When Viṣṇu Sleeps: Pralaya, 4.32 Billion Years/432 Crore years, and the Science of Cosmic Time
Hindu Cosmology, Yugas, Kalpas, Pralaya, Carl Sagan, and the Limits of Scientific Comparison
© Dhinakar Rajaram, 2026
Bibliotheque Series — Science, Wonder, and the Indian Gaze
Foreword
A short social-media Reel recently made a striking assertion: whenever Viṣṇu falls asleep, the world ends, and His sleep lasts 4.32 billion years. It is the sort of sentence designed for the modern attention span—dramatic, memorable, and irresistible to anyone fascinated by cosmic time.
Yet the underlying Hindu cosmological tradition is considerably more subtle than the Reel suggests.
The figure of 4.32 billion years is indeed part of the traditional Hindu reckoning of cosmic time. But it is properly associated with a Kalpa, the day of Brahmā. Brahmā's night is of equal duration, making one complete day-and-night cycle 8.64 billion years/ 864 Crores Years ( 1 billion is 100 Crore) . Hindu cosmological literature further describes periods of dissolution and subsequent re-manifestation, with Viṣṇu, Nārāyaṇa, or Hari appearing in the imagery of cosmic repose.
That distinction matters. Otherwise a profound cosmological tradition is reduced to an attractive but inaccurate slogan.
The more interesting question is not whether an Instagram Reel has got every detail right. It has not. The interesting question is this: Why did Indian cosmological thought imagine time on such prodigious scales, what exactly does Pralaya mean in the traditional framework, and what—if anything—can modern cosmology legitimately say alongside it?
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வாசகர்கள் தங்களுக்கு விருப்பமான மொழியில் இந்தக் கட்டுரையைப் படிக்க, தங்கள் உலாவியில் உள்ள Translate / மொழிபெயர்ப்பு வசதியைப் பயன்படுத்தலாம். இக்கட்டுரையின் ஆங்கிலப் பதிப்பே அசல் மற்றும் அதிகாரப்பூர்வப் பதிப்பாகும்.
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Approximate reading time: 12–15 minutes
Constitutional Requirement — Article 51A(h)
The Constitution of India places upon every citizen the duty to develop scientific temper, humanism, and the spirit of inquiry and reform under Article 51A(h).
A respectful examination of Hindu cosmological ideas through the lens of modern science is entirely compatible with that duty. Scientific temper does not require contempt for inherited traditions, nor does respect for tradition require us to suspend critical examination. The two can meet in the same intellectual space: curiosity without credulity, and criticism without contempt.
About the Author
I am an independent science writer, science communicator, amateur astronomer, and outreach volunteer based in Chennai. My engagement with astronomy formally dates from 2010, although my fascination with the night sky began much earlier. I am a founding member of the Tamil Nadu Astronomical Society and have been involved in astronomy outreach, public observation, and discussions concerning the relationship between science, astronomy, history, and Indian culture.
I approach subjects such as Hindu cosmology from two positions that need not be enemies: as a practising Hindu who regards the tradition with respect, and as a science communicator who believes that claims about the physical universe should be examined with evidence, proportion, and intellectual honesty.
For me, the most rewarding questions are often found at the boundary between what we know, what we infer, what our ancestors imagined, and what remains unknown.
Preface — A Reel, a Number, and a Very Large Question
The modern social-media version goes roughly like this: Viṣṇu goes to sleep, the universe ends, His sleep lasts 4.32 billion years, and the human lifetime is therefore little more than a cosmic camera flash.
There is a kernel of genuine Hindu cosmological arithmetic buried inside that formulation. The problem lies in the joining of the pieces.
The number is real within the traditional system. The cosmic sleep is a real theological and literary motif. Pralaya is a genuine concept of dissolution. Cyclic creation and re-manifestation are deeply embedded in Hindu cosmological thought.
But these are not interchangeable terms.
It would be rather like taking several pages from a long astronomical almanac, shuffling them together, and announcing that the resulting pile is the original calendar. The ingredients may be authentic; the arrangement may not be.
So let us put the pieces back in their proper order.
1. The Number: Why 4.32 Billion Years? [ 432 Crores years]
The traditional calculation begins with four Yugas: Satya, Tretā, Dvāpara, and Kali.
Their traditional durations are:
| Yuga | Traditional duration |
|---|---|
| Satya Yuga | 17,28,000 years |
| Tretā Yuga | 12,96,000 years |
| Dvāpara Yuga | 8,64,000 years |
| Kali Yuga | 4,32,000 years |
| One Mahāyuga | 43,20,000 years |
One thousand such Mahāyugas constitute a Kalpa, traditionally described as one day of Brahmā.
Therefore:
43,20,000 years × 1,000 = 4,320,000,000 years
That is 4.32 billion years/ 432 Crore.
The Bhagavad Gītā describes Brahmā's day and night as being of equal duration. Thus:
Day of Brahmā = 4.32 billion years
Night of Brahmā = 4.32 billion years
Day + night = 8.64 billion years
This is the first essential correction to the Reel.
2. So, Does Viṣṇu Sleep for 4.32 Billion Years?
Here the matter becomes more delicate.
Hindu literature contains several overlapping cosmological descriptions, theological perspectives, and layers of interpretation. Viṣṇu, Nārāyaṇa, Hari, Brahmā, Śeṣa, cosmic waters, creation, preservation, dissolution, and re-manifestation appear in different relationships depending upon the text and theological tradition under consideration.
The Viṣṇu Purāṇa, for example, describes the dissolution at the end of Brahmā's day and depicts Nārāyaṇa in the imagery of repose upon the serpent couch while the cosmic process lies in suspension. At the end of the corresponding night, manifestation resumes.
The popular image is therefore not invented out of thin air. But the arithmetic must not be attached to the wrong entity or the wrong interval.
The safer formulation is:
That is rather more precise than saying, “Viṣṇu takes a 4.32-billion-year nap.”
The latter is catchy. The former is cosmologically defensible.
3. Pralaya Is Not Simply “The Universe Has Been Destroyed”
The Sanskrit term Pralaya is frequently translated as dissolution, destruction, or absorption. Yet “destruction” can give a misleadingly terminal impression to a tradition whose cosmological imagination is fundamentally cyclical.
The Purāṇic tradition distinguishes different forms of dissolution. The Naimittika Pralaya, associated with the end of Brahmā's day, is not identical to the more comprehensive Prākṛtika Pralaya, nor are either identical to other senses of dissolution discussed in Hindu philosophical literature.
This distinction is scientifically interesting even before one asks whether the cosmology is physically true.
A modern reader accustomed to a binary vocabulary—exists or does not exist—may find the traditional vocabulary more graduated. Manifestation can cease without the philosophical framework requiring absolute annihilation of everything whatsoever.
In some Purāṇic descriptions, living beings and the material manifestation enter a latent condition and are subsequently manifested again. The metaphor is therefore not necessarily a cosmic rubbish bin into which existence is thrown once and for all. It is closer to a vast cycle of manifestation, withdrawal, latency, and re-manifestation.
That is a theological and cosmological description, not a laboratory observation. But it is an important distinction when discussing the tradition accurately.
4. The Human Lifetime Becomes Almost Invisible
The Reel's “camera flash” comparison is rhetorical, but the underlying point about scale is legitimate.
Suppose, merely for illustration, that a human life lasts one hundred years.
100 ÷ 4,320,000,000 × 100 = approximately 0.0000023%
A century is therefore an extraordinarily small fraction of a traditional Kalpa.
There is a useful scientific lesson here. Human intuition is calibrated by terrestrial experience. We are comfortable with seconds, days, seasons, generations, and perhaps centuries. Geological time already strains ordinary imagination. Stellar evolution moves the goalposts again. Galactic evolution makes the human calendar look positively parochial.
Cosmology goes further still.
The traditional Hindu system did not merely enlarge the human calendar by a few orders of magnitude. It constructed an elaborate hierarchy of temporal scales extending through Yugas, Mahāyugas, Kalpas, and the immense lifespan attributed to Brahmā.
That does not make the traditional arithmetic a scientific measurement. It does, however, make it an extraordinary example of humanity's attempt to escape the tyranny of everyday time.
5. A Remarkable Numerical Coincidence — But Only a Coincidence Unless Evidence Says Otherwise
This is where intellectual discipline becomes particularly important.
The Earth is approximately 4.54 billion years old. The present observable universe is about 13.8 billion years old according to the standard cosmological framework and measurements of the cosmic microwave background, expansion history, and other observations.
A traditional Kalpa of 4.32 billion years therefore falls surprisingly close to the age of the Earth in numerical scale.
This is fascinating.
But fascinating is not the same thing as causally connected.
There is no accepted scientific evidence that the composers or transmitters of the Purāṇic cosmological system possessed a measurement of the Earth's age comparable to modern geochronology.
Nor does the numerical proximity demonstrate that Hindu cosmology anticipated modern geological dating.
To make such a claim would be to cross the line from comparison into retrofitting: taking a modern number and searching an ancient numerical system for something sufficiently close to it.
A good science essay must know where to stop.
The honest statement is more modest, and consequently more durable: the traditional Indian cosmological timescale is astonishingly large and, in some instances, numerically comparable in order of magnitude with timescales that modern science later established for the Earth and cosmos.
6. When Carl Sagan Looked Towards India
Carl Sagan's treatment of Hindu cosmology in Cosmos remains one of the best-known modern encounters between popular science and Indian cosmological thought.
In the tenth episode, The Edge of Forever, Sagan considered the origin and fate of the universe and discussed the Hindu conception of vast cosmic cycles. He drew attention to the enormous timescales assigned to Brahmā's day and night and to the broader idea of repeated cosmic manifestation and dissolution.
Importantly, Sagan did not present this as a laboratory confirmation of Hindu cosmology. He explicitly observed that the resemblance between the traditional timescales and modern scientific cosmology was, in his words, “no doubt by accident”.
That qualification is easily lost when Sagan is quoted on social media.
What Sagan found intellectually striking was the scale of the Hindu conception of cosmic time. At a period when popular Western culture had often been accustomed to much shorter chronologies, Hindu cosmological literature offered a conceptual universe in which billions of years were not an absurdity but part of a systematic temporal hierarchy.
Sagan also discussed the then-current scientific possibility of an oscillating universe, in which cosmic expansion might eventually give way to contraction and perhaps another expansion.
That was an important scientific idea in its historical context.
It is equally important to recognise that the scientific picture has changed considerably since 1980.
7. The Oscillating Universe: What Has Changed Since Sagan?
When Cosmos was first broadcast, cosmologists were seriously considering whether gravity might eventually halt cosmic expansion and reverse it into a contraction—the celebrated Big Crunch scenario.
The discovery in the late 1990s that the expansion of the universe is accelerating altered the landscape dramatically.
Modern cosmology attributes this accelerated expansion to what is conventionally called dark energy, although its physical nature remains one of the great open questions of cosmology.
The standard cosmological picture describes a universe about 13.8 billion years old that has expanded from an extremely hot, dense early state. Measurements of the cosmic microwave background and other observations have placed the model on remarkably firm observational foundations.
That does not mean that every question has been answered.
We still do not possess a complete physical theory describing the earliest conceivable moment of cosmic history. Nor do we know with certainty whether our observable cosmic history represents the absolute beginning of all reality.
Several theoretical frameworks have explored bouncing or cyclic cosmologies. These are serious areas of theoretical investigation, but they remain distinct from the traditional Hindu cosmological cycle.
The distinction is crucial:
Hindu cosmology says something about cosmic cycles within a religious-philosophical framework.
Scientific cosmology constructs mathematical models whose predictions are tested against observation.
The two may be compared. They should not be conflated.
8. “Big Bang” Does Not Mean “Creation from Absolute Nothing”
Another common misunderstanding deserves attention.
The Big Bang model does not, by itself, amount to the philosophical statement that the universe was created from absolute nothingness at a particular instant.
The scientific model describes the evolution of the observable universe from an extremely hot, dense early state and successfully accounts for several major observations, including cosmic expansion, the cosmic microwave background, and the observed abundance of the light elements.
Questions concerning what, if anything, preceded the earliest describable phase lie beyond what the standard model alone can presently answer.
This is one reason cosmology remains such an intellectually fertile field. The farther back we look, the closer physics approaches questions at the frontier of its own explanatory machinery.
Science is at its most impressive not when it pretends to possess every answer, but when it marks the boundary between established knowledge and an unanswered question.
9. The Difference Between Brahman and “Star-Stuff”
My earlier essay, Cosmos in India — When Carl Sagan Met the Vedas, discussed a possible resonance between the philosophical idea of Brahman and the scientific fact that the chemical elements in living beings have cosmic origins.
That comparison can be evocative, but it needs a sharper boundary.
In Hindu philosophy, Brahman is not simply “the material from which stars and people are made”. In the Upaniṣadic and Vedāntic traditions, Brahman concerns ultimate reality, existence, consciousness, and the ground or principle of being, although different schools articulate the concept in different ways.
Modern astrophysics, meanwhile, tells us that many of the elements essential to life were forged through stellar nucleosynthesis and other cosmic processes. Carbon, oxygen, nitrogen, calcium, iron, and many heavier elements have histories that reach into stars and stellar explosions.
These are two different propositions.
One is metaphysical.
The other is physical and observational.
They may be placed in conversation with one another. One should not be quietly substituted for the other.
This is precisely where a science-and-culture essay should resist the temptation to score a rhetorical point. A comparison becomes more interesting when its boundaries are visible.
10. Cosmic Time as a Change of Perspective
There is perhaps a deeper lesson in all this.
Human beings habitually measure reality by the scale of their own lives. We speak of childhood, adulthood, generations, dynasties, civilisations, and recorded history. Geology then arrives like a stern schoolmaster and reminds us that continents move, mountains rise and erode, and species appear and disappear over intervals vastly exceeding recorded history.
Astronomy is more ruthless still.
Stars are born. They burn. They change. They die. Galaxies collide. Planetary systems emerge from discs of gas and dust. Elements forged inside stars become part of rocks, oceans, atmospheres, organisms, and eventually human bodies.
Against such a backdrop, a human lifetime really is a fleeting interval.
The Hindu cosmological tradition takes that intuition and pushes it to a staggering extreme. Its great units of time are not merely large numbers; they are an intellectual device for making human beings relinquish the assumption that their ordinary clock is the measure of the cosmos.
Modern cosmology performs a similar act of intellectual decentring by observation rather than scripture.
The methods differ.
The enlargement of perspective can nevertheless be profound.
11. A Small Human Life Inside an Enormous Universe
Consider the numerical contrast once more.
A century is tiny beside 4.32 billion years.
The age of the Earth is tiny beside the full hierarchy of traditional Brahmā time.
And the age of the present observable universe is itself only one chapter in the larger question of whether physical reality has an absolute beginning, a prior phase, a cyclic history, or some structure that our present theories have not yet captured.
The point is not to manufacture a contest between Hinduism and physics.
Nor is it to turn physics into a footnote to scripture.
The point is to recognise that human beings have always asked questions larger than the immediate horizon:
- How did the cosmos arise?
- Does it have a beginning?
- Does it have an end?
- Is time linear, cyclic, or something stranger?
- Does dissolution mean annihilation, transformation, or a transition into latency?
- What does “before” mean if time itself has a physical history?
The ancient question and the modern question may sound similar. The methods used to answer them are not.
12. What Science Can—and Cannot—Say About Pralaya
Modern science cannot presently confirm Pralaya as a physical event corresponding to the traditional Hindu timetable.
There is no observational evidence establishing that the universe undergoes a 4.32-billion-year cycle of manifestation followed by a 4.32-billion-year period of dissolution.
Nor is there scientific evidence that the sleep of a divine being constitutes a physical mechanism governing cosmic expansion, contraction, or re-creation.
At the same time, science has not settled every philosophical question concerning cosmic beginnings and ultimate endings.
The universe is observed to be expanding. The expansion is accelerating. Dark energy remains poorly understood. The early universe raises profound questions about the relationship between gravity and quantum physics. Cyclic and bouncing models continue to appear in theoretical cosmology, although they are not established descriptions of our universe.
Thus, the scientifically proper position is neither “modern physics has proved Pralaya” nor “science has disproved every form of cosmic cyclicity”.
The evidence permits something more disciplined: some modern cosmological models explore cyclic behaviour, but no established observation identifies such a cycle with the Hindu cosmological system.
13. The Real Meeting Point Between Tradition and Science
Perhaps the most productive meeting point is not a numerical coincidence at all.
It is the willingness to think beyond ordinary human scale.
The Hindu cosmological imagination asks us to contemplate cycles lasting billions and trillions of years. Modern astronomy asks us to contemplate stellar lifetimes, galactic evolution, cosmic expansion, and a universe whose observable history stretches across 13.8 billion years.
Neither perspective permits the human ego to remain comfortably at the centre of the stage.
Yet there is an important difference in intellectual method.
A cosmological tradition derives meaning from sacred texts, theological concepts, metaphysical reasoning, and inherited systems of thought. Science derives physical knowledge through observation, measurement, mathematical modelling, prediction, testing, replication, and revision.
When these are kept distinct, conversation becomes possible.
When they are collapsed into one another, confusion follows.
14. From an Instagram Reel to a Serious Question
The Reel that prompted this essay is therefore neither wholly right nor wholly without substance.
It has taken a genuine traditional number, a genuine Hindu image of divine repose, and a genuine conception of cosmic dissolution, and compressed them into a sentence suitable for a few seconds of social-media viewing.
The compression is where the trouble begins.
The 4.32 billion years belongs to the traditional reckoning of a Kalpa, Brahmā's day.
The corresponding night is another 4.32 billion years.
The complete traditional day-and-night therefore totals 8.64 billion years.
Viṣṇu or Nārāyaṇa's cosmic repose belongs to the theological imagery surrounding dissolution and re-manifestation, but reducing the whole system to “Viṣṇu sleeps and the universe switches off” does not do justice to the literature.
And modern cosmology does not provide a laboratory confirmation of the traditional timetable.
What it does provide is something no less magnificent: observational evidence that the universe is ancient on a scale that dwarfs human history, that space itself expands, that the cosmic microwave background preserves information from the young universe, and that much about the ultimate nature of cosmic reality remains unresolved.
In that sense, the old question remains alive.
Does the cosmos merely exist, or does it undergo a deeper rhythm of becoming and withdrawal?
Hindu cosmology has one family of answers.
Physics continues to investigate another.
And the proper response to both is not premature certainty, but curiosity.
15. Addendum — The “Camera Flash” and the Human Scale of Time
There is a final irony in the Reel's comparison.
Calling a human life a “camera flash” beside a Kalpa is scientifically inexact as a literal comparison, because a camera flash is a physical event lasting a tiny fraction of a second, whereas a human life is a biological interval of decades.
As a metaphor, however, it works.
The universe has no obligation to conform to the scale of human attention.
We are creatures who count birthdays. Stars count their existence in millions or billions of years. Galaxies evolve over still larger spans. Cosmology forces us to abandon the parochial clock in our heads and confront time as a physical dimension of astonishing depth.
Perhaps that is the most valuable lesson to retain from the Reel.
Not that social media has discovered a secret equation connecting Viṣṇu's sleep with modern physics.
Rather, that an ancient Indian civilisation developed a cosmological imagination capacious enough to speak comfortably of billions and trillions of years—and that modern astronomy has independently revealed a universe in which such numbers are not fanciful at all, but part of the ordinary vocabulary of cosmic history.
That is wonder enough.
Expanded Glossary
- Brahmā
- The creator deity within the familiar Hindu Trimūrti framework. Brahmā should not be confused with Brahman, the philosophical term for ultimate reality in several Hindu traditions.
- Brahman
- A central philosophical concept in Hindu thought concerning ultimate reality. Its interpretation differs among Hindu philosophical schools. It is not simply a scientific synonym for matter, energy, or the universe.
- Kalpa
- A traditional cosmological period identified with one day of Brahmā and conventionally reckoned as 4.32 billion human years.
- Mahāyuga
- A complete cycle of the four Yugas—Satya, Tretā, Dvāpara, and Kali—traditionally lasting 4.32 million years.
- Yuga
- A great chronological age within Hindu cosmological reckoning. The four principal Yugas are Satya, Tretā, Dvāpara, and Kali.
- Pralaya
- A term generally translated as dissolution. Hindu texts distinguish several forms and scales of dissolution rather than treating every Pralaya as identical.
- Naimittika Pralaya
- The periodic dissolution associated with the end of Brahmā's day in Purāṇic cosmology.
- Prākṛtika Pralaya
- A more comprehensive dissolution involving the material elements or Prakṛti in traditional cosmological thought.
- Yoganidrā
- Literally, “yogic sleep” or a state of profound divine repose. In Vaiṣṇava imagery it is associated with the Lord's cosmic repose and the interval surrounding manifestation and dissolution.
- Nārāyaṇa
- A major name of the Supreme in Hindu traditions, particularly prominent in Vaiṣṇava theology, with meanings and theological interpretations varying across textual and philosophical contexts.
- Viṣṇu
- A major Hindu deity associated with preservation in the Trimūrti and worshipped as the Supreme Reality in Vaiṣṇava traditions.
- Śeṣa / Ananta
- The cosmic serpent upon whom Viṣṇu or Nārāyaṇa is traditionally depicted reclining upon the cosmic waters.
- Cosmology
- The scientific study of the origin, large-scale structure, evolution, and fate of the universe.
- Big Bang
- The scientific model describing the evolution of the universe from an extremely hot, dense early state. It is not, by itself, a complete philosophical account of creation from absolute nothingness.
- Cosmic Microwave Background
- The relic electromagnetic radiation from the early universe, released when the universe became sufficiently cool for light to travel freely over great distances.
- Dark Energy
- The name given to whatever physical component is responsible for the observed accelerated expansion of the universe. Its underlying nature remains unknown.
- Oscillating Universe
- A family of cosmological ideas in which expansion may be followed by contraction and potentially another expansion. Such models have a history in theoretical cosmology but are not equivalent to Hindu cosmology.
- Big Crunch
- A hypothetical cosmic future in which expansion reverses and the universe contracts. Current observations of accelerated expansion do not establish that this will occur.
- Cosmic Cycle
- A recurring sequence of manifestation and dissolution. In Hindu cosmology this belongs to a religious and philosophical framework; in physics, cyclic cosmology refers to specific mathematical and physical models.
- Scientific Temper
- A disposition towards inquiry, evidence, critical examination, reasoned doubt, and willingness to revise conclusions in the light of evidence.
References & Further Reading
- Bhagavad Gītā, Chapter 8, Verse 17. The principal scriptural reference for the duration of Brahmā's day and night.
- Bhagavad Gītā, Chapter 8, Verses 18–19. Discusses manifestation and dissolution associated with Brahmā's day and night.
- Viṣṇu Purāṇa, Book I, Chapter III. Contains a detailed traditional reckoning of cosmic time, Yugas, Manvantaras, Brahmā's day, and the associated dissolution.
- Viṣṇu Purāṇa, Book I, Chapter IV. Discusses the nature of dissolution and the cosmic repose of Hari.
- Śrīmad Bhāgavata Purāṇa. Relevant passages concerning Brahmā's day, cosmic manifestation, dissolution, and the role of Nārāyaṇa.
- Carl Sagan, Cosmos: A Personal Voyage, Episode 10, “The Edge of Forever”, 1980. Sagan's discussion of Hindu cosmological timescales, cosmic cycles, and the scientific question of an oscillating universe.
- Carl Sagan, Cosmos, Random House, 1980. The companion book to the television series, including the discussion of cosmic time and Hindu cosmology.
- NASA Science — Universe and Big Bang resources. For the modern scientific account of cosmic expansion, the early universe, the cosmic microwave background, and the approximately 13.8-billion-year age of the observable universe.
- European Space Agency — Planck mission science. For measurements of the cosmic microwave background and the age and composition of the universe.
- Stanford Encyclopedia of Philosophy — Vedānta and Śaṅkara. For philosophical discussions concerning Brahman, Īśvara, manifestation, and dissolution.
These sources should be consulted in their original editions or authoritative online versions for detailed textual, philological, theological, and scientific study.
Copyright & Sharing
© Dhinakar Rajaram, 2026. All rights reserved.
This article is an original work by Dhinakar Rajaram, prepared for educational, scientific, cultural, and public-outreach purposes. Readers are welcome to share the article's link, title, author attribution, and contextual information for non-commercial educational circulation, provided that the work is not altered in a manner that changes its meaning or misrepresents the author's views.
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The Hindu religious concepts discussed here are presented respectfully as part of India's intellectual, philosophical, and cultural heritage. Scientific interpretations are distinguished from theological or traditional claims wherever necessary. Mention of religious, historical, scientific, or public figures does not imply endorsement by those persons or institutions.
© இரா. தினகர், 2026. அனைத்து உரிமைகளும் பாதுகாக்கப்பட்டவை.
கல்வி, அறிவியல், பண்பாட்டு மற்றும் பொதுமக்கள் அறிவியல் பரப்புரைக்காக இக்கட்டுரையின் இணைப்பை, தலைப்பை, ஆசிரியர் பெயரை மற்றும் உரிய சூழல் தகவலுடன் வணிகரீதியற்ற முறையில் பகிரலாம். முழுக் கட்டுரையை மறுபதிப்பு செய்வது, வணிகரீதியாகப் பயன்படுத்துவது, பெருமளவில் மாற்றுவது அல்லது வேறு வெளியீட்டில் பயன்படுத்துவது ஆசிரியரின் முன் எழுத்து அனுமதிக்கு உட்பட்டது.
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