Indra's net

 Indra's net (also called Indra's jewels or Indra's pearls or इंद्रजाल in Sanskrit) is a metaphor used to illustrate the concepts of emptiness, dependent origination, and interpenetration in Buddhist philosophy.

The Buddhist version of the metaphor of Indra's net was developed by the Mahayana Buddhist school in the 3rd-century scriptures of the Avatamsaka Sutra, and later by the Chinese Huayan school between the 6th and 8th centuries.

Explanation

Buddhist concepts of interpenetration hold that all phenomena are intimately connected.
Huayan school

For the Huayan school, Indra's net symbolizes a universe where infinitely repeated mutual relations exist among all members of the universe.

This idea is communicated in the image of the interconnectedness of the universe as seen in the net of the Vedic god Indra, whose net hangs over his palace on Mount Meru, the axis mundi of Vedic cosmology and Vedic mythology. Indra's net has a multifaceted jewel at each vertex, and each jewel is reflected in all of the other jewels.

Francis Harold Cook describes the metaphor of Indra's net from the perspective of the Huayan school in the book Hua-Yen Buddhism: The Jewel Net of Indra:

    Far away in the heavenly abode of the great god Indra, there is a wonderful net which has been hung by some cunning artificer in such a manner that it stretches out infinitely in all directions. In accordance with the extravagant tastes of deities, the artificer has hung a single glittering jewel in each "eye" of the net, and since the net itself is infinite in dimension, the jewels are infinite in number. There hang the jewels, glittering "like" stars in the first magnitude, a wonderful sight to behold. If we now arbitrarily select one of these jewels for inspection and look closely at it, we will discover that in its polished surface there are reflected all the other jewels in the net, infinite in number. Not only that, but each of the jewels reflected in this one jewel is also reflecting all the other jewels, so that there is an infinite reflecting process occurring.

Proto-Sāṃkhya and early Buddhism

The earliest materials from which this metaphor were drawn can be found throughout the oldest of India's works, the Vedic Saṃhitās, Brāhmaṇas, Upaniṣads, and Sūtras, so that "Indra's net" has its philosophical roots in early "proto-forms" of Samkhya, one of the six Hindu darshanas. Indeed, some scholars have speculated that Buddhism is itself a branch of these early forms of Sāṃkhya.
Cognition

The Sāṃkhya hermeneutics, associated with Yoga practices, teach that cognition is a vertical "protam" (the thread that runs vertically through a loom, the "warp"), and that phenomenal (i.e., spatial) nature is the dynamic, visible, horizontal "woof" or "weft" that shuttles back and forth across the loom, in three "colors", of white/yellow/gold, red/brown, and blue/black. These three colors correspond to the triguṇa theory, composed of sattva (goodness/quiescence/existence), rajas (passion/activity), and tamas (darkness/morbidity).
Indra

Earlier phases of this view are less discrete, but can be glimpsed in descriptions of Indra as the vertical skambha, or world column, which is also associated with the motionless timeless center of the universe, the axle of the world-wheel. Approaching the felly of the wheel, one experiences the passing of Time, but approaching the center, no experience passes at all, a state in Sāṃkhya called [[[kaivalya]]] (isolation). This isolation is said to free one from duḥkha (literally, a 'broken or disjointed axle', but which comes later to simply signify "suffering" in all its varieties).

It is also likely that Indra's depiction as a chariot-driver, reins in hand, helped to reify the image of the threads that comprise the net, since it is with these reins that Indra causes the world to revolve.
Buddhism

During the Upaniṣadic period (c. 1000 BCE – 200 AD), the Vedic god, Indra, was semiotically displaced by Viṣṇu (often translated as "the all-pervading one") and Śiva (the "auspicious one"), likely owing to the former's early associations with both the cyclical year (the felly of the world-wheel of Time) and its central axis, probably owing to the latter's early associations with Mt. Kailāśa, a beautiful, but virtually unscalable mountain in the Himalayan region thought to be the world-pillar.

But as Buddhism branched off and developed its sectarian form about halfway through this period, later Buddhists, such as those living in the 3rd century CE, tended to identify more so with the older elements than with any that rose up afterward. This is, of course, not universally true, and numerous other elements of Hindu/Buddhist philosophy continued to interpenetrate throughout the course of South Asian history even up to the contemporary period.
Time

The "vertical" element of Time (kāla) emerges from the tendency to regard the north as identical to the celestial north (uttara, literally, "upper"). For Indians, living in the lower half of the Northern hemisphere, the world was regarded as a mountain, around which the sun travelled on its daily course. This revolution constituted one of the fellies of the wheel of Time, and designated the northern axis as the universal axis, sometimes called the world column, or "spine" (skambha). This vertical direction was at some point associated with the pinnacle (kūṭa) of reality, a motif that can be seen again in the Tibetan Buddhist kāla-cakra-tantra, the "loom of the wheel of time".

As Time is here regarded in its psychological sense, that of having a notion of past, present, and future, and as "persistently standing in the present", this vertical column was also associated with consciousness (the Sāṃkha system uses the term, kṣetra-jña (knower of the field), or just jña (knower)). In effect, this allowed for the identification of psychological time with World-Time (mahākāla). For South Asian metaphysicians, this explained how the soul (ātman) was able to live eternally, being but one of the measureless strands of eternal Time. Yet in Buddhist metaphysics, Time's non-phenomenality, along with its role as a limiting, destructive factor with respect to all spatial entities implied that the ātman itself was "empty of any permanent phenomenal content" (śūnyata).

Conceptually, the "vertical world axis", understood previously as the un-fatiguable, eternal master of mortality, gave way to "Time, the emptiness of all phenomenality".
Western interpretations
Gödel, Escher, Bach

Douglas Hofstadter uses Indra's net as a metaphor for the complex interconnected networks formed by relationships between objects in a system—including social networks, the interactions of particles, and the "symbols" that stand for ideas in a brain or intelligent computer.
Vermeer's Hat

In Vermeer's Hat, a history book written by Timothy Brook, the author uses the metaphor:

    Buddhism uses a similar image to describe the interconnectedness of all phenomena. It is called Indra's Net. When Indra fashioned the world, he made it as a web, and at every knot in the web is tied a pearl. Everything that exists, or has ever existed, every idea that can be thought about, every datum that is true—every dharma, in the language of Indian philosophy—is a pearl in Indra's net. Not only is every pearl tied to every other pearl by virtue of the web on which they hang, but on the surface of every pearl is reflected every other jewel on the net. Everything that exists in Indra's web implies all else that exists.

Writing in The Spectator, Sarah Burton explains that Brook uses the metaphor, and its interconnectedness,

    [T]o help understand the multiplicity of causes and effects producing the way we are and the way we were [...] In the same way, the journeys through Brook's picture-portals intersect with each other, at the same time shedding light on each other.