Notes on the Bhadra-sūkta, a hymn for felicity to the Vedic pantheon

For better type-setting and Devanagari rendering use PDF version.
Introduction
Several sub-schools of the Taittirīya school of the Kṛṣṇayajurveda possess their own collections of mantra-s distinct from their saṃhitā-s known as the mantra-pāṭha-s. These include mantra-s that are often found in other traditions but not in their own saṃhitā or brāhmaṇa. Additionally, they also include some mantra-s which are unique to these mantra-pāṭha-s. For example the famous Yajurvedic version of the Śrīsūkta is found in the mantra-pāṭha of Bodhāyana. However, most practitioners in South India do not correctly use the accents of this sūkta and seem to be unaware that the KYV version of this sūkta occurs in this text. The mantra-paṭha-s of the Āpastamba, Bodhāyana, Vaikhānasa and Hiraṇyakeśin sub-schools have come down to us. The mantra-s in them are typically deployed in gṛhya rituals directed by instructions from their gṛhyasūtra-s. However, the Vaikhānasa-mantra-pāṭha is distinctive in having a late terminal part that is used in the iconic worship of Viṣṇu by the Vaikhānasa-s. The Hiraṇyakeśī-mantra-pāṭha has a ṛk-saṃhitā as part of it and is used by the Hiraṇyakeśīn-s of Maharashtra and Tamil Nadu in their rituals to this date. In someways this is reminiscent of the hautra-pariśiṣṭha of the Āpastamba-s that is used by Yajurvedin-s to supplement the role normally performed by the Ṛgvedin hotṛ.

Other than these mantra-pāṭha-s of the KYV, we also have comparable supplementary mantra-collections among the Sāmavedin-s in their Mantrabrāhmaṇa and the famous khila of the RV. Further, beyond the Taittirīya school, the Kaṭha school, which was once widespread in the northern parts of the subcontinent like Kashmir and the Panjab, had it own mantra-paṭha that went along with their gṛhyasūtra, namely that of Laugākṣi. While the original form of this mantra-paṭha does not survive to my knowledge, a version of it with accretions of tāntrika and paurānika material used in smārta practice by the brāhmaṇa-s of Kashmir and some of their counterparts in Himachal Pradesh has come down to us. This text was published by the Kashmirian brāhmaṇa-s Keśava Bhaṭṭa and Kāśīnātha-śarman in the first half of the 1900s. I had earlier examined a defective version of this text but thanks to the massive text-scanning effort of the eGangotri Trust of the texts at the Kashmir Research Institute, Srinagar we can now examine a better version of this text.

The melding of tāntrika and vaidika tradition found in this Kashmirian text has a long history in Hindu tradition. Indeed, as we have pointed out before[footnote 1], a small mantra-saṃhitā comparable to the mantra-pāṭha-s is found preserved in the śākta-purāṇa, the Devī-purāṇa, which might preserve a distinct vaidika tradition. Similarly, the Āṅgirasakalpa of the Paippalāda school of the Atharvaveda preserves a combined mantra-deployment of Paippalāda AV mantra-s along with tāntrika-vidyā-s. With regard to the AV tradition one may also point to the Tripurārṇava-tantra, an authoritative mūla-tantra of the Śrīkula tradition. The 20^{th} taraṃga of this text preserves a combined tāntrika-vaidika mantra-deployment for the Indramahotsva (the great festival of Indra) which associates itself the AV. This association is likely genuine for the AV is the one vaidika tradition that has clear injunctions for the Indramahotsva in its pariśiṣṭha-s. This section of the Tripurārṇava-tantra specifies several vaidika mantra-s that are to be used in the worship of Indra and other deva-s, which are combined with the worship of the Bhairava of the Śrīkula tradition under the tāntrika scheme.

Our Kashmirian text associated with the Kaṭha school, like other mantra-paṭha-s, has some unique Vedic material. One such is the Bhadrasūkta which is the topic of this note. This sūkta of 22 ṛk-s is to our knowledge not found in any other saṃhitā. It is mostly comprised of regular jagati-s (12-12-12-12), with the last ṛk being a triṣṭubh (11-11-11-11). There may be some hypermetrical verses like ṛk-17 (12-12-12-13). The sūkta has a form rather similar to the RV7.35 of Vasiṣṭha Maitrāvaruṇi. Like that one it is a vaiśvadeva-sūkta, which invokes the entire pantheon for luck or felicity. In RV7.35 the word for luck or felicity is the indeclinable śam. In our sūkta the word bhadra is used instead. It is used as an adjective that declines as the deity being invoked with the dative pronoun naḥ (“for us”) being used just like in RV7.35. Hence, in this sūkta we translate bhadra as auspicious (or can be taken in the sense of the deity being luck-granting). This word bhadra is also found in multiple ṛk-s of another famous vaiśvadeva-sūkta, RV1.89 of Gotama Rāhūgaṇa, in an equivalent sense (ā no bhadrāḥ; devānām bhadrā sumatir ṛjūyatāṃ; bhadraṃ karṇebhiḥ śṛṇuyāma devā bhadram paśyemākṣabhir yajatrāḥ). More generally, the pattern of the repetitions of bhadra is seen on multiple occasions in the RV albeit not in vaiśvadeva-sūkta-s (e.g. RV8.62 of Pragātha Kāṇva) and also in the AV saṃhitā-s. A ṛk of Sobhari Kāṇva (RV8.19.19) also uses the word bhadra repeated in a sense similar to this sūkta:
bhadro no agnir āhuto bhadrā rātiḥ subhaga bhadro adhvaraḥ | bhadrā uta praśastayaḥ ||
For us auspicious Agni when he is made an offering, the auspicious gift, the auspicious ritual, you the giver of good luck, [for us] auspicious hymns of praise.
Another comparable word is svasti (“well-being”) used in a similar sense by the Atri-s in their vaiśvadeva-sūkta, RV5.51.11-15 and also by Gotama Rāhūgaṇa in RV1.89. Indeed, in the Kashmirian tradition the Bhadrasūkta is used on conjunction with RV1.89 and RV5.51.11-15. This style continues into the epic period where we observe Kausalyā confer a blessing on Rāma using a comparable incantation with svasti ( in R2.25).

The pantheon of the Bhadrasūkta is entirely Vedic with no paurāṇika features. This squarely places the sūkta within the classic vaidika tradition and it was perhaps even originally attached to some now lost saṃhitā. However, in ṛk-2 we encounter the god Prajāpati. He is not found in the comparable RV7.35 or other core RV vaiśvadeva-sūkta-s. He appears to have entered the Vedic tradition relatively late from a para-Vedic tradition [footnote 2]. His position in the sūkta suggests that he has not superseded the old aindra system as it happened in the even later Vedic layers. In this regard his position is comparable to that found the camaka-praśna of the Yajurveda tradition. This suggests that the sūkta indeed belongs to a comparable relative temporal layer and was a relatively late composition with the Vedic tradition, perhaps consciously mirroring the RV7.35 and RV1.89. The final ṛk has the refrain: “tanno mitro varuṇo mā mahantām aditiḥ sindhuḥ pṛthivī uta dyauḥ” (Mitra and Varuṇa, Aditi, the river, the Earth and also Heaven should grant this to us), which is characteristic of the Kutsa-s of the RV (e.g. RV1.94). Kutsa also has a certain predilection for composing low complexity sūkta-s, which is also seen rather plainly in this one. Importantly, his two vaiśvadeva-sūkta-s, RV1.105 and RV1.106, have characteristic low-complexity style with repetition. Notably, his sūkta to the Sun (RV1.115) uses the word bhadra repeatedly as in this sūkta. Together, these indicate that the composer of the Bhadrasūkta was a member of the Kautsa clan.

Some notable features of the Bhadrasūkta are:

1. Venas is implored to be ever-desirous (uśan…sadā) of the worshiper. This furnishes a link between Venas and the later name of Venus in Sanskrit tradition, Uśanas. Thus, it further strengthens the identification of Venas with Venus and suggests an early IE origin for this planetary name.

2. Mātariśvan is explicitly identified with Vāyu in this sūkta. In the RV Mātariśvan is often mentioned as bringing Agni to the Bhārgava-s and humans at large (evidently from Vivasvat). In RV3.29.11 Viśvāmitra clarifies this identification by stating: “mātariśvā yad amimīta mātari vātasya sargo abhavat sarīmaṇi ||”: [He is called] Mātariśvan when he measures out [the space] in his mother; he became the rush of the wind in flowing out. Thus, we translate Mātariśvan as “he who grows in his mother” meaning “he who grows in the world-womb.

3. In ṛk-10 we seen an invocation of various physiological processes. This is unique for a vaiśvadeva-sūkta and not seen in RV sūkta-s of this type. In this regard it has a flavor more typical of the AV.

4. In ṛk-s 12-14 we encounter a great diversity of devatā-dvandva-s, which is unprecedented in any other vaiśvadeva-sūkta elsewhere in the śruti.

The emended text of the sūkta is presented below with an approximate translation.

Bhadrasūkta: text
bhadro no agniḥ suhavo vibhāvasur bhadra indraḥ puruhūtaḥ puruṣṭutaḥ |
bhadraḥ sūrya urucakṣā uruvyacā bhadraś candramāḥ samitheṣu jāgṛviḥ ||1||
For us the auspicious Agni, well-invoked and abounding in light, the auspicious Indra much-invoked and much-hymned; the auspicious Sun, wide-seeing and wide-ranging, the auspicious Moon keeping an eye [on us] in the battle.

bhadraḥ prajā ajanayannaḥ prajāpatir bhadraḥ somaḥ pavamano vṛṣā hariḥ |
bhadras tvaṣṭā vidadhad rūpāṇy adbhuto bhadro no dhātā varivasyatu prajāḥ ||2||
For us the auspicious Prajāpati [who] progeny-generated, the auspicious Soma, the purified one [Footnote 3] and the manly yellow one; the auspicious Tvasṭṛ giving wondrous forms [to things]. May the auspicious Dhātṛ show favor to [our] progeny.

bhadras tārkṣyaḥ suprajastvāya no mahām̐ ariṣṭanemiḥ pṛtanā yudhā jayan |
bhadro vāyur mātariśvā niyutpatir veno gayasphāna uśan sadā ‘stu naḥ ||3||
For us the auspicious Tārkṣya Ariṣṭanemi for the sake of good progeny and for conquering the hostile army by means of battle; the auspicious Vāyu, expanding within the world-womb [footnote 4], the lord of the team of horses. May Venus, the wealth-increaser, be always desirous of us.

bhadro mitro varuṇo rudra id vṛdhā bhadro ‘hirbudhnyo bhuvanasya rakṣitā |
bhadro no vāstoṣpatir astv amīvahā bhadraḥ kṣetrasya patir vicarṣaṇiḥ ||4||
For us the auspicious Mitra and Varuṇa, and Rudra verily with augmentation, and Ahirbudhnya the protector of the universe; the auspicious guardian of the homestead: may he be the destroyer of illness and the auspicious guardian of the field, ever-full of activity.

bhadro vibhur viśvakarmā bṛhaspatir bhadro dviṣastapano brahmaṇaspatiḥ |
bhadraḥ suparṇo aruṇo marut-sakhā bhadro no vāto abhivātu bheṣajī ||5||
For us the mighty, all-maker Bṛhaspati, the auspicious foe-scorcher and lord of the ritual; the auspicious falcon, reddish-brown and the friend of the Marut-s [footnote 5]. May the auspicious Vāta [footnote 6] blow medicines towards us.

bhadro dadhikrā vṛṣabhaḥ kanikradad bhadraḥ parjanyo bahudhā virājati |
bhadrā sarasvām̐ uta naḥ sarasvatī bhadro vaśī bhadra indraḥ purūravaḥ ||6||
For us the auspicious [horse] Dadhikra, the neighing stallion, the auspicious Parjanya [who] manifoldly shines forth; the auspicious Sarasvat and also Sarasvatī, the auspicious cow and the auspicious Indra, the loud-roarer.

bhadro naḥ pūṣā savitā yamo bhago bhadro ‘graja ekapād aryamā manuḥ |
bhadro viṣṇur urugāyo vṛṣā harir bhadro vivasvām̐ abhivātu nastmanā ||7||
For us the auspicious Pūṣaṇ, Savitṛ, Yama and Bhaga, and the auspicious first-born Ekapāt, Aryaman and Manu; the auspicious Viṣṇu, the wide-strider and the manly lion. May indeed the auspicious Vivasvat blow towards us.

bhadrā gāyatrī kakub uṣṇihā virāḍ bhadrānuṣṭup bṛhatī paṅktir astu naḥ |
bhadrā nas triṣṭub jagatī purupriyā bhadrāticchāndā bahudhā vibhūvarī ||8||
For us the auspicious Gāyatrī, Kakubh, Uṣṇihā [Footnote 7] and Virāṭ. May Anuṣṭubh, Bṛhatī, Paṅkti each be auspicious to us. For us the auspicious Triṣṭubh, the much-loved Jagati [footnote 8] and the auspicious long meters manifold and of many treasures.

bhadrā no rākānumatiḥ kuhūḥ suhṛd bhadrā sinīvāly aditir mahī dhruvā |
bhadrā no dyaur antarikṣaṃ mayaskaraṃ bhadro ‘śvo dakṣastanayāya nas tuje ||9||
For us the the auspicious Rākā, Anumati and friendly Kuhū, auspicious Sinivālī, Aditi, and the firm Earth goddess. For us the auspicious Heaven goddess, the atmosphere giving pleasure, the auspicious horse, and Dakṣa for extending for us [our] lineage.

bhadro naḥ prāṇaḥ sumanāḥ suvāgasad bhadro apānaḥ satanuḥ sahātmanā |
bhadraṃ cakṣur bhadram icchotram astu no bhadraṃ na āyuḥ śarado asacchatām ||10||
For us the auspicious life-process with a good mind and good speech unmanifest, the auspicious excretory process with the body and the consciousness; indeed may the vision be auspicious and hearing be auspicious for us. For us the auspicious life with autumns, a 100 yet to manifest.

bhadrendrāgnī no bhavatām ṛtāvṛdhā bhadrā no mitrāvaruṇā dhṛtavratā |
bhadrāśvinā no bhavatāṃ navedasā bhadrā dyāvā-pṛthivī viśva-śaṃbhuvā ||11||
For us the auspicious Indra and Agni fostering the Law; for us the auspicious Mitra and Varuṇa maintaining the Laws. May the two auspicious Aśvin-s be cognizant [of us]. For us the auspicious Heaven and Earth benevolent to all.

bhadrā na indrāvaruṇā riśādasā bhadrā na indrā bhavatāṃ bṛhaspatī |
bhadrendrāviṣṇū savaneṣu yāvṛdhā bhadrendrāsomā yudhi dasyu-hantamā ||12||
For us the auspicious Indra and Varuṇa, devourers of foes. May Indra and Bṛhaspati be auspicious to us. [For us] auspicious Indra and Viṣṇu who augment [us] during the soma libations. May the auspicious Indra and Soma slay the dasyu in battle.

bhadrāgnāviṣṇū vidadhasya prasādhanā bhadrā no ‘gnīndrā vṛṣabhā-divaspatī |
bhadrā no agnīvaruṇā pracetasā bhadrāgnīṣomā bhavatāṃ navedasā ||13||
For us the auspicious Agni and Viṣṇu, the ornaments of the gift-distribution. For us the auspicious Agni and Indra, the bulls, the lords of heaven. For us the auspicious Agni and Varuṇa, the ever-mindful ones. May the auspicious Agni and Soma be cognizant of us.

bhadrā sūryā-candramasā kavikratū bhadrā somā bhavatāṃ pūṣaṇā naḥ |
bhadrendrāvāyū pṛtanāsu-sāsahī bhadrā sūryāgnī ajitā dhanañjayā ||14||
For us the auspicious Sun and Moon, the two full of insight. May Soma and Pūṣaṇ be auspicious. [For us] the auspicious Indra and Vayu conquering in battle and the auspicious Sūrya and Agni unconquered and winning wealth.

bhadrā naḥ santu vasavo vasuprajā bhadrā rudrā vṛtrahaṇā purandharā |
bhadrā ādityāḥ supasaḥ sunītayo bhadrā rājāno maruto virapśinaḥ ||15||
May the auspicious Vasu-s be wealth and progeny [giving]. For us the auspicious Rudra-s who slay Vṛtra and smash the [hostile] forts and the auspicious Āditya-s, well-seeing and well-guiding, and the auspicious kings, the Marut-s, the exuberant ones [footnote 9].

bhadrā na ūmā suhavāḥ śataśriyo viśvedevā manavaś carṣaṇīdhṛtaḥ |
bhadrāḥ sādhyā abhibhavaḥ sūracakṣaso bhadrā naḥ santv ṛbhavo ratna-dhātamāḥ ||16||
For us the auspicious helper-[gods], well-invoked and with a 100 riches, all the gods and the Manu-s, supporters of the folks. For us the auspicious Sādhya-s, the overpowerers, radiant as the Sun. May the auspicious Ṛbhu-s be gem-givers for us.

bhadrāḥ sarve vājino vājasātayo bhadrā ṛṣayaḥ pitaro gabhastayaḥ |
bhadrā bhṛgavo ‘ṅgirasaḥ sudānavo bhadrā gandharvāpsarasaḥ sudaṃsasaḥ ||17||
Auspicious the racers, winners of booty; auspicious the sages, [our] ancestors, the sun-beams. Auspicious the Bhṛgu-s and Aṅgiras-es, the liberal givers; auspicious the Gandharva-s and Apsaras-es, the powerful ones[footnote 10].

bhadrā āpaḥ śucayo viśvabhṛttamā bhadrāḥ śivā yakṣmanudo na oṣadhīḥ |
bhadrā gāvaḥ surabhayo vayovṛdho bhadrā yoṣā uśatīr devapatnyaḥ ||18||
For us the auspicious waters, pure and the foremost supporters of all, the auspicious, benign, disease-repulsing herbs; the auspicious cows, charming and invigorating, the auspicious nymphs and loving wives of the gods.

bhadrāṇi sāmāni sadā bhavantu no bhadrā atharvāṇa ṛco yajūṃṣi naḥ |
bhadrā nakṣatrāṇi śivāni viśvā bhadrā āśā ahrutāḥ santu no hṛdi ||19||
May the Saman-s forever be auspicious to us. For us the Atharvaṇ spells, the ṛk-s and the yajuṣ-es. May the auspicious asterisms [be] all benign and [may the] directions, the coordinate lines be auspicious at their conjunction.

saṃvatsarā na ṛtavo mayobhuvo yo vā āyuvāḥ susarāṇy uta kṣapāḥ |
muhūrtāḥ kāṣṭāḥ pradiśo diśaś ca sadā bhadrā santu dvipade śaṃ catuṣpade ||20||
The years [of the 5 year cycle] and the seasons be gladdening to us, be they productive, easy-going or drought-ridden. May the muhūrta (=48 minutes)-s and kaṣṭa (=3.2 seconds)-s, the directions and the inter-directions be ever-auspicious and [may there be] welfare for the bipeds and quadrupeds.

bhadraṃ paśyema pracarema bhadraṃ bhadraṃ vadema śṛṇuyāma bhadram |
tanno mitro varuṇo mā mahantām aditiḥ sindhuḥ pṛthivī uta dyauḥ ||21||
May we see auspiciousness. May we perform auspiciousness. May we speak auspiciousness. May we hear auspiciousness. Mitra and Varuṇa, Aditi, the river, the Earth and also Heaven should grant this to us.


Footnotes
1. https://manasataramgini.wordpress.com/2010/06/25/the-mantra-samhita-of-the-devi-purana/
2. Note the presence of a comparable deity among the Greeks in the form of Phanes or Protogonos
3. Emended pāvamāna to pavamāna
4. Mātariśvan: literally growing within the mother: the mother implies the world-womb or the world-hemisphere
5. Later tradition clarifies him to be the charioteer of the Sun
6. The wind deity
7. Another form of Uṣṇih meter
8. The composer seems to indicate his love for the Jagati, the meter in which he has composed most of the sūkta
9. Emended virapsin to virapśin keeping with the form found in the RV
10. Emended sudamśas to sudamsas

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Liṅga-kāmādi-sūtrāṇi

Devanagari PDF version

.. Sūtrapāṭhaḥ ..

atha liṅga-kāmādi-sūtrāṇi vyākhyāsyāmaḥ . jīvasūtrāṇunām anukrameṣu parimeyā vikārā jīvā-paramparāyā+avaśyam . jantvoḥ saṃgrāmas tasya paramaṃ kāraṇam . tasmād ajāyata jīvasūtrāṇunāṃ vyūḍhīkaraṇam . RecA-nāma jīvakāryāṇu-kulaṃ jīvasūtrāṇunāṃ vyūḍhīkaraṇaṃ karoti . mukhyaśo ‘nagnijīvasūtrāṇunām . anagnijīvasūtrāṇu-mithunayor maithunāt . idam eva jantūnāṃ maithunasya rahasyam ..

anābhikānām pranābhikānāṃ ca+ anagnijīvasūtrāṇunām maithunaṃ naimittikā prakriyā . nābhikānām maithunaṃ nirūpitā prakriyā . vyūḍhīkaraṇa-dvirbhāvāt . tasmād ajāyanta liṅgāni . nābhikeṣu bahuśo liṅge dve . kecid bahuliṅgāni pradarśayanti . kyākūni yukta-kāmarūpiṇa romakoṣṭhakāś cety udāharaṇāni . eteṣu liṅga-koṣṭhānām parimāṇa-bhedo bahuśo nāsti . dviliṅgasthitau liṅga-koṣṭhyor parimāṇa-bhedaḥ sadaivodeti (sadaiva udeti) . mahattaro liṅga-koṣṭhaḥ strīti (strī+iti) . sā bahuśas tiṣṭhati . kanīyaḥ pumān iti . sa bahuśo gacchati . liṅgakoṣṭhānāṃ nirmāṇasya dattāṃśa-bhedāj jāyate liṅgayoḥ saṃgrāmaḥ . kiṃ tu paramparā-santatyai viparītayor liṅga-koṣṭhayor ākarṣaṇaḥ saṃgamanaṃ saṃyogaś ca+avaśyam . etaddhi mūla-kāraṇaṃ kāmasya . dattāṃśa-saṃgrāmo ‘karṣaṇaś ca dvayoḥ pratidhrājyoḥ sammelanād dvayor ekaḥ pūrṇaṃ vijayaṃ nāpanoti . tasmāj jāyate ‘nantā spardhā liṅgayoḥ ..

bahukoṣṭha-jantuṣu liṅgakoṣṭhā anyebhyaḥ koṣṭhebhyo bhinnāḥ saṃvṛtāḥ . tasmād viviktā upasthāḥ . udāharaṇāny oṣadhīṣu paśuṣu ca . teṣu viparītānāṃ liṅgakoṣṭhānāṃ sammelanāya vividhā upastha-lakṣaṇāny avartanta . śepo yoniḥ puṣpañ cety udāharaṇāni . paśuṣu trividhā maithuna-vyavasthā . dhṛṣṭa-vratam bahupatnī-vratam ekapatnī-vratam vā . bahupatnī-vratam dvividham . krameṇa bahupatnayaḥ sadyo bahupatnayo vā . pṛṣṭhadaṇḍa-paśuṣu prāyeṇa 25 jīvasūcanāḥ prabhavanty ekapatnī-vratam . Gibbons, orangutans, gorillas, chimpanzees and bonobos te sarve nṛbandhavaḥ . alpa-nṛbandhuṣv ekapatnīvratam bahuśaḥ prakṛtimat . anyeṣu nṛbandhuṣu bahupatnī-vrataṃ ca dhṛṣṭa-vrataṃ ca sāmānyam . mānaveṣu ca . prāyeṇeyaṃ nṛbandhunām mūla-sthitiḥ . strī-balātkāro rakta-nṛbandhunām eko maithunopāyaḥ . rakta-nṛbandhunāṃ vṛṣa-jātir dvividhā . nemivantaś ca+anemivantaḥ . nemivanta ugrā dhunimantaś ca parasparaṃ yudhyante ca . anemivantaḥ śāntāḥ pracchanam maithunaṃ kurvanti ca . bhūriretāḥ pumān nityam bahūn maithunāvakāśān mṛgayate . svāṇḍānāṃ niṣekāya strī su-jīvasūcanā-dhāriṇam puruṣam pratīcchaty anyān nirākaroti ca . tasmād bahuvidhāḥ spardhāś ca pradarśanāni ca ..

manuṣyāṇām prāyeṇa sahajā vyavasthā dhṛṣṭa-vrataṃ ca krameṇa bahupatnī-vrataṃ ca sadyo bahupatnī-vrataṃ ca . lubdhaka-vanagocarāvasthāyāṃ te sarve ‘vartanta . kiṃ tu manuṣyānām upavāsitāyāṃ sthiteḥ prabhūtyās te sarve vaighnakā abhavan . kāsmat? strībhyaḥ parasparam puruṣāṇāṃ naiṣṭhikāt saṃgrāmāt . idaṃ kāraṇaṃ ekapatnī-vratasya+āvaśyakatā dhruvāyopavāsita-jīvanāya . kiṃ tu mānavānāṃ netṛtvam pratibhā sāhasaṃ ca nṛdravyād udiyanti . ataḥ strīniyāna-bhedo nṛṇāṃ svābhāvikaḥ . tasya virodhāt upadravam pravartituṃ śaknoti . paraṃ tu śāntyai dhruvāyopavāsita-jīvanāya ca sarvebhyo puruṣebhyo nyūnātinyūnam eka-patnyā saha vivāham avaśyam . strīṣu vyābhicāriṇī-vratam bhrūṇahatyā +adhipuruṣānudhāvanaṃ cetyādi pravṛttīnāṃ codanāt puruṣa-viṣādo’pi vardhate . ādhunikatā nūtana-kṛtrima-mānava-samājasya nirmāṇaṃ vā tāni sarvāṇi codanti . ataḥ prāyeṇa + ādhunikatāḥ paura-saṃskṛteḥ pratiṣṭhām pratirundhate . yadīdam tatvaṃ tarhi manuṣyāṇām ādhunika-paura-saṃskṛtiś cirāyur nāsti ..

namaḥ somārudrābhyāṃ namaḥ prajāpataye ..

.. Nighaṇṭhuḥ ..
jīvasūtrāṇuḥ : nucleic acid molecule
anagnijīvasūtrāṇuḥ : DNA
jīvakāryāṇuḥ : protein
vyūḍhīkaraṇam : recombination
vyūḍhīkaraṇa-dvirbhāva : meiosis
anābhikaḥ : bacteria
pranābhikaḥ : archaea
nābhikaḥ : eukaryotes
kyāku : fungus
yukta-kāmarūpin : plasmodial slime mold
kāmarūpin : amoebozoan
romakoṣṭhakaḥ : ciliate
liṅgakoṣṭhaḥ : gamete
dattāmṣaḥ : investment
bahu-koṣṭha-jantuḥ : multicellular organism
jīvasūcanā : gene
nṛbandhuḥ : ape
alpa-nṛbandhuḥ : gibbon
rakta-nṛbandhuḥ : orangutan
nṛdravyam : testosterone

Posted in History, Politics, Scientific ramblings | Tagged , , , , ,

Creating patterns through matrix expansion

People who are seriously interested in emergent complexity and pattern formation might at some point discover matrix expansion for themselves. It is a version of string rewriting that allows one to create complex patterns. For me, the inspiration came from the way Hindu temples were constructed in their phase of maturity and this led me to walking on a path, which probably several others have done before me. While matrix substitution is perhaps common knowledge to students of pattern formation, I am still recording it here in part due to the great beauty of the patterns that can be generated by this very simple mechanism.

Let us consider the following substitution rules:

0 \to \begin{bmatrix}{} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \\ \end{bmatrix}; \; 1 \to \begin{bmatrix}{} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{bmatrix}

We initialize the process with M_0=[0]

What these substitution rules do is to replace every 0 and 1 respectively by the indicated matrices in each round. As a consequence we get an expanding matrix which starts with a single element, which then grows to a M_2: 3 \times 3 matrix and then to a M_3: 9 \times 9 matrix and so. Once we arrive at a matrix of a certain size we can render it as an image by assigning a color to each cell based on the number it contains. In this case we have two numbers 0,1; hence, our image would be two-colored. We could have done the same with a 2 \times 2 or a 4 \times 4 matrix as the substituent. However, these either have too little diversity or grow too fast to produce nice images on a typical laptop screen. Unless indicated otherwise all images in this note were produced by expanding the matrix to a size of M_5: 243 \times 243

Applying the above rules initialized as indicated above leads us to the famous box fractal (Figure 1).

fig1_carpetFigure 1

Applying the below rules yields us the famous Sierpinski box (Figure 2):

0 \to \begin{bmatrix}{} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix}; \; 1 \to \begin{bmatrix}{} 1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 1 \\ \end{bmatrix}

Initialized with M_0=[1]

fig2_carpet_sierFigure 2

Applying the below rules yields us a variant, the box-dot fractal (Figure 3):

0 \to \begin{bmatrix}{} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix}; \; 1 \to \begin{bmatrix}{} 1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 1 \\ \end{bmatrix}

Initialized with M_0=[0]

fig3_carpet_boxdot_0Figure 3

Applying the below rules yields us a more complex variant of the box fractal (Figure 4):

0 \to \begin{bmatrix}{} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \\ \end{bmatrix}; \; 1 \to \begin{bmatrix}{} 1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 1 \\ \end{bmatrix}

Initialized with M_0=[0]

fig4_carpet_sierbox_0Figure 4

Applying the below rules yields us an interesting carpet weaving (Figure 5). A search on the internet reveals that a certain Jeff Haferman was the first to announce this: in any case he arrived at it before my times.

0 \to \begin{bmatrix}{} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{bmatrix}; \; 1 \to \begin{bmatrix}{} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \\ \end{bmatrix}

Initialized with M_0=[1]

fig5_carpet_1Figure 5

We can get many other weavings; however, to realize the full potential of this mechanism we shall now go to higher number of colors. For 3 colors we need to formulate 3 rules and so on. We show below some carpets and fractals obtained with 3 and 4 colors.

The below rules yield a 3 color carpet (Figure 6):

0 \to \begin{bmatrix}{} 1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 1 \\ \end{bmatrix}; \; 1 \to \begin{bmatrix}{} 2 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 2 \\ \end{bmatrix}; \; 2 \to \begin{bmatrix}{} 2 & 0 & 2 \\ 0 & 2 & 0 \\ 2 & 0 & 2 \\ \end{bmatrix}

Initialized with M_0=[1]

fig6_carpet3_1Figure 6

The below rules yield a 3 color carpet (Figure 7):

0 \to \begin{bmatrix}{} 1 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 1 \\ \end{bmatrix}; \; 1 \to \begin{bmatrix}{} 2 & 1 & 2 \\ 1 & 2 & 1 \\ 2 & 1 & 2 \\ \end{bmatrix}; \; 2 \to \begin{bmatrix}{} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix}

Initialized with M_0=[1]

fig7_carpet3_1Figure 7

The below rules yield a 3 color carpet (Figure 8):

0 \to \begin{bmatrix}{} 1 & 0 & 1 \\ 0 & 2 & 0 \\ 1 & 0 & 1 \\ \end{bmatrix}; \; 1 \to \begin{bmatrix}{} 2 & 1 & 2 \\ 1 & 2 & 1 \\ 2 & 1 & 2 \\ \end{bmatrix}; \; 2 \to \begin{bmatrix}{} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix}

Initialized with M_0=[0]

fig8_carpet3_0Figure 8

The below rules yield the liṅga-ṣaṭkoṇa fractal (Figure 9). The inspiration for this comes from the matrix figure used in the śaiva tradition known as the liṅga-yantra. One can see the fractal liṅga-s combined with a hexagonal gasket fractal with an inner skewed Koch’s curve:

0 \to \begin{bmatrix}{} 1 & 0 & 1 \\ 1 & 1 & 1 \\ 1 & 0 & 1 \\ \end{bmatrix}; \; 1 \to \begin{bmatrix}{} 0 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix}; \; 2 \to \begin{bmatrix}{} 0 & 2 & 2 \\ 2 & 0 & 2 \\ 2 & 2 & 0 \\ \end{bmatrix}

Initialized with M_0=[2]

fig9_linsh_2Figure 9

The below rules yield a 4 color carpet (Figure 10):

0 \to \begin{bmatrix}{} 2 & 2 & 2 \\ 2 & 0 & 2 \\ 2 & 2 & 2 \\ \end{bmatrix}; \; 1 \to \begin{bmatrix}{} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix}; \; 2 \to \begin{bmatrix}{} 1 & 3 & 1 \\ 3 & 2 & 3 \\ 1 & 3 & 1 \\ \end{bmatrix}; \; 3 \to \begin{bmatrix}{} 3 & 0 & 3 \\ 0 & 3 & 0 \\ 3 & 0 & 3 \\ \end{bmatrix}

Initialized with M_0=[2]

fig10_carpet4_2Figure 10

The below rules yield a 4 color H-fractal with repeating copies of the letter “H” (Figure 11):

0 \to \begin{bmatrix}{} 2 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 2 \\ \end{bmatrix}; \; 1 \to \begin{bmatrix}{} 0 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 0 & 0 \\ \end{bmatrix}; \; 2 \to \begin{bmatrix}{} 1 & 3 & 1 \\ 3 & 2 & 3 \\ 1 & 3 & 1 \\ \end{bmatrix}; \; 3 \to \begin{bmatrix}{} 3 & 2 & 3 \\ 2 & 0 & 2 \\ 3 & 2 & 3 \\ \end{bmatrix}

Initialized with M_0=[3]

fig11_carpet4_h_3Figure 11

The below rules yield a 4 color hexagonal gasket fractal with a Koch curve boundary and some further complex structure. This image was created after 6 rounds of matrix expansion i.e. a M_6: 729 \times 729 matrix (Figure 12):

0 \to \begin{bmatrix}{} 1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}; \; 1 \to \begin{bmatrix}{} 1 & 1 & 0 \\ 1 & 3 & 1 \\ 0 & 1 & 1 \\ \end{bmatrix}; \; 2 \to \begin{bmatrix}{} 1 & 2 & 2 \\ 2 & 2 & 2 \\ 2 & 2 & 1 \\ \end{bmatrix}; \; 3 \to \begin{bmatrix}{} 0 & 3 & 3 \\ 2 & 1 & 2 \\ 3 & 3 & 0 \\ \end{bmatrix}

Initialized with M_0=[2]

fig12_carpet4_hexadFigure 12

The below rules yield a 4 color carpet. This image was created after 6 rounds of matrix expansion i.e. a 729 \times 729 matrix (Figure 13):

0 \to \begin{bmatrix}{} 2 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 2 \\ \end{bmatrix}; \; 1 \to \begin{bmatrix}{} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ \end{bmatrix}; \; 2 \to \begin{bmatrix}{} 1 & 3 & 1 \\ 3 & 2 & 3 \\ 1 & 3 & 1 \\ \end{bmatrix}; \; 3 \to \begin{bmatrix}{} 3 & 0 & 3 \\ 0 & 3 & 0 \\ 3 & 0 & 3 \\ \end{bmatrix}

Initialized with M_0=[2]

fig13_carpet4_2Figure 13

Not all matrices yield aesthetically worthwhile images. A reader might experiment with this method to obtain further images.

Posted in art, Scientific ramblings | Tagged , , , , , ,

An apparition of Mordell

Consider the equation:

y^2=x^3+k

where k is a positive integer 1, 2, 3… For a given k, will the above equation have integer solutions and, if yes, what are they and how many?

We have heard of accounts of people receiving solutions to scientific or mathematical problems in their dreams. We have never had any such dream; in fact we get most of our scientific or mathematical insights when we are either in a semi-awake but conscious reverie or at the peak of our alertness. However, we on rare occasions we have had dreams which present mathematical matters. On the night between the 18th and 19th of Jan 2019 we had such dream. It was a long dream which featured human faces we do not recall seeing in real life: we forgot their role in the dream on waking. However, what we remembered of the dream was the striking and repeated appearance of the above equation along with its solutions for several k. On waking, we distinctly recall seeing the cases of k=1, 8, 9 though there were many more in the dream. There was a degree of discomfort from the dream for in the groggy state of waking from it we knew that some of these solutions had slipped away. So, at the first chance we got, we played a bit with this equation this on our laptop. We should mention that we have not previously played with this equation and have given it little if any thought before: we glanced at it when we had previously written about the cakravala but really did not give it any further consideration then or thereafter. Hence, we were charmed by its unexpected and strong appearance in our dream.

This equation is known as Mordell’s equation after the mathematician who started studying it intensely about a century ago. The equation itself was know before him to a French mathematician Bachet and is sometimes given his name. It has apparently been widely studied by modern mathematicians and they know a lot about it. As a mathematical layman we are not presenting any of that discussion here but simply record below our elementary exploration of it.

mordell_fig1

Figure 1.

These equations define a class of elliptic curves and by definition given the square term they are symmetric about the x-axis (Figure 1). Since, we are here only looking at the real plane, the curve is only defined starting from y=0, x=-\sqrt[3]{k} where it cuts the x-axis. From there on it opens symmetrically towards \infty in the direction of the positive x. So, essentially we are looking for the lattice points which lie on this curve. Given the square term, we will get symmetric pairs of solutions about the x-axis (Figure 1). In geometric terms, it might be viewed as the problem of which squares and cubes with sides of integer units are inter-convertible by the addition of k units.

Mordell had pointed out about a century ago that these equations might have either no integer solutions or only a finite number of them. Let is consider a concrete example with k=1: y^2=x^3+1. One can get three trivial solutions right away: by setting x=-1 we get y=0. Similarly, setting x=0, we get y=\pm 1. Further, we see that if x=2 we get y=\pm 3. Thus, we can write the solutions as (x,y) pairs: (-1,0); (0,1); (0, -1); (2,3); (2,-3): a total of 5 unique integer solutions. Can there be any more solutions than these? To get an intuitive geometric feel for this we first observe how these solutions sit on the curve (Figure 1). We notice that the 3 distinct ones are on the same straight line (also applies to their mirror images via a mirrored line). This is a important property of elliptic curves (being cubic curves) that allows us to understand the situation better.

mordell_fig2Figure 2.

In Figure 2, we consider the curve y^2=x^3+1 in greater detail. We observe that if we have two integer solutions points P and Q and we connect them we get the third one R_1. Thus, if we have two solutions, in this case the trivial ones, we can easily find the third one by drawing a line through them and seeing where it cuts the elliptic curve. The point where it cuts it gives a further solution. We also see from the figure that joining symmetric solutions like points Q, Q^\prime will not yield any further solutions because the resultant line will be parallel to the y-axis. Thus, we might get an additional solution only if the slope of the line joining the 2 prior solutions is neither 0 (coincident with the x-axis) nor \infty (parallel to y-axis). Hence, we may ask: now that we have R_1 can we get a further solution? We can see that joining Q^\prime to R_1 yields a line that will never again cut the curve y^2=x^3+1. Thus, we can geometrically see that there can be no more than the 5 above solutions we obtained.

In algebraic language the slope of the line joining the first two solutions can be written as:

m=\dfrac{y_P-y_Q}{x_P-x_Q}

From this one can calculate the coordinates of the third point R as:

x_R=m^2-x_P-x_Q, \; y_R=y_P+m(x_R-x_P)

Thus, in Figure 2, if we were to join Q^\prime to R_1 we have:
m=2; thus for the “new point” we get: x=4-0-2=2, \; y=-1+2(2-0)=3.
We simply get back R_1 indicating that there are no further integer solutions than the 5 we have.

One may also see a parallel between this procedure of obtaining a third integer solution by joining two points and the process of obtaining a composite number by multiplying two prime numbers. If we know the two starting points it is easy to get the third but if we were to only know the third point for a large number getting its precursors would be a difficult task. This relates to the use of elliptic curves as an alternative for primes in cryptography (see our earlier note on the use of prime numbers in the same).

Figure 3.

In the case of y^2=x^3+1 the joining procedure terminated after the first new point we got but can there be cases where this yields more points? To see an example of this let us consider y^2=x^3+9. One can get a trivial solution by simply placing x=0 to get y=\pm 3. Further, it is also easy to see that by taking x=-2 we get y=\pm 1. Thus, we get four points that we may call P, P^\prime, Q, Q^\prime(Figure 3 panel 1). By joining P to Q we get a further point R_1. Similarly, joining P^\prime to Q or Q^\prime to R_1 we get yet another point R_2. We can likewise obtain their mirror images R^\prime_1 and R^\prime_2 (Figure 3). Finally, by joining R^\prime_1 to R_2 we get yet another point R_3 and likewise we can get its mirror image point R^\prime_3. Beyond this the joining procedure yields no further points. Thus, we are left with a total of 10 integer solutions for y^2=x^3+9: (-2,1); (0,3); (3,6); (6,15); (40,253); (-2,-1); (0,-3); (3,-6); (6,-15); (40,-253)

mordell_fig4Figure 4.

We can then systematically explore the solutions for all k=1..1000. If we plot all solutions and zoom in close to origin we find a dense clustering of the solutions forming a swallow-tail like structure whose outline is an integer approximation of an elliptic curve (Figure 4).

mordell_fig5Figure 5.

We further find that some k have no integer solutions at all. There is a formal way to use modulos and factorization to prove this for particular k. The sequence of k for which no integer solutions exist can be computationally obtained and goes as: 6, 7, 11, 13, 14, 20, 21, 23, 29, 32… Figure 5 shows how the n^{th} term of this sequence grows. We find empirically that it appears to be bounded by or at least approximated by the shape of a scaled form of the logarithmic integral: y=\pi^2 \textrm{Li}(x). Whether this is true or what the significance of it may be remains unknown to us.

mordell_fig6Figure 6.

We can also look at the sequence defined by the number of integer solutions by k. This is plotted in Figure 6 and remarkably shows a structure with no obvious regularity. A closer look allows us to discern the following patterns:
1) The most common number of solutions is 0. For k=1..1000 this happens with a probability of 0.549, i.e. more than half the times there are no integer solutions for Mordell’s equation. The next most frequent number of solutions is 2. This happens with a probability of 0.306 in this range. These are the cases when you just have two symmetric solutions differing in the sign of their y value.

2) An odd number of solutions is obtained only when k is a perfect cube. This is because only in this case we get the unpaired solution of the form (-\sqrt[3]{k},0). The cubic powers of 2 are particular rich in solutions. E.g. k=2^9=512 yields 9 solutions: (-8,0); (-7,13); (4,24); (8,32); (184,2496); (-7,-13); (4,-24); (8,-32); (184,-2496).

3) If k is a perfect square then for k, k+1 and k-1 we will have at least 2 solutions: for k we have (0,\pm\sqrt{k}); for k-1 we have (1,\pm\sqrt{k}); for k+1 we have (-1,\pm\sqrt{k}). This would predict that, taken together, perfect square k and their two immediate neighbors on either side would have a higher average number of solutions than an equivalent number of k drawn at random in the same range. This is found to be the case empirically (Figure 7).

mordell_fig7Figure 7.

The mean number of solutions of the square k and their immediate neighbors is 4.108696 (red line in Figure 7) as opposed to the mean number of solutions of 1.522 (black line) for all k=1..1000. The former is 10.42 standard deviations away from the mean for equivalently sized samples drawn randomly from k=1..1000.

4) Further some squares and square-neighbors show what we call a lucky cubic conjunction (LCC), i.e. they generate a significantly larger number of perfect squares when summed with cubes than other numbers. One such square showing a LCC is 15^2=225. It shows the record number of solutions (26) for k=1..1000: (-6,3); (-5,10); (0,15); (4,17); (6,21); (10,35); (15,60); (30,165); (60,465); (180,2415); (336,6159); (351,6576); (720114, 611085363); (-6,-3); (-5,-10); (0,-15); (4,-17); (6,-21); (10,-35); (15,-60); (30,-165); (60,-465); (180,-2415); (336,-6159); (351,-6576); (720114, -611085363). One can right away see that:
-6^3+225=3^2\\ -5^3+225=10^2\\ 4^3+225=17^2\\ 6^3+225=21^2\\ 10^3+225=35^2 and so on.

A square neighbor with a LCC is 17=16+1 which has 8 cubic conjunctions leading to its 16 solutions: (-2,3); (-1,4); (2,5); (4,9); (8,23); (43,282); (52,375); (5234,378661); (-2,-3); (-1,-4); (2,-5); (4,-9); (8,-23); (43,-282); (52,-375); (5234,-378661).

1025=32^2+1 is an even more monstrous square neighbor with a LCC outside the range that we systematically explored. This number has a whopping 16 cubic conjunctions giving rise to 32 solutions: (-10,5); (-5,30); (-4,31); (-1,32); (4,33); (10,45); (20,95); (40,255); (50,355); (64,513); (155,1930); (166,2139); (446,9419); (920,27905); (3631,218796); (3730,227805); (-10,-5); (-5,-30); (-4,-31); (-1,-32); (4,-33); (10,-45); (20,-95); (40,-255); (50,-355); (64,-513); (155,-1930); (166,-2139); (446,-9419); (920,-27905); (3631,-218796); (3730,-227805). In my computational exploration of these elliptic curves I am yet to find any other that out does 1025.

5) While k which are squares and square neighbors have at least 2 solutions guaranteed, in principle a non-square or non-square neighbor number can show a LCC and give rise to a large number of solutions. One such as k=297 which shows 9 cubic conjunctions to give 18 solutions: (-6,9); (-2,17); (3,18); (4,19); (12,45); (34,199); (48,333); (1362,50265); (93844,28748141); (-6,-9); (-2,-17); (3,-18); (4,-19); (12,-45); (34,-199); (48,-333); (1362,-50265); (93844,-28748141). k=873 also shows a similar LCC, again with 9 cubic conjunctions. Outside the range we systematically explored, we found k=2089 to show a remarkable LCC with 14 conjunctions yielding 28 solutions: (-12,19); (-10,33); (-4,45); (3,46); (8,51); (18,89); (60,467); (71,600); (80,717); (170,2217); (183,2476); (698,18441); (9278,893679); (129968,46854861); (-12,-19); (-10,-33); (-4,-45); (3,-46); (8,-51); (18,-89); (60,-467); (71,-600); (80,-717); (170,-2217); (183,-2476); (698,-18441); (9278,-893679); (129968,-46854861). This is the second highest number of solutions we have seen for the Mordell’s equations we have studied.

A reader might explore these and see if he can find a k with bigger number of solution

Posted in Scientific ramblings | Tagged , , , , , , , , , , ,

A dinnertime conversation

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Things had cooled off a bit on the political front after the assassination of the Prime Minister Pratap Simha by a massed drone attack. On the personal front Vidrum had sort of come to terms with the mysterious disappearance of his lover Meghana. He was now ever more convinced that she had been killed by a marūnmatta. He had the vague sense of possible trouble in the future but for now his duties as a young physician kept him busy. He had set aside the anger and indignation that resulted from the disappearance of Meghana and was following the advise of his precocious junior and professor at the medical school, Vrishchika, to chart out an appropriate niche for himself. Thus, he was watching less cricket and spending his free time looking more closely at the relevant clinical data along the lines Vrishchika had suggested to him. While he initially found it boring, he soon felt he was seeing patterns he had never before seen in his life. This perked him up and he started putting in effort to try to understand them. In the process he was for the first time trying to read a wider array of literature with a more critical mind. Suddenly, it was dawning on him that several of the practices and larger framework of the modern medical practice itself was mostly not grounded in alleviating the problems of the patients.

That afternoon Vidrum felt his mind crowded with less-comforting issues. Perhaps, deep within he had some foreboding that there was only a temporary lull in the troubles and that a great clash of men was waiting to happen in the future. Due to the company of his old friends Vidrum was moving away from the alluring but fallacious ideas of his other friends like Samikaran, Gardabh and Mahish. In fact it seemed as though they could no longer be counted as his friends, especially if the terminal life and death struggle between the dharma and rākṣasavāda was to play out as Lootika had witnessed in her dream. To add to this general background of thoughts, the experience with a patient he had to handle the previous day was playing on his mind for some reason – he was hardened to such sentiments but for some reason this case kept returning to his mind. It specifically raised a question which had always bothered him in the background – the thought of whether he had really been a success in life. He had faithfully followed what his parents had charted for him. After all the trials and tribulations he had been successful at that but it was not entirely clear to him if had truly met ‘success’. His parents were extremely proud of where he had reached in life but he was not entirely sure if that meant anything at all. On the other side his mind wandered with some happy expectation to the impending appointment he had early that evening. He had been invited to Indrasena and Vrishchika’s house for hanging out for the evening and dinner. Indrasena’s younger brother Pinakasena and his wife Shallaki were also on vacation with them for a few days. But then his mind wandered back to the issue of achievement when he thought of the people whom he was slated to visit that evening. Nevertheless, knowing that time was short he quickly made some lemonade and ṣaṭṭaka -s that he wished to take to his hosts to accompany the dinner they were spreading.

On arriving at Indrasena and Vrishchika’s house he was introduced to Pinakasena and Shallaki. He had met Pinakasena once before when he and his brother had come from their city, the sprawling Kshayadrajanagara, to stay with his classmate Somakhya for a fortnight. They refreshed their acquaintance. Vidrum: “I recall the the great game of cricket we had at Somakhya’s house.” IS: “Ah you have a good memory. It seems to have faded away from my ken but perhaps one us might have have a record of it in our scrapbooks.” Vrishchika: “Indrasena, it has to be your scrapbook for certainly me and my sisters did not join you all to play cricket of all things. I guess the only thing Indrasena remembers from that trip was meeting me, the walk we all took to that gigantic palm tree – a rare one for our city, which even towered over the specimens we might encounter in Visphotaka or Samudragandha, and that memorable dinner we had at the Hotel Yadava.” Vidrum chuckled: “I can imagine.” Pinakasena: “Ah, I do recall the good cricket we played. Indra batted badly and bowled badly throughout. Vrishchika, while you were not physically there, I think you can take the blame for that. I think bro was only thinking of you even while playing.”

Vrishchika: “You guys are always blaming me even when I was not there!”
Vidrum: “But, Vrishchika, you were always a big trouble-maker, right?”
Vrishchika: “I thought that title squarely belonged to my elder sister: after all on the last day of school our principal said that I was such a well-behaved girl, unlike Lootika who was always into mischief. OK Vidrum, why don’t you now inaugurate the goṣtḥī with some of the nimbūka-rasa that you have so kindly brought us. Lootika and Varoli always considered me and Jhilleeka as inferior cooks. Even if that were true, you would not have to suffer that, for much of the bhakṣa for today has been handled by honorary sister Shallaki, whom even Somakhya’s mother has certified as good. So let me get you some of her modaka-s to get going.” Vidrum: “It is very brahminical indeed: bhojya and bhakṣya with no trace of onions or garlic. I get that feel of purity.”

Vidrum: “So Shallaki do you also originally hail from Kshayadrajanagara?” Shallaki: “No, I come from Shodasharajya (Ṣoḍaśarājya).” Vidrum: “Hmm, an out of the way, quite place. I am sure the air was cleaner than our mahānagara and the life much quieter. How did you find the education there?” Shallaki: “You are right with regard to the air and the charming life of a small town but regarding education there was nothing much to speak of. I must say that situation was quite nice too for in the village of the blind the one eyed woman is queen.” Vidrum: “What do you mean?” Shallaki: “In our town there was not much competition as you had in your city, both in school and in college. So by just putting in a little effort I could stand out in academics and boost my ego. That might have done me some bad but thankfully I got see reality upon running into the caturbhaginī.”

Vidrum: “Interesting that you say so. I have been thinking a lot on related issues. Imagine being in school with likes of Vrishchika and all her sisters! It certainly was a stiffer situation for us, especially given all the pressure my parents put on me to catch up with the rest. I understand from Indrasena that Kshayadrajanagara was only a little less stressful than our city.”

Pinakasena: “Indrasena and I made the following estimates using data from Vrishchika and Shallaki. The average IQ of your college class, Vidrum, was approximately 120-123; in Kshayadrajanagara in mine and Indra’s class it was 116-119. In Shallaki’s college in Shodasharajya it was just 103. But remember in the general populace across all our regions the average is at best around 90. Now you can also look at it this way. In a mahānagara like yours there were roughly around 250000 students who would have written the university entrance exam with you. If we take all the medical colleges in the district there are approximately 200 government rate seats available to the open category. Now not all students with that high IQ threshold needed to make it to the medical school will want to study medicine. About the same number will want to go to engineering school. About ¼ that number will go to law school. Another ¼ will want to become accountants and go on the track peculiar to them. It will only be a rare minority like Indra or me, say one each year, who would drop out of those professional courses to take a pure science degree. That will give us approximately 500 students and a fraction of 0.002 who pass the IQ threshold to potentially enter medical school. Now with that mean of 90 for our general populace and a typical standard deviation of about 15 we can calculate that threshold to be an IQ of 133.17. Now, if we take your class of about 120 students with a mean IQ of 121.5 and a similar s.d then you have about 26 students in your class alone who could go for those seats. Whereas in my class there would have been roughly 18 such students and in Shallaki’s only 3. So indeed it must have been quieter for her than for you.”

Vidrum: “Yes. I would believe your estimates are anecdotally in the right range given that there were little over 10 people in my class with IQs in the 140 and above range. In that range the difference really tells with respect to even the mean for our class. It can be demoralizing to those lower down the ladder as the case of a patient I had to deal with yesterday illustrates.”
Vrishchika: “Talking of patients, I was itching to ask about patient #29?”
Vidrum: “Yes, I was thinking of that very case. First, I must thank you for your spot on and immediate diagnosis: as I had expected it was something up your alley.”

Vrishchika: “So you confirmed mycetism ?”
Vidrum: “Yes indeed: it was an attempt of suicide.”
IS: “Mycetism from a suicide attempt?”
Vidrum: “The young patient was brought it with an emergency situation. He presented profuse perspiration, miosis in an eye, accompanied by ptosis in the other, convulsions and difficulty breathing among other things. After we struggled a bit since the patient was not orally communicative, I went over to check with your wife, as such are the things she might see through instantly. She was quick to take a shot that it was likely muscarinic intoxication. I acted on it and found that he had attempted suicide by shroom and stabilized him with atropine.”
Vrishchika: “Tragic, but that’s an unexpected twist. Why would someone attempt suicide via something so inefficient and painful as a muscarinic mushroom? He could have tried something else that might have brought his end more efficiently. Remember the guy in our school who committed suicide with Mercuric Chloride?”
Vidrum: “It is indeed an odd choice of an agent. But regarding the pain, remember that when people are in such a state they do not care much about the pain: leave alone that HgCl_2 case, we have had people make an attempt at suicide by drinking phenol, right?”
Indrasena: “Remember, access to agents is not always easy. HgCl_2 is not exactly something easily available unless he has access to the chemistry lab in school or college. Moreover, Vrishchika, you might be underestimating the potential of muscarinic intoxication. During my peregrinations in the ghats with a naturalist, I came across a large bolete mushroom, likely of the Rubinoboletus group. My naturalist companion informed me that it was particularly toxic and used by the hill pastoralist-subsistence farmers to impose a death-punishment on life-stock thieves whom they caught. They were said to prepare a mushroom-bun using this fungus and administer it to the thieves who would then die in their custody within 24-36 hours. I collected a sample and it is in my storage. However, after realizing its active principle is mostly just muscarine, I did not investigate it further.”

Vidrum: “That’s most interesting. This young patient’s surname suggests that he might have his origins in the shepherd community. I wonder if after all he was doing what he did with some prior knowledge but simply did not dose correctly. That’s why we were able to salvage him with atropine and adrenaline.”
Vrishchika: “Vidrum, here you have a notable case at hand, which you should carefully work up into a manuscript and publish it – it will make an interesting case report. Hope you have all the material well-documented.”
Vidrum: “Yes I do. I hope we can get Indra’s sample to do some tests for muscarine.”

Shallaki: “Very outre indeed but perhaps it is my low IQ that I am unable to see what was it in this case that made Vidrum think of ones place on the IQ ladder.”
Vidrum: “This is the time of the results for the university entrance exams. The patient very badly wanted to make it to medical school but he came up against mathematics and pulled the plug in it. The failure led him to attempt self-destruction. It reminded me of my own situation years ago. My parents had, from an early age, nailed it into me that the way forward in life was to have the initials MD engraved next to my name. They placed tremendous pressure on me to reach that goal. It was tough struggle at the university entrance exam: just as Pinakin laid out I was competing with the IQ heavyweights in our class for that small slice of seats. My parents naively thought that just putting in hours of work would take me there. Looking at your clans-folk, Somakhya and Lootika, or the nasty Hemaling, I got my first doubts about whether this was true. I asked Somakhya one day about this and he said: ‘don’t slack on your studies but it will mostly boil down to what your parents gave you at conception. Knowing you well, I think you will survive but then the gods or luck can always have other plans.’ Thankfully, that turned out to be true and the gods favored me, But then time and again I was reminded that there is something more than just the work you put in or the education your receive. In fact it was this which lead to me breaking up with Manjukeshi and falling out with Samikaran.”

Vrishchika: “That’s where realism might help. It might be a good idea not to force people who do not have it in them to become doctor but do something else which could still make their lives cheerful. An animal can lead a happy life without having fathomed the depths of calculus or the arcana of enzymology or peering down infected orifices: it is good to know that early rather than commit yourself to misery with no pay off.”
Vidrum: “Vrishchika, I have always felt you speak from a lucky vantage point, as you were undoubtedly the most intelligent student, male or female, in your class all the way from school to the end of your education. How else could we explain that just in your first year you aced several of the exams of the last year? Should you not be having some consideration for the aspirations of the less-fortunate rather than seeking to write them off at birth?”
Shallaki: “I think we all are speaking from a lucky vantage point. Otherwise we would not all be here together chatting in this manner. I think neither you nor anyone else in this room can really experience the world in first person as it is for someone at the mean IQ of a 90 or for that matter even someone at 100, at or below which 3/4th of our people are situated. Hence, I would not at all say that Vrishchika has no consideration for the less-endowed in this dimension due her advantage. She is only advocating a realistic approach where we do not whip a horse without the wherewithal in hope that it might win a difficult race. That said, I do acknowledge that slotting someone as a low intellectual achiever, perhaps at birth itself, might discourage them from achieving even what their potential might hold for them. This is a difficult dilemma when you want implement something humane for society at large.”

PS: “In fact the likelihood of a more humane approach stems from first recognizing the ground situation. We need not follow fashions of the Abrahamistic Occident to shape our thoughts, which is one of a fear of reality and multiplicity – be it in religion or in society at large. In our tradition we have always had a more realistic picture of human nature and the existence of substructure within it. Hence, we need to work from what is actually there rather than what we think should be there. In fact, rather than write off people, such a realization might help provide the right kind of assistance and prior information for them, rather than make them play to lose.”

IS: “Then we were in school, the wife and I were part of a study to use a gigantic compendium of single nucleotide polymorphisms to study the correlations of each with a bunch of quantitative traits like muscle mass, tendency for psychotic conditions, brain size and general intelligence. By adding these correlations we were able to develop polygenic scores that were rather predictive of these quantitative traits. In fact, due to the enormity of the data sets that were accrued we were able to do better than any prior attempt.Vrishchika is now extending the results to include observations such as the orientation of the dendritic arborizations in smart people with big brains with respect to the duller folks with smaller brains. The regular arrangement of the dendritic arborizations is a reasonable predictor of intelligence. As you know, to be a commissioned officer in the army requires you to pass a pretty rigorous test that probes general intelligence. However, the non-commissioned servicemen are not selected via such stringent procedure. But the army as part of its modernization plan was very interested in probing this aspect for it has serious bearing on performance and training. In fact general intelligence is a good indicator of whether a javān will be trainable for something important in modern warfare than just being a grunt. Thus, the army supported this study and has made good use of our polygenic scores as a predictive tool. Of course they do not substitute it for an intelligence test; however, it informs them of the kind training they should invest in for a javān and the specialization they should deploy them in. The predictions of psychoses also informs them regarding the types of posting and training that might be appropriate for an individual. They way it works is they all have equal opportunity but if they fail the given opportunity then this information guides the future steps that are taken.”

Vidrum: “That is interesting. I can see something like this being of use in the army. Due to your wife, I am also beginning to utilize in civilian medicine for a variety of conditions and it does help me sharpen the advise I can give my patients. But can we really do any good using the brain size and polygenic scores for general intelligence among civilians?”
Vrishchika: “The army project is the one which actually convinced me of the utility of the polygenic scores for intelligence in civilian medicine. I actually advised the army in this matter and saw it to have utility in routine military medicine. I am sure you would agree Vidrum that the medical counsel for a patient is not a simple matter. Many actions a patient needs to follow in order to successfully take modern medication is not exactly simple for someone closer or below the mean in the IQ distribution. For instance, take management of diabetes or something even simpler like taking the medications appropriately as instructed. In such a situation the polygenic score gives us a non-invasive prediction of whether the patient’s intelligence would be sufficient to handle the task of auto-administration of the prescribed medication. If we get an indication that it might not then we might recommend some extra training, take some extra-steps and provide some additional aids to ensure that they correctly auto-administer.”

Vidrum: “Alright that sounds like something interesting and certainly I need to take a closer look at it. But I am still worried about the misuse of tests of general intelligence or still worse a polygenic prediction, especially in education. What about the psychological trauma derived from knowing that you have bad polygenic score? We could have more cases like our patient #29 then.”
Shallaki: “While I completely see your point of the potential educational misuse, as Indrasena pointed out regarding the army, I think there is also potential for the converse. One thing that has emerged from intelligence studies is that a person with a certain low IQ score can be trained by repeated specific instruction by example to do certain tasks as well as a higher IQ person. However, such people are less-likely to benefit much from abstract higher general education unlike a higher IQ person. Hence, I would say that we should start by giving an equal opportunity and encouragement for all. However, if a person is showing a tendency of failure we can make an assessment of the prospects and the potential training course for the individual based on the polygenic score. Learning Newton’s laws of motion or principles of linguistic evolution are not for everyone. If we have a good evidence that their intelligence is not up to it then it is actually a good thing not to make them waste their time with such things but train them more practically for matters likely to give them a respectable life.”

PS: “This brings to mind a conversation we had with Indra’s classmate and Somakhya’s cousin Saumanasa who used to think everyone has the same intelligence, sort of like your evil Samikaran. The results of Indra and Vrishchika really shook her up. She was wanting to go the route of ‘the universal right to income for these people until robotics and artificial intelligence would aid them catch up with the rest.’ We were trying to tell her that a typical man values a dignified life coming from being employed more than an income without work and those who do go that route are likely to end up with serious issues like substance abuse and violent crime.”
Vidrum: “It is interesting she puts a positive spin on robotics. The away automation and robotics are entering our lives I think life is getting harder for these people with lower IQ. I sort of started sympathizing with a famous advaitācārya, who is sort of a Luddite, who felt that we should turn our backs to mechanization and automation.”
PS: “I think the jury is still out on that one. I see that smarter ones projecting even greater power with the artificial computer minds. Yet I admit that those with lower intelligence are to an extent doing certain otherwise difficult tasks by themselves nowadays because of the computer’s mind aiding them with information in a more ready to use form. For example, in our parents days people had to look up a table for the very normal distribution we have been talking about but we can simply do it at the tap of a finger due to the extended mind in the form of our computers. Looking up some of those tables was a more complex task making such information less accessible to many.”

Vidrum: “Well, Samikaran is still laboring under the belief that all are equal in the upper story and is probably still serious about his ‘Trash the Brahmin’ project since he believes that all inequality has arisen in our people dues ‘brahmins’ denying them education. But what about engineering the genetics to change the potential? Did Saumanasa not suggest that one?”
Indrasena: “I am sure Saumanasa would love to see samatvam too. But she is a human geneticist herself and actually knows a thing or two of the matter. The main issue is that we are dealing with a quantitative trait with a polygenic foundation. So it is not something for easy intervention. There are some loci that we ourselves uncovered from a study of a big-brained family of V1s, which by themselves can give a good effect but even those are tricky. In fact the Qin Shi Huang’s merry successors acted almost immediately after we published our analysis of PAX6 and delta-catenin to engineer a Crispr Ding and a Crispr Dong based on those variants. They were big-headed no doubt but apparently the intended super-Hans are not doing too well according to the latest intelligence we have gotten. There is something about the genetic background, which we figured out only much later, that they missed. All this, I guess, is keeping Saumanasa from venturing in the direction of a genetic intervention and preferring an artificially engineered one. ”

Vidrum: “If this were the scenario on the ground I wonder what this might mean for the differences that have been claimed for the IQ of nations in their mean IQ. Evidently the data is already there to explore its causes and consequences, right?”

Posted in art, Life | Tagged , , , ,

Newton’s cows

Cultures with an Indo-European background have had a long history of symbiosis with the bovine animal since they started herding on the steppes in the Black Sea-Caspian region. Indeed, the very emergence of the modern steppes of Eurasia is likely a result of human-animal action to foster a certain pattern and type of grass growth. Hence, not surprisingly, cows frequently appear even in their mathematical literature, like the Greek problem of cows attributed to Archimedes or Nārāyaṇa’s cow population problem. Even after the destruction of Indo-European tradition by Abrahamism in groups like the English such problems persisted and once such example is Newton’s problem from his Arithmetica universalis. It goes thus:
a_1 cows graze b_1 fields bare in c_1 days,
a_2 cows graze b_2 fields bare in c_2 days,
a_3 cows graze b_3 fields bare in c_3 days,
What relationship exists between the 9 quantities from a_1, a_2, a_3...c_3?

To solve this we must make some assumptions that Newton indicates in his work: 1) On an average the cows and fields are equivalent. That would mean that we can take each cow to eat the same amount w daily and each field to have the same type and amount of grass x. Grass, ungulates and fungi are in a complex relationship. Grass “hire” fungal symbionts to produce toxins like the ergot alkaloids to deter ungulates. There is some empirical evidence that grazing by ungulates triggers grass growth. So we get the assumption: 2) The grass is not at standstill while being grazed daily but is growing back at a daily rate of y.

From the above, for the first set of cows and fields, we have b_1x as the total amount of grass at the start of the grazing. The amount of grass growing on the field in day 1 would be b_1\times 1 \times y. The total amount of grass eaten by the a_1 cows at the end of the day would be a_1\times 1 \times w. Thus, we can calculate the amount of grass at the end of day 1 as: b_1x +1b_1y - 1a_1w
At the end of day 2 we get: b_1x+2b_1y-2a_1w and so on.
Hence, when the fields are grazed bare in c_1 days we get: b_1x+c_1b_1y-c_1a_1w=0. By writing z=-w we get the equation: b_1x+b_1c_1y+a_1c_1z=0. Similarly, for the other two sets of cows and fields we get the equations: b_2x+b_2c_2y+a_2c_2z=0 and b_3x+b_3c_3y+c_3a_3z=0. We thus have as set of 3 simultaneous equations in 3 variables:

b_1x+b_1c_1y+a_1c_1z=0
b_2x+b_2c_2y+a_2c_2z=0
b_3x+b_3c_3y+c_3a_3z=0

We can hence eliminate x, y, z using the determinant of the system \det A= 0

\det A = \begin{vmatrix} b_1 & b_1c_1 & a_1c_1 \\  b_2 & b_2c_2 & a_2c_2 \\  b_3 & b_3c_3 &  a_3c_3 \end{vmatrix} =0

If one wants to avoid \det A=0 being obtained for solutions: 1) where the amount of grass eaten by a cow is w=0; 2) negative values for y or w we have to set further constraints:
a_1, a_2, a_3...c_3>0
c_2>c_1
a_1b_2-a_2b_1>0
a_2b_1c_2-a_1b_2c_1>0

In Newton’s original numerical example the matrix of the 9 values is:

\begin{bmatrix} a_1 & b_1 & c_1 \\  a_2 & b_2 & c_2 \\  a_3 & b_3 &  c_3 \end{bmatrix} = \begin{bmatrix}  36 &  10 & 12 \\   63  & 30 & 27 \\  162 & 108 & 54 \end{bmatrix}

This yields:

\det A = \begin{vmatrix}  10 &  120 &  432 \\   30 & 810 & 1701 \\  108 & 5832 &8748  \end{vmatrix} =0

There seems to have been a error in Newton’s original copy of this problem which he corrected late in his life. A question that comes to us is: Is there some easy algorithm for generating such valid integer nonads and is there some pattern to them?

Posted in Scientific ramblings | Tagged , , , ,

A novel discrete map exhibiting chaotic behavior

The map proposed by R. Lozi over 40 years ago is one of the simplest two dimensional maps that exhibits chaotic behavior and generates a wide range of interesting structures. The map may be defined thus:

x_{n+1}=1-a_1|x_n|+y_n
y_{n+1}=a_2x_n
where a_1, a_2 are real parameters.

We have discussed our explorations of this map at some length before. At that point we also discussed our discovery of certain generalizations of this class of maps. In course of the quest for the generalized Lozi-like maps, we discovered another generalization remarkable for the diversity of forms it produced. We held back from presenting it because we wished to investigate its geometry further. As a result we are now able to explain the basic geometric determinants of the map; however, there are other aspects that still remain mysterious to us. Hence, we are presenting this map with whatever understanding we could arrive at along with the issues that are still open questions for us. They are maps of great beauty; hence, they are also worth beholding for the sake of their aesthetics.

This map is defined thus:
x_{n+1}=a_1+a_2x_n+a_3|x_n|+y_n
y_{n+1}= a_4-x_n

Alternatively, we can also define it as:
x_{n+1}=a_1+a_2x_n+a_3|x_n|-y_n
y_{n+1}= a_4+x_n

Here a_1, a_2, a_3, a_4 are real parameters: a_2 + a_3 is in the range (-2,2) (see below for explanation), while we explored a_1, a_4 primarily in range (-1.01, 1.01) as this is the range where there is higher tendency for the map to yield chaotic attractors. If we take the first definition then the map is bilaterally symmetric about the axial line y=-x. It we take the second definition it is similarly symmetric about the axis y=x. For the below discussion we refer only to the first definition because the second one shows comparable behavior with change in signs of some of the parameters.

To explore the basic geometric principles behind the attractors generated by this map we started with a set of 200 equally spaced starting points x_0, y_0 all of which lie on the circle x=\cos(t), y=\sin(t). With each starting point the map was iterated 2000 times. Then the evolution of each starting point under the map was plotted in one of 11 different colors. We were able to determine the following rules for the structure of these attractors:

1) The most important determinant of the geometry of the attractor is sum of the parameters a_2, a_3. When a_1, a_4 are closer to 1 and a_2+a_3 \rightarrow 2\cos\left(\tfrac{2\pi}{p/q}\right) where p,q are mutually prime integers, we get a p-ad structure in the attractor (e.g. Figure 1).

lozi_figure01_4heptasFigure 1. Emergence of a heptad structure for 4 different values of a_2, a_3

In Figure 1, a_1=a_4=1, and a_2, a_3 takes successively, by row, the following 4 pairs of values (-0.5450419, 0.1); (-0.2450419; -0.2); (-0.7450419, 0.3); (-0.9450419; 0.5) . In each case a_2+a_3 \rightarrow 2\cos\left(\tfrac{2\pi}{7/2}\right). This determines the central heptad structure but the rest of the attractor shows considerable variability that cannot be explained in a straightforward way from these parameters.

If a_2 \rightarrow 0 and a_3 \rightarrow 2\cos\left(\tfrac{2\pi}{p/q}\right) then there is greater tolerance for the range of values a_1, a_4 can take.

lozi_figure02_a2_0_nonaFigure 2. a_1=0.3, a_2=0, a_3=2cos\left(\tfrac{2\pi}{9/2}\right), a_4=0.7

With a_3=2\cos\left(\tfrac{2\pi}{9/2}\right) we get a nonad structure even though a_1 is relatively low.

When a_1 \rightarrow 0 and a_4 \rightarrow 1 a_2+a_3 \rightarrow 2\cos\left(\tfrac{2\pi}{p/q}\right) the p-ad structure in the attractor tends to be retained (Figure 3).

lozi_figure03_hepta.8Figure 3. a_1=-.001, a_2=-.22, a_3=-.22, a_4=0.8

In this case a_2+a_3 is close to 2\cos\left(\tfrac{2\pi}{7/2}\right); thus, we get a core with a heptad structure.

Irrespective of a_1, a_4, if a_2, a_3 are close to each other in magnitude but opposite in sign then we get square structure in the core of the attractor (Figure 4). This is because a_2+a+3 \rightarrow 0, which is 2\cos\left(\tfrac{2\pi}{4}\right)

lozi_figure04_oppositesFigure 4. a_1=-0.33, a_2=-0.34, a_3=0.34, a_4=0.8

2) For low values of both a_1, a_4 or at least a_4, including when they are below 0 and a_2+a_3 \rightarrow 2\cos\left(\tfrac{2\pi}{p/q}\right) we see a loss of the p-ad structure or a flattening of the attractor along the axis of symmetry from the top left side (Figure 5).

lozi_figure05_pentaFigure 5. Effect of low a_1, a_4

In both panels a_2=a_3=0.305; a_2+a_3 \approx 2\cos\left(\tfrac{2\pi}{5}\right). In the left panel a_1= a_4=1 and the expected pentad structure is retained. In the right panel a_1=a_4=0.1; we notice that the pentad structure has been lost with flattening along the axis of symmetry.

These are the basic rules by which the core structure of the attractors can be accounted for. The parameters a_1, a_4 can be seen as scaling parameters whereas a_2, a_3 can be seen as rotational parameters. The latter pair sets up the rotational structure based on how close their sum approaches 2\cos\left(\tfrac{2\pi}{p/q}\right). This is the cosine principle that occurs as the primary structure-determinant in other chaotic maps, including maps which we discovered and discussed before (1, 2). As in the previously discussed cases, this sets the limits of a_2+a_3 as (-2, 2); however, here there is greater scope for diversity because the cosine principle is distributed over the sum of two separate parameters. Beyond the cosine principle,n these attractors exhibit additional features whose origins remain mysterious to us:

1) For example, In Figure 3 in addition to the core heptad one can see additional harmonics like a 11-ad, 18-ad and 29-ad structures. What is the explanation for them? In the previous examples of chaotic maps we could explain higher harmonics based on other \tfrac{p}{q} whose \cos\left(\tfrac{2\pi}{p/q}\right) might lie close to the primary cosine. However, in this case the 11-ad, the 18-ad and 29-ad correspond to no such cosines. Hence, their emergence remains a mystery.

2) Further, there is an effect of the starting x_0, y_0, which can result in the appearance of a n-ad structure independently of the cosine principle (Figure 6).

lozi_figure06_hepta_anomalFigure 6. The effect the initial points.

In both the above cases the four parameters a_1, a_2, a_3, a_4 are respectively, -0.567115129642189, -0.761931293738961, -0.347582505706949, -0.069076271019876. However, the left panel was initialized with a circle of radius 0.9 and the right panel was initialized with a circle of radius \tfrac{1}{3}. Only in the second case we see the emergence of a heptad structure, which cannot be accounted for by the sum of a_2, a_3. The role of the starting x_0, y_0 in generating n-ad structures is another open question.

To further explore the diversity of chaotic attractors within the above-stated parameter space for the 4 parameters of the map we set up our code to search for such maps thus:
1) We set x_0=0.3\cos(\pi/4), y_0= 0.3\sin(\pi/4) as the starting point.
2) We then randomly generated a set of parameters a_1, a_2, a_3, a_4 in the range (-1.01, 1.01).
3) For each such set we allowed (x_0, y_0) to evolve for 1000 iterations under the map. If the evolution resulted in the divergence to \infty or convergence to one or few point attractors, then we discarded those parameters.
4) A heuristic for chaos is that very small differences in the initial conditions can result in very different end results upon evolution under the map after a certain number of iterations. Hence, for those parameters which survived the above filters we tested if a second x_0, y_0, which differed from the first by 10^{-7} in each coordinate diverged from the evolutionary path of the original x_0, y_0 after the same number of iterations. If it did so, the parameter set was retained as it was the sign of being on a chaotic attractor.
5) These surviving parameter sets were then explored more fully by studying the evolution of 200 equally spaced points on the circle x=0.3\cos(t), y=0.3\sin(t) for 2000 iterations of the map.

We present below few examples of attractors emerging from the above procedure with parameters listed in order from a_1 to a_4.

lozi_figure7_27Figure 7. -0.21947270759847, 0.33858182718046, 0.610763950282708, 0.315013298424892
This attractor assumes of the form reminiscent in some ways of the gingerbread man seen in the classical Lozi attractor. The gingerbread man has a “heart” in the form a five-lobed structure with a ellipse within it.

lozi_figure8_28Figure 8. -0.344555260520428, -0.59013448276557, 0.523313496601768, 0.713451223839074
This attractor has parallels to the previous one in being somewhat like the gingerbread man with 4 hands.

lozi_figure9_18Figure 9. -0.26224758869037, -0.488204386695288, 0.879756956817582, 0.301065913862549
This attractor takes the form of a fish or some crustacean naupilus larva.

lozi_figure10_2Figure 10. -0.309558889106847, -0.797052371953614, -0.980002021430992, 0.34323377257213
This attractor looks somewhat like an echinoderm larva.

lozi_figure11_3Figure 11. -0.482874695337377, -0.819933905000798, -0.381899853958748, 0.100596646997146
This is a representative of a prevalent type of attractor generated by this map that may be termed the “Sombrero hat” type. The central heptad structure in this attractor is a mystery because it is not explicable by the cosine conditions above-presented .

lozi_figure12_7Figure 12. -0.442830848232843, 0.0637841782858595, 0.495140142193995, 0.24925118080806
This form of the attractor resembles the butterfly attractor generated by the square-root modification of the Lozi map that we had described earlier.

lozi_figure13_11Figure 13. -0.170671575525775, -0.671315040569752, 0.563780032042414, 0.24513541710563
Another version of the “Sombrero hat” type.

lozi_figure14_rad1Figure 14. -0.517300257436, 0.795868384265341, 1.00516145080794, 0.493695715968497
For this attractor the radius of the x_0, y_0 circle was changed from 0.3 to 1, though it is stable and similar at the former value too. It vaguely resembles some Cambrian animal.

lozi_figure15_41Figure 15. -0.307423420087434, -0.0870326082082465, -0.337442451966926, 0.766096443301067
This attractor assumes a bun-shaped morph that is seen quite often under these maps. An explanation for the multiple harmonics of this attractor remains as yet mysterious to us.

lozi_figure16_31.4Figure 16. -0.0823114865785465, -0.766779959597625, 0.868352874149568, -0.0100218075420707
Another Sombrero hat type form. The radius of the x_0, y_0 circle was changed from 0.3 to .4 for this attractor. How one accounts for the central pentagonal zone of restriction and the triad of octagonal restriction zones around it remains mysterious.

lozi_figure17_12.31Figure 17. -0.16997885087505, -0.0931982280220837, -0.631216614013538, -0.248738911962137
Here the radius of the x_0, y_0 circle was changed from 0.3 to 0.31. It shows a central pentad surrounded by 9 further pentads. An explanation for the emergence of these pentads and the 9 fold harmonic however remains elusive.

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