Srimad-Bhagavatam and Conformal Mappings

itihāsapurāṇābhyāṃ vedaṃ samupabṛṃhayet bibhetyalpaśrutādvedo māmayaṃ prahariṣyati The meaning of Vedas is to be understood with the help of purāṇas and the itihāsas. Vedas themselves are afraid of those who are unaware of the itihāsas and purāṇas for these ignorant people may end up attacking the Vedas incorrectly.

Conformal cyclic cosmology (CCC) is a cosmological model that suggests the universe is the same at all times and places, while still allowing for some evolution. This model was proposed by Roger Penrose in 2004 and has since been further developed by several researchers. CCC challenges traditional notions of time and space, proposing a cyclic universe in which Big Bangs are conformally related to Big Crunches. This concept provides an intriguing explanation of the origin and evolution of the universe, and has implications for our understanding of dark energy, black holes, inflationary models, quantum gravity, and even string theory. It is important to note that Conformal cyclic cosmology (CCC) has been met with considerable criticism. Proposed by Roger Penrose, one of the Nobel Laureates for Physics in 2020, the theory suggests that the universe goes through an infinite cycle of expanding and collapsing universes. Critics argue that CCC cannot explain core cosmological questions such as why the universe is so homogeneous, why the universe appears to be accelerating in its expansion and why certain fundamental physical constants have the values they do. However, our goal here is not to develop a new physical theory or challenge Sir Roger Penrose’s claims or the claims of his critics. Instead, we will explore a very pleasing analogy between the underlying proposed geometrical structure behind this theory of the cosmos and a very popular story of śrīmadbhāgavatam. The underlying geometry is that of conformal mappings.

Conformal Mappings - The Geometric Idea Behind CCC. Conformal mappings are a powerful tool used to transform complex geometries and intricate shapes from one “domain” to another. By using these mappings, mathematicians can visualize infinity inside a finite boundary. Conformal mappings also allow us to map one geometric figure onto another while preserving angles and shapes. Practical examples of conformal mappings can be found in fields such as architecture, engineering, and cartography. By using these mappings, architects can create more efficient and aesthetically pleasing structures, engineers can design innovative products with fewer constraints, and cartographers can produce accurate maps with a greater level of detail. One of the most beautiful examples used in engineering to calculate impedances of transmission lines is called a Smith-chart.

We can now extrapolate the idea into 3 dimensions and can enjoy the analogy of this concept with the incident between jagadguru śrīkṛṣṇa and mātā yaśodā found in the śrīmadbhāgavatam.

Śrīkṛṣṇa and Mātā Yaśodā - An Allegorical Representation of Conformal Maps. In the previous section, we have explored the beauty of consistency in our rigorous philosophical systems but is the same level of enjoyment present in allegorical forms within our culture? Can we intelligently and more importantly without making any compromises, relate to the latest mathematical and physical concepts using analogies that fuse the insights of physical theories while also connecting with our core human experience?

We find that in the case of Conformal mappings, the answer is a yes. The essence of the parable is captured in these 10 beautiful verses composed by Melpatūr Nārāyaṇabhaṭṭādiri in his brilliant nārāyaṇīyam -

ayi deva purā kila tvayi svayamuttānaśaye stanandhaye।

parijṛmbhaṇato vyapāvṛte vadane viśvamacaṣṭa vallavī॥ 46-1॥

punarapyatha bālakaiḥ samaṃ tvayi līlānirate jagatpate।

phalasañcayavañcanakṛdhā tava mṛdbhojanamūcurarbhakāḥ॥ 46-2॥

ayi te pralayāvadhau vibho kṣititoyādisamastabhakṣiṇaḥ।

mṛdupāśanato rujā bhavediti bhītā jananī cukopa sā॥ 46-3॥

ayi durvinayātmaka tvayā kimu mṛtsā bata vatsa bhakṣitā।

iti mātṛgiraṃ ciraṃ vibho vitathāṃ tvaṃ pratijajñiṣe hasan॥ 46-4॥

ayi te sakalairviniścite vimatiścedvadanaṃ vidāryatām।

iti mātṛvibhartsito mukhaṃ vikasatpadmanibhaṃ vyadārayaḥ॥ 46-5॥

api mṛllavadarśanotsukāṃ jananīṃ tāṃ bahu tarpayanniva।

pṛthivīṃ nikhilāṃ na kevalaṃ bhuvanānyapyakhilānyadīdṛśaḥ॥ 46-6॥

kuhacidvanamambudhiḥ kvacit kvacidabhraṃ kuhacidrasātalam।

manujā danujāḥ kvacitsurā dadṛśe kiṃ na tadā tvadānane॥ 46-7॥

kalaśāmbudhiśāyinaṃ punaḥ paravaikuṇṭhapadādhivāsinam।

svapuraśca nijārbhakātmakaṃ katidhā tvāṃ na dadarśa sā mukhe॥ 46-8॥

vikasadbhuvane mukhodare nanu bhūyo’pi tathāvidhānanaḥ।

anayā sphuṭamīkṣito bhavānanavasthāṃ jagatāṃ batātanot॥ 46-9॥

dhṛtatattvadhiyaṃ tadā kṣaṇaṃ jananīṃ tāṃ praṇayena mohayan।

stanamamba diśetyupāsajan bhagavannadbhutabāla pāhi mām॥ 46-10॥

The english translations of the above verses taken from Jaishree Iyer are as follows:

1. Oh Lord! Once, when, sucking at Thy mother’s breast, Thou lying flat on her lap, opened Thy mouth to yawn, the entire Universe was seen by yaśodā, in it.
2. Oh Lord of the universe, at another time, you were playing with your friends, plucking fruits, cheated them and ate the fruits gathered by them. Angry with Thee, they went and complained to your mother that you ate mud.
3. O Lord! At the time of the deluge Thou do consume everything earth water etc. Yet Thy mother was frightened that Thou may fall sick by eating mud and so she became angry with you…
4. Oh Lord! When your mother asked you with concern, “Is it that you have eaten mud??”. Thy mother questioned Thee, for a long time, Thou kept on denying and smilingly kept silent, making her feel that her anger is without any effect.
5. Thy mother, not believing Thee, because all the other children had convinced her of it, asked Thee “Little one, if you do not agree with what all these children are telling with certainty, please open your mouth, let us see.” And without any hesitation, you opened, Your mouth which was like a lotus flower blooming.
6. Thy mother was eager to see just a trace of mud in Thy mouth. As though to please her, and to give her abundant satisfaction Thou showed her in Thy mouth not only this whole earth but the entire universe.
7. At that time, in Thy mouth what not was seen by yaśodā? She saw in your open mouth, in one place forests, In another place sea, yet another place the sky, In yet another place nether world, men, asuras and devas, And there was nothing that she was not able to see in your mouth, For all the fourteen worlds and their things were there.
8. Yaśodā saw Thy manifold forms in Thy mouth; In your mouth again she saw you as the one Who sleeps in the ocean of milk, As the divine one who lived in vaikuṇṭha, And also you as a baby just standing in front of her. And was there any form of yours that she did not see there?
9. Seeing in your mouth, where the whole universe shone clearly, she saw yet another similar face and open mouth, with the whole universe shining in it; within that yet another mouth, displaying the whole universe and thus an endless revelation of Thy infinite quality. And this clearly explained to her that, All this world is limitless and temporary. Wonderful indeed!
10. At that time for a moment, yaśodā returned to the world of realty, You charmed her with the very affectionate illusion, And asked from her more milk and lay on her lap, Oh Lord ! who took the form of that wonderful child, please protect me.

The bold sections are of particular relevance in the above since they describe two fundamental aspects of the Conformal mappings - one being projecting an infinite structure into a finite space and the other being the inherently recursive and cyclic quality. This brings our explorations to connect our cultural moorings to the ongoing advances in our understanding of the physical world to an end for now. Having had a taste of what a consistent OET does for strengthening our understanding of Physics, we will now discuss some implications from this brief study.

Implications on Bhārata

Two things should now be clear to us.

1. Adding the metaphysical layer from our śāstras to physics enriches our experience of learning about our physical world in a meaningful way and helps us reconcile some of the questions posed by quantum mechanics.
2. The artificial separation between theology and physics is an outdated artefact from a model of thinking that is prevalent in the world but not necessarily helpful in processing the nature of infinities in our universe.

These statements should not be a surprise to anyone subscribed to the dhārmik worldview. We already know this intuitively because we are surrounded by this message of oneness. We see it in the seam- less integration of architecture, material science, beauty in form, engineering, and mathematics in our temples that still stand before us after millennia. We see it in how our mathematicians invented the kaṭapayādi system to express themselves on their own terms and without any forced restrictions on their potential. All the examples in this article have been compiled with the hope of creating excitement within the dhārmik fold and starting a conversation around the question of “We might be free but are we independent?” in our minds. This question was posed by pūjya śaṅkarācārya of the kāñci maṭha in 1947. In the domain of natural philosophy, we must address this question at least in 2023.

Authored by CSK Chaitanya. Published by Brhat.