FractalPark
ExoticMedium

Spider

A rational map that adds c/z to a quadratic orbit; the pole at zero pulls out long legs and web-like filaments.

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The Spider fractal: long legs and abyssal filaments radiating from a dark central core
The Spider fractal rendered by FractalPark: long legs and web-like filaments emerge from a dark core, in deep abyssal blues and blacks.

Overview

The Spider formula joins the familiar squared term z² to c/z. It is a degree-2 rational map with two competing pulls. Far from zero, z² runs the orbit much as it does in a quadratic Mandelbrot escape. Close in, c/z takes over and points toward the pole at the origin.

That contest stretches the parameter boundary into branching filaments, not merely a shifted escape threshold. A narrow centre, radiating legs, and dark gaps explain the name.

Because the rule divides by z, z = 0 is singular. Renderers must handle values extremely close to zero with a numerical fallback. c itself may be zero; then the map becomes z² and ordinary quadratic Julia dynamics return.

The Mathematics

Spider iteration

z(n+1) = z(n)^2 + c / z(n)

The reciprocal term gives a quadratic orbit a sharp tug around the pole at zero.

Start with c = 0. The reciprocal term vanishes, leaving z², the standard quadratic Julia map; initial values inside the unit disk can still have bounded orbits.

Now take c = 1 and z₀ = (1, 0). The first iteration gives z₁ = 1² + 1/1 = 2, already far from the origin; the next gives z₂ = 4 + 1/2 = 4.5, and escape is rapid. Begin near zero instead—say z₀ = (0.01, 0)—and 1/0.01 = 100 dominates, throwing the orbit out at once. The spider filaments live on the boundary between those regimes.

Visual Characteristics

Long legs, dark filaments, and web-like junctions run out from a narrow centre. The origin's singularity makes a dark core; from it the threads branch and thin.

The canonical FractalPark rendering uses deep blues and blacks. The rational term breaks full symmetry, leaving the set slightly twisted rather than neatly balanced.

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 ExplorerOpen Formula Editor

Frequently Asked Questions

What happens at z = 0?

At z = 0 the reciprocal term asks for division by zero, which is undefined. The implementation uses a stable fallback for extremely small values, by clamping the magnitude or substituting a small constant. This does not change the visible structure: points entering that near-zero zone are normally flung outward by the large reciprocal value on the next step.

What creates the spider‑like legs?

The legs form where quadratic expansion and reciprocal attraction are nearly balanced. There an orbit can alternate between a pull toward zero and a throw outward, drawing the boundary between bounded and escaping behaviour into long threads. The effect is strongest near the origin, where the reciprocal term is largest; farther out, z² regains control and the threads taper.

Source walkthrough

  1. frm-v1:bf1462942af10c52a5adbecf8856b4e34a8a5110637c84fa613728b76a1d6b4a:init/0

    This cited initialization node performs only: if ismand: z = 0; otherwise z = pixel.

  2. frm-v1:bf1462942af10c52a5adbecf8856b4e34a8a5110637c84fa613728b76a1d6b4a:loop/0

    This cited first loop node performs only: z2 = z * z.

  3. frm-v1:bf1462942af10c52a5adbecf8856b4e34a8a5110637c84fa613728b76a1d6b4a:bailout

    This cited bailout node tests only: |z| <= 256.

Syntax features

initialization-flow

The initialization annotation isolates if ismand: z = 0; otherwise z = pixel rather than a later initialization statement. This is specific to spider.

iteration-flow

The iteration annotation isolates the first loop assignment z2 = z * z; spider squares z before the later magnitude branch chooses an update. This is specific to spider.

bailout-test

The termination annotation identifies the exact continuation comparison |z| <= 256. This is specific to spider.

Parameter experiment

Steps

  1. Render spider with the pinned parameter-plane baseline: center (0, 0), zoom 0.25, rotation 0, and 96 iterations.
  2. Keep center (0, 0), zoom 0.25, rotation 0, and 96 iterations fixed except for iterations, render once at 48 iterations, and compare it with the 96-iteration baseline.

What to observe

Record whether the 48- versus 96-iteration comparison changes anything, and note the source limit |z| <= 256. This is the spider comparison.

Exercise

For spider, use the three annotations to distinguish the single-node operations if ismand: z = 0; otherwise z = pixel, z2 = z * z, and |z| <= 256; explain why no neighboring statement belongs to the loop annotation.

Completion check

Name all 3 annotation IDs, then state exactly that the cited loop node is z2 = z * z and the cited bailout node is |z| <= 256. This completes the spider reading.

Sources and rights

Grounded in runtime source revision bf1462942af10c52a5adbecf8856b4e34a8a5110637c84fa613728b76a1d6b4a and the publication decision for a9263b21-23c1-5d94-ba30-f7bef1a66629.

rightsStatus is project-owned; implementationBasis is project-owned; publicationDecision is publish; decisionReason is publish-project-owned-native-recipe. These records describe this project decision only and do not assert broader rights in any third-party original.

Formula Record

spider

This canonical FractalPark implementation is published and runnable.

Deterministic preview of spider

Canonical source

Language
frm-like/1
Standard library
1
Profile mode
parameter-plane
Profile center
0, 0
Profile zoom
0.25
Profile iterations
96
Profile evidence
family

Parameters

No declared parameters

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Canonical source

This verified canonical revision is read-only. Remix creates a separate editable fork.

Loading canonical source…

Source and implementation

Formula ID
a9263b21-23c1-5d94-ba30-f7bef1a66629
Canonical name
spider
Original name
spider
Historical source
FractalPark
Current implementation
Project-owned

This is a FractalPark project-owned implementation. The linked file is the pinned canonical Definition.

Rights or attribution concern?

Send the Formula ID and supporting evidence. The maintainer can hold or withdraw the implementation while the claim is reviewed. contact@fractalpark.com