The Ayurveda of Ecology

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.^[6] 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 eight 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 Patient as Found: Ground and Light

Before any rule is set running, the patient should be looked at. Vāgbhaṭa puts examination ahead of everything he has to say about place and time: darśana-sparśana-praśnaiḥ parīkṣeta ca rogiṇam, by looking, touching and questioning let the patient be examined.^[4] So the players do not start from an even split, or from any mix I might find convenient. They start from the Earth as it can be measured, and the question that has to be settled first is what there is to measure.

The five elements are not quantities in any catalogue of the Earth sciences, and they are not all the same kind of thing. Two of them are what the ground is made of. Land is earth and the sea is water, down to the sea’s floor, and a map and a sounding line are enough to tell them apart. The other three cannot be found that way. There is no map of fire, or of moving air, or of open sky. But there is a measure of what those three do, and it is the sunlight, with the wind it raises. Of the light that reaches the ground at a place, some is thrown back unused, some is handed to the air and sets it moving, and some leaves as heat. So I count each place twice: once as ground, for what it is made of, and once as light, for what the sun does on it.

It should be said what kind of claim that is. No text says that ākāśa is light thrown back, or that fire is the heat a surface radiates away. Those are my readings, made so that the five could be measured at all. They are not made from nothing. Caraka gives each element a single mark by which it is known.

Each element on this page is counted by something that bears its mark: the solid of the ground and the liquid in it, the heat that leaves a surface, the light that meets no resistance and is thrown back unused, and, for air, movement. Everything after this section follows from those readings, and none of it can be better than they are. Where this essay says that a place is pitta or kapha, it means as read by these five measures and by the weave of section five, and nothing wider.

The ground

• Earth is land: rock, soil and what stands in it, less the water it holds. It is also the floor of the sea and the rock under the ice.
• Water is the sea, the lakes and the ice, and the water that is in the ground.

Ground has depth, and how deep to count it has to be decided. Counted at the surface alone the sea is water and nothing else, and that cannot be right. Every sea has a floor, and a shelf sea a hundred metres deep is not the same ground as an abyss. So I count the ground of each place as a column, from the surface down to a fixed depth, and for that depth I take the mean depth of the ocean, which is about 3,700 metres.^[19] On land the column is ground all the way down. At sea it is water as far as the floor and ground below that, so that a shallow sea is mostly floor and a sea deeper than the column is all water. The ice sheets are counted as 2,000 metres of ice lying on rock. That is a round figure and not a map of the ice. Antarctica’s averages about two thousand metres,^[20] and Greenland’s is thinner.

How much water is in the ground has to be got from somewhere, and for the land I get it from the rain. Take the rain that falls on a piece of ground in a year, and set it against what the sun there could dry: the energy the ground absorbs and does not radiate, counted as the water it would evaporate if it were spent on nothing else. Ground that gets as much rain as the sun could dry is soaked, and I count soaked ground as half water, which is about what a saturated soil is. Ground that gets less is wet in proportion. So land is never less than half earth. The ground under the sea and under the ice lies under water, and I count it as soaked. So the wettest land and the shallowest sea meet at half and half, and nothing jumps at the coast.

The box in the middle of the Sahara is 97 parts earth in a hundred, and the one on the Amazon 50. By this rule 31 parts in a hundred of the world’s soil are water. The column under the open sea is on average 12 parts floor, and so the open sea is 6 parts earth in a hundred. About 65 parts in a hundred of its floor lie below the column, and in the boxes where all of it does, which are 24 parts in a hundred of the open sea, the ground is still all water. Under the ice the column is 46 parts rock, and the ice sheets are 23 parts earth. Taken together, the planet’s ground is 75 parts water and 25 earth. The count cannot tell fresh water from salt, or soil from bedrock, but it does know where its water and its earth lie. Of the water, 76 parts in a hundred are the sea, 2 the ice, 11 the water of the soaked ground beneath them both, and 11 the water in the land. Of the earth, 68 parts are the land, 29 the floor of the sea and 3 the rock under the ice.

The light

• Space is the part of the sunlight thrown back unused. Sand, snow and ice return much of the light that lands on them, and the sea very little. This is ākāśa in its plainest sense: what offers no resistance and takes no part.
• Air is counted twice over: as the wind, and as the part of the sunlight handed straight to the air as warmth. The first is air’s own mark, its moving. The second is where warmed air begins to rise, and on dry ground it is where most of the sun’s heat goes.
• Fire is the part that leaves the surface as heat, radiated away.

Those three are not all of the sunlight. Of every hundred parts that reach the planet’s surface, 13 are thrown back, 8 warm the air and 25 leave as heat. The other 54 go into the ground itself, most of them to lift its water as vapour and the rest to warm it. I do not count those a second time. The ground they go into has been counted already, as what it is.

Air needs a word more, because it is counted by two things that are not the same. One is the wind: the mean speed of the air ten metres above the ground, from the same reanalysis as the light. Movement is the mark of air in the verse above, and it is vāta’s as well, though in Caraka’s long lists it is the one quality of vāta that his table of the elements gives to none of them.^[3][6] The other is the warmth the ground hands to the air. How the heat a piece of ground absorbs is shared between warming the air and lifting water is a measure of its dryness; how much there is to share is the sun’s. So this second count is greatest where the ground is both dry and bright, and dry is the first word in Caraka’s list for vāta and stands in his list for air.

The air of a place is half the one and half the other. The wind is set against the planet’s mean of 6.3 metres a second and given the planet’s mean warming for its unit, so that over the planet as a whole the two halves come to the same total. That makes them equal in sum and not in sway. From place to place the warming strays from its mean by 16 watts on the square metre and the wind by 6, so that where the two disagree the warming mostly carries it. I have no derivation for the half, or for counting the wind by its speed against the planet’s mean. They are the plain choices. The wind is no part of the sunlight’s account; it is the sun’s work at one remove. Over the land it blows at 3.8 metres a second on average and over the open sea at 7.5, so this half gives the sea an air that the warming alone would not.

Joining the two

The two counts have to be joined, and there is no natural rate of exchange between a share of ground and a share of light. Here I have had to choose a number, as I did for the depth, and what follows leans on the choice, so here it is plainly. The ground of every place counts for a hundred, however much of its column is water. The light of a place counts for as much as there is of it: its three kept parts, the wind’s half of the air among them, in watts, against the sunlight the planet’s surface receives on average, which is 187 watts on the square metre. So the light of the whole planet counts for 46, the light of the Sahara under its fierce sun for 141, and the light of the North Atlantic for 27. The two counts are added, and the total is called a hundred. Fig. 3 lets you count the ground at half a hundred or at two, and to other depths, the air by either of its halves alone, and fire by how warm the place is, and section six says which readings survive that and which do not.

The outlines of land, ice and lakes are Natural Earth’s, and so are the depths of the sea, which it draws as contours at 200 metres, at 1,000, and at every thousand below.^[18] The rain is from the Global Precipitation Climatology Project, which joins rain gauges to satellites.^[25] The light and the wind are from ERA-Interim, the reanalysis of the European Centre for Medium-Range Weather Forecasts,^[21] in the copy prepared for comparing climate models against observation and read off at points ten degrees apart: 19 bands of latitude by 36 of longitude.^[22] Each point stands for the box of ten degrees around it. A reanalysis is a weather-forecasting model corrected every few hours by the world’s observations, run over the whole record. That needs saying, because the warming of the air is measured directly at very few places on Earth; over most of the planet it is computed from winds and temperatures that were measured. “Today” on this page is the average of the ten years 2009 to 2018, the last full decade the reanalysis has.

Counted over the whole planet, the answer is this. In a hundred parts, 51 are water, 17 fire, 17 earth, 9 space and 6 air.

The planet as found, 2009–2018: the five in a hundred parts.

The light can be checked against the standard account of the Earth’s energy. Wild and colleagues, working from direct observations, give for the oceans 185 watts of sunlight reaching each square metre, averaged over the year, of which 170 are absorbed, 53 leave as heat, 16 warm the air and 100 go to evaporation; and for the land 184, 136, 66, 32 and 38.^[23] The reanalysis, read every ten degrees, with each box counted as sea or land by which most of the ground near its centre is, gives 186, 172, 51, 13 and 102 for the sea, and 189, 141, 68, 29 and 42 for the land. As a second check I replaced the reanalysis’s three radiation terms with those the CERES satellite team derive from orbit,^[24] over the years 2001 to 2018 that the two share. The planet’s five shares move by 0.7 points at most. The ground can be checked more simply. Land comes to 28 parts in a hundred of the surface as these boxes read it, against the 29 of the atlases, and 3 of those parts are under ice. The contours give the sea a mean depth of 3,539 metres as these boxes read them, against 3,682 by the best sounding.^[19]

From here on, land means a box where everything within a degree and a half of the centre is land, and sea a box where none of it is. That makes 20 parts in a hundred of the planet’s surface land and 61 sea. The other 19 are coast, part earth and part water, and coast is left out whenever land is compared with sea.

Two things in this count should be pointed at before they are found. The first is that the depth is my choice, as the hundred is. A column of 3,700 metres reaches the floor under the shelves, the ridges and the shallower basins. It does not reach it under the deeper basins and the abyssal plains, which are about two-thirds of the open sea, and in about a quarter of the boxes of open sea it reaches no floor anywhere, so that they have no earth in them at all. A shallower column would leave more of the sea without a floor, and a deeper one less. The second is that ice is water. Greenland and Antarctica are land on the map. In the count they are a little more than half a column of water, with rock beneath. Both have consequences for the doṣas, and section six gives the readings at five other depths and does not hide what moves.

The table gives the same map by bands of latitude, each band taken around the globe. Land is the share of the band that is land, ice included. Sunlight is what reaches the ground, in watts on each square metre, averaged over the year.

And by bands of longitude thirty degrees wide, between sixty south and sixty north.

The belts of the Earth can be read straight off the first table. Water is the largest of the five in every band, and it is 69 to 75 parts in a hundred from forty south to sixty south, where there is hardly anything but ocean. Earth is greatest at sixty and seventy north, where the land is widest and the seas between are shallow, and there is little of it from forty south to sixty south, 3 to 6 parts in a hundred, and as little over the deep water at the north pole. Fire is strongest under the high sun of the tropics and subtropics. Air is spread more evenly, 6 or 7 parts in a hundred from fifty south to forty north but for a dip at the equator, where the winds are light: its warming is the dry subtropics’ and its wind the southern ocean’s. Space is small everywhere but over the ice, where it is 18 to 32 parts in a hundred. The second table shows what longitude adds: the band through Africa and Arabia and the band through the middle of the Pacific lie under the same sun and are quite different places.

What this measure leaves out. The ground is counted to one depth everywhere, and the depth of the sea is known here only by classes a thousand metres wide. The ice is given one thickness. The whole column takes the wetness of its top: land is counted as wet all the way down as the rain makes its soil, and the floor of the sea as soaked all the way down, though rock a mile under either holds far less water than that. The water in the land is inferred from rain and sun and was not measured in the soil. Rivers, snow lying on the ground and the water standing in plants are not counted separately. The light is the ground’s account and not the sky’s: of the sunlight arriving at the top of the air, a little more than a fifth is turned back by cloud and air before it reaches the ground, and about as much again is absorbed on the way down,^[23] and both are left out. A box is one point, read at its centre, and in the tropics it stands for more than a million square kilometres around it. The figures are a ten-year average. The year’s wetness is reckoned from the year’s whole rain against its whole sun, which sets it too high where the rain comes in one season; Part Two tries a soil that carries water from month to month, and under it the world’s soil is 25 parts water in a hundred and not 31. And five numbers and one unit in the measure are my choice: the hundred that the ground counts for against the light’s forty-six, the 3,700 metres, the 2,000 metres of the ice, the half of soaked ground that is water, and the half of the air that is its wind, with the counting of the wind by its speed against the planet’s mean. The depth is the sounded mean of the ocean, the thickness is near what the ice sheets average, and the half of the soil is about what a saturated soil holds; the hundred and the half of the air have nothing behind them but plainness. Nothing in the measure is fitted, and the scripts that make it search over nothing. Section six gives the readings with the hundred, the depth and the air’s half set otherwise, and the foot of the page with the soil’s.

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 thousands 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.

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. It came to the same rest from the measured start, from an even one, and from each of two hundred starts drawn at random.

Set that beside the patient as found: 9, 6, 17, 51 and 17. The game’s constitution is not the planet’s. It agrees that water is the largest of the five, and it is within a point on fire and on earth. But it has little more than half the planet’s water, 28 against 51, and has given the difference to the two lightest: 13 points more of air and 8 more of space. So a game started from the measured planet does not stay there. It closes half the distance to its own rest in about 4.5 units of model time, and nine-tenths in about 70. I have not tuned the table to remove the difference. With twenty sizes free and four numbers to match, a fit would be easy and would prove nothing; the mismatch is the honest measure of a table that was built from couplings and a wish for balance, and never shown the planet. A table built to keep five players even was never going to rest at a planet that is half water.

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,^[33] and it has been used before to say what each species adds to the diversity of a whole.^[34] 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 two or three thousand times the cost each.

The three doṣas, read from the five

The verse at the head of this section speaks of doṣas, and the model has elements. The texts give the bridge between them in a line.

The quickest reading of this is arithmetic. Add space to air and call it vāta, call fire pitta, add water to earth and call it kapha. Two things are wrong with the sums.

The first is that a sum cannot tell a mixture from a heap. A place with a great deal of water and no earth would score as high for kapha as a marsh does. The sentence itself says that vāta comes from the two, in the dual number, and so does kapha. A dual would serve a sum as well, and I do not rest much on it. But I take it at its plainest: a doṣa is what two elements make together, and where one of the two is missing there is nothing for the other to make it with.

The second is pitta. “Fiery” is what the Saṅgraha says, and it is easy to take that to mean fire alone. Pitta’s own description will not allow it, and the description is Vāgbhaṭa’s as much as anyone’s. In the Hṛdayam he gives its qualities as sa-sneha-tīkṣṇoṣṇaṃ laghu visraṃ saraṃ dravam: unctuous, sharp, hot, light, rank-smelling, flowing, liquid.^[4] Caraka’s list is the same in substance, sasneham uṣṇaṃ tīkṣṇaṃ ca dravam amlaṃ saraṃ kaṭu, with sour and pungent added,^[3] and Suśruta’s has it sharp, liquid and hot.^[9] A thing that is hot and also wet is not fire alone. Suśruta asks outright whether there is any fire in the body apart from pitta, and answers that none is found: pitta is fiery by nature, and is spoken of as the inner fire when it burns and cooks.^[9] So the body’s fire is never met alone. It is met as pitta, and pitta is a liquid. That is why the table I give elsewhere on this site, and the teaching I learned, have pitta as fire and water.^[15][16]

So on this page a doṣa is a weave of two strands. I ask three things of a weave. When its two elements stand in the doṣa’s own proportion, it is as large as their sum. As they depart from that proportion it is smaller. When either is absent it is nothing. Many formulas would do that; the one I use is given at the foot of the page.

The proportions, counted from Caraka

A weave needs a proportion, and the texts that give the pairs give no ratios. But Caraka gives the means of counting one. In his first chapter he lists the qualities of each doṣa.^[3] In the twenty-sixth he lists the qualities that abound in substances of each element,^[6] and that second list is tabulated in section seven. So each of a doṣa’s qualities can be looked up: which elements carry it? The table does that for all three. A quality that two elements share is counted for both.

The count gives vāta three parts of air to one of space, pitta equal parts of fire and water, and kapha two parts of water to one of earth. Those are the proportions used on this page and in Part Two. Kapha comes out mostly water, with earth in it for weight and steadiness, and that agrees with Suśruta’s one word for it: pitta is fiery, he says, and kapha is saumya, of the nature of soma, the cool and the moist.^[9]

The count is exact, but the way of counting is mine, and four things about it should be said. Every quality weighs the same, which no physician would quite accept. Pitta’s unctuousness is one I have rounded up: Caraka’s word is sa-sneha, with some oiliness in it, and I counted it as a whole quality of water’s. At half weight, pitta would be four parts of fire to three of water. The tastes in the lists are left out, because Caraka derives tastes from the elements by a rule of their own;^[6] counted in by that rule, sour and pungent would bring pitta to two parts of fire for one of water, and sweet would bring kapha to five of water for two of earth. And the count is untidy about vāta. Four of vāta’s qualities are fire’s as well as air’s: dry, light, subtle and clear. By Caraka’s own table vāta has twice as much in common with fire as with space. I have kept to the two elements the texts name for each doṣa, and the lists themselves are not so strict.

The plate below starts from the counted proportions and lets you change them. Every doṣa reading on the page follows the sliders: the measured globe of Fig. 3, the small table in Fig. 4, its globe when it is coloured by doṣa, and the profile by latitude in Fig. 7. The figures quoted in the text are for the counted proportions.

Two things follow at once. Water belongs to two doṣas, so pitta and kapha are bound to each other through it, and whatever takes water from a place takes something from both. And fire without water is not pitta. Slide the water in Fig. 6 to nothing and pitta goes with it, however fierce the fire. Dry heat, in the texts’ own calendar, is vāta’s.

Now the patient can be read. The planet as found, 9 of space, 6 of air, 17 of fire, 51 of water and 17 of earth, weaves to vāta 8.2, pitta 42.9 and kapha 48.9 of a hundred. Taken whole, the Earth is a kapha-pitta planet: a great deal of water, with earth enough for the one and fire enough for the other, and not much of the dry, moving and empty.

Taken whole is how the game takes it, as one pot. Woven place by place and then averaged, the same planet reads vāta 9.5, pitta 51.8 and kapha 38.6. Pitta and kapha have changed places. The reason is the plainest fact on the map: the planet’s earth and its water are mostly not in the same places. The earth is in the continents and under the shallow seas, and most of the water is in the open sea, where the count finds little floor or none. A weave needs both strands where it is made. That difference between the pot and the map is the subject of the next section, and it is larger for the measured planet than for any of the model’s scenarios that settle.

That vāta is the least of the three holds however the count is made. Which of the other two stands first does not. It turns on whether the planet is taken whole or place by place, as just seen, and for the planet taken whole it turns a little on the count as well. With the tastes counted in, the whole reads 9.0, 36.9 and 54.1, with kapha well ahead. With every weave at equal parts it reads 10.8, 44.4 and 44.8, with the two level. With the satellite’s radiation in place of the reanalysis’s it reads 8.2, 42.3 and 49.4. On the map, by the counted proportions, pitta leads over 58 parts in a hundred of the surface and kapha over 41. With the tastes counted in it is 49 and 51, so that kapha leads the larger part, and the satellite’s radiation changes the leading doṣa on 3 parts in a hundred. Two more things move the map, and move it further: the weight given to the ground against the light, and the depth to which the ground is counted. A third, the half of the air that is counted by its wind, moves the dry land. The first of them turns the planet taken whole as well. With the ground at half a hundred it reads 13, 48 and 39, with pitta ahead. All three are taken up in the next section.

The game’s resting mix weaves to vāta 26, pitta 37 and kapha 37. It has three times the vāta of the planet as found, because it has three times the air and twice the space. The small table in Fig. 4 reads every scenario the same way. For the game it weaves the planet’s shares. For the map it makes the weave at each place from what is there, and averages the places.

Suśruta makes the comparison himself, and in the direction this essay takes it: moon, sun and wind, he says, uphold the world by giving, taking and scattering, as kapha, pitta and vāta uphold the body.^[9] The sun takes, and what it takes is water: the heat spent lifting it comes to 45 parts for every hundred of sunlight that reach the ground. Part Two quotes the verse.

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,^[6] and Vāgbhaṭa arranges the twenty qualities as ten pairs of opposites.^[7] So for any two elements one can count, as section five counted for the doṣas: 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.

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. 4.

Run it. The game starts from a planet that is 54 parts water in a hundred, and loses all of it. 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 slow ones are unlike nearly everything, including each other, and have no one to raise them. Read as doṣas, vāta is the whole of what the three weaves can find. There is no kapha, because there is neither water nor earth. And there is no pitta, though fire holds a third of the planet, because pitta is fire with water in it and the water is gone. That fire is left over. It belongs to no doṣa: heat with nothing wet in it.

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.^[3] A world run on likeness alone has nobody administering the opposite.

The map does not soften the verdict. One might have expected the cold poles to shelter water, since fire is weak there, or the measured oceans to hold out where they began. They do not, though a planet that begins three-fifths ocean takes its time about it. The sun that feeds fire is the same sun that lifts water, and where it is weak both are weak. Water and earth are gone from the map as they are from the game, and the two never differ by more than two points in a hundred. By 13 units of model time less than half of the surface still has the leading doṣa it was measured with, and at rest only the deserts have, which were vāta to begin with.

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.

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, the measured start along with the sun, and the map follows the game to within the error of the arithmetic. 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,^[28] 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 a two- or three-thousandth of the computing.

Place changes the pattern, and sometimes everything

Each scenario was run for 600 units of model time from the measured map, with the game started from the map’s planet-wide average. The last row is the five-way contest run again from a scattered start: an even mix made patchy, with no measured map in it. The button marked Scatter in Fig. 4 starts any scenario that way, and each press draws a different scattering. The first press is the one in the table.

In four of the five the game’s totals stay near the map’s on average, within six points, though early in the run, while the measured map is still being rearranged, they part by as much as nine. 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. Started from the measured planet, the map does the same. The whole globe is soon moving together, and it lurches as one pot would. But the map can do something the game cannot. From a scattered start, spiral waves may form and chase one another round the globe, and then all five persist for as long as I have run it. That space can sustain a diversity which a well-mixed model destroys was seen in real bacteria on a dish in 2002^[31] and worked out mathematically in 2007;^[30] the lurching itself was described by May and Leonard in 1975.^[29] When coexistence itself depends on place, a model without place is not approximately right. It is wrong.

Whether the waves start is another matter, and I have no rule for it. Of twenty scattered starts that differ only in the random numbers that made the patches, fifteen made waves and five lurched as one pot. The measured start lurches. So does the measured map with its averages evened out, so that every form is as common over the planet as every other and only the pattern is kept. It is not a matter of fineness, since the scattered patches are broader and smoother than the measured boxes. Place can rescue the five. It does not always, and the game cannot say when.

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.

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.^[37] 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,^[35] 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.

How It Works

Each subcomponent a has a composition c_a: mostly its own element, partly the one it leans toward. With shares x, the planet’s elemental make-up is y = Σ x_b c_b, and the payoff to a is

π_a = v_a + c_a · M y − κ x_a

where M is the table of rules, v_a 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

dx_a/dt = x_a (π_a − π)

with π the average payoff. The map carries a density u_a at every grid point, computes the same payoffs from the local shares, and adds spreading:

∂u_a/∂t = D_a ∇²u_a + u_a (π_a − π + 1 − ρ)

Here ρ is the total density at the point, held near 1 by the last term, and D_a 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, switches the sun gradient off and lifts the handicap on human combustion, so that none has an advantage.) The sun gradient adds to the payoff of anything with fire, air or water in it, by the amount of the three it contains, in proportion to how far a year of sunlight at its latitude, on a sphere tilted 23.44 degrees, 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. The scattered start multiplies each form’s even share, at each place, by a smooth random factor made of five broad waves, each rippling between one and four times in a turn of the globe. Nothing present ever falls below one part in a billion, so anything can return if conditions change.

The light. For each month at each point, with S the sunlight reaching the ground, R the part thrown back, L the net heat radiated upward, E the heat spent on evaporation, H the heat handed to the air, and G = S − R − L − E − H what goes into ground or sea, write x⁺ for x where it is positive and zero otherwise, and

f = (S − R) / (L⁺ + E⁺ + H⁺ + G⁺)

which is 1 in a month when the surface is taking heat in, and less than 1 when heat that was stored, or heat brought down by the air, is paying part of what leaves. The three kept parts are space R, the warming of the air f H⁺ and fire f L⁺, in watts on the square metre; f E⁺ and f G⁺ go into the ground and are not counted. Over the ten years used here the sun itself pays for 83 parts in a hundred of the evaporation, warming of air and radiated heat, and stored and imported heat pay for the rest. Applied to whole years and not to months, the formula would give the light thrown back, the warming of the air and the fire as 13.0, 9.5 and 28.4 parts of the planet’s sunlight and not 13.0, 8.5 and 25.0.

The air. With v the mean speed of the wind ten metres above the ground in a month, v₀ = 6.3 metres a second its mean over the planet in 2009–2018, and A₀ = 15.8 watts the planet’s mean of f H⁺ over the same years, the air of a box in a month is ½ f H⁺ + ½ A₀ v/v₀, in the same watts. The planet’s air is therefore the same whichever half is counted, and so are the planet’s five shares; what changes is where the air is. The rows of section six with one half alone put the whole weight on it.

The ground. Around each point, twenty-five places within a degree and a half are looked up on Natural Earth’s outlines and depth contours: the share that is land is λ, the share that is land under ice ι, the share that is lake l, and the rest is sea, sorted into classes of depth by the contours at 200 metres, at 1,000 and at every thousand below. The soil is s = λ − ι − l. With P the rain over a period and D the drying power of the sun over the same period, which is the sum of (S − R − L)⁺ divided by the heat it takes to turn a kilogram of water to vapour, the water in the soil is w = ½ min(1, P/D). With C = 3,700 metres the depth of the column, a sea of depth d has floor in the share (1 − d/C)⁺ of its column, averaged across its class of depth; lakes are counted with the shallowest sea; and the ice, taken as T = 2,000 metres thick, has rock in the share (1 − T/C)⁺. Call the sum of those shares over the box u, the ground under water. It is counted as soaked, half water. The ground of the box is then earth s (1 − w) + ½ u, and water the rest. The other depths of section six are the same count with C changed, and the surface alone is u = 0.

The join. With S₀ = 187.2 watts the planet’s mean sunlight at the ground, the five at a box are space R/S₀, the air as just given over S₀, fire f L⁺/S₀, and the ground’s water and earth each times g = 1, all divided by their sum. Over a region the same is done with the region’s mean fluxes and its ground averaged by area, each box weighted by the area of its band of latitude. The sphere’s places take their starting shares from the box they lie in. Four other ways of joining are computed beside this one, and for the three lands of section six they give, in the order dry, ordinary, wet and vāta, pitta, kapha: with the light counted as shares of each place’s own sunlight and not against the planet’s, 30, 37, 33; 14, 40, 47; 9, 36, 55. With the ground fixed at two-fifths of every place and the light at three, 39, 37, 24; 23, 44, 33; 20, 45, 35. With soaked ground counted as a third water, 40, 35, 25; 16, 38, 46; 10, 35, 54; and as two-thirds, 31, 38, 30; 12, 41, 47; 8, 39, 53.

A box is land when every one of the twenty-five places is land, sea when none is, and coast otherwise. The doṣas of a region are woven box by box and averaged by area. The one exception is the planet taken whole, which is the weave of the planet’s pooled shares. Both are given in section five.

The map’s places are the corners of an icosahedron with each face divided in four, four times; each has five or six neighbours. Spreading uses the standard cotangent form of the Laplacian on that mesh, scaled so that a mobility of 1 carries a thing about four and a half degrees of arc in a unit of time. Every planet-wide figure counts each place by the area it stands for.

A weave of two strands a and b, with p the first strand’s share of the recipe, is

W = (a/p)^p (b/(1 − p))^{1 − p}

which equals a + b when a : b is p : 1 − p, is less otherwise, and is zero when either is zero. Vāta is the weave of air and space with p = ¾, pitta of fire and water with p = ½, kapha of water and earth with p = ⅔. These are the counted proportions of section five. The three are then given as shares of their own total.

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.^[13]

The scripts

Plain JavaScript with no dependencies; the test scripts run under Node. The two scripts that read the source files are Python; both need numpy, and the second xarray.