A thermograph with two bends

{ {5 | {3 | 1}} | 0} is a fight whose Left option is a fight whose Right option is a fight. Left moving takes the position to {5 | {3 | 1}}, which settles nothing; Right’s answer there goes to {3 | 1}, which settles nothing either; only the move after that reaches a number. Right, on the other hand, ends the whole thing at 0 in one move.

Three levels on one side and one on the other. The thermograph of that position has a Left wall that changes direction twice, at heights 1 and 3/2, before meeting the other wall at 7/4.

Most thermographs drawn on this site bend once or not at all, and a reader who has seen a dozen of them could be forgiven for treating the single bend as the general case. It is not. The number of bends in a wall is a measurement of how deep the fight underneath it goes, and it is the part of a thermograph that survives being asked a question the temperature cannot answer.

Where the bends come from #

The construction is the previous rung’s and is not re-derived here. A thermograph is built from its options’ sets it out: Left’s wall of a position is the largest, over Left’s options, of that option’s Right wall shifted down by the tax, and Right’s wall is the smallest, over Right’s options, of that option’s Left wall shifted up by it. The sides swap at every level.

This rung is what that construction produces when it is run more than twice.

Follow the chain for { {8 | {5 | 3}} | 0}, a position with the same shape and rounder numbers. Left’s wall is built from the Right wall of {8 | {5 | 3}}. That Right wall is built from the Left wall of {5 | 3}. And {5 | 3}'s options are numbers, so the chain stops there.

Three positions in the chain, and each has its own temperature: 3 for the whole thing, 2 for {8 | {5 | 3}}, 1 for {5 | 3}. The two bends in the drawn wall sit at 1 and 2 — the temperatures of the two positions below the top of the chain, in the order the recursion met them.

That is the rule the whole essay rests on. A bend is inherited: it is a height at which some position further down the chain has finished cooling and stopped contributing slope. The chain alternates sides, its temperatures descend, and every one of them below the first appears as a bend.

The temperature is not a bend #

It is tempting to read a two-bend thermograph as a position with two temperatures, the lower of which is “the real one”. That reading is wrong twice over, and getting it right is most of what this rung is for.

The temperature is where the two walls meet, and a meeting is not a bend. For { {8 | {5 | 3}} | 0} the walls meet at 3, and neither wall changes direction there — above the meeting point there is no wall to change direction, only the mast. The bends are at 1 and 2, both strictly below 3.

Every bend is strictly below the position’s own temperature, always. The chain’s temperatures descend, so the highest bend belongs to the immediate follow-up and each lower one to something deeper. The single number a player writes down is therefore the largest of the family, not the smallest, and the bends are what the single number discards on the way down.

Three positions make the progression visible, and all three have temperature 3 and mean value 3. The first is the floor: a fight between two numbers, with nothing underneath it for a bend to be inherited from.

Now replace Left’s option by a fight and ask what the wall does. The question is whether the two numbers move: the swing widens from six points to seven, and a reader who expects a wider swing to mean a hotter position is about to be corrected by one level of nesting.

Put a fight inside that fight and the same thing happens again, one level lower. The chain is now three deep, so there are two temperatures below the top rather than one, and the rule above says both of them should be visible as corners in the same wall.

The flat stretch in each is the sente signature the previous rung named: a segment where the option’s own falling wall is cancelled exactly by the tax the shift adds. What the two-bend case adds is that the cancellation can start and stop and start again, once at each level of the tree.

Three positions one number cannot tell apart #

Grouping the whole census below by the pair (temperature, mean value) turns those three figures into a demonstration rather than a sequence. {6 | 0}, { {7 | 5} | 0} and { {8 | {5 | 3}} | 0} agree on both numbers — temperature 3, mean value 3 — and have no bends, one bend and two bends respectively.

So two positions of the same temperature can have different bend structure, and three of them do. Anything that summarises a position by its temperature and its mean value cannot separate these, and every technique on this site that works by summarising a region does exactly that.

Set the three drawings above side by side and the point is a visual one. All three carry the same pair of numbers in their footers, and the walls are three different shapes: a straight wedge, a wedge with one corner, a wedge with two. The pair of numbers is what a summary keeps and the difference between the three pictures is what it throws away.

The cost of throwing it away is measurable. The endgame account adds up the regions’ mean values and then takes their temperatures in turn, largest first, alternating sign — an estimate that is exact when every region is a plain switch and drifts when one is not.

The account is out by 1 because it read one number off a region that needed two. That is not a defect of the arithmetic; it is the arithmetic being handed a summary that had already discarded the answer.

What the census found #

The claim that two bends need a three-deep chain is checkable by exhaustive search, and it was checked before it was written down.

Family A is a census rather than a selection. Every position {X | Y} was written out with X and Y drawn from the integers −4 to 10 and from the switches {a | b} between them, with at least one of the two a fight: 50,190 positions, of which 17,255 are hot. Of those, 11,690 have two straight walls, 3,787 have a bend in Left’s wall, 2,765 have a bend in Right’s, and 987 have a bend in both. The four counts reconcile exactly, since the 987 are counted twice.

And the number that matters: the most bends in any single wall, over all 17,255, is one.

Two bends in one wall are therefore not rare at this depth — they are impossible. The chain from a wall of {X | Y} runs through at most one hot position, so at most one temperature is available to be inherited. A second bend requires a third level, and putting one there produces them immediately: taking {A | n} for A among the 2,765 whose Right wall bends and n an integer in the same range gives 16,152 hot positions, of which 4,513 have two bends in one wall. The shortest are { {3 | {2 | 1}} | 0}, { {4 | {2 | 1}} | 0} and { {4 | {3 | 2}} | 0}.

That makes a second bend a piece of evidence about the tree rather than about the drawing. A wall that bends twice is a certificate that the position has a follow-up with a follow-up, and no shallower position can forge it.

One count in family A deserves scepticism rather than interpretation. Left’s wall bends in 3,787 positions and Right’s in 2,765, and that asymmetry is a fact about the family and not about the theory: the range −4 to 10 is not symmetric about zero, so the family is not closed under negation, and a lopsided family gives lopsided counts.

Those are the 987 of family A, and they are a different object from the essay’s subject. A position with a fight on each side gets one inherited temperature per side at depth two; a position with a fight inside a fight on one side gets two on that side. The counts separate the two cases and the picture does not, which is a theme of the next two sections — and separating them took an exhaustive sweep of the family rather than a look at any drawing.

The other cause of a bend, and how rare it is #

There is a second thing that can bend a wall, and every account of thermography states it first, this site’s included: a bend is where the option achieving the extremum changes. Left’s wall is an upper envelope over Left’s options, and an envelope bends where the winning option changes hands.

That is true, and over the positions this subject actually contains it is close to never.

Family D is {A, B | 0} with A and B drawn from the integers −2 to 8 and the fights over them: 813,450 positions written, of which 53,306 still have two Left options after reduction to canonical form — the rest lose one to domination or reversibility, and a dominated option can never take a wall over because it never held it. Of those 53,306, 11,117 have a bend somewhere. And of the 11,117, exactly 206 have a bend at which an option changes hands.

Nearly two per cent of one per cent of the family. Every other bend in the subject is an inherited temperature from one level down.

The rarity is easy to underestimate in the other direction too. A first exhaustive search, over 141,570 positions of a slightly different shape, found zero — which is how comfortable it would have been to conclude that an option never changes hands at all, and to write the opposite of the textbook claim with a census behind it.

What the picture cannot show #

Set that last figure beside the hero and the problem is plain. Both are drawn at temperature 7/4 with mean value 7/4, both have two bends, and both bend at exactly 1 and 3/2. One wall shape, two entirely different explanations: in the hero, both bends are inherited temperatures from a chain three deep; in the other, the lower one is an option changing hands between two siblings at the same level.

The picture cannot tell them apart. The heights are identical, the slopes are identical, and nothing in the geometry records which mechanism produced which corner. The figures on this page print the cause in a footer, and that footer is text computed from the option tree, not something read off the drawing.

The second thing a thermograph cannot show is a bend that is above the temperature. {5 | {4 | 0}} has a follow-up {4 | 0} whose temperature is 2, and the position’s own temperature is 1. The bend that follow-up would contribute stands at 2, the walls have already met at 1, and the diagram is clipped there. Drawn beside the four bent walls above, it is the one position on this page whose picture withholds what the others display.

That is not a shortcoming of this drawing but of every drawing of the position, and it is why the census above counts bends in walls rather than fights in trees: the two quantities come apart exactly here. This site’s generator refuses to pretend otherwise. Asking thermograph for the bends of {5 | {4 | 0}} throws rather than drawing an unmarked diagram, because a figure captioned “the bends” with no bends on it would be a claim that the position has none. The right answer is that the diagram cannot report a follow-up hotter than the position, and the honest way to say so is to refuse to draw it. That refusal is also the whole subject of an environment made of coupons, where the play in such a position stops at the follow-up’s temperature and not at the position’s.

Third and smallest: the option set. A wall records only the winner at each height, so a position with six Left options and one dominant one draws the same wall as a position with one. The 53,306 canonical positions above were counted by asking the value how many options it had, not by looking at any picture.

Where the bends go in a sum #

A bend is a local fact, and the question a player actually has is what happens on a board with several regions on it.

The unsatisfying part first: the sum’s temperature is not obtainable by any arithmetic on the parts’ temperatures, which is what temperatures do not add is about. It is bounded by the hottest part’s and is usually below it. Put the one-bend position { {7 | 5} | 0} beside a plain {2 | 0} and the mean values add exactly — 3 and 1 make 4 — while the temperatures do not: the sum’s is 3, the larger of the two rather than their total. The bend is not what fails there; the arithmetic on temperatures fails whether or not anything bends, and where in the permitted range a sum actually lands is the rung that measures it.

What is true, and is the next rung of this ladder rather than this one, is that the thermographs compose better than the temperatures do. The bends carry information the temperatures do not, and a sum theorem that respects them exists, under a condition on the components. Where the condition holds the whole apparatus becomes exact and the alternating account degenerates out of it; where it fails the account is left estimating, as it was above.

Who built the diagram, and the convention it rests on #

Thermography is Conway’s, in On Numbers and Games (1976), and it reached a wide audience through Winning Ways (1982) by Berlekamp, Conway and Guy. The mean value it computes is older than the diagram: Milnor studied positional games with a mean in 1953 and Hanner gave the bound on how far a sum can stray from its parts’ means in 1959, both before there was a wall to draw.

The bends became a practical matter rather than a curiosity with Berlekamp and Wolfe’s Mathematical Go (1994), where thermographs of real Go endgame regions were computed and the regions with structure below the temperature were exactly the ones an orthodox count got wrong. A Go player’s word for a region whose fight has a fight inside it is not “two bends” — the vocabulary there is sente, gote and the follow-up — but the object is the same and the correction is the same size.

Normal play throughout, and it is doing more work here than usual. Both walls are extrema — a maximum for Left and a minimum for Right — and that pairing is the normal-play convention written as arithmetic. Misère play is a different game with the same rules, it has no such pair of envelopes, and nothing on this page transfers to it.

Two smaller conventions decide what the drawings say. The walls are clipped at the temperature, which is why a follow-up hotter than the position leaves no mark at all; the clipping is a drawing decision and the missing bend is a real feature of the position that the decision hides. And a number’s temperature is printed as −1 on this site, a label for an absence rather than a measurement, since a number freezes at every tax and there is no crossing to report.

Every height quoted on this page is exact. The walls are piecewise-linear functions with slopes 0 and ±1 and breakpoints at dyadic rationals, computed by cooling the position symbolically rather than by evaluating it at a series of taxes. A sampled thermograph would look identical and would put every bend wherever the sampling happened to land, which for a bend at 3/2 on a grid of tenths is nowhere.

Where the ladder goes next #

The thermograph of a sum, and the condition its construction needs. The bends are precisely what makes a sum theorem possible, since the temperatures alone are known not to compose. The rung is the statement of that theorem, the hypothesis it carries, and a worked case where the hypothesis fails and the composed diagram is wrong.

Thermographs of loopy games. Positions that can return to themselves have no bottom for the recursion to reach, so the walls cannot be built upward from numbers. Generalised thermography answers this with a pair of diagrams — an optimistic and a pessimistic wall — and the rung is what a bend means when there are two of them and they disagree.

What a bend means in an orthodox Go account. The count a strong player makes is an alternating sum over regions, the same shape as this site’s account, and it goes wrong on exactly the regions whose walls bend. The rung sets a real endgame beside its thermographs and asks how large the correction is in points.

Bends and the ambient temperature. Which component to play in depends on which side of a bend the board is currently on, so a bend is a scheduling instruction as much as a shape. The rung is a board whose correct move order changes as it cools, with the crossing computed.

And the bend that is a boundary, not a corner. The switch a player is imagining shows that a bent-walled position cannot be replaced by any plain switch, because the mismatch is a whole function rather than an offset. The rung after this one is what a sequence of switches — one per bend — can and cannot replace it with, which is the reduced canonical form arriving from the direction of the diagram.