The Ayurveda of Ecology
The five great elements as players in a game for the planet, the differential equations that game competes against, and what Āyurveda already knew about the difference between the two.
यावन्तो हि लोके भावविशेषास्तावन्तः पुरुषे,
यावन्तः पुरुषे तावन्तो लोके ॥
yāvanto hi loke bhāva-viśeṣās tāvantaḥ puruṣe,
yāvantaḥ puruṣe tāvanto loke ||
Published 7 October 2026
The World as Patient
Caraka opens his chapter on the person with a sentence that a physician can spend a career inside: the person is commensurate with the world. Agniveśa, hearing it, says what any student would say — we do not follow; say more. Ātreya answers that the parts of the world are beyond counting and so are the parts of a person, but that the gross ones can be named. Six constituents gathered together are called “world”: earth, water, fire, air, space, and the unmanifest brahman. The same six gathered together are called “person.” In the person, earth is form, water is moisture, fire is heat, air is the breath of life, space is the hollows of the body, and brahman is the inner self.[1]
The likeness is usually read in one direction. We take what we know of the world — seasons, rivers, wind, the heat of the sun — and use it to understand a body. That is most of what Āyurveda’s chapters on regimen do. But Ātreya states the sentence both ways round: as many as in the person, so many in the world. If that is meant, then the physician’s way of reasoning ought to work on the world as well. There should be an Āyurveda of the loka, and the modern name for the health of the world’s body is ecology.
This essay asks whether that can be made exact enough to compute. I take the five mahābhūtas as five participants in the life of a planet. I give them a rule taken from the first chapters of Caraka and of Vāgbhaṭa — what is like increases, what is unlike decreases — and write the rule out as a table. The mathematics that fits a table of who gains in whose presence is game theory, so the planet becomes a game with five players. Then I run the same rule a second way, as planetary science would, with partial differential equations on a map, and set the two side by side.
Everything on this page is live. The plates run in your browser, every number in the rules can be changed, and the players can be taken apart into subcomponents and put back together differently. Nothing here is a forecast, and I will say why more than once. It is an essay in method: what a physician’s categories can and cannot see when the patient is a planet.
The five in the order in which the Taittirīya Upaniṣad has them arise, each from the one before, with Ātreya’s reading of each in the body (Caraka, Śārīrasthāna 5.5). Each is drawn throughout this page with the figure commonly used for its tattva: an ovoid, a circle, a triangle, a crescent, a square.[4]
Five Players, and What They Are Made Of
A player needs a name and a share. The names are the five elements, read at the scale of a planet. Space is openness: the clear sky that heat escapes through, and room for anything to be at all. Air is movement: winds, and the gases that ride them. Fire is heat and transformation, from sunlight landing to a forest burning to a cell digesting. Water is the oceans, ice, rain and cloud. Earth is rock, soil and the standing mass of living things. A share is how prominent each is in the planet’s affairs. The five shares always add to one, so that one player’s gain is somebody’s loss.
No physician would leave it there, because no real substance is one element. Caraka says so in the chapter on tastes: every substance, for the purposes of medicine, is made of all five.[5] A thing is called watery or fiery by what predominates in it. Ice is water, but water that has taken on something of the open sky: it is still, and bright, and sends the sun’s heat back out. Wildfire is fire that behaves partly like wind. Soil is earth that holds water.
So each player here is made of subcomponents, and a subcomponent can lean toward a second element. A lean is a proportion. Ice, in the starting arrangement, is seven parts water and three parts space, and whatever the rules say space does to the others, ice does three-tenths of. A subcomponent can also be given a push of its own — a head start, or a handicap, that owes nothing to the other players. That second control looks minor. It turns out to be the most consequential one on the page, and section seven is about it.
The seventeen subcomponents I start with are one reasonable way to cut a planet, not the only one. Human combustion is among them, as a kind of fire that leans toward air, and it starts with a handicap large enough to keep it out. I wanted its arrival to be something a reader does deliberately.
The Rule: Like Increases, Unlike Decreases
Vāgbhaṭa needs half a verse for the law on which the whole of treatment rests.
Caraka has the same law with its philosophical names attached: sāmānya, likeness, is always and for all things the cause of increase; viśeṣa, difference, is the cause of decrease; and each does its work only when it is brought to bear.[2] Heat a hot condition and it grows. Cool it and it shrinks. Every prescription is an application of this.
To use the law on five players, write down, for each pair, whether more of one raises or lowers the other, and by how much. That gives a table of five rows and five columns. Read along a row: this is how that player fares when there is more of each player named across the top.
helps harms. The dashed diagonal is how much each player crowds itself.
Select a cell to see the reasoning behind it.
Now the rule of motion. At any moment each player has a payoff: its own row of the table, weighted by how much of everyone is present. Whoever is doing better than the planet’s average grows, in proportion to how much better. Whoever is doing worse shrinks. That is all. In symbols it is called the replicator equation, and it is the founding equation of evolutionary game theory.[11]
The word “game” needs a caution. Wind does not choose, and rock has no strategy. The games of economics are played by choosers; this kind is not. John Maynard Smith and George Price showed in 1973 that the mathematics of games holds for players that never decide anything, so long as whatever does well becomes more common.[12] Nothing more is assumed here, and Āyurveda’s law asks nothing more either. Vṛddhi and kṣaya, increase and decrease, are what happen; nobody has to intend them.
Where the numbers come from. In the scenario the page opens on, which I call the Earth system, the signs in the table follow familiar physical couplings: water quenches fire, wind feeds it, heat evaporates water, water grows forests. Select any cell and the reasoning appears beside it. The sizes are another matter. They are not measured. I searched for sizes, close to my first guesses, at which all five players persist in the game, and kept those. A table that begins from balance is a choice — the physician’s choice to describe health before disease — and it should be read as a choice. Section six replaces my judgment with a table computed from the texts, and the result is instructive in a different way.
The Same Rule, Run Twice
On the left of the plate below, the planet is one well-stirred pot. There are seventeen numbers, one share for each subcomponent, and there is no such thing as where. This is the game.
On the right, the same rule runs separately at each of 3,200 points on a grid of latitude and longitude. Neighbouring points leak into one another — space and air quickly, fire and water slowly, earth hardly at all — and the sun favours fire toward the equator. A rule of local growth joined to spreading across a surface is a system of partial differential equations,[17] which is the form in which climate and ocean science are actually done.[21] This panel has 54,400 numbers.
Āyurveda would not be surprised that the second panel is needed. A few verses after defining health, having said how the patient and the disease are to be examined, Vāgbhaṭa turns to place.
जाङ्गलं वातभूयिष्ठमनूपं तु कफोल्बणम् ।
साधारणं सममलं त्रिधा भूदेशमादिशेत् ॥
jāṅgalaṃ vāta-bhūyiṣṭham anūpaṃ tu kapholbaṇam |
sādhāraṇaṃ sama-malaṃ tridhā bhū-deśam ādiśet ||
Deśa, place, means both the country and the patient’s body, and the country is itself classified by doṣa. A medicine without a place is not yet a prescription. The left panel is the planet considered without deśa. The right panel puts it back.
One caution about the word. The fifth player, ākāśa, is not this kind of space. Āyurveda keeps the element, the openness in which things can be, apart from diś, direction and location, and what the map supplies is the second.[9]
The game
The equations
The game The map, averaged over the planet
| Read as doṣas | Vāta space and air | Pitta fire | Kapha water and earth |
|---|---|---|---|
| The game | |||
| The map |
- Gap between game and map now
- 0.0 points
- Largest gap this run
- 0.0 points
- Unevenness of the map, which the game cannot see
- 0.0 points
- Computing the map costs
- … times the game
Five scenarios are on the buttons. Earth system is the table described above. Combustion era is the same table with one subcomponent pushed. Emanation chain takes its rules from the order in which the Upaniṣad has the elements arise: each is fed by the subtler one before it and drawn down by the grosser one after.[4] Like increases like computes its table from the qualities Caraka assigns to each element. Five-way contest has nothing to do with Āyurveda; it is a known hard case from mathematical ecology, included because it is where the two panels disagree most.
Two things to try before reading on. Click anywhere on the map to drop a patch of fire, and watch whether it spreads or dies where it landed; the game receives the same disturbance, but only as a slightly larger total. Then set the sun gradient to zero and press Remove geography. The dashed lines fall onto the solid ones and stay there. With nothing to tell one place from another, the equations and the game are the same model.
What the Game Sees: Sāmya
Health, in this definition, is not a quantity of anything. It is a proportion that holds. The game is very good at questions about proportions that hold, because it is small enough to be run hundreds of times in the moment it takes the map to run once. The four answers below are recomputed whenever a rule or a player changes.
Working…
Where does it settle?
Does it come back after a shock?
Who is shut out?
Who makes room for the others?
A score of 1 means a player carries exactly its own weight. Above 1, others persist because it is there. Below 1, it crowds others out.
Each of the four is a physician’s question. Where does it settle asks for the patient’s prakṛti, the proportion this constitution returns to. In the Earth system the game comes to rest with all five present: space 17, air 19, fire 18, water 28 and earth 17 in a hundred. Read as doṣas by the usual composition — vāta of space and air, pitta of fire, kapha of water and earth[10] — that is vāta 36, pitta 18, kapha 46.
Does it come back after a shock is the question of resilience, and the game answers it by nudging the settled mix and timing the recovery. In the Earth system a small disturbance loses half its size about every 55 units of model time. Who is shut out lists what cannot hold a place and how fast it would fade if reintroduced; in the language of the theory, what cannot invade.
The fourth question comes from a different branch of game theory, in which the players form coalitions and the task is to say what each one contributes to the whole. Lloyd Shapley’s answer of 1953 is to average a player’s contribution over every order in which the coalition could have been assembled,[18] and it has been used before to say what each species adds to the diversity of a whole.[19] Here a coalition is worth the number of elements it keeps in play. In the Earth system water scores 1.20 and earth 0.86: water is the player whose presence most lets the others stay, and earth, which takes up room and gives least back, the one that most crowds them.
All of this costs almost nothing. The four answers take thirty-four runs of the game and arrive in well under a second. To get them from the map would take the same thirty-four runs at about two thousand times the cost each.
What Only the Map Sees: Deśa
The game has one number for each player. It cannot say that fire holds one part of the planet and water another, because it has no parts. The plate below takes the map from Fig. 3 and averages it around each circle of latitude.
- At the pole
- At mid-latitude
- At the equator
In the Earth system the planet sorts itself into bands. At the poles earth and water hold more than four-fifths between them and fire is all but absent. At the equator fire leads with two-fifths, space and air take most of the rest, water is down to a fourteenth and earth is gone. Switch the plate to doṣas and the same picture reads: kapha 83 at the pole; vāta 53 and pitta 40 at the equator; and in the middle latitudes a band where vāta and kapha cross and neither leads by much.
That is Vāgbhaṭa’s threefold land. A country that runs to kapha, a country that is mostly vāta, and between them a middling one with its doṣas nearer even. I did not build it in. The rules table says nothing about doṣas or latitude; the bands come from a sun that is stronger at the equator acting through the table.
What this does not show. The resemblance is to the shape of the teaching, three kinds of land distinguished by which doṣa leads. It is not a map of the Earth. Vāgbhaṭa’s jāṅgala is dry scrub country and his ānūpa is marsh; on the real planet the wettest land lies along the equator, where this toy puts its driest. The model has no rain belt, no continents and no seasons. What it does show is narrower and still worth having: a rule with no geography in it, once it is given a place to act, produces lands of different constitution. The game alone cannot produce them.
And place changes the totals as well as the pattern. In the Earth system the game gives water 28 in a hundred; the map gives it 23, because fire’s hold on the tropics is stronger than any planet-wide average of sunlight would suggest. The game is wrong by five points while being right about everything it was asked.
Like Increases Like, Taken Alone
The Earth system table was my judgment. The texts offer a way to do without it. Caraka lists the qualities that predominate in substances of each element,[5] and Vāgbhaṭa arranges the twenty qualities as ten pairs of opposites.[6] So for any two elements one can count: a point for every quality they share, a point against for every pair on which they are opposed. Likeness increases, difference decreases, and the table writes itself.
| Element | Qualities that predominate (Caraka, Sūtrasthāna 26.11) |
|---|---|
| Space | soft (mṛdu), light (laghu), subtle (sūkṣma), smooth (ślakṣṇa) |
| Air | light, cold (śīta), dry (rūkṣa), rough (khara), clear (viśada), subtle |
| Fire | hot (uṣṇa), sharp (tīkṣṇa), subtle, light, dry, clear |
| Water | liquid (drava), unctuous (snigdha), cold, slow (manda), soft, slimy (picchila) |
| Earth | heavy (guru), rough, hard (kaṭhina), slow, stable (sthira), clear, dense (sāndra), gross (sthūla) |
| Shared less opposed | Space | Air | Fire | Water | Earth |
|---|---|---|---|---|---|
| Space | · | +1 | +2 | +1 | −4 |
| Air | +1 | · | +3 | −1 | 0 |
| Fire | +2 | +3 | · | −4 | −2 |
| Water | +1 | −1 | −4 | · | −2 |
| Earth | −4 | 0 | −2 | −2 | · |
Fire and air share four qualities — light, dry, clear, subtle — and are opposed on one, hot against cold: three. Fire and water share nothing and are opposed on four. Each score is multiplied by 0.15 to give the rule, every player crowds itself as before, and the scenario is on the button marked Like increases like in Fig. 3.
Run it. In the game, space, air and fire divide the planet between them, about 30, 35 and 36 in a hundred. Water and earth are shut out completely. The three light, subtle elements are alike, so each raises the others; the two heavy ones are unlike nearly everything, including each other, and have no one to raise them. Read as doṣas the game’s planet is vāta and pitta with no kapha at all: hot, dry, moving, and without moisture or substance.
Likeness alone is a recipe for runaway. That is why the half-verse has a second half.
A physician will recognise this. The same law that explains how a disorder grows is the law by which it is treated, and the two halves are not of equal use. Sāmānya is how one gets into trouble: the hot person who craves heat, the dry season that dries. Viśeṣa is the way out, and Caraka says it three times over in the verses that give the qualities of the doṣas: each is calmed by substances of opposite quality.[2] A world run on likeness alone has nobody administering the opposite.
The map softens the verdict a little, and in an interesting place. Water, gone from the game, survives on the map: about a twentieth of the planet in all, but a fifth of the land at the poles, where fire is too weak to drive it out. Place does for water what no rule in the table does. It gives the unlike somewhere to be. Earth gets no such refuge, and is gone from the map as well.
I do not take this as a finding about the planet. It is a finding about the law, and it is the reason I trust the Earth system table more than this one even though this one is better sourced. Real earth and water are not held in the world only by their resemblance to other things. They are fed. Rain is delivered by wind; soil is laid down by water; fuel is supplied to fire by earth. Those are relations in which one gains from the presence of another that is quite unlike it, and a table of likeness cannot contain them. The Earth system table can, which is why its entries are not symmetric: water helps earth more than earth helps water, and fire takes from earth while earth gives to fire.
One Player with a Push of Its Own
Āyurveda has a chapter on what happens when the world itself is deranged. In the third chapter of his section on measure, Caraka turns from the diseases of persons to the diseases of whole populations, people of different constitutions, diets and habits who nonetheless fall ill together. What they have in common, Ātreya says, is four things: air, water, place and time. When those are deranged, a country is destroyed.[7] Agniveśa asks the obvious next question: and what deranges them?
तन्मूलं वाऽसत्कर्म पूर्वकृतम्; तयोर्योनिः प्रज्ञापराध एव ।
tan-mūlaṃ vā ’sat-karma pūrva-kṛtam; tayor yoniḥ prajñāparādha eva |
Prajñāparādha, the offence against one’s own understanding, is Caraka’s name for the cause that does not come from the elements. Air does not go wrong by itself. Something acts on it that is not part of the balance, and the text is plain that the something is us.
The model has exactly one control of that kind: a subcomponent’s push of its own. Every other influence in the table is a relation, one player’s effect on another. The push is not. It is a term in the payoff that does not depend on who else is present. In the Combustion era scenario I take the Earth system and change one number: the push on human combustion goes from a handicap of 0.25 to an advantage of 0.1.
The game’s response is out of all proportion to the change. Fire rises from 18 to 46 in a hundred, and all of that fire is now human combustion; sunlight’s warmth, wildfire and the fire of metabolism are shut out by it, because the new fire is fire they cannot compete with. Air falls from 19 to 12. Earth falls from 17 to 1. The number of elements effectively in play drops from 4.9 to 3.7. And the response is steep in a way worth noticing: with no push at all, human combustion holds about a twentieth of the planet; at 0.05 it holds nearly a third; at 0.1, nearly half.
The map tells the same story with one difference, and it is the difference of section five again. Earth, which the game writes off at 1 in a hundred, holds 12 on the map. It survives in the high latitudes, where fire is weakest, at nearly two-fifths of the land. The game says earth is finished. The map says earth has retreated to where it can still live.
I want to be careful about what is being claimed. This is not a projection of anything. The steepness belongs to this model, and a different table would give a different curve. The claim is structural, and it is Caraka’s: a system of relations can absorb a great deal of rearrangement among its own members, and is far more exposed to a cause that stands outside the relations and pushes. He adds one more thing in the same chapter that the model, so far, cannot represent at all. Of the four common factors, he says, water is weightier than air, place than water, and time weightier than place, each being harder to escape than the last.[7] This model has air, water and place. It has no time: no seasons, no day and night, no kāla.
Game or Equations?
The question I set out with was how a game among the five elements compares, as a way of modelling a planet, with partial differential equations. Having run both on the same rules, I think the answer divides cleanly.
Without place, they are one model
Take away geography and the map follows the game exactly. This is not a quirk of my code. It is a known result that the replicator equation and the Lotka–Volterra equations of ecology are the same system in different coordinates,[13] and the map is those equations with spreading added. So the game is the equations with deśa removed. On accuracy it can tie them and can never beat them. When it ties, it does so for about a two-thousandth of the computing.
Place changes the totals, and sometimes everything
Each scenario was run for 600 units of model time from the same patchy start, with the game started from the map’s planet-wide average.
| Scenario | Average gap | Largest gap | In play, game | In play, map | What the game missed |
|---|---|---|---|---|---|
| Earth system | 5.1 points | 5.3 points | 4.9 of 5 | 5.0 of 5 | Water: 28 in a hundred in the game, 23 on the map, because fire holds the tropics. |
| Combustion era | 9.5 points | 10.7 points | 3.7 of 5 | 4.2 of 5 | Earth: 1 in the game, 12 on the map, surviving at high latitudes. |
| Emanation chain | 1.2 points | 1.5 points | 4.6 of 5 | 4.6 of 5 | Almost nothing in the totals; only the banding by latitude. |
| Like increases like | 5.1 points | 5.5 points | 3.0 of 5 | 3.5 of 5 | Water: gone from the game, 6 on the map, a fifth of the land at the poles. |
| Five-way contest | 34.5 points | 53.7 points | 2.2 of 5 | 4.9 of 5 | Coexistence itself. The game lurches between near-monopolies; on the map, spiral waves keep all five in play. |
In four of the five the game never strays more than eleven points from the map’s planet-wide totals, and is blind only to where. The fifth is different in kind. In a contest where each player beats the next and loses to the one before, the game lurches from one near-monopoly to another, and on average barely two elements are in play. On the map, spiral waves form and chase one another, and all five persist indefinitely. That space can sustain a diversity which a well-mixed model destroys was seen in real bacteria on a dish in 2002[16] and worked out mathematically in 2007;[15] the lurching itself was described by May and Leonard in 1975.[14] When coexistence itself depends on place, a model without place is not approximately right. It is wrong.
The game answers what the equations make expensive
Whether a balance holds, what can invade it, who is carrying whom: each of these needs many runs, and the game does all of them in less than a second. That makes it the right instrument for deciding which costly simulations are worth running, and for the structural questions that a physician asks first.
| Approach | Keeps track of | Answers well | Cannot see | Cost |
|---|---|---|---|---|
| Evolutionary game left panel of Fig. 3 | One share for each participant | Who persists, who can invade, whether the balance holds | Where, how fast, how much in real units | Microseconds |
| Cooperative game Shapley value, Fig. 4 | The worth of every coalition | Whom the whole depends on | Mechanism and timing | 31 runs of the game |
| Strategic game Nash equilibrium | The options and payoffs of those who choose | What nations, firms or foragers will do, and why agreements hold or fail | The physics the choices act on | Small |
| Box model ordinary differential equations | A few stocks and flows in real units | Planet-wide budgets of carbon, heat and water | Where. It is the game with units attached | Microseconds |
| Reaction and diffusion right panel of Fig. 3 | Every participant at every place | Zones, refuges, fronts, speed of spread | Choice | The game, times the number of grid points |
| Earth system models | Winds, currents, radiation and chemistry on millions of cells | Forecasts in degrees and millimetres that can be checked against instruments | Choice, unless joined to an economic model | Supercomputers |
| Agent-based models | Individuals and chance | Local extinction, small numbers, luck | General laws; every answer needs many runs | Moderate to large |
| Statistical and learned models | Patterns in the record | Near-term prediction where data are rich | Causes, and anything the record has never shown | Varies |
Neither panel is a forecast
The models used to project the real climate and biosphere are partial differential equations built on the conservation of mass, energy and momentum, with constants measured in laboratories. Their output has units, and can be wrong in ways a thermometer will show. For simulating the physical state of the planet, game theory is not their rival, and nothing on this page should be read as suggesting it.
Game theory earns its place in ecology where something chooses or is selected: strategies that evolve, a commons that is shared, treaties that must enforce themselves.[22] A game among the five elements sits at the edge of that territory. Its payoffs are judgments without units. Small models of a whole planet have an honourable history, of which Daisyworld is the best known,[20] and their use has always been to show what kind of thing can happen, never what will. What it gives in return is a map of the relations: who depends on whom, what can enter, what the whole can absorb. That is the kind of knowledge Āyurveda has always claimed, and the comparison lets me say exactly what kind it is. It is knowledge of sāmya. It is not knowledge of deśa, and the tradition never said it was; it listed place separately and told the physician to examine it.
What Part One Has and Has Not Done
It has taken Ātreya’s sentence at its word in the less usual direction and found that it computes. The five elements can be made players; the first law of treatment can be written as their rules; health as a proportion that holds becomes something a small model can test for, perturb and attribute. The exercise also returned three things I did not put in. Likeness taken alone runs away to a world with no kapha in it. A cause that stands outside the relations and pushes does far more than any rearrangement within them. And a rule with no geography in it, given a sun and a surface, makes lands of three constitutions.
It has not made a model of the Earth. The limits are worth setting down plainly.
- The players do not decide anything. This is an evolutionary game, in which what does well grows; bargaining, foresight and promises have no part in it.
- Shares measure prominence. They are not kilograms or joules, and nothing is conserved.
- The sizes in the Earth system table are chosen, not measured, and small changes to them can have large effects.
- The map is a flat cylinder with no continents, currents or weather. Wind exists only as fast spreading.
- The doṣas appear only as sums of elements. A doṣa is more than that: it has seats, functions and a daily and yearly course, none of which is here.
- There is no time. No seasons, no day and night, no ages of life — and kāla is the factor Caraka calls hardest of all to escape.
The last two are where the second part will begin.
How It Works
Each subcomponent a has a composition ca: mostly its own element, partly the one it leans toward. With shares x, the planet’s elemental make-up is y = Σ xb cb, and the payoff to a is
πa = va + ca · M y − κ xa
where M is the table of rules, va is the subcomponent’s own push, and κ = 0.15 is a little crowding that keeps near-twins from being exactly tied. The game is then
dxa/dt = xa (πa − π)
with π the average payoff. The map carries a density ua at every grid point, computes the same payoffs from the local shares, and adds spreading:
∂ua/∂t = Da ∇²ua + ua (πa − π + 1 − ρ)
Here ρ is the total density at the point, held near 1 by the last term, and Da is mobility: space 1, air 0.8, fire 0.3, water 0.2, earth 0.03, in the order of subtlety, and scaled by the mixing slider. (The five-way contest gives all five the same mobility so that none has an advantage.) The sun gradient adds to the payoff of anything with fire in it, in proportion to how far the cosine of its latitude sits above or below the planet-wide mean, so that the game and the map receive the same sunlight in total. The game is integrated by fourth-order Runge–Kutta and the map by Heun’s method, both with a time step of 0.1, on a grid of points that wraps east to west and is closed at the poles. Nothing present ever falls below one part in a billion, so anything can return if conditions change.
The ordering of the mobilities is the one piece of Āyurvedic theory in the map’s machinery: the subtler the element, the further it reaches. I have written about that principle elsewhere on this site.[8]
The scripts
Plain JavaScript with no dependencies; the test scripts run under Node.
References
The texts
- Caraka Saṃhitā, Śārīrasthāna 5 (puruṣa-vicaya), 3–5. Sanskrit as in the Charak Samhita New Edition of the Charak Samhita Research, Training and Development Centre, carakasamhitaonline.com. The epigraph is the opening of 5.3, omitting the bracketed mūrtimanto (“embodied”) that some editions read before bhāva-viśeṣāḥ. The six constituents of world and person, 5.4; earth as form (mūrti), water as moisture (kleda), fire as heat (abhisantāpa), air as breath (prāṇa), space as the hollows (suṣirāṇi), brahman as the inner self, 5.5. The translations on this page are mine.
- Caraka Saṃhitā, Sūtrasthāna 1.44: sarvadā sarva-bhāvānāṃ sāmānyaṃ vṛddhi-kāraṇam | hrāsa-hetur viśeṣaś ca, pravṛttir ubhayasya tu; 1.45 defines the two terms. The qualities of the three doṣas and their calming by substances of opposite quality (viparīta-guṇair dravyaiḥ), 1.59–61. Text at carakasamhitaonline.com.
- Vāgbhaṭa, Aṣṭāṅgahṛdayam, Sūtrasthāna 1: the law of increase and decrease, 1.14ab; health and disease as proportion, 1.20ab; the two kinds of place and the three kinds of land, 1.23cd–24ab. Sanskrit as in the SARIT e-text, sarit.indology.info, whose verse numbers are followed here.
- Taittirīya Upaniṣad 2.1: tasmād vā etasmād ātmana ākāśaḥ sambhūtaḥ | ākāśād vāyuḥ | vāyor agniḥ | agner āpaḥ | adbhyaḥ pṛthivī — “From that Self arose space; from space, air; from air, fire; from fire, the waters; from the waters, earth.” The order used in the emanation-chain scenario.
- Caraka Saṃhitā, Sūtrasthāna 26 (ātreya-bhadrakāpyīya): sarvaṃ dravyaṃ pāñcabhautikam asminn arthe, “every substance, in this context, is made of the five elements,” 26.10; the qualities abounding in earthy, watery, fiery, airy and spacious substances, 26.11, from which the table in section six is taken. Each list in the text ends with the element’s own sense-quality (smell, taste, form, touch, sound), which I leave out of the scoring because no two elements share one. Text at carakasamhitaonline.com.
- Vāgbhaṭa, Aṣṭāṅgahṛdayam, Sūtrasthāna 1.18: the twenty qualities as ten with their opposites — heavy and light, slow and sharp, cold and hot, unctuous and dry, smooth and rough, dense and liquid, soft and hard, stable and mobile, subtle and gross, clear and slimy.
- Caraka Saṃhitā, Vimānasthāna 3 (janapadoddhvaṃsanīya, “on the destruction of communities”): the four factors common to a population, vāyur udakaṃ deśaḥ kāla iti, 3.6; their order of gravity, vātāj jalaṃ jalād deśaṃ deśāt kālaṃ svabhāvataḥ | vidyād duṣparihāryatvād garīyastaram arthavit, 3.9–10; the root of their derangement, 3.20. Text at carakasamhitaonline.com.
On this site
- Renay Oshop, The Strength of Subtlety: Sūkṣmatā and the Locus of Power in Āyurveda (AyurAstro, 8 July 2026). On this site.
- Renay Oshop, Two Skies, One Word: Ākāśa and Diś in Āyurveda (AyurAstro, 8 July 2026) — on why the “space” that is a player here, ākāśa, is not the “space” of direction and location, diś, which is nearer to what the map supplies. On this site.
- Renay Oshop, The Nine Causative Substances (AyurAstro, 16 May 2026), for the composition of the doṣas from the elements: vāta of space and air, pitta of fire with a little water, kapha of water and earth. The sums on this page count pitta as fire alone. On this site.
Games, ecology and equations
- Peter D. Taylor and Leo B. Jonker, “Evolutionary stable strategies and game dynamics,” Mathematical Biosciences 40 (1978), 145–156 — the replicator equation.
- John Maynard Smith and George R. Price, “The logic of animal conflict,” Nature 246 (1973), 15–18.
- Josef Hofbauer and Karl Sigmund, Evolutionary Games and Population Dynamics (Cambridge, 1998), ch. 7, for the equivalence of the replicator and Lotka–Volterra equations.
- Robert M. May and Warren J. Leonard, “Nonlinear aspects of competition between three species,” SIAM Journal on Applied Mathematics 29 (1975), 243–253 — cyclic contests that lurch between monopolies, the behaviour of the game in the five-way contest.
- Tobias Reichenbach, Mauro Mobilia and Erwin Frey, “Mobility promotes and jeopardizes biodiversity in rock–paper–scissors games,” Nature 448 (2007), 1046–1049.
- Benjamin Kerr, Margaret A. Riley, Marcus W. Feldman and Brendan J. M. Bohannan, “Local dispersal promotes biodiversity in a real-life game of rock–paper–scissors,” Nature 418 (2002), 171–174.
- R. A. Fisher, “The wave of advance of advantageous genes,” Annals of Eugenics 7 (1937), 355–369 — growth joined to diffusion makes travelling fronts, the mathematics of the right-hand panel.
- Lloyd S. Shapley, “A value for n-person games,” in Contributions to the Theory of Games II (Princeton, 1953), 307–317.
- Claus-Jochen Haake, Akemi Kashiwada and Francis Edward Su, “The Shapley value of phylogenetic trees,” Journal of Mathematical Biology 56 (2008), 479–497 — the Shapley value applied to biodiversity.
- Andrew J. Watson and James E. Lovelock, “Biological homeostasis of the global environment: the parable of Daisyworld,” Tellus B 35 (1983), 284–289 — a planet regulated by competing life, as a small model.
- Gerald R. North, “Theory of energy-balance climate models,” Journal of the Atmospheric Sciences 32 (1975), 2033–2043 — climate as a diffusion equation in latitude.
- Scott Barrett, “Self-enforcing international environmental agreements,” Oxford Economic Papers 46 (1994), 878–894 — the strategic kind of game theory, applied to the global commons.
Written in October 2026, from a question about whether the five elements could be made to play a game: the first law of treatment written out as a table, a planet run twice from the same rules, the two runs measured against each other, and the verses checked against their editions.
Five players · One rule · Two planets
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