The Geometry of the Equation

Reality = Actual / Expectation

Equations can feel cold on the page. So in this chapter I want to turn this one into shapes, because the shapes are easier to hold, and because they will carry us through the rest of the book.

A point

Start with Actual, the numerator. Picture it as a point.

A point, in geometry, has no size and no direction. It does not stretch left or right, up or down. It simply is, at one location, without dimension. That is a good picture of Actual. It is the finished past, collapsed. It is fixed. It does not move. It is given to you whole.

A line with a dot on it

Now the result, Reality. Picture it as a straight line with a dot somewhere along it.

The dot can slide. It can move toward one end or the other. That sliding is the feeling of your life: the moment the good news comes and something in you lifts, the moment the phone rings at the wrong hour and something in you drops. Where the dot sits is how the moment registers.

Unlike the point, the dot has somewhere to go. It is variable. And what moves it is entirely what lies beneath it.

A rectangle

Now the denominator, Expectation. Picture it as a rectangle.

The rectangle has two dimensions, a base and a height, and they are independent. You can make a rectangle wider without making it taller, or taller without making it wider. Neither side controls the other.

The base of the rectangle is the real part of Expectation, Prediction: the forecast of habit and history. It runs left to right, along the ground of ordinary life.

Independent does not mean unrelated. As we will see, when your bearing crosses a condition you have a name for, the forecast bends toward it for a while. But no amount of accuracy in the forecast ever turns into orientation, and no orientation is a forecast. They are two different kinds of thing, measured along two different directions.

The height of the rectangle is the imaginary part, the Ideal: the lean of ideas. It rises perpendicular to the base, into a dimension that ordinary forecasting does not reach.

Putting them together

Now draw the whole thing. A line with a dot. Then a horizontal bar beneath it, like the bar of a fraction. Above the bar, a point. Below the bar, a rectangle.

Line with a dot = Point / Rectangle

That is the geometry of your experience. A fixed point divided by a two-dimensional shape produces a line along which a dot slides.

The diagonal

There is one more thing to draw, and in the first edition I did not draw it.

Run a line from one corner of the rectangle to the opposite corner. That diagonal is the whole of your Expectation in a single stroke. It has a length, and it has an angle.

The length of the diagonal is how much you are expecting, all things considered. A long diagonal is a large Expectation; a short one is small. When Actual arrives and is compared against it, it is the length that sets where the dot lands on the line. Expect a great deal and receive a little, and the dot drops. Expect little and receive much, and the dot lifts.

The angle of the diagonal is something else. It is the direction your Expectation is pointing: how much of it leans along the ground of Prediction and how much rises toward the Ideal. Two rectangles can have diagonals of exactly the same length and point in very different directions. One is long and flat, nearly all Prediction. Another is tall and narrow, nearly all Idea.

What you can feel, and what you cannot

Here is the part I find most remarkable, and the rest of this book will keep returning to it.

You feel the length. You do not feel the angle.

The dot on the line responds to how much Actual arrived compared with how much was expected. That comparison is the jolt in your body, the lift or the drop. But the direction your expectation was pointing does not register as a feeling at all. There is no instrument in you that reads it. It is real, and it shapes which way a moment turns you, and you cannot feel it turn.

I will spend a whole chapter later on what this means. For now, just keep the picture. A point. A line with a dot. A rectangle with a diagonal. The dot moves with the length. The angle is the part of your life that is happening to you in a direction you cannot sense.

A word about honesty

I want to be clear about the status of this geometry, because a picture can be mistaken for a proof.

The point, the line and the rectangle are ways of seeing. They are not measurements of your brain. When I say Expectation is a rectangle, I mean that it behaves like a quantity with two independent parts, and that picturing it this way lets us reason about it carefully. When I say you feel the length and not the angle, I am making a proposal: the mathematics separates the two, and I am proposing that feeling reads only one of them. As far as I can tell it matches what lived experience reports, but it could fail. It is a proposal about experience, offered as precisely as I can make it. You should test it against your own life, which is the only laboratory that matters here.