Start with a family
Open the binary explorer, choose a named family, and move its parameter. The fractal is a dense numerical rendering of the attractor. The family controls describe the actual slice being explored; the canonical coordinates show where that slice sits in the larger parameter space.
Try the Koch–wiggle family at 30°, the Heighway dragon at 45°, and the polynomial complex-tree families from the 2018 thesis. Use the free DD, DO and OO controls to leave these special slices.
The DD survey laboratory explores the precomputed angular survey and its linked specimens. The higher-arity explorer opens ternary complex trees. The specimen plates produce deterministic vector comparisons.
The four-dimensional charts
A normalized ordered pair consists of two strict contractions with translations −1 and +1:
f+(z) = +1 + λ+ Cε+(z)
C0(z) = z, C1(z) = conjugate(z)
0 < |λ−|, |λ+| < 1
The two complex multipliers provide four real coordinates. D means direct orientation; O means opposite orientation. DD, DO and OO are three orientation charts. An OD pair is represented after exchanging its ordered branches; symbolic addresses must be exchanged with them.
λ+ = (2(1−w)/ρ) exp(−i(α+γ))
0 < w < 1, ρ > 2 max(w,1−w)
The angular coordinates have the identification (α,γ) ∼ (α+π,γ+π). A torus is a convenient way to draw these periodic coordinates. Its appearance does not establish that the connectedness locus, or its boundary, is itself a torus.
Equal contraction means |λ−| = |λ+|, equivalently w = 1/2. A common linear part imposes the stronger condition λ− = λ+ in DD. These describe different slices.
How an original pair enters the chart
For maps Fj(z) = tj + ujCεj(z), let q be the unique fixed point of their average and d = F+(q) − F−(q). If d ≠ 0, the similarity h(z) = 2(z−q)/d sends the pair to the normalized chart. Direct multipliers remain uj; reversing multipliers become uj conjugate(d)/d. The atlas evaluates this conjugacy explicitly, retaining the original family maps.
If d = 0, both branches fix q and the attractor is a singleton. Such a degenerate pair has no chart with distinct translations ±1.
Curves, marked families and exact slices
A binary zipper joins two endpoint images at an intermediate point p. Its contraction lens is |p| < 1 and |1−p| < 1. Choices of traversal direction and planar orientation give several marked presentations. Endpoint coupling proves connectedness throughout this domain; it does not prove that the parametrized curve is injective.
On the isosceles line p = 1/2 + (i/2) tan θ, both contraction ratios equal 1/(2 cos θ), with −60° < θ < 60°. Different endpoint traversals recover the Lévy, Heighway and Koch–wiggle families. At θ = 30°, two binary levels of the Koch presentation recover its familiar four-piece generator.
Calegari’s Wiggle Island studies the embedded members of the corresponding direct family. The domain of embedded curves is a stricter object than the domain of connected attractors. The atlas preserves that distinction.
| Object | Meaning in the atlas |
|---|---|
| Family / slice | A specified map from one or more parameters into a similarity chart. |
| Connectedness locus | Parameters whose limiting compact attractor is connected. |
| Marked contact | A pair of symbolic addresses representing the same attractor point. |
| Boundary / frontier | Relative to a stated ambient parameter space. A point interior in a slice need not be interior in the full chart; a boundary point in a slice supplies an ambient exterior sequence. |
| Similarity dimension | The number s solving Σrjs = 1. Overlaps can prevent it from equalling Hausdorff dimension. |
The registry includes proved real and product slices, alongside families governed by contact equations and numerical exploration. The family reference records exact formulas, contact identities and thesis pages. Its source notes state the mechanism for each case; an attractive shape is not used to assign a topological classification.
From binary to ternary complex trees
In the master’s-thesis convention, the tree T{z1,…,zm} has branch maps Fj(x) = 1 + zjx. Its tipset is the compact invariant set of these maps. The finite drawn tree records branch segments; it is a different object from the limiting tipset.
The thesis’s ternary family T{z,1/2,1/(4z)} is defined on the strict contraction annulus 1/4 < |z| < 1. The companion with middle ratio −1/2 offers a contrasting deformation. The explorer evaluates these original formulas, rather than treating every ternary system as a binary pair.
An unrestricted list of m complex branch ratios has 2m real coordinates in this rooted tree convention. A named one-complex-parameter family traces a two-real-dimensional slice of that larger space. The current higher-arity interface explores source-grounded slices; it is not a complete survey of the full six-dimensional ternary locus.
The thesis distinguishes tipset-connected and root-connected families and develops contacts from symbolic equivalence relations. See the 2018 master’s thesis for those definitions and their hypotheses.
Read the evidence beside the image
| Label | What it establishes |
|---|---|
| Exact family theorem | A conclusion under the displayed mathematical hypotheses. Coefficients on screen are floating-point evaluations of the stated formula. |
| Numerical exclusion | A finite floating-point calculation found separation. It is not an exact-arithmetic certificate. |
| Finite survivor | The chosen depth and bounds did not exclude the parameter. Connectedness remains unresolved by that calculation. |
| Budget reached | The computation stopped at a resource cap. This is not a geometric conclusion. |
| Exact rational certificate | A separate verifier checked a complete finite cover and a positive separation bound for explicitly specified rational multipliers and an explicit disconnected parameter neighborhood. |
For a binary IFS, connectedness is equivalent to intersection of its two first-level pieces. Finite cylinder enclosures test this condition at increasing depths. The difference view helps inspect possible contacts; a finite graph, point cloud or apparent touch does not prove the topology of the limiting set.
The downloadable rational-verifier documentation explains the independent certificate checker. A certificate for rational coefficients cannot be assigned to a nearby slider parameter without a verified containment bound.
Display resolution and research resolution
The dense renderer samples the attractor with deterministic addresses and refines the drawing after interaction. GPU rendering and a software fallback share the map definitions. A larger point budget or PNG increases visual detail; it does not increase a mathematical proof depth.
The DD laboratory retains 43,173 survey rays: 13 weights, each with a 41 × 81 angular grid. High and ultra meshes interpolate those samples. Their smooth appearance does not add new measurements, prove radial monotonicity or resolve narrow bands missed by the survey.
Exports carry the relevant parameter state and finite calculation settings. Save a session alongside a figure to revisit its coordinates. Vector specimen plates are finite addressed constructions and include their own depth metadata.
Sources and attribution
Mathematical research, complex-tree families and original interactive explorations by Bernat Espigulé. This atlas brings together the 2017–18 research, subsequent papers and the 2026 Canvas and research-atlas prototypes.
- Bernat Espigulé, Complex trees and their families of connected self-similar sets, master’s thesis, Universitat de Barcelona, 2018. Supervisors: Núria Fagella and Xavier Jarque.
- Miwa Aoki, Masayo Fujimura and Masahiko Taniguchi, The shape of the dust-likeness locus of self-similar sets, Journal of Fractal Geometry 1 (2014), 335–347.
- Danny Calegari, Sarah Koch and Alden Walker, Roots, Schottky semigroups, and a proof of Bandt’s conjecture, ETDS 37 (2017), 2487–2555.
- Danny Calegari, Wiggle Island, arXiv:2205.11442, revision of 7 September 2026.
- Danny Calegari and Alden Walker, Laminations and External Angles for Similarity Pairs, arXiv:2608.12774, August 2026.
Individual family entries identify their source and scope. This guide is software documentation; the evolving companion research manuscript remains a separate publication project.