Chapter fifteen

The two body problem, and why three of anything is where physics gives up.

In 1687, in a book that is still, on most measures, the most consequential single book on the physical universe ever published, Isaac Newton wrote down a system of equations that completely solved a problem nobody had been able to solve before.

The problem was: how do two objects move when they pull on each other by gravity?

The answer, in Newton's hands, turned out to be one of the most beautiful results in the history of mathematics. Two objects attracting each other by gravity, given any reasonable starting positions and velocities, move in orbits that are described by one of three precisely defined curves: an ellipse, a parabola, or a hyperbola. The motion is exactly predictable, forever, by an algebraic formula a moderately bright fifteen-year-old can write down on a single sheet of paper. The earth and the sun. The moon and the earth. A pair of binary stars, alone in deep space. Two bodies, two equations, one closed-form solution. The system is, in technical terms, integrable, which is the mathematical word for obeys, forever, a clean rule that can be written down.

This is what physics looks like at its best. A clean rule. A complete answer. Two objects, dancing in perfect predictable orbits, forever.

Now add a third object.

The system you have just created, by adding one more object to the two that were so beautifully behaved, has no general closed-form solution. None. Not one that has not been discovered yet. None, in principle, in the entire mathematical universe. The three-body problem in gravitation is, in technical terms, non-integrable. The motion of the three objects can be computed numerically, step by step, with arbitrary precision, but no algebraic formula exists that tells you where the three objects will be at an arbitrary future time. The system was shown, by Henri Poincaré in the 1880s, to exhibit what we now call chaotic behavior: small differences in the starting conditions produce, after enough time, completely different trajectories. Poincaré's work on this problem is the historical origin of chaos theory. The universe, in moving from two bodies to three, has crossed a line, and on the other side of the line, the rules are different in a way no amount of additional mathematics can fix.

I want you to register this carefully, because it is not just an inconvenience. It is one of the most consequential discoveries about the structure of reality that humans have ever made. The universe is willing to write down clean rules for two objects. The universe will not write down clean rules for three. The fact that this is true is, on inspection, also the reason for an enormous amount of suffering in human life, which is what the rest of this chapter is about.

I want to draw the difference, because the figure is worth more than three paragraphs of prose.

Look at the picture, because it tells you something important about what happens when you go from two to three. In the left panel, the orbit is closed. The bodies pass through the same configurations again and again, on a schedule. You could, in principle, mark a wall with the position of each body and check, a thousand years from now, that the marks were still accurate. The right panel does not behave this way. The trajectories cross, loop, almost collide, escape briefly, return. There is no schedule. There is no repeating pattern that a wall could record. The three bodies, given an infinite amount of time, will produce an infinite number of distinct configurations, none of which is exactly the same as any other.

This is the difference between two and three. It is not a difference in degree. It is a difference in kind.

Now I want to apply this, carefully, to human life, because the application is, in my honest opinion, one of the most useful things this book has to offer.

A relationship between two people is, in the relevant abstract sense, a two-body system. There are two participants. There is one relationship. There are forces between them, in both directions. The forces are simple in the sense that they have a single object on either end. I love this person. This person loves me. We are sometimes too close, sometimes too far, but we are pulling on each other and only on each other, and the dynamics, once we have learned them, become reasonably predictable.

This is true whether the two-body system is a marriage, a close friendship, a parent with a single child, two siblings in isolation, or two close colleagues. The dynamics of two people in any sustained relationship are, in technical terms, tractable. They can be learned. They can be predicted, in their broad outlines. Most adults, given enough time, do learn the dynamics of their two-body systems, and the relationships become, over time, more stable and less surprising. This is the version of relationships that books and films like to present, because it has a clean shape that you can write a clean story about.

Now add a third person.

The system you have just created has no general closed-form solution. Not because human beings are particularly complicated, although they are, but because three of any reasonably interacting things have this property. The three of you now have three pairwise relationships (you and A, you and B, A and B) instead of one. But more importantly, you have, for the first time, the possibility of coalitions. Any two of you can be temporarily allied against the third. The allegiance can shift. The shifting can be conscious or unconscious. The third person, perceiving themselves as currently outside the coalition, will, by every available means, try to disrupt the coalition or form a new one. The dynamics are not just more complicated. They are qualitatively different, in the same way that a three-body gravitational system is qualitatively different from a two-body one.

This is why three people in any sustained relationship, whether a family of three, a friendship of three, a workplace team of three, a love triangle, or any other configuration, will tend, by default, toward what feels from the inside like drama. The drama is not a moral failure of the participants. The drama is the system following its own equations of motion, which, for three bodies, do not admit a stable solution.

I want to be careful here, because there is a piece of folk wisdom that says three is a crowd, and I am not arguing for that. Three of anything can work. Three has been the foundation of many beautiful institutions throughout human history. Three can, however, only work if it is structured. Three bodies in space, left to gravity alone, produce chaos. Three bodies in space with the additional constraint of, say, two of them being in fixed orbit around a much larger central body, can produce stable patterns. The structure is what stabilizes the system. Without structure, three bodies will, with mathematical reliability, produce trajectories that nobody can predict and that nobody, in retrospect, will have wanted.

I owe you the autobiographical version of this, because Chapter 2 contains the original three-body problem of my life and I have not yet examined it through the right lens.

When I was four, the system I lived inside contained four bodies: my mother, my father, my sister, and me. The system was, by the standards of family life, working. My mother was the gravitational center of the household. My father, even at his worst, was held in orbit by her presence. My sister and I were small and not yet adding much to the dynamics. The system was a four-body problem, but the masses were so unequal that, in practice, it functioned as a stable hierarchy.

My mother died. The largest mass left the system. My father, no longer held in any orbit, began to drift in the direction he had always been inclined to drift in, which was downward and outward, toward drink, gambling, women, and eventually toward functional absence. My sister and I, now in a system that had collapsed in the middle, were placed into a new system, which was the system of my uncle's house. The new system contained, in the relevant sense, three bodies: my uncle, my sister, and me. My father, although technically alive, was not in the system. He was the gravitational ghost of a fourth body, his presence still felt even though he was not in the room.

This was, in technical terms, the three-body problem I grew up inside. It was not stable. It was not capable of being stable. My uncle, holding two children he had not asked for, with the children's father technically alive somewhere out of frame, was running an equation that had no closed-form solution. The forces did not balance. The ledger could not be balanced, because the ledger was, in the precise mathematical sense, ill-posed.

I do not say this to indict anyone. The whole point of the chapter is that nobody could have produced a stable solution to that system, because the system did not have one. My uncle was not a bad man trying to be good and failing. My uncle was a finite person running an equation that does not admit a steady-state solution. The drift, the resentment, the slow accumulation of debt on the ledger, the final closing of the account when I turned eighteen, were not, in the relevant sense, anyone's fault. They were the system following its own dynamics.

This is, in some quiet way, the kindest thing the mathematics has ever told me. The childhood I had was not, in retrospect, the result of any particular adult failing me. The childhood I had was what happens when three bodies, of which one is a small child, are placed in orbit around the gravitational absence of a fourth, in a culture and an economy that does not provide the structural scaffolding that three-body systems require to be stable. The forces were what they were. The trajectories were what they were. No one, given that starting configuration, could have produced a different outcome.

I make this argument with care, because it can be misused. It can be misused by people who do, in fact, hurt children in ways that are clearly their personal responsibility. The argument is not nobody is ever responsible. The argument is, more narrowly, that some configurations of family are non-integrable, and that the participants in such configurations, including the children, are not the cause of the instability. They are inside it.

Here is a small simulation, because watching the difference between a two-body and a three-body system is, in numbers, more striking than any prose argument I can make.

def step(bodies, dt=0.01):
    """
    Advance a system of point masses one small time step.
    Each body has position (x, y), velocity (vx, vy), mass m.
    Standard Newtonian gravity, with a tiny softening to avoid
    division by zero when bodies pass close to each other.
    """
    new_bodies = []
    for i, b in enumerate(bodies):
        ax, ay = 0.0, 0.0
        for j, other in enumerate(bodies):
            if i == j:
                continue
            dx = other["x"] - b["x"]
            dy = other["y"] - b["y"]
            r2 = dx * dx + dy * dy + 0.001
            r = r2 ** 0.5
            a = other["m"] / r2
            ax += a * dx / r
            ay += a * dy / r
        new_b = dict(b)
        new_b["vx"] += ax * dt
        new_b["vy"] += ay * dt
        new_b["x"]  += new_b["vx"] * dt
        new_b["y"]  += new_b["vy"] * dt
        new_bodies.append(new_b)
    return new_bodies


# Two-body: a clean orbit
two_bodies = [
    {"x":  1.0, "y": 0.0, "vx": 0.0, "vy":  0.5, "m": 1.0},
    {"x": -1.0, "y": 0.0, "vx": 0.0, "vy": -0.5, "m": 1.0},
]

# Three-body: the same starting positions for two of them,
# plus a third body slightly offset.
three_bodies = [
    {"x":  1.0, "y":  0.0, "vx":  0.0, "vy":  0.5, "m": 1.0},
    {"x": -1.0, "y":  0.0, "vx":  0.0, "vy": -0.5, "m": 1.0},
    {"x":  0.0, "y":  1.5, "vx":  0.7, "vy":  0.0, "m": 1.0},
]

# Run both for 5000 steps and look at the final positions.
for _ in range(5000):
    two_bodies = step(two_bodies)
    three_bodies = step(three_bodies)

print("Two-body system, final positions:")
for b in two_bodies:
    print(f"  ({b['x']:+.2f}, {b['y']:+.2f})")
# Approximately the same as 5000 steps earlier, on a stable orbit.

print("Three-body system, final positions:")
for b in three_bodies:
    print(f"  ({b['x']:+.2f}, {b['y']:+.2f})")
# Wildly different. Bodies far apart, one likely flung outward.
# Run it again with a tiny perturbation to one starting velocity
# and the final positions will be completely different from this run.

You can run this. The two-body system, after five thousand steps, will be in approximately the same configuration it started in, give or take some orbital phase. The three-body system, after the same number of steps, will be somewhere unrecognizable. If you change the initial velocity of one of the three bodies by one part in ten thousand, the final position, five thousand steps later, will be entirely different from what it was on the first run. This is what sensitive dependence on initial conditions looks like in code, and it is what chaos theory describes, mathematically, in formal terms.

The unpredictability is not a bug. The unpredictability is a structural property of the system. The three bodies do not know the rules they are following. They are just following them. The rules, given three interacting bodies, simply do not have a closed-form solution. The chaos is not their fault. It is just what physics does on the other side of two.

The practical advice I want to extract from this is gentle and limited, because the chapter is making a structural claim about the universe, and the universe is not, in general, susceptible to advice.

First, when you find yourself inside a three-person system that feels like it is producing more drama than the participants can account for, do not look for the villain. The drama is the system following its equations of motion. There may also be a villain. There usually is not. There is usually a system that, given three reasonably good people, is producing patterns that none of them, taken individually, would have wished for. The triangulation, the alliances, the shifting coalitions, the sense that nothing ever resolves, are not, in most cases, evidence that someone is being a bad person. They are evidence that you are in a three-body system without structure.

Second, three-body systems can be made stable by adding structure. A family of three works, often beautifully, when the structure is explicit: clear roles, clear lines of responsibility, regular and predictable interaction patterns. A workplace team of three works when the hierarchy is clear or when the roles are deliberately complementary. A friendship of three works when the participants have, often without ever saying so, agreed that the three of them are all friends with the same group, rather than three separate friendships sharing a room. The structure is the additional constraint that turns chaos into orbit. Without it, the system has the dynamics it has, and no amount of goodwill from the participants will alter the underlying mathematics.

Third, sometimes the kindest thing you can do for a three-body system is to step out of it. Not because anyone has done anything wrong. But because the system, with you in it, is producing trajectories that nobody, including you, is willing to keep absorbing. The two-body system that remains after your departure will, in most cases, find its own stable orbit much faster than the three-body system did. Your absence is not, in the relevant sense, abandonment. Your absence is, sometimes, the structural change the system needed to become integrable.

Chapter 16 takes a turn into information theory, and into the specific question of why telling the story of your own life is, in technical terms, a lossy compression problem. The chapter will introduce Shannon's notion of entropy as it applies to messages and signals, and it will explain, with some care, why every autobiography, including the one in this book, is necessarily an approximation, and why some approximations are nevertheless better than others.

For now, the page closes here. Two bodies produce orbits. Three bodies produce drama. The drama is not your fault. The structure is what stabilizes three-body systems, and where the structure does not exist, the trajectories are doing what trajectories on the other side of two have always done, which is to refuse, with mathematical certainty, to be predictable.