The universe is wholly contained in one's own Self like a city seen in a standing sidewalk mirror.
Due to Maya, this inner universe appears to be externally manifested, like an inner world appearing to be an outer one while dreaming.
To the One who knows this at the time of awakening as one's nondual self, O to that divine teacher, the one who is facing south.
Commentaries
One of the forms in which Lord Siva is pictured is that of the youthful world-teacher seated beneath the sacred fig-tree, facing South, and teaching elderly disciples through silence. The significance of this form is the supreme identity of God (Iśvara), Teacher (Guru), and Self (Atman). It is the same Self that appears as God and as the soul, as the teacher and as the taught.
In the first verse, it is shown that the South-facing Siva, the supreme Lord who is the Self, is the substrate of the entire universe. The manifold universe consisting of the elements and the elementals exists in one's Self. The Self is one; the world is manifold. The Self is unchanging; the world is subject of ceaseless change. The Self is infinite; the world consists of finite entities. The Self is pure; the world is full of impurities. How can the former be the substrate of the latter? This is explained with the help of an example. The world appears in the Self like the city with its streets and buildings, market-places and business-houses, tanks and towers, vehicles and living beings in a clear, unsoiled mirror. The mirrored city is not real; and yet it appears in the mirror.
An example for the appearance, outside, of what is within is dream-experience. The dream-world is what is superimposed on the Self which is the witness; yet it appears as if outside the Self. Similarly in the state of waking, one sees the world which is superim-posed on one's Self, as if it exists outside. Because of the superimposition of identity with the body, etc., which are superimposed on the Self, one sees the world as if it exists outside. No relation of the world which is inert, such as conjunction and inhe-rence, is possible with the Self which is consciousness. If it were not for superimposition on the self-lumi-nous Self, the world would not even be manifest. Therefore, the world is only a superimposition on the Self.
~Mahadevan
This opening stanza can be considered as containing the invocation of the Supreme by the author, in all devotion, for the successful culmination of his great work. It considers the south-faced Lord2 as none other than the manifested form of the Infinite. The objects of the world, in fact, exist and play about only in the observer’s mind and yet, due to māyā, they are seen, as though outside, just as the reflection of oneself in a mirror.
The example is used to emphasise that the reflected image has no reality apart from the object reflected. Though the reflected image is of the same form as the object reflected, there is a lateral inversion of details in the reflection. The right side of the onlooker will be seen as the left side of his own reflection. Thus, sitting in front of a mirror, if you write with a pencil in your right hand, the reflection will look as though it is writing with the left hand. This ‘lateral inversion’ in our personality is that which causes confusions and sorrows of life, called the saṁsāra. The supreme Existence-Knowledge-Bliss (Sat-Cit-Ᾱnanda), when It gets reflected in the mind, is expressed as the world – non-existent, inert and sorrowful.
~Chinmayananda
This vast universe with its countless objects, moving and unmoving entities, manifests externally as the objects we see and internally as feelings, perceptions, and emotions. In this first verse, Shankara inquiries into the nature of the world (jagat), the nature of the Creator (Iśvara), and the individual (jīvā) who feels helplessly trapped between the world and the Creator.
Shankara focuses on jagat (world) in the very first stanza because jagat is distinctly visible and ever present, while Iśvara and jīvā are not visibly present. From what is present and visible, the invisible and the absent can be inferred.
~Rao
Shankaracharya presents this scenario to explain four key principles...the first principle: wherever a reflection exists, regardless of its nature, the mirror must be present. No reflection can exist without the mirror. Second, examining the mirror’s contents—buildings, roads, parks, skyline, vehicles, people—we discover that actually no such entities exist within the mirror...The mirror is real (satyam); the city is an appearance (mithyā). Third, nothing within the reflected city can affect the mirror...No element of the reflection influences the mirror. The mirror stands unattached (asaṅga) to the reflection...The fourth point—crucial and subtle—is that the mirror is non-dual with respect to the city. Although the reflected city contains countless variety of elements, if we count only the mirror, nothing exists but the mirror. This leads to the central teaching of the entire Dakshinamurti Stotram and Advaita Vedanta: The city, with its thousand or million different features, is not a second reality, separate from the mirror. Considering only the mirror, there is only one reality, though it appears manifold. This is called advaitam. ~Sarvapriyananda
Translations
Chinmayananda 1
viśvaṁ darpaṇa-dṛśyamāna-nagarī-tulyaṁ nijāntargataṁ paśyannātmani māyayā bahirivodbhūtaṁ yathā nidrayā, yaḥ sākṣāt-kurute prabodha-samaye svātmānamevādvayaṁ tasmai śrī-guru-mūrtaye nama idaṁ śrī-dakṣiṇāmūrtaye. (1) विश्वं – the universe of names and forms; दर्पणदृश्यमाननगरीतुल्यं – like a city seen in a mirror; निजान्तर्गतं – within one’s own Self; पश्यन् – recognising; आत्मनि – in the Self; मायया – due to the power of māyā; बहिः – outside; इव – as though; उद्भूतं – produced; यथा – as; निद्रया – in dream; यः – he who; साक्षात् कुरुते – experiences directly; प्रबोधसमये – at the time of Realisation; स्वात्मानम् – his own Self; एव – alone; अद्वयं – non dual (immutable); तस्मै – to Him; श्रीगुरुमूर्तये – the divine Teacher; नमः – prostration; इदं – this; श्रीदक्षिणामूर्तये – to Śrī Dakṣiṇāmūrti 1. He who experiences, at the time of Realisation, his own immutable Self, in which the Self alone plays as the universe of names and forms, like a city seen in a mirror, due to the māyā power, as though produced outside, as in a dream, to Him, the divine Teacher, Śrī Dakṣiṇāmūrti, is this prostration.
Mahadevan 1
To Him who by mayā as by dream, sees within Himself the universe which is inside Him, like unto a city that is seen in a mirror, (but) which is manifes-ted as if without: to Him who apprehends, at the time of awakening. His own non-dual Self: to Him, of the form of the Preceptor, the blessed Dakshinamurti may this obeisance be!
trans of 1a
The universe is like a reflection seen in a mirror, appearing like a city within one’s own Self, but seeming to be outside, through the illusory power of Māyā, as if dreaming in a sleep. ~Denton
through the illusion of the Ātman as through sleep, sees the universe existing within himself like a city reflected in a mirror, as though it were manifested externally. ~Sarvapriyananda