Why Sarasvatī/AGNI was called त्रिषधस्था - triṣadhasthā (one who pervades 3 worlds)?

 

Rig Veda is a pure spiritual treatise, conveyed in esoteric sense, by the rishis (seers) of that era.

The formless ENERGY/BRAHMAN pervades all 3 regions, ie., the foundation level, the mid-space or antariksha and the Heaven and beyond, which is called त्रिषधस्था - triṣadhasthā - pervading in three regions.

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If we carefully study the Rig veda, we can find that त्रिषधस्था - triṣadhasthā was mentioned not only in respect of Sarasvatī but AGNI also.

Sarasvatī - सरस्वती/ADITI - अदिति was described in Rig Veda, in an esoteric manner, the feminine form of the formless ENERGY/BRAHMAN, while AGNI was mentioned as the Masculine form of the formless ENERGY/BRAHMAN. 

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Sarasvatī or AGNI aid practitioners in progressing spiritually.

• The work of Iḻā/Iḍā (इळा)/AGNI is at foundational level,ie., at the beginning of spiritual journey.
• The work of Sarasvatī - सरस्वती/INDRA is at the mid-level or antariksha, ie., Sarasvatī/Indra will aid the practitioner at the mid-level of spiritual journey.
• The work of Mahī/Surya or Savitṛ is in the final stages of the practitioner in spiritual journey.

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a) Sarasvatī - Rig Veda 6.61.12

त्रिषधस्था सप्तधातुः पञ्च जाता वर्धयन्ती । वाजेवाजे हव्या भूत् ॥
triṣadhasthā saptadhātuḥ pañca jātā vardhayantī | vāje-vāje havyā bhūt ||

English translation:

Having pervaded in three regions and seven dhātus, strengthening the five units, at every spiritual practice, she is worthy to be invoked.

b) Agni - Rig Veda 6.12.2

आ यस्मिन्त्वे स्वपाके यजत्र यक्षद्राजन्त्सर्वतातेव नु द्यौः । त्रिषधस्थस्ततरुषो न जंहो हव्या मघानि मानुषा यजध्यै ॥
ā yasmin tve sv apāke yajatra yakṣad rājan sarvatāteva nu dyauḥ | triṣadhasthas tataruṣo na jaṃho havyā maghāni mānuṣā yajadhyai ||

English translation:

You in whom heaven in its entirety [=all the gods] will now perform sacrifice, as it were—you the very clever, the means of sacrifice, the king— (you) with your three regions, like the plumage of (a bird) that has traversed , are to sacrifice the oblations, the bounties of the sons of Manu.

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Interestingly, in Puranas the River Ganga was mentioned in Mahabharata, in the episode of Bhagiratha bringing Ganga from Heaven to the Earth, and to Netherworld, as Tripathagā - त्रिपथगा (mover in 3 worlds).

पितॄणां चोदकं तत्र ददौ पूर्णमनोरथः ।

एतत्ते सर्वमाख्यातं गङ्गा त्रिपथगा यथा ॥

Bhagiratha offered libations of water to the names of his forefathers; thus was his heart's wish fulfilled. As Ganga descended from Heaven, landed on the Earth and finally reached the Netherworld, she has come to known as Tripathagā - त्रिपथगा (mover in 3 worlds).