Reading a thermograph

The thermograph is one diagram carrying a position’s entire hot behaviour: what it is worth, how urgent it is, and how both change as the cost of moving rises.

It looks like two lines leaning towards each other. Reading it takes five minutes and repays them.

The axes #

Horizontal: value. Positions to the right are better for Left.

Vertical: temperature. Zero at the bottom, increasing upwards. Height $t$ shows the position as it is after cooling by $t$ — that is, with every move taxed $t$.

At each height there are two marks: the best Left can guarantee at that tax, and the best Right can. Stacking those marks over all heights gives the two walls, left and right.

The left wall is on the right-hand side of the diagram and vice versa, which reads backwards until it is seen once. Left’s wall marks how far right Left can push the value; Right’s marks how far left Right can push it. So the walls face each other with the truth between them.

The bottom #

At temperature zero there is no tax, and the two walls sit at the extremes the players can reach outright.

For $\{5 \mid 1\}$: Left, moving, reaches five; Right reaches one. So the walls start at five and one, four apart. The gap at the bottom is the confusion interval — the range of numbers the position is confused with. $\{5\mid1\}$ is confused with every number strictly between one and five, and greater than every number below one, and less than every number above five.

That gap has a direct reading: it is the range over which the position’s outcome depends on who moves. Wide gap, much at stake.

The two numbers at the bottom have names of their own — the stops — and it is worth seeing them marked on the diagram that has just been drawn without them, because everything above the bottom is these two numbers being taxed.

The walls closing #

As the tax rises, both players’ reachable extremes shrink. Left, taxed $t$ per move, reaches only $5 - t$; Right reaches $1 + t$.

So the left wall runs from five leftwards at slope one, and the right wall from one rightwards at the same slope. They approach and meet.

$$5 - t = 1 + t \implies t = 2$$

They meet at height two, at value three. Height two is the temperature; value three is the mean value. Above the meeting point the walls coincide and rise vertically together — the mast — because a position cooled past its temperature is a number and stays that number however much more it is taxed.

The whole diagram, then: a wedge from a gap at the bottom to a point, and a vertical mast above it. The mast’s horizontal position is the mean; the wedge’s height is the temperature.

Vertices, and why the walls bend #

For a switch the walls are straight, because there is only one option each way. Positions with several options, or options that are themselves hot, produce walls with vertices.

A vertex is a height at which the best option changes. Below it, one option is best; above it, taxed more heavily, another is. The wall bends where the crossover happens.

That is why a thermograph carries more than two numbers. A position with a bent wall behaves differently at different temperatures — an option that is best on a quiet board is not best on a busy one — and the bends record exactly where the switchover lies.

Reading them: a vertex on the left wall at height $t$ means the option holding that wall up has stopped holding it up at board temperature $t$. That is directly actionable, and it is what the diagram gives that a pair of numbers cannot. In practice the option almost never changes hands; what almost always happens is that the option is itself a fight and has just finished cooling, so the height of the bend is that follow-up’s own temperature — which is what a wall with two bends measures.

A bent wall also does something to the bottom of the diagram that a switch cannot, and it is easiest to see on a position whose braces promise a great deal more than its feet deliver.

Recursive construction #

The walls are built from the options’ walls, which is why the diagram works for positions of any depth.

The left wall of $G$ at temperature $t$ is the maximum over Left’s options of that option’s right wall at $t$, minus $t$. The switch of side is the essential part: after Left moves, it is Right to play, so what matters is what Right can then guarantee.

Symmetrically for the right wall, with minimum and plus $t$.

That recursion bottoms out at numbers, whose thermographs are a single vertical mast at their own value with no wedge at all — nothing at stake, so nothing to draw. Every thermograph is therefore a stack of masts and wedges assembled from below, and a bent wall is where two of the sub-diagrams handed over.

Two facts about the construction are worth carrying before it is used. The first is that the shift of $\pm t$ is the entire content of the tax, so a wall’s slope is set by how deep the fight beneath it goes rather than by how large the numbers in it are. The second is that a wall records only the winning option at each height, so a position with six options and one dominant one draws the same wall as a position with one. The construction in full is a rung of its own, and everything below here can be read without it.

A thermograph built by hand #

The recursion is easier to trust after building one, so here is a two-level position taken from the bottom up.

$$G = \{\, \{4 \mid 2\} \mid 0 \,\}$$

The inner switch first. $\{4 \mid 2\}$ has mean three and temperature one. Its left wall runs from four leftwards at slope one, its right wall from two rightwards, and they meet at height one, value three. Above that, a mast at three.

Now the outer position’s right wall. Right’s only option is $0$, a number, whose thermograph is a mast at zero. Right’s wall of $G$ at temperature $t$ is that mast plus $t$: a line from zero running rightwards at slope one.

And the left wall. Left’s only option is the inner switch, so the left wall of $G$ at $t$ is the inner position’s right wall at $t$, minus $t$. Below height one that right wall is at $2 + t$, so the outer left wall is at $2 + t - t = 2$ — vertical. Above height one the inner right wall is the mast at three, so the outer left wall is at $3 - t$, sloping left.

The vertex. At height one the left wall bends, from vertical to sloping. That bend is the signature of the inner switch resolving: below temperature one the inner fight is live and Left’s option is worth what Right can hold it to; above it the inner fight is settled at three and the tax starts to bite.

The meeting. The walls meet where $3 - t = t$, at $t = \tfrac32$, value $\tfrac32$. So $G$ has mean $\tfrac32$ and temperature $\tfrac32$.

Neither number is guessable from the braces. The bend is what makes the difference, and it exists because one of the options was itself hot.

Common misreadings #

Four, in the order they usually occur.

The walls are swapped. Left’s wall is on the right. It marks how far right Left can push the value, and Left prefers larger values, so it is the right-hand boundary of the wedge. Everybody reads this backwards the first time.

The meeting point is not the value. It is at height equal to the temperature and horizontal position equal to the mean. Two different quantities read off one point, and the vertical coordinate is not a value at all.

The bottom gap is not an error bar. It is the confusion interval — the numbers this position is genuinely confused with. Not a range of uncertainty about a true value; the position really does behave differently against different numbers in that range.

The mast is not “the value of the position”. The position is not a number, and the mast is the number it becomes when cooled enough. A position with mean three is not worth three; it is worth three when the moves in it are shared, and worth five or one when they are not.

That last one is the important one. The mean is a summary that is exactly right for adding up and exactly wrong for playing.

The confusion interval is worth one more sentence, because it connects the diagram to the coarser vocabulary the subject starts with. A position is confused with a number when neither is greater than the other and the outcome turns on who moves — the situation the four outcome classes describe with a single fuzzy case. A hot position is confused with a whole interval of numbers rather than just with zero, and the gap along the bottom of the diagram is that interval drawn to scale.

What the solver computed #

thermograph(g, tMax) implements that recursion directly, returning the two walls as lists of vertices — exact piecewise-linear functions rather than samples.

This matters for what the figures are allowed to claim. A sampled wall would let the temperature be read as “where the two drawn curves appear to cross”, which is a statement about the picture. Computed walls let wallAt(w, t) evaluate either wall exactly at any height, and the temperature is the solved intersection of two line segments.

The values were checked by hand before any figure was drawn. $\{5 \mid 1\}$: mean three, temperature two, walls straight with no vertices. $\{4 \mid 2\}$: mean three, temperature one. $\{2 \mid 0\}$: mean one, temperature one. Each matches the switch formula $m = (l+r)/2$, $t = (l-r)/2$, and assertValue re-checks each one before it is drawn.

For a number the routine returns coincident walls and a negative temperature, which is the correct report of a cold position rather than a failure. That case was the one that needed care in the implementation: a search for a crossing finds one immediately at the bottom, and the code must recognise that as “no wedge” rather than “temperature zero”.

Reading one quickly #

In practice a thermograph is scanned rather than analysed, and there are four things to take from it.

Where is the mast? That is the mean value — what the position contributes to the total.

How high is the meeting point? That is the temperature — how urgent the position is.

How wide is the bottom? That is the confusion interval — the range of numbers this position is confused with, and therefore how much the outcome swings on who moves.

Are the walls bent? If so, the best move here depends on the board temperature, and the position needs re-examining as the game cools.

Four glances, and they answer where this component sits in the ordering of moves and what it is worth when the dust settles.

The scale of the axes #

A practical note about drawing these, since the two axes carry different things.

The horizontal axis is a value, measured in whatever units the game uses — moves, points, counters. The vertical axis is a temperature, measured in the same units, since a tax on moving is subtracted from a value.

So the axes have the same units and the walls have slope one. That is a real constraint rather than a drawing convention: a wall can only have slope $\pm 1$ or be vertical, because a wall’s motion with temperature is either “taxed at the full rate”, giving slope one, or “already settled”, giving vertical.

A wall that appeared to have slope one-half would be a drawing error, and the generators here enforce it. The vertices are computed and the segments between them are drawn at the slope the mathematics dictates, so a bend in a wall is always a change between the two allowed slopes rather than a smooth curve.

This makes thermographs easy to check by eye. Every segment is at forty-five degrees or vertical; anything else is wrong. It is the same kind of visual invariant that makes the number line placement of Hackenbush stalks checkable — the picture has a property that a mistake would break.

The mast is the mean, and the mean adds #

One structural fact makes the diagrams worth stacking up.

Mean values add: the mean of a sum is the sum of the means. So a board of components has a total mean equal to the sum of the mast positions, and that total says who is ahead once every hot fight has been shared out fairly.

Temperatures do not add, and there is no operation on thermographs that produces the thermograph of a sum. That is the standing limitation of the whole method, and it is why the diagrams are read one at a time and compared, rather than combined.

The comparison is still worth a great deal. Ordering components by the height of their meeting points orders the moves, and summing the mast positions says who wins if the ordering is respected. Those two facts together are essentially the whole practical content of temperature theory, and the endgame account is what they come to when the arithmetic is written out on a board of several regions.

Two readings of one picture, and only one of them is safe #

A thermograph carries two kinds of information and a reader picks them up in the same glance, so it is worth separating them once, because everything above this rung depends on which of the two is being used.

A height — the temperature, the height at which a wall bends, the height the two walls meet — is a threshold. It says at what tax a decision changes, and a threshold is exactly the kind of quantity that moves when anything is added to the position. Nothing about a height composes: temperatures do not add, and two positions with the same temperature can behave completely differently in company.

A number on the horizontal axis — the mast, the mean, the two stops — is a value. The mean adds across a sum, exactly and always, and the stops translate under addition of a number. Those are the readings a player can carry from one board to another.

So the diagram’s vertical readings are local and its horizontal readings are not, and every result on this anchor is an instance of that split. The mean composes and the temperature does not; the mast of a sum is exact and the feet are wrong; the score adds and the schedule does not.

The habit that follows is worth adopting early. Whenever a thermograph is used to justify something about a board rather than about a position, check whether the quantity being used is a height or a coordinate. A height is a fact about this position alone, and carrying it across a sum is the error this anchor spends four of its rungs measuring.

What the diagram omits #

Worth saying, since a thermograph is a complete description of a position’s hot behaviour and an incomplete description of the position.

It says nothing about which move achieves a wall. The wall records how far a player can push the value; the move that does it is not in the diagram, and two positions with identical thermographs can require entirely different play.

It says nothing about infinitesimal differences. Two positions differing by $\uparrow$ have the same thermograph, because cooling washes out everything below temperature zero. In a game where the numbers cancel, that difference is the whole outcome and the diagram is silent on it. The clearest way to see how little the diagram has to say there is to ask it for $\uparrow$ itself.

It says nothing about the position’s outcome class directly, though the class can be recovered from where the walls sit relative to zero.

And it says nothing about length — how many moves the fight takes — which matters for whether a component gets resolved before the game ends.

So the thermograph answers “how much and how urgent” thoroughly and everything else not at all. That is a good trade for endgame planning and a bad one for the last few moves, which is exactly the point at which the small values take over.

Where it came from #

The thermograph is Conway’s, in On Numbers and Games, and the picture has not changed since. The vocabulary — mast, wall, wedge, confusion interval — is from Winning Ways.

The applied development is Berlekamp’s. Mathematical Go, with David Wolfe, uses thermographs on real endgame positions and introduces refinements needed for Go specifically, including handling of ko and of positions where the sum’s temperature must be estimated rather than derived.

The diagram’s persistence is a small point in favour of it. Forty years of a subject that rewrites its notation regularly, and the thermograph is drawn the way it was drawn first.

Where the model stops #

No thermograph of a sum. The most-wanted operation does not exist. Thermographs describe components.

Exact walls need exact evaluation. The construction is recursive over the options’ thermographs, so it inherits the ceiling on exact computation. Every diagram here is of a position small enough to build exactly.

The temperature of a real board is estimated. Applications use an ambient temperature estimated from the components, not derived. That estimate is where practical Go analysis puts its judgement.

Cold positions have degenerate diagrams. A number’s thermograph is a bare mast. That is correct and uninformative, and it means the diagram is a tool for hot positions only.

Normal play, as everywhere.

Where the ladder goes next #

thermograph opens here with how to read the diagram, and six rungs stand above it.

The endgame accounted for is what the diagram is for: a board of components, each measured, played hottest-first, with the total computed rather than estimated. Temperatures do not add is the first thing that goes wrong with that — the temperature of a sum is bounded by the hottest part and is often far below it, so two hot fights can add to something neither player wants to touch.

A thermograph is built from its options opens the construction rather than the reading, and a thermograph with two bends is what happens when a wall turns more than once — which is the shape every position with a follow-up beneath a follow-up has, and the case most accounts draw and none analyses.

The last two rungs ask what survives a sum. How cold a sum of hot games can be measures where in the permitted range a sum’s temperature actually lands, and finds the collapse to nothing happening only when the two temperatures are exactly equal — commonest when the components are identical. And when two thermographs can be added tries the repair of adding the diagrams pointwise: the mast always comes out right, the walls always bound the true ones, and the whole construction is exact only when at most one component is hot.

So the anchor runs from a reading to a construction to a warning: the diagram is exact about one position, and almost nothing about it composes except the number at the top.