ConnectednessBernat Espigulé About the author

Fractal geometry · A research companion

An atlas of
connectedness.

Two similarities generate a fractal. Four continuous parameters reveal how its pieces meet, separate, and form structures that persist.

Mathematics and programme by Bernat EspiguléUniversitat de Girona

Enter the explorer Manuscripts & poster
Two pieces. One contact criterion.
Kp = f−(Kp) ∪ f+(Kp)
Kp connected 2 ∈ Dp

The marked difference set turns the meeting of two fractal pieces into a question about a single point.

Follow the construction

The companion explorer

One parameter. Linked geometries.

Move a locator or travel to a new point. Rotate each view independently, inspect the first pieces, and follow contact through the marked difference.

TypeDirect w0.500 α0.000π γ0.250π ρ1.4142
Numerical radial survey Angular parameter chart Drag to orbit · scroll to zoom · double-click to travel

Select a locator, then drag it on the surface. Double-click a new position to travel there.

A numerical radial survey at the selected weight. Smooth display refinement preserves the sampled evidence; a finite survivor remains unresolved.
Preparing the numerical survey…
Try this example
Synchronized witness

The planar attractor

finite survivor
1×
Drag to pan · scroll to zoom
Deterministic finite approximation; display markers are distinct from audit enclosures.

Preparing the linked attractor and marked difference…

Precise controls

Choose, inspect, and record a parameter

Vary both complex multipliers independently, or enter a named family to explore its own coordinates.

Orientation and family key

Direct: (f₀, f₁), M = Mff; mixed: (f₀, g₁), Mfg; reversing: (g₀, g₁), Mgg.

f₀(z) = −1 + az, f₁(z) = 1 + bz; g₀(z) = −1 + az̄, g₁(z) = 1 + bz̄. Here a = λ₋ and b = λ₊.

A slice imposes parameter equations within an orientation type. A marked stable family preserves the full coding relation on its stated domain; a selected contact alone does not establish stability. Methods and notation ↗

Four-dimensional coordinates

λ₋ = (2w/ρ)eⁱ⁽ᵅ⁻ᵞ⁾   ·   λ₊ = (2(1−w)/ρ)e⁻ⁱ⁽ᵅ⁺ᵞ⁾

Angles in radians. ρ₂(w) = 2√(w² + (1−w)²) is the area-critical reference, not a connectedness test.

rad
rad
Enter complex multipliers

Apply all four components together. The selected orientation type is retained; numerical entries are not exact certificates.

Drawing overlays

Research notebook 0 saved

Save named parameter states in this browser and export a portable JSON collection.

Numerical evidence
Computing selected parameter…

The finite test and image resolve independently.

Display depth —Test depth —Max. tail radius —

Finite survival means unresolved. Browser arithmetic supplies numerical evidence. The separate rational verifier checks explicit cylinder covers and disconnected parameter neighbourhoods.

Reproducible figures

Export current view

The figure is rendered again at the chosen resolution. Exact numeric parameters, computation limits and view settings are embedded in the PNG as JSON.

High-resolution geometry uses the displayed deterministic points and depth. Other depth options rebuild finite attractor cylinders. Inspection bounds are preserved when exporting the current region. Angular surveys, marked differences and contact graphs retain their actual computed resolution. Export never upgrades evidence to a theorem.
Classified collection

Landmark catalogue

Load a reference parameter or compare up to three finite approximations.
Evidence dossier

Finite survivor

Reference classification and finite contact data

Finite computation

Canonical specialization

Source and caveats

What the interface may say

  1. Exact theoremAnalytic classification or exact landmark.
  2. Numerical exclusionAll tested enclosures are pruned; exact replay is required.
  3. Finite survivorUnresolved — never relabeled connected.
  4. ProgrammeStructural interpretation under explicit hypotheses.
Field guide

Definitions, controls and sources

Controls

  • Drag the torus to orbit; use Phase map for explicit angular coordinates and sampled-radius colour.
  • In the dynamical plane, drag to pan, scroll or use +/− to zoom, and Fit or the 0 key to reset. The arrow keys pan. Target 2 inspects the marked-difference neighbourhood.
  • The Research notebook stores named parameters locally and imports or exports a JSON collection.
  • Use the six lenses to keep one parameter while changing representation.
  • Press ← and → to move through the journey.
  • Press 1–6 for Angular chart, Attractor, Marked difference, Symbolic structure, Catalogue and Evidence.
  • Use Learn for the introduction, Focus for a larger figure, or Workspace for precise controls.
  • Press E for the parameter inspector, ⌘/Ctrl-K for navigation, and ? for this guide.

Terminology

Sources and scope
Structural comparison

Compare without losing the shared coordinates
Navigate the atlas

Go to a concept, view or landmark

From geometry to structure

Read the views together.

Each view describes the same pair of contractive similarities. The parameter chart locates the system; its attractor and difference set show what that system creates.

01 / The system

Four continuous coordinates

Weight w, inverse scale ρ, branch angle α, and phase γ determine two complex multipliers. Direct, mixed, and reversing specify whether both, one, or neither of the maps preserve orientation.

f±(z) = ±1 + L±(z)

Here L(z) is λz or λz. The normalized family has four real parameters for each orientation.

02 / The contact

Two pieces, red and blue

The attractor is the union of its two first-level pieces. Their meeting is encoded by the marked difference set, where the target point is always 2.

Dp = L−Kp − L+Kp

Point opacity reveals sampled density. Finite hulls and address depth help interpret the approximation at the selected scale.

03 / The coding

A structure behind the image

Symbolic addresses describe repeated choices of the two maps. Contact fibres record which addresses meet; stable coding quotients organize identifications that persist within a parameter set.

Definitions and exact families

Advanced tree layers lift finite addresses into three dimensions, adding a view of the construction to the planar geometry.

Research and publications

The mathematics behind the atlas.

The main article and research companion are in preparation. The catalogue records their provisional titles, the submitted BMD 2026 poster, and related publications.

Results, evidence, and sources Source code and reproducible tools

Continue your exploration

A connected collection of tools.

Interactive atlas

Six places to begin

Travel across the full angular direct chart with movable locators and high-resolution specimens.

Research laboratory

Inspect the sampled landscape

Study a dense direct-similarity sector survey, finite contact signatures, and reproducible parameter sessions.

Methods and data

Follow the mathematical record

Find definitions, numerical conventions, exact families, sources, and documented data formats.

Authorship and reuse

Explore, record, and cite.

This companion is created and maintained by Bernat Espigulé. Individual publications retain their listed authors. When sharing a figure or a parameter discovery, include its coordinates, orientation type, computation settings, and the relevant source.

Cite the companion

Bernat Espigulé. Four-Dimensional Connectedness Loci: interactive research companion. https://complextrees.com/4D/

BibTeX
@misc{espigule_connectedness_companion,
  author = {Bernat Espigulé},
  title = {Four-Dimensional Connectedness Loci: Interactive Research Companion},
  url = {https://complextrees.com/4D/}
}