Trigonometric tangles-3: the fractals

•April 30, 2017 • Leave a Comment

See also: https://manasataramgini.wordpress.com/2016/05/06/the-astroid-the-deltoid-and-the-fish-within-the-fish/

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

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

epicycloids_complex

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

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

Mandelbrot_zcubedFigure 2: Mandelbrot set for z^3 showing nephroid

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

circle2ellipse

Figure 3: Mapping a unit circle on to ellipses.

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

circle_cassinian_mapFigure 4

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

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

weierstrass

Figure 5

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

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

weierstrass2

Figure 6

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

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

weierstrass_tangles03

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

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

weierstrass_tangles01

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

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

weierstrass_tangles04Figure 9: Further fractal maps with fractional c.

Marching onward in the American spring but where to?

•April 23, 2017 • Leave a Comment

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

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

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

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

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

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

continued…

Euler’s squares

•April 20, 2017 • Leave a Comment

On account of our fascination with the geometry of origami (albeit not well-endowed in mathematical capacity) we discovered for ourselves shortly after our father had taught us trigonometry that,
\arctan(1)+\arctan(2)+\arctan(3)=\pi
We had earlier shown the origami proof for that. But it was only a little later while drifting away from one of those trigonometric identities that you routinely faced in those annoying college exams we stumbled upon a beautiful relationship that was apparently first discovered by the great Leonhard Euler. This is a relationship parallel to the above one:
\arctan\left(\frac{1}{2}\right)+\arctan\left(\frac{1}{3}\right)=\arctan\left(1\right)=\frac{\pi}{4}
The proof for this, like the origami proof for the above, can be achieved from a self-evident construction of Euler — what the Hindus of yore would have called an upapatti or mathematicians today term “wordless” proof. It is illustrated below but I add several words for the benefit of the non-geometrically oriented reader.

Euler_squares

1) Draw square ABCD and triplicate it so that the three squares share a side.
2) Draw diagonals AC and CG of first two squares and use them to draw square ACGF and duplicate it.
3) Draw \overline{AE}: from the construction it is apparent that \angle GAE=\arctan\left(\frac{1}{3}\right)
4) From the construction it is clear that \angle EAC=\arctan\left(\frac{1}{2}\right)
5) We thus see: \angle GAE+ \angle EAC = \angle BAC= \frac{\pi}{4}= \arctan(1)=\arctan\left(\frac{1}{2}\right)+\arctan\left(\frac{1}{3}\right)

This relationship is one of a class of strange trigonometric relationships that interestingly bring in the meru-średhī (called in western literature as Fibonacci sequence):

\arctan\left(\frac{1}{M_{2n}}\right)=\arctan\left(\frac{1}{M_{2n+1}}\right)+\arctan\left(\frac{1}{M_{2n+2}}\right)

M=1,1,2,3,5,8,13,21..., the meru-średhī; thus for n=1, we get M_2=1; M_3=2; M_4=3.
This leads us to a formula for \pi based on the odd terms of meru-średhī starting from M_3:

\pi=4\displaystyle \sum_{n=1}^\infty \arctan \left(\frac{1}{M_{2n+1}}\right)

Shown below is the convergence of the above series to \pi: We reach an accuracy of 6 decimal places for n=18.

pi_meru

Trigonometric tangles-2

•April 15, 2017 • Leave a Comment

We had earlier described our exploration of the spirograph, hypocycloids, epicycloids and related curves. In course of our study of the śaiva tantra-s of the kaula tradition we started thinking about a remarkable piece of imagery mentioned in them. Tantra-s like the Virūpākṣa-pañcākśika and Nityāṣoḍaṣikārṇava talk of the waves and spinning cakra-s of Śakti-s. The Virūpākṣa-pañcākśika describes these innumerable Śakti-waves and wheels of Śakti-s as emanating the universe:

svāñge cid-gagana+ātmani dugdhodadhi-nibhaḥ sva-śakti-laharīṇām |
sambheda-vibhedābhyāṃ sṛjati dhvaṃsayati cai(e)ṣa jagat || 2.13

In his (Śiva’s) own body, in the void of the conscious-self, resembling the milk-ocean, through the constructive and destructive interference of the waves of his own śakti-s this universe is generated and destroyed.

rūpādi-pañca-viṣayātmani bhogya-hṛṣīka-bhoktṛ-rūpe ‘asmin |
jagati prasarad ananta-sva-śakti-cakrā citirbhāvyā ||2.14

In the contents of the five sense-streams starting with form, in the objects being sensed, in the sense organs and the first-person experiencer, in the world generated by the innumerable wheels of his Śakti-s consciousness should be conceived.

The Nityāṣoḍaṣikārṇava states:
procchalan-mada-kallola-pracalaj-jaghanasthalām |
śakti-cakrocchala-cchaktivalanā-kavalīkṛtām | 2.24-25

The goddess Tripurā is worshipped as: The gushing intoxicant waves setting in vibration the vulval receptacle, setting in motion the śakti wheels, spinning the Śakti wheels and devouring all.

This account of the waves of Śakti-s and their wheels led us to conceive the imagery as related to the more general epicycloid problem: Imagine a point on the circumference of a wheel rotating at some speed either clockwise or anticlockwise. That point is the center of another wheel, which is likewise rotating at a distinct speed. On the circumference of the second wheel is a point which in turn is the center of yet another wheel rotating at yet another speed and so on. Now what curve would a point on the rim of the terminal wheel of this system would trace out? [See link for animation

More than 20 years ago mathematician Frank Farris, who has a great eye for symmetry and beauty, showed that this can be described rather simply by a function in the complex plane thus:
z(t)=a_1e^{k_1it}+a_2e^{k_2it}...+a_ne^{k_nit} ; \; z(t) \in \mathbb{C}
a_j=a_1,a_2...a_n represent the relative radii and k_j=k_1, k_2...k_n represent the relative speeds of rotation of the n wheels in the system. If the wheel is rotating clockwise then k_j<0 and a_j=i\cdot a_j; where i=\sqrt{-1}

spirographic_6tangle1Figure 1: The curve generated by a system of 9 wheels of relative radii: 1.0000000, 0.6180340, 0.3197333, 0.1227690, 0.2256262, 0.2233212, 0.4724658 and rotation speeds: -1, 5, 11, 17, 23, 29, 35.

Thus more generally we can define a complex function which would generate “trigonometric tangles” related to those generated by a n-wheel system:
z(t)=\displaystyle \sum _{j=1}^{n}a_je^{k_jit}
where k_j is real number and a_j might be real or imaginary. Now these functions are symmetric and closed as t takes all values [0,2\pi] when the following conditions are satisfied:
k_j=l\times n+m; \; l \in \mathbb{Z}; \; n,m \in \mathbb{N}_1\; and \; \textrm{gcd}(n,m)=1
i.e if l is any integer and n,m are relatively prime, positive integers then the resultant curve is closed and has n-fold symmetry. Thus, keeping to symmetric closed curves this equation can literally produce nava-nava-chamatkAra.

spirographic_8tangles2Figure 2: A 12 term system with 8-fold symmetry allowing only real a_j

spirographic_4tangles1Figure 3: The svastika-system with 4-fold symmetry, random number of terms up to 10, and allowing both real and imaginary a_j

spirographic_24guillocheFigure 4: The 17 term system with 24-fold symmetry allowing both real and imaginary a_j. It somewhat resembles the guilloche security patterns used on bank notes.

Śarabha vidhi

•April 9, 2017 • Leave a Comment

The account of this rite continues from the prefatory narrative.

Lootika: “There are several ways in which Rudra is worshipped as Śarabha conceived as the great dinosaur. We shall follow the way which is appropriate for us brāhmaṇa-s who observe the śruti as our supreme religious authority rather than the āgama-s, even though we accept the latter as specific authorities. The purely āgamika methods are practiced by śaiva-s, śivabhakta-s and liṅgin-s.”

sharabha2Somakhya then rolled out a canvas and showed Jhilleeka and Prachetas a picture of Śarabha: “One shall accordingly visualizing Śarabheśāna as having five dinosaur-like heads each with with 3 eyes and a heavy, sharp beak. He has 12 arms two of which will be wings. I shall talk of the weapons and other objects held in the arms later. Enclosed in his two wings are the goddesses Pratikriyā-śūlinī and Pratyaṅgirā. In his heart we visualize the fierce Bhairava of the Dakṣiṇasrotas, Svacchanda, associated with the pada ghora-ghoratrebhyaḥ of the bahurūpin. In his belly we visualize the seven-flamed Agni Vaḍavānala. In this two dinosaur-like feet we visualize Jvareśvara and the ferocious lion-headed Haribhadra, even as they were emitted by Rudra to slay Dakṣa.

One begins by invoking Rudra the lord of animals thus:
rudraḥ paśūnām adhyakṣaḥ
sa māvatv asmin brahmaṇy
asmin karmaṇy asyāṃ purodhāyām
asyāṃ devahūtyām asyām ākūtyām
asyām āśiṣi svāhā ॥

Make an offering in the aupāsana fire at the svāhā.

Having visualized Śarabha as stated earlier you shall first perform the worship the five faces conjoining the Brahma-mantra-s thus:

oṃ hauṃ tatpuruṣāya pakṣirājāya namaḥ । tatpuruṣaya vidmahe…||
oṃ hauṃ vāmadevāya gauryai gāndhāryai garuḍyai namaḥ । vāmadevāya namo… ||”
oṃ hauṃ sadyojātāya śālvāya namaḥ । sadyojātaṃ prapadyāmi… ||
oṃ hauṃ aghorāya ākāśabhairavāya namaḥ । aghorebhyo’tha… ||
oṃ hauṃ īśānāya Śarabheśānāya namaḥ । īśānaḥ sarva-bhūtānām… ||

Then you shall practice the japa of the mūlamantra:
vāsudeva ṛṣiḥ । triṣṭubh chandaḥ । kālāgnirudraḥ Śarabheśāno devatā ॥
khaṃ bījam । khaṃ śaktiḥ । khaṃ kīlakam ॥
khaṃ Śarabheśānāya khaṃ śālvāya khaṃ pakṣirājāya hum phaṇ namaḥ ॥

Then you pacifies the deity with:
oṃ Śarabhaṃ tarpayāmi ।
oṃ śālvaṃ tarpayāmi ।
oṃ pakṣirājaṃ tarpayāmi ॥

Thereafter one worships the aṅga-devatā with their mantra-s. Lootika will give you those mantra-s of the śakti-s. Hearing it from her mouth you stand the chance of instantaneous success.”

Lootika: First one should worship Pratikriyā-śūlinī with the mighty Vairocanī-ṛk: tām agnivarnāṃ tapasā… closing one’s eyes. If the image of a Durgā riding a lion holding a trident in her hand appears while uttering the mantra then it is a sign of success. She has six-arms and it colored like a flame with a bluish tinge. You pacify her with the mantra:
oṃ maheśānīṃ mahāŚarabhaśūlinīṃ tarpayāmi ॥

Then you worship Atharvaṇabhadrakālī-Pratyaṅgirā with the mantra:
oṃ hrīṃ pratyaṅgire dhāma dhāma jyotir jyotir bhūr-bhuvas-suvar-mahar-vṛdhat karad ruhan mahat tac cham oṃ lakṣmyai namo vaḥ ॥ She is fourteen-armed, black-colored, with dense free-flowing hair, wearing bright osseous ornaments and evoking erotic sentiments. She is Lakṣmī because she is Siddhilakṣmī. The special vyāhṛti-s of the mantra are from the Gopatha-brāhmaṇa of the veda of our ancestors the Bhṛgu-s and the Añgirasa-es. If when uttering the mantra with closed eyes she appears before you with a hallow like an eclipsed moon around her head then it is a sign that you might be on the path of success. You must see her upraised sword. If erotic sentiments arise in you at that point it is a good sign. But you must not get lost in them. Instead you should take in a breath with the oṃ hrīṃ and retaining it even as you slowly utter the rest of the mantra till the vyāhṛti-s you must locate your mūlādhāra-cakra and perform the jālandhara and mūla-bandha. You pacify her with the mantra:
oṃ pratyaṅgirāṃ viśvalakśmīṃ somaguptāṃ mahāŚarabhīṃ tarpayāmī ॥

Now Somakhya would impart to you the remaining male devatā-s.”

Somakhya: “Having worshiped the śakti-s you shall invoke in the heart of Śarabheśāna the fierce Bhairava with the following mantra:
aiṃ kṣāṃ kṣīṃ kṣūṃ kṣaḥ kṣmlvryūṃ svacchandabhairavāya sphuratātmane huṃ phaṇ namaḥ ॥
He appears as five-headed and ten-armed riding a white bull.

Then you worships the fierce Agni Vaḍavānala with his seven tongues which correspond to the heptagon of the Śarabha-yantra with this mantra:
oṃ raṃ hiraṇyāyai kanakāyai raktāyai kṛṣṇāyai suprabhāyai atiraktāyai bahurūpāyai namaḥ kṛṇuśva pājaḥ prasitiṃ…||
The terminal of the mantra is the rakṣohā-ṛk of Vāmadeva Gautama.

Then you worship jvareśa or the jvarāstra of triadic form, which Rudra hurled at Kṛṣṇa Devakīputra with this mantra:
oṃ namo bhagavate rudrāya jvararūpāya trikāyarūpāya vo namaḥ ॥

You then worship Haribhadra accompanied by a hundred Rudra-s of blue-hued necks as has been described in the śruti with the mantra: “nilagrīvāḥ śitikaṇṭhāḥ…”. He is worshiped by the mantra combining the utterance from the Skandam and Acintya-viśva-sādākhyam:
oṃ jaya jaya rudra mahāraudra bhadrāvatāra oṃ śrīṃ hrīṃ mṛgendrāya haribhadrāya namaḥ । oṃ jūṃ saḥ oṃ ॥
Now Lootika shall impart to you the worship of the zoocephalous āvaraṇa-yoginī-s.”

Lootika: “First in the four directions having completed the worship of Śarabheśāna and his aṅga-s you shall worship the following yoginī-s thus starting east in pradakṣiṇa order:
oṃ gajānanāṃ pūjayāmi tarpayāmi namaḥ । elephant-headed.
oṃ siṃhamukhīṃ pūjayāmi tarpayāmi namaḥ । lion-headed
oṃ gṛdhrāsyāṃ pūjayāmi tarpayāmi namaḥ । vulture-headed
oṃ kākatuṇḍīṃ pūjayāmi tarpayāmi namaḥ । crow-headed.

Then you shall worship the mighty maṇḍala-yoginī with the following mantra:
namāmi mahāyoginīṃ sarvaśatruvidāraṇīṃ ।
aṣṭavaktrīṃ koṭarākṣīṃ vakrāṃ vikaṭalocanām ।
uṣṭragrīvāṃ hayagrīvāṃ vārāhīṃ Śarabhānanām ।
ulūkīṃ ca śivārāvāṃ mayūrīṃ vikaṭānanām ॥

She bears the following 8 heads: 1) camel, 2) horse, 3) sow, 4) dinosaur, 5) owl, 6) jackal, 7) peacock, 8) lion. She is the guardian of the Śarabha-maṇḍala.

Then you pacify her:
aṣṭavaktrīṃ tarpayāmi pujayāmi namaḥ ।

Now Somakhya would take you through the concluding pacifications.”

Somakhya: “The concluding part of the worship is conducted with the mantra of the Bhṛgu-s of yore:
tubhyam āraṇyāḥ paśavo
mṛtyā vane hitās tubhyaṃ
vayāṃsi śakunāḥ patatriṇaḥ ।
tava yakṣaṃ paśupate ‘psv antas
tubhyaṃ kṣaranti divyā āpo vṛdhe ॥

You tie a rakṣa with the mantra:
śune kroṣṭre mā śarīrāṇi
kartam ariklavebhyo gṛddhrebhyo
ye ca kṛṣṇā aviṣyavaḥ ।
makṣikās te paśupate vayāṃsi
te vighase mā vidanta ॥

Then worship his weapons:
triśulaṃ tarpayāmi ।
daṇḍaṃ tarpayāmi ।
pinākaṃ tarpayāmi ।
pāṣupataṃ tarpayāmi ।
kuliśaṃ tarpayāmi ।
cakraṃ tarpayāmi ।
musalaṃ tarpayāmi ।
khaḍgaṃ tarpayāmi ।
khaṭvāṅgaṃ tarpayāmi ।
paraśuṃ tarpayāmi ॥

i.e. the trident, rod, pināka bow, pāśupata missile, thunderbolt, discus, pestle, sword, skull-brand and axe.

Then he concludes by reciting the mantra-s:
tasmai prācāyā diśi antardeśād bhavam iṣvāsam anuṣṭhātāram akurvan ।…ya evaṃ veda॥

Thus the rite is concluded.”

A prefatory narrative

•April 8, 2017 • Leave a Comment

Jhilleeka and Prachetas were visiting Lootika and Somakhya. Prithika, the daughter of the latter two was much excited as she resembled and was to resemble her youngest aunt in more than one way – they seemed to have a some special connection. Thus, she clung to Jhilleeka until she was put in bed. Then the adults went up to the terrace of Somakhya and Lootika’s house, where Jhilleeka started narrating the special matters pertain to their visit even as the pleasant prathamī moon shone above them: “A while back when you and Vrishchika were away Varoli and me had conspired with Prachetas and Mitrayu to do our summer internships at labs in Turushkarajanagara. One evening after dinner we went up to the terrace of the house in which were staying and started plying the planchette with the hope of snaring a vetāla. I must confess that I wished to test Prachetas’ mettle to see if he was really up to my level in prayoga matters. agrajā I wonder if you did something like that with Somakhya? Lootika merely smiled and said: “I don’t want to break your narrative with our exploits; pray continue.”

Jhilleeka: “After a while a vetāla did seize our bhūtacakra. He announced his name Pṛthuroman. I immediately, deployed the mantra known as Harahuṃkāram and made the vetāla seize Prachetas. He seemed to be totally taken over by it. Varoli and Mitrayu wanted to intervene right away but I bound both them with the Sarasvatī-bandha and they were unable to intervene. Only late that night Varoli broke free and greatly upbraided me. She even wanted to call Vrishchika and tell her what I had done but I convinced her that I would ensure that no real harm comes to Prachetas in the event he is unable to defend himself. He will now tell you what happened next.”

Prachetas: “Thus, attacked by surprise by my future patnī I succumbed to the seizure by the vetāla. I took a bus in that state to seizure and went to what in those days was a little hamlet at the outskirts of Turushkarajanagara. There, I walked at the dead of night to the boundary of a cemetery. Despite being a just a hamlet there seemed to be a brisk flow of corpses which were being fed to Kravyāda. I wandering under the possession I sneaked in through a hole in fence of the cemetery to enter the grounds. Evidently the vetāla was guiding me there for some reason. But whatever the reason, I suddenly realized what had happened to me. I realized that it was my future patnī who had caused this possession – but at that moment I was livid with anger and wanted to perform a powerful prayoga to strike back at her. But first I had to get myself free of the vetāla. My eyes had by then adjusted themselves to the dark and I saw a corpse of a young man lying in the vicinity. He seemed to have a noble bearing. I wondered who had left the corpse there without it being cremated. With some effort I deployed the gandharva mantra and performed svāveśa with a gandharva, who then drove out the vetāla. Intended to dispatch the gandharva then at Jhilli so that she might seized by him like Viśvāvasu seizing a young woman. The vetāla entered the corpse that I had just seen. The corpse underwent a strange transmogrification thereafter. The young man aged rapidly and his hair fell off and what remained whitened. Then the corpse seated itself cross-legged and began to speak: ‘That girl Jhilleeka who is your friend is going to be your wife in the not so distant future so don’t let any harm reach her due to your pratikriyā. I realized that when the vetāla speaks we get a prognostication. To make him speak more I deployed the Sarasvatī-pravāha.

He pranced about a bit and then grabbing one hand of his with another he said sat down and scrapped some ashes and applied it on himself. I observed him more closely and he looked strangely familiar giving me a fright for the first time. He then said: “I was a brāhmaṇa like you. I descended from the clan of a bhārgava or an aṅgirasa. I performed many vaidika rituals and drank soma. One day when I traveled to Turushakarajanagara to visit the #$*& institute. [Prachetas: strangely that happened to be the institute next to where Jhilli and I worked]. As I was talking to my host there suddenly a centrifuge spun out of control in the floor below us and it smashed through our floor and struck me. Thus, I expired. As I had no close relatives, my corpse was dumped by the cops after their investigation at the cremation ground for the general cremation which would take place at 4:00 AM. I glad you have heard my story. In pocket lies a letter for a girl who lives in the Bhatahata sector the city. If you post it so that it might reach her then good would come to you. But I tell you a death not different from mine could be yours and your wife’s too unless you become the master of tīkṣṇatuṇḍa. He then handed me the letter in his pocket and shrieking loudly ran into the darkness and dropped down about 25 meters away from me. I snapped out of the āveśa and withdrew the gandharva I had intended to seize pretty Jhilli. I took the first bus I could get in the morning and returned to our rental after posting the letter of the animated corpse. I was brooding over the prognosis of our deaths and wondering what tīkṣṇatuṇḍa was as I entered our dwelling. I and Mitrayu lived on the ground floor while Jhilli and Varoli lived on the first floor. Even as I came in Mitrayu had freed himself from the bandha and had joined Varoli in screaming at Jhilli. I calmed them all down and even as I did so I forgot about the strange prognosis.

Jhilleeka: “Last night I had a terrifying dream in which I saw a spinning object come flying through the floor and kill a young man. I then saw a young woman jump off a building and die. I narrated it to Prachetas who told me it was a very deadly sign and mentioned that only if we figured out what tīkṣṇatuṇḍa was we could be saved. You agrajā and Somakhya are our elders and guides hence we place ourselves at your feet so that you may help us cross the noose of Vaivasvata that the young brāhmaṇa in Turushkarajanagara could not.

Somakhya: “By tīkṣṇatuṇḍa what is meant is the mantra-vidhi of Śarabheśāna that we shall we reveal to you. It is hard to master even by a good brāhmaṇa. Many have tried and failed but he who does so shall be saved like Mārkaṇdeya of the race of the Bhṛgu-s when confronted by Antaka at the dead of night when the powers of the Dānava-s are exalted.”

Trigonometric tangles

•April 2, 2017 • Leave a Comment

Let us define a define the trigonometric tangle as the following parametric function:
x=\cos(t/k)-a\cos(b\cdot t)\sin(t)
y=\sin(t/k)-a\cos(b\cdot t)\cos(t)
where k can be a rational number \frac{p}{q} or an irrational number. a and b are any real number. If k is a rational number and |b|=1 then we get a tangle c petals defined thus:
k=\dfrac{p}{q}

\dfrac{c}{d}=\dfrac{2p+q}{q}, such that c,d are relatively prime.

When |b|\gg 1 then the envelop of the tangle converges to a figure with e-fold symmetry defined thus:
\dfrac{e}{f}=\dfrac{2p+2q}{q} such that e,f are relatively prime.

For irrational k for the above conditions we get figures coming close those of k=\frac{p}{q} corresponding to the continued fraction approximations of the irrationals. This provides us an interesting way of visualizing the irrationals. Interestingly, visual distinction can be made between algebraic and transcendental irrationals with these curves.

trignots4Figure 1. Convergence from c-lobed initial to e-fold symmetry

trignots5Figure 2. Curves for small and large b for integral k

trignots6Figure 3. Curves for small and large b for fractional rational k

trignots7.1Figure 4. Curves for various irrational k=\sqrt{2},\phi,\sqrt{3}, e, \pi, \sqrt{10}

 
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