaninaṃ vicarṣaṇiṃ
rudrasya sūnuṃ havasā gṛṇīmasi |

The agile, pure, winning, all-seeing son of Rudra we extol with an invocation.

Everywhere else in the RV the term is always in plural — rudrasya sunavaḥ except for the rather notable similar singular usage by the other Āṅgirasa clan, the Bharadvāja-s, in their parallel sūkta to the Marut-s (RV 6. 66):

rudrasya sūnuṃ havasā vivāse |

This growing Māruta, with a blazing spear, the son of Rudra I bring with with an invocation.

Hence, it does seem that these rare singular usages for the Marut-s in the śruti point to the latent tendency for consolidation of the Marut-s into a single form. This process was indeed likely on the pathway to the emergence of Skanda in the para-Vedic Indo-Iranian borderlands. I could also point to a more circumstantial connection via the peculiar word hṛṇīyasā in the AV mantra. Its root is of early Indo-European vintage and can be related to the fury of the gods such as Rudra and Varuṇa. Hence, it is notable that in the late Atharvaṇic tradition the hymns to fury, the Manyu-sūkta-s are associated with a hapax ṛṣi named Brahmāskanda, which is none other than the name of our patron deity. Further, those Manyu hymns are used in the Skanda ritual of the late AV tradition.”

ayajvanaḥ sākṣi viśvasmin bharo3m ||

Posted in Heathen thought, Life |

## Sequences related to maps based on simple fractional functions

One of the pleasures of an unstructured youth in the pre-computer era was what we called calculator games. As our father took his prized calculator with him to work we only got a little time with it in the evenings. However, we got our break when a relative gifted us a Japanese solar-powered calculator. The roots (pun intended) of this note lies in the games that followed: One of those was determining the consequences of simple functional iterations. We soon realized that these are intimately related to some familiar and not so familiar sequences. Some of them were mysterious and beyond our calculator efforts. We finally cracked them and obtained a more formal understanding of the convergences only when our mathematical knowledge improved and we were in possession of a computer in the late phase of our youth. Consider maps which are iterates of simple fractional functions of the form:

$x_0=0; \; x_n=\dfrac{1}{1+k x_{n-1}}$

If $k=1$, $n \to \infty, \; x_n \to \dfrac{1}{\phi}$, where $\phi$ is the Golden Ratio

One can see that the iterates of this map are sequence of fractions: 0, 1, 1/2, 2/3, 3/5, 5/8, 8/13, 13/21, 21/34. One notices that its terms are derived from the famous mātrā-meru sequence.

If $k=2, n \to \infty, \; x_n \to \dfrac{1}{2}$

$x_n=\dfrac{1}{1}, \dfrac{1}{3}, \dfrac{3}{5}, \dfrac{5}{11}, \dfrac{11}{21}, \dfrac{21}{43}, \dfrac{43}{85}, \dfrac{85}{171}...$

This sequence of fractions can be derived from the famous Jacobsthal sequence: $2f[n]+(-1)^n; f[1]= 1$: 1, 3, 5, 11, 21, 43, 85, 171…
Like the mātrā-meru, the Jacobsthal sequence appears in numerous seemingly unrelated places in mathematics.

If $k=3$, $x_n \to \dfrac{\sqrt{13}-1}{6} \approx 0.43425854591$

Here $x_n= 0, \dfrac{1}{1}, \dfrac{1}{4}, \dfrac{4}{7}, \dfrac{7}{19}, \dfrac{19}{40}, \dfrac{40}{97}...$

One observes that these fractions are related to the sequence $f[n]=f[n-1]+3f[n-2]; f[1]=1, f[2]=1$:
1, 1, 4, 7, 19, 40, 97…

If $k=\dfrac{1}{2}$, $x_n \to \sqrt{3}-1 \approx 0.73205080756888$

Here $x_n= 0, \dfrac{1}{1}, \dfrac{2}{3}, \dfrac{3}{4}, \dfrac{8}{11}, \dfrac{11}{15}, \dfrac{30}{41}, \dfrac{41}{56}, \dfrac{112}{153}...$

These fractions are related to the sequence $f[n]=2f[n-1]; f[n+1]=f[n]+f[n-2]; f[n+2]=f[n+1]+f[n-1]; f[1]=1, f[2]=1$:
1, 1, 2, 3, 4, 8, 11, 15, 30, 41, 56, 112, 153…

If $k=\dfrac{1}{3}$, $x_n \to \dfrac{\sqrt{21}-3}{2} \approx 0.79128784747792$;

Here $x_n=0, \dfrac{1}{1}, \dfrac{3}{4}, \dfrac{4}{5}, \dfrac{15}{19}, \dfrac{19}{24}, \dfrac{72}{91}, \dfrac{91}{115}, \dfrac{345}{436}, \dfrac{436}{551}...$

These are related to the sequence $f[n]=f[n-1]+f[n-3]; f[n+1]=f[n]+f[n-2]; f[n+2]=3f[n+1]; f[1]=1, f[2]=1, f[3]=3$:
1, 1, 3, 4, 5, 15, 19, 24, 72, 91, 115, 345, 436, 551…

Now, one can use the substitution $x_n=y_n-\tfrac{1}{k}$ to convert our map into an equivalent but alternative map:

$y_{n+1}=\dfrac{1}{k}\left( 1+\dfrac{1}{y_n} \right)$

By writing $y_n=\tfrac{w_n}{w_{n-1}}$ the above map becomes a linear difference equation:

$w_{n+1}=\dfrac{1}{k}(w_n+w_{n-1})$

Such a linear difference equation defines the quadratic equation: $y^2-\tfrac{y}{k}-\tfrac{1}{k}=0$ with roots:

$y=\dfrac{1\pm \sqrt{4k+1}}{2k}$

Thus, the map in the variable $y_n$ will be attracted to the greater root $\tfrac{1+ \sqrt{4k+1}}{2k}$ and repelled by the lesser root $\tfrac{1- \sqrt{4k+1}}{2k}$ for any starting value of $y_n$, except when $y_0=\tfrac{1- \sqrt{4k+1}}{2k}$; interesting aspects of the convergence of this type of map was discussed earlier. Now, given that $x_n=y_n-\tfrac{1}{k}$, we get the attractor $A$ of our original maps in $x_n$ to be:

$A= \dfrac{\sqrt{4k+1}-1}{2k}$

Now if $k < -\tfrac{1}{4}$ then $A$ will be a complex number. Given that our original maps of the form $x_n=\tfrac{1}{1+k x_{n-1}}$ are defined in real space $\mathbb{R}$, what will happen to the iterates in such a case? Consider an example of $k=-1$ and $x_0=-1$. Here the map does not converge to a single attractor $A$ but cycles between 3 values: -1, 1/2, 2. Now, if we use a different $x_0=0$ we get 0, 1, $\infty$. Thus, the 3 values that the iterates cycle through depends on $x_0$ but the cycle length is always $3$.

Consider another case where $k=-\tfrac{1}{3}$ and $x_0=0$. Here again the map does not converge to a single value but cycles through 6 different values: 0, 1, 3/2, 2, 3, $\infty$. Again the cycle length is always 6 though the actual values change with $x_0$. This cycling in the above examples immediately suggests a connection between the arithmetic operation of our map and trigonometry. This connection becomes clear from the form $A$ takes:

If $k<-\dfrac{1}{4}, \; A= -\dfrac{1}{2k}+i\dfrac{\sqrt{-4k-1}}{2k}$

If we have a complex number $z=x+iy$ then it defines an angle $\theta=\textrm{Arg}(z)=\arctan\left(\tfrac{y}{x}\right)$ in the complex plane. Thus we get the angle corresponding to $A$ to be:

$\theta = \textrm{Arg}(A)= \arctan\left(-\sqrt{-4k-1}\right)$

Since for our purposes the sign of the angle does not matter we can work with $\theta= \arctan\left(\sqrt{-4k-1}\right)$. As an aside, we can also express this angle as $\theta= \arccos\left(\tfrac{1}{2\sqrt{-k}}\right)$.

Now when $A$ is real, $\theta = \textrm{Arg}(A)=\pi$. Thus, $\tfrac{\pi}{\theta}=1$ and our map converges to a single value, i.e. it has cycle length 1.

Instead, when say $k=-1, \; \theta= \arctan\left(\sqrt{3}\right)=\tfrac{\pi}{3}$. Thus, $\tfrac{\pi}{\theta}=3$ and our map has cycle length 3. Similarly, when $k=-\tfrac{1}{3}, \; \theta = \tfrac{\pi}{6}$. Thus, $\tfrac{\pi}{\theta}=6$ and our map has a cycle length 6.

What if $k$ does not generate a $\theta$ which is a rational sector of a circle? The simplest such example that is easily apprehended is $k=-2$. In such a case the map would never converge to single value or a cycle. However, we observe remarkable behavior where it oscillates through several overlapping cycles which are defined by “explosive'', much greater than average values (Figure 1-4).

Figure 1 illustrates the cycle of length 13 (marked by red dots). One of these starts at $n=13$ and decays in terms of absolute magnitude at every 13th iterate thereafter: 26, 39, 52… Another starts at $n=5$ and increases in absolute value every 13th iterate thereafter: 18, 31, 44, 57… If one observes closely one see that there are smaller barely discernible cycles of 5, 3 and 2 nested within those of length 13.

Figure 2 shows the next cycle of length 213 again increasing or decreasing in absolute magnitude at every 213th iterate (marked by red dots). Within each cycle of length 213 one can discern 16 cycles length 13.

Figure 3 shows the next set of cycles of length 1291 (orange) and 1504 (red). One can discern 6 cycles of length 213 in each of the 1291 length cycles. There is a single cycle of length 1291 in that of length 1504.

Figure 4 shows even larger cycles of length 4299 (green) and 10102 (red). The former includes 2 cycles of length 1504 within it. Likewise, the cycle of length 10102 includes 2 cycles of length 4299.

How do we derive these cycles lengths with no clear cut pattern from first principles? For the map defined by $k=-2$, the characteristic angle $\theta=\arctan\left(\sqrt{7}\right)$. Thus, the cycles correspond to the numerators of the successive rational approximation fractions from the continued fraction representation of $\tfrac{\pi}{\arctan\left(\sqrt{7}\right)}$. Thus, we get the sequence of the cycle lengths to be:
2, 3, 5, 13, 213, 1291, 1504, 4299, 10102, 135625, 145727, 28135, 989783…

More generally, when $\theta$ is not a rational sector of a circle the map shows cycles of increasing length that are specified by the numerators of the successive rational approximations obtained from the continued fraction for $\tfrac{\pi}{\theta}$.

We also notice the following curious pattern typical of rational fraction approximations for the successive cycle lengths:
\begin{aligned} 5 & = & 1 & \; \times & 3 & \; + & 2\\ 13 & = & 2 & \; \times & 5 & \; + & 3\\ 213 & = & 16 & \; \times & 13 & \; + & 5\\ 1291 & = & 6 & \; \times & 213 & \; + & 13\\ 1504 & = & 1 & \; \times & 1291 & \; +& 213\\ 4299 & = & 2 & \; \times & 1504 & \; + & 1291\\ 10102 & = & 2 & \; \times & 4299 & \; + & 1504\\ 135625 & = & 13 & \; \times & 10102 & \; + & 4299 \end{aligned}

Let $C_n$ be the length of the cycle $n$. Thus,

$C_n=kC_{n-1}+C_{n-2}$

Hence, $k$ will give the number of repeats of the previous cycle $C_{n-1}$ which are contained in $C_n$.

Finally, let us consider the intertwined maps of the form where we alternate through $\pm k$:

$x_n=\dfrac{1}{1+ky_{n-1}}; \; y_n=\dfrac{1}{1-kx_{n-1}}$

This alternation forces a convergence to a point $(C_x, C_y)$. Doing the algebra we can obtain,

$C_x=\dfrac{2k+1-\sqrt{4k^2+1)}}{2k} \\[10pt] C_y=\dfrac{2k-1+\sqrt{4k^2+1)}}{2k}\\[10pt] C_x+C_y=2$

Thus, as examples we have:
$k=1; n \to \infty, \; x_n \to 2-\phi; \; y_n \to \phi$, where $\phi$ is the Golden Ratio.

$k=2; n \to \infty, \; x_n\to \dfrac{5-\sqrt{17}}{4} \approx 0.21922359359558;\; y_n \to \dfrac{3+\sqrt{17}}{4} \approx 1.78077640640442$

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