A novel rendering of the first 20 prime numbers


Posted in art

Fancies of the parabola and hyperbola

Many revolutions-of-the-sun ago we were seeking a device that could construct a parabola. We made a discovery in this connection that is exceedingly elementary Euclid for mathematicians, but for us it was an insight-giving philosophical revelation. A parabola can be made thus:1) Draw a line L1;
2) Take a point P on it;
3) Draw perpendicular line P1 through P;
4) Take a random fixed point F, which will be the focus of the future parabola;
5) Draw the segment FP;
6) Draw the perpendicular bisector line P2 of segment FP;
7) Mark the point Q where P2 intersects P1;
8) Track the locus of Q as P moves on P1, i.e. the locus of the intersection of the perpendicular bisector P2 of FP with P1 as P moves on L1;
This locus is a parabola with focus at F. All the lines P2 are tangents of this parabola.

We can obtain the equation of the parabola thus:
Place origin O (0,0) at the minimum of the parabola;
Then the focus F is at (0,a);
Because P2 is the perpendicular bisector of FP we can show that the equation of line L1 is y=-a using similarity of triangles;
Coordinates of the point on the generated locus are Q(x,y);
Hence, P(x,-a);
By bhuja-koTi-karNa-nyAya: FQ^2=PQ^2; hence, (x-0)^2+(y-a)^2=(x-x)^2+(y+a)^2;
Thus we have y=x^2/(4*a) as the equation of the parabola (or proof that the locus of Q is a parabola).

We then found that if you repeat the above procedure with a circle C1 in place of line L1, and a radial line P1 in place of the P1 of the above example we get a hyperbola:

Posted in Life, Scientific ramblings

The kAlI-krama

The khachakra as described in the KhPS has an octagonal circuit in which are seated the 8 yoginI-s who are rejoicing at the great festival of the sport of yoga at the graveyard in karavIra. Inside their circuit are the 5 circuits of the 17-sided polygons (the Gaussian polygon) which shine with the 85 yogini-s. From them emanate another circuit of 85 yogini-s, together forming the 170 yogini-s of the khachakra. These yoginI-s behold in the middle the mighty kAlI, who is known as mahAma~NgalA in the midst of an unimaginable blaze. One of her 9 syllabled mantra-s is hence “namaH shrImahAma~NgalAyaI”. They praise the supreme kAlI, the mistress of the khachakra, who is full of wonder (mahAshcharya) thus:

khaphrekAra-dhAmasthe dhAma-rUpe nira~njane |
chid-dhAma para-dhAmAkhye mahAdhAma pare kale ||
gaganAmbara-saMkAshe nirAdhAra-padoj-jhite |
nabhovyomAntaraakAre manthAnAnte namo.astu te ||
You who reside in the abode of the bIja khaphreM, whose form is the abode [of all] , the stainless, the abode of consciousness known as the primal abode, the great abode and primary unit of time. You who are like the garment of the space, one who is free from even the state without any supporting foundations, you who is within space and vacuum, you who is within the churning one [*1], salutations to you!

[*1] manthAna: this is an allusion to the great bhairava who in a different kula tradition, i.e. that of the pashchimAmnAya is called the most great manthAna bhairava.

Finally mahAma~NgalA manifests as 5 sequences: The first two are series of the 13 kAlI-s, and the remaining 3 are the kAlI-s of dikchara, gochara and bhuchara circuits – this forms the heart of the kAlI kula. In the last circuit she is seen as the vaiShNavI shakti-s — kAlasaMkarShiNI and nArasiMhI, and as the deva-shaktI-s too .

The two 13 kAlI sequences:


Posted in Heathen thought

The lineage of kAli-kula as presented by arNa-siMha

paNDita arNa-siMha is today a poorly known teacher. From his two works the khachakrapa~nchaka stotra and mahAnaya prakAsha, we glean that he was once part of an illustrious kAli-kula lineage, which began with shivAnanda-nAtha-I who he calls j~nAna-netra or j~nAna-chakShu. He even suggests that the khachakrapa~nchaka’s mantrashAstra was first “brought down” by shivAnanda. He names a lineage of kAlikula teachers to which he belongs which appears to intersect with the lineage mentioned by jayaratha the commentator on abhinavagupta. arNasimHa’s lineage is:

          female teacher keyUravatI
                  /         \
 female teacher IshAnikA | prabodha
            nandaka      | jaiyaka
            sajjana      | pa~Nka
            someshvara   | nAga
                   \        /

After chakrabhAnu the kAli-kula tradition branched into two lineages which unified again under arNasimha.
khachakrapa~nchaka stotra is stotra full of power – good stotra-s are of different kinds. Some have good poetry and are easily followed by all sAdhakas, such as those attributed to sha~Nkara, some have deep mantra embeddings, and some are only for advanced users and convey rahasya-s of a tantric system to them. The KhPS is of the latter type and also conceals within it an immense mantra power. It is not easy to expound it but it needs to be “experienced” as a first person experience.
This is sensed right in the 2nd verse itself:
ekA devI parasarati parA vyomahaMkAra-ghoShA
chakraM raudraM gagana-sadR^ishaM rashmi-pu~njaM samagraM |
bhAShAtItaM vismaya-viShayaM devI-chakraM namAmi ||
The one goddess, the supreme, spreads the sound the ahaMkAra [*1] of space – the terrifying chakra [*2] which appears like the sky, the collection [of all things], the mass of rays. Whose form is like pure space, formidable with the light of the bliss of the unmanifest spot (or embryo) [*3]. I salute the chakra of the goddess whose sphere is wondrous and beyond language.
[1] The term ahaMkAra here should be understood in the sAMkhya sense as expounded in the bhR^igu smR^iti.
[2] It must be understood that the chakra referred to twice in this verse is multifunction in that it refers to the maNDala or yantra of bhAnavI-kaulinI and its projection in the form of the world with the sun coursing through the sky and the projection into consciousness.
[3] kalana often means a spot or the zygote in embryology. Both these meanings are woven together in this case –the bindu of the yantra as the spot and the unmanifest existence is likened to the zygote which represent the “unmanifest” form of the organism that will develop from it. It also represents the basic bIjamantra of the kula: khaphreM.
This is a succinct the description of bhAnavI-kaulinI the mistress of the khachakra. As we noted above the stotra itself uses the term “bhAShAtItaM” beyond language. Then starting with vinAyaka and his shakti the stotra outlines the complex maNDala of the devI with the hosts of yoginI-s in each circle.

Posted in Heathen thought

Musings on some alternative grammarians

durgasiMha the commentator on the nirukta of yAska states:
vyAkaraNaM aShTadhA |”.
The physician and grammarian vopadeva from the medieval yadava court expands on the same theme naming the eight forms of saMskR^ita grammar:
indrash-chandraH kAshakR^itsn.a-ApishalI shAkaTAyanaH |
pANiny-amara-jainendrA jayantya aShTAdishAbdikAH || (in dhAtupATha discussion)

Despite the tremendous productivity of saMskR^ita thinkers on grammar it appears that the rest have not matched pANini’s success and have to greater of lesser degrees been washed aside. somadeva, the author of that great literary masterpiece the kathAsaritsAgara apocryphally but colorfully narrates this process in his story on the debate of kAtyAyana and pANini. Of course tale goes back to somadeva’s source material the bhR^ihat-katha.

Of these, jinendra, despite his name, is a bauddha and the last in the old line of grammarians. He is supposed to have lived in the va~Nga country and been a bodhisattva. He appears to have been patronized the pAla kings around 725-750 CE. He is cited by the great vedic commentator haradatta mishra from the drAviDa country and criticized by the much latter bhaTTojI dIkShita from mahArAShTra who appears to have clearly been aware of jinendra’s bauddha biases in interpreting the vaidika aspect of pANini. In any case this is still illustrative of his widespread influence as a major grammarian both in spatial and temporal terms. jinendra himself cites kAshakR^itsna and Apishali and thereby preserves fragments of their lost works. Another renowned nAstika paNDita, chandragomin, is usually identified as the author of the chandra grammar by tradition. bauddha tradition holds that the founder of chandravyAkaraNa was a kShatriya from varendra in the va~Nga country, and probably lived roughly around the same time as jinendra. He is supposed to have debated other tAthAgata-s like the nAstika mahApaNDita chandrakIrti with the aid of his mantra prayoga-s to invoke avalokiteshvara. He is reputed to have been a mantravAdin of great siddhi-s, who journeyed from nAlandA to the Andhra country, where he was locked in prayoga battles with Astika mantravAdin-s who deployed saiddhAntika, bhairava and kubjikA mantra-s to counter him. He is said to have composed several long stotra-s to ma~njughoSha, avalokiteshvara and the kula goddess sitAtapAtrA. While visiting gavalakuNDa, R1’s father brought out an old file of handwritten papers and showed me a prayoga stotra to pratya~NgirA composed by chandragomin along with some nAstika mantra-s to the same deity.

chandragomin identifies pratya~NgirA with his chosen deity the kula-mistress sitAtapAtrA, whom he calls the queen of chandradvIpa. A tale is narrated where an irate ruler tries to drown chandragomin, but he is carried by the devI to chandradvIpa and saved. This is of considerable interest because chandradvIpa is also the primal kShetra of kubjikA. kubjikA too is identified with pratya~NgirA as ghora rudrakubjikA-atharvaNa-bhadrakAlI in the tradition of the uttarAmnAya and the kubjikA upaniShad. chandragomin begins his invocation to pratya~NgirA by using the compound name uShNISha-sitAtapAtrA-pratya~NgirA, which is parallel to another stotra in which he identifies aparAjitA with sitAtapAtrA. Another text of chandragomin (of course given as a dialog between shuddhodhana-putra and his disciples) showed by R1’s father was the dhAraNI of mahApratya~NgirA, which had a picture of the devI with numerous arms holding all kinds of weapons. Another text of chandragomin had a long nAstika-pratya~NgirA mAlA mantra which seeks protection against all kinds of dangers.

Posted in Heathen thought, History

AdhAra dhyAnaM

indrAya namaH

Posted in art, Heathen thought

AtatAyin revealed

We wondered who the AtatAyin was,
the ari who was being a vR^itra to our kratu-s,
the vile kulu~ncha who was prowling among our possessions.
We sought the aid of the son of agni, the slayer of tAraka,
to make visible to us the unseen DAmArikA.

We spread forth the mantra of the one with spear.
With the passing of the day the son of rudra uncovered the veil,
and showed to us who the ari was!

It was the same vile DAmArikA that had haunted us for long.
We launched the operation patra-trAta.
We repeatedly called the 6-headed one to perform uchChATana
of the shishNa-devatA-s who were vR^itra-s to our kratu-s.

Posted in Life