Why is 2 the point of no return for z² + c?

Pick one number, c, and hold it fixed. Start a second number, z, at 0. Then repeat one rule: square z, then add c. Does z stay near 0, or fly away? That question sits behind the Mandelbrot set, and One Point Orbit lets you test it one step at a time.

▶ Try it yourself in One Point Orbit.

Make a prediction

Take c = 1. The tool draws a circle of radius 2 around 0. Before you press anything, guess: how many steps until z lands outside that circle?

Try it

1. Choose the preset Quick escape · c = 1.
2. Drag the Steps knob all the way down, or press Home on it, so 0 steps are allowed. Only z₀ = 0 is drawn.
3. Press +1 step twice. Each step squares the point, then adds the same arrow, c. You get z₁ = 0² + 1 = 1, then z₂ = 1² + 1 = 2.
4. Look at the stage and the card. z₂ sits exactly on the circle. The callout says “exactly 2: not outside”, and the result reads “Not escaped through 2”.
5. Press +1 step once more. Watch 2 square to 4, then shift by c to 5. The result changes to “Escaped at step 3”.

Why one step outside settles it

The test is strict: a point escapes only when its distance from 0 is greater than 2. Exactly 2 is still inside. Past 2, squaring wins. If |z| is greater than 2 and at least |c|, then |z² + c| is at least |z|² − |c|, which is more than |z|. Every later step lands farther out, so z never comes back. For c = 1 you can continue by hand: 5 becomes 26, then 677. The tool stops at the first escaped point, so those two values are arithmetic here, not something it draws.

What staying inside does not tell you

At two steps, the computer could only say “not escaped yet”. One more step changed the answer. That is the honest limit of any finite run: if z stays within radius 2 for N steps, the result is “not escaped through N”, and nothing more. The Mandelbrot set is the set of values of c whose orbit from 0 stays bounded forever. A finite run can show that a value of c is outside the set. It can never show that one is inside.

Some orbits can be settled by hand instead. With c = −1 the point goes 0, −1, 0, −1 and repeats forever. With c = −2 it goes 0, −2, 2, 2, 2…, sitting on the circle without ever crossing it. The preset c = −0.12 + 0.75i is different. The tool labels it a computed 3-step repeat, because its rounded values repeat exactly. That is a fact about rounded arithmetic, not proof of an exact cycle.

Where the model stops

The tool uses ordinary double-precision arithmetic, shows values to three decimal places, and allows at most 500 steps. It does not draw the Mandelbrot set, zoom into its edge, or show the Julia set for each c. Its one job is to make a single orbit visible: c held fixed, one point moving.

Where the idea comes from

The set is named for Benoit Mandelbrot, whose 1980 paper iterated z → λz(1 − z) for complex λ and z. The z² + c form used here is the standard modern way to write that family. The bounded-orbit definition follows the Mandelbrot-set section of Curtis T. McMullen’s Complex Dynamics and Renormalization.

A next question

When the page opens, the default c = −0.75 + 0.1i has 12 steps allowed and has not escaped yet, but its chip says “Slow escape”. How many steps does it need? Raise the budget and count. The short film for this tool walks through the c = 1 experiment above.