Collinear Fractals · Explore

This explorer requires JavaScript. See the project documentation for mathematical background and reference implementations.

Explorer controls

Example

Number of maps n

3

Selected parameter c

Re(c)

Im(c)

|c|

arg(c) · degrees

Direct mode: |c| > 1, including real parameters. For 0 < |c| < 1, calculations use 1/c. Zero and |c| = 1 are excluded.

Dynamical view

Search limits

Max depth k_max

Max nodes L_max

Maximum depth bounds the selected-point search and capture-level search at pixel centers. Depth 0 tests only the initial trap. Boundary depth separately controls escape coverage. Higher limits may resolve more points and take longer.

Rendering engine

Rendering engine

Initializing renderer…

Automatic mode shows a bounded GPU preview, then refines pixel searches in double precision. Selected search records always use the full numerical settings. Prefix and histogram drawings use their own renderer.

Attractor rendering

Collinear attractor renderer

Boundary depth · 0 = automatic

Increase detail as you zoom

Escape coverage starts at depth 16 for two maps and 12 otherwise, increasing with zoom. Capture levels use the separate maximum search depth k_max.

Color first-level pieces

Collinear attractor opacity

100%

Sharp boundary evaluates the attractor at pixel scale and outlines each colored first-level piece in black. Piece colors follow the first digit. These visual settings do not change the selected search limits.

Layers

Difference attractor ½ E(c, N) Collinear attractor E(c, n) Trap ½ P_c(S, V) Enclosure ½ P_c(S_E, V_E) Inverse-search tree nodes Capture / escape orbits Escape strata

N = 2n − 1. The difference attractor is displayed at half scale.

Colors & finite capture

Capture display

Reveal minimum capture at pixel centers through the maximum search depth. Parameter and Sharp boundary coverage use whole pixels; the half-difference uses point samples. Piece hues and black contours remain. This choice changes colors, not the computed sets.

Capture-depth cycle q

3

Minimum capture levels repeat modulo q in all numerical set layers. Solid color means capture was found but its minimum is not yet established; pale color marks finite escape coverage.

Search palette

This palette colors the difference attractor, single-color attractors, unresolved results and background. Capture shading preserves parameter-set and first-piece hues; overlaps blend their colors.

Search and background colors In-lens capture Undetermined Exterior / background

Companion software for collinear fractals, finite capture, and restricted polynomial roots.

B. Espigule · 2026 v0.2.0-alpha

project repository

Parameter plane c

Refining…

Digits 0/5

F_n,t = {c : c ∈ t + c⁻¹E(c,n)}

Overlay any first-digit subsets. Every later digit uses A_n.

Original digits A_n · together M⁰_n

Complementary digits · together M¹_n

Each checkbox is independent. The M buttons above remain separate overlays. Overlapping selections blend their colors.

Legend

Parameter lens X_n

Selected parameter sets

Minimum capture level, mod 3

Known membership · no assigned capture level

Finite escape coverage

Off-lens trap hit

Unresolved search / precision limit

Exterior

Minimum capture at pixel centers; boundary coverage uses whole pixels. A fixed first digit counts as one step. The selected-point record is separate.

Dynamical plane z

Refining…

Legend

Difference ½ E(c, N)

Collinear E(c, n)

Minimum capture level, mod 3

Known membership · no assigned capture level

Finite escape coverage

Unresolved search / precision limit

Trap ½ P_c(S, V)

Enclosure ½ P_c(S_E, V_E)

Evaluated parameter

Origin 0

N = 2n − 1. The difference attractor is displayed at half scale.

Sharp boundary shades finite-capture levels and retains every first-piece contour.

Capture levels use pixel centers and the maximum search depth. E(c,n) boundary coverage uses whole pixels; the half-difference uses point samples. Whole-attractor capture starts at depth 0.