ExoticHard

Rational Map 1

A degree-2 rational iteration, z²/(z + c), with one zero and a pole that travels as c changes.

Rational Map 1 rendered by FractalPark: a slow-turning fan of curved blades radiates from the pole, with narrow bands recording near-pole orbits.

Overview

Rational Map 1 iterates z² / (z + c). A polynomial such as z² + c has no place where its denominator fails; this map does. At z = −c sits a moving pole. An orbit that brushes it can be hurled outward, then take a quite different route on the following steps.

There is one zero, at z = 0, and one pole, at z = −c. The function still has degree two, yet that pole redraws the plane. Far out, z² / (z + c) grows roughly like z (for large |z|, z² / (z + c) ≈ z), so infinity is not the one commanding fixed point. Basins collect around a finite set of attracting cycles while the pole cuts curved channels and fan-shaped sectors between them.

FractalPark shows parameter planes, where c moves, and Julia views, where c is held still. This guide image is a Julia close-up: curved blades open around the pole, while fine colour bands follow orbits that skim the singularity.

The Mathematics

Rational iteration

z_{n+1}=\frac{z_n^2}{z_n+c}

Each step squares z, then divides by z + c. The numerator is zero at z = 0; the denominator is zero at z = −c, which makes that point a pole. The map has degree two: (z²)/(z + c) is degree‑2 over degree‑1, and after cancelling a common factor it has one zero and one pole, each counted with multiplicity.

Try the pole at close range. Take c = 0.5, so the pole is at z = −0.5, and start at z₀ = −0.49, just to its right. Then z₁ = (−0.49)² / (−0.49 + 0.5) = 0.2401 / 0.01 = 24.01. One step has thrown the orbit far out. Since |z₁| is large, the next step gives z₂ ≈ z₁² / z₁ = z₁ = 24.01: the large value persists instead of snapping back as in the magnet family. Pixel colour records the steps before the orbit crosses the escape threshold.

Unlike polynomial maps, where infinity always attracts escaping orbits, this map is nearly linear far out (z → z). Thrown-out orbits may drift. The escape-time colours therefore spread into broad, gradual bands rather than the abrupt jumps familiar from z² + c.

Visual Characteristics

This is a Julia close-up. Around the pole, curved blades open into sweeping channels that follow the map's local direction. Tight colour bands between them mark orbits that graze the pole and need many steps to cross the escape threshold.

Because large-z motion is nearly linear rather than explosive, the bands change colour slowly and broadly. It reads as a turning fan with the pole for its hub. Move c and the pole moves too; the whole fan reorganizes.

Remix and Examples

Start from the canonical Document

Open the same approved formula state used by this guide, then change the view, coloring, transforms, or animation in the interactive Explorer.

Remix in Explorer Open Formula Editor

Sapphire Fan

View artwork

Frequently Asked Questions

What makes this a rational map?

Its rule is a quotient of two complex polynomials, z² divided by z + c, not one polynomial alone. That permits poles: places where the denominator vanishes and the value becomes infinite. Polynomial maps have no such places, so their dynamics lack this particular detour.

Where is the pole?

Set z + c to zero: the pole is at z = −c. It moves with c and helps draw the curved fan and blades. An orbit passing close by is thrown to a large value, then may drift because the map grows only linearly far from the origin.