I (Muni Mahendra Kumar) have tried to resolve the problem on the basis of a modern mathematical methodology, viz., Integral Calculus.^{[1]} When we try to study the problem of the shape of *loka *in the light of this methodology, we can know the exact shape of the *loka*, in which not only all the fundamental beliefs of both Digambara and Shvetambara traditions would be fulfilled but it would also be in perfect consistency with the modern mathematical norms. Through this effort, we reach to a shape (as given in Appendix IV) of the *loka *which has the volume of 343 cubic *rajjus *as well as which keeps intact all the fundamental beliefs (stated in Shvetambara tradition and acceptable to the Digambara on the basis of the original conditions given in their works). The shape we get thus is:

Now let us critically examine this shape in the light of the Digambara and Shvetambara traditions:

- As we have seen that the shape (see fig. 4 on p. 137) is conceived by Virasenacharya in
*Dhavalā*for proving that (i) the total volume is 343 cubic*rajjus*, (ii) the volume of the upper universe is 196 cubic*rajjus*and (iii) the volume of the lower universe is 147 cubic*rajjus*. In this shape, he had to assume that the width (one dimension) in the north-south direction should be 7*rajjus*throughout, because if this is not assumed, in no other shape we could obtain the total volume as 343 cubic*rajjus*, and volume of lower and upper universes as 196 cubic*rajjus*and 147 cubic*rajjus*respectively.^{[2]}

Now, in our new shape, even without the assumption of 7*rajjus*width throughout, we get the required volumes.

- All the fundamental beliefs of the Digambara as well as Shvetambara traditions are kept intact in this new shape. Now, let us further critically deliberate over the fundamental beliefs of the Digambara tradition.

It has already been amply proved that the belief of “7 *rajjus *width throughout in north-south direction” is not a fundamental one, as accepted by Virasenacharya himself.^{[3]}

Besides this the belief of ‘7 *rajjus*’ breadth throughout in north-south direction” does not comply with the fundamental beliefs which the author of *Dhavalā *himself has referred to in *Dhavalā*; whereas there is no discrepancy of our new shape with them. The three verses quoted by *Dhavalākāra *are:^{[4]}

“The universe has a shape of *Vertrāsana *in the lower part, *jhallarī *in the middle and *mṙdaṅga *in the upper part; its height is 14 times the dimensions in the middle (i.e., 1 *rajju*).”

“The universe is indeed *akṛtrima *(not made by anyone), beginningless and endless (with respect to time), formed by nature, pervaded by the sentient and non-sentient substances and has an over all shape of a *tālavṛkṣa *(palmyra tree).”

“Its *vi*ṣ*kambha *(length and breadth) is of four types: seven, one, five and one respectively (at the end of the lower universe, in the middle (the *madhyaloka*), in the middle of the upper universe near *Brahmaloka *and the upper end of the upper universe).”

Now, there is an obvious discrepancy in the *āyatacatursrakāra *(the shape as accepted by *Dhavalākāra*- as given in fig. 4) which is based on the belief that “one dimension (breadth) of the universe in the north-south direction is 7 *rajjus *throughout”, with the above three verses; in order to get rid of this discrepancy, the *Dhavalākāra *pleads:^{[5]} “The universe (*āyatacatursrā-kāra*) has no discrepancy with the first verse, because the shapes of *etrāsana *and *mṙdaṅga *can be seen in one direction. If it is argued that in the *loka *described here, (in the middle part) the shape of *jhallarī *is not found, then (the refutation is that) it is not so, for, in the *madhyaloka *the shape of that region which is surrounded on all sides by Svayaṃbhūramaṇa Ocean and which is innumerable *yojanas *in length and breadth and one lakh *yojanas *in height, appears like *jhallari *similar to the moon’s *manḍala *(circular shape), and again, there cannot be complete sameness in the *dṛṣṭānta *(example) and the *dārṣṭānta *(the thing which is compared), otherwise there would arise the occasion of total extinction of both. Again, if it is argued that there is no possibility of the shape of palmyra tree in the *loka *described here, then (the refutation is that) it is not so, for, seen from one direction the *loka *appears like the palmyra tree. And there is no discrepancy with the third verse also, because here also in the east-west directions the dimensions of the universe at four places are still in conformity with the four types of *viṣkambha *described in the last verse......”

Thus, the far-fatchedness of the arguments put forward by the Dhavalākāra in defence of the *āyatacaturasrākāra *(to prove that his innovative imagination of “7 *rajjus*” breadth throughout north-south direction is not against the fundamental/original beliefs) can easily be gauged. In case of the shape suggested on the base of modern mathematics, these doubts do not arise at all. For, in the new shape, the universe appears perfectly symmetrical from all sides. Another thing is that the Dhavalākāra’s arguments are not much satisfactory, for, nowhere in the verses quoted, it is mentioned that the *loka *is akin to *vetrāsana *etc. only in the one direction. On the contrary, the general implication of the quoted verses would be that in all directions, the shape of the *loka *is that of *vetrāsana *etc.. The *āyatacatursrākāra *is also not in perfect conformity with the second verse which explicitly mentions that the shape at the middle part of universe is a circular shape like the moon’s *maṇḍala*. This is only possible if both the dimensions - length and breadth (or the diameter) of the middle universe is 1 *rajju *only (see fig. A), but in proposed shape of *Dhavalākāra*, it is oblong, on account of one dimension (length) in east-west direction being 1 *rajju *and the other (breadth) in north-south direction being 7 *rajjus*. (See fig. B).

As far as the third verse is concerned, it is quite evident from it that the four types of *viṣkambha *at four places mean both length and breadth should be 7, 1, 5 and 1 at respective places; then how one can conclude that it is only so in east-west direction and 7 *rajjus *throughout in south-north direction? There is no reference to a particular direction in the verse. *Viṣkambha*, generally denote both length and breadth^{[6]} and when there is no explicit separation of both, it definitely would mean that both length and breadth should be 7, 1, 5, and 1 at respective places, and not that in one direction it (i.e., say, length) is 7, 1, 5 & 1 at respective places, while in another direction it (i.e., breadth) may not be so. Therefore, from the third verse, the meaning which becomes clear is that the universe is 7, 1, 5 and 1 *rajju *in both east-west and north-south directions at the respective places.

In our new shape we get direct solution to all such doubts, because the shape therein is perfectly symmetrical and in perfect conformity with the shape (*vetrāsana *etc.) and dimensions (both length and breadth) mentioned in the verses. We have already mentioned the fundamental beliefs of the Shvetambara tradition and we find that our new shape is in perfect conformity with all of them. Thus, our new shape is in perfect conformity with both Shvetambara as well as Digambara traditions.

- The new shape is based on the accepted mathematical methodology in modern mathematics.

- The curvature of
*lokakāśa*is also comparable to the curvature of space in modern science. It is also not impossible that the non-Euclidean geometry may hold true in the*lokākāśa*(*dharmāstikāya*& adharmāstikāya).^{[7]}

Apart from these four specialties, there are also some other beliefs regarding the *loka *and mathematics, which are corroborated more perfectly in our new shape.^{[8]}

vidhippaḍisedhā bhāvādo. - Ṣaţkhaṇḍāgama, p. 22

majjhimavitthāreṇa ya coddasaguṇamāyado logo..

logo akaţţimo khalu aṇāiṇihaṇo sahāvaṇivvato.

jīvājīvehi phuḍo ṇ icco talarukkhasaṃţhāṇo..

loyassa ya vikkhaṃbho cauppayāro ya hoi ṇāyavvo.

sattekkago ya paṃcekkago ya rajju muṇeyavvā.. - Ibid., p. 11.

ṇa ca ettha jhallarīsaṃţhāṇaṃ ṇatthi, majjhamhi sayabhuramaṇodahiparikkhittadeseṇa cadamaṇḍalamiva samamlado asaṃkhejjajoyaṇaṃrudeṇa joyaṇalakkhabāhalleṇa jhallarīsamāṇāttādo.

ṇa ca diţţhaṃto dāriţţhaṃtieṇa savvahā samāṇo, doṇhaṃ pi abhāvappasaṃgādo.

ṇa ca tālarukkhasaţhāṇamettha ṇa saṃbhavai, egadisāe tālarukkhasaṃţhāṇadaṃsaṇādo.

ṇa ca taīyāe gāhāe saha viroho, ettha vi dosu disāsu cauvvihavikkha ṃbhadaṃsaṇādo.

- Ṣaţkhaṇḍāgama, pp. 21-22.