Find your way into the space
Follow six live specimens
Drag a locator, travel to a new point, and rotate the linked attractor and difference views. The coordinates and finite calculation stay with the specimen.
Move through exact families
Choose direct, mixed or reversing similarities, then enter a named curve or contact family. Its original maps and canonical coordinates explain where it sits in four dimensions.
Investigate a sampled feature
Use the direct-similarity laboratory to inspect stored cells, compare numerical depths, and save a state alongside an image.
Connect the image to a result
Follow a named theorem or a complete finite certificate. The guide distinguishes a proved family from a numerical survey.
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 direct, mixed and reversing controls to leave these special slices.
The Direct-similarity 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
Orientation types and manuscript notation
A normalized ordered pair consists of two strict contractions with translations −1 and +1:
h1(z) = +1 + b Cε₁(z)
C0(z) = z, C1(z) = conjugate(z)
0 < |λ−|, |λ+| < 1
Write f₀(z) = −1 + az and f₁(z) = 1 + bz for direct similarities, and g₀(z) = −1 + az̄ and g₁(z) = 1 + bz̄ for reversing similarities. The coefficients are a = λ₋ and b = λ₊, with 0 < |a|, |b| < 1.
| Orientation type | Ordered generators | Connectedness locus |
|---|---|---|
| Direct | (f₀, f₁) | M = Mff |
| Mixed | (f₀, g₁) | Mfg |
| Reversing | (g₀, g₁) | Mgg |
Each type has four continuous real parameters. The names describe the similarities; they do not assign a topology or orientation to the attractor. The mixed convention places the direct generator first. Exchanging branches also exchanges their symbolic addresses. Portable records retain the codes DD, DO and OO, respectively, and the parity pairs (0,0), (0,1) and (1,1).
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). Toroidal coordinates display their periodicity; the projection below specifies how phases and ρ determine a point in the three-dimensional view.
Equal contraction means |λ−| = |λ+|, equivalently w = 1/2. A common linear part additionally requires a = b and equal orientation parity. Equal contraction is a three-real-dimensional slice; a common linear part gives a two-real-dimensional slice.
The attractor and its contact difference
Dp = λ−Cε−(Kp) − λ+Cε+(Kp)
Kp is connected ⇔ 2 ∈ Dp.
The attractor’s first-level pieces are red and blue. The difference view uses the selected specimen colour and marks 2. Both primary explorers use this convention. A legacy laboratory diagnostic may instead display Δ = h0Kp − h1Kp = Dp − 2, with target 0; its label identifies that translation.
The difference is a projection of Kp × Kp. With independent multipliers, its construction must retain two contraction products and two orientation parities. It is not automatically the attractor of a closed four-map planar IFS.
Use reciprocal coefficients outside the unit disk
|c0| > 1, |c1| > 1.
These exterior coefficients express exactly the same strict contraction domain. For a reversing map, conjugation remains part of its inverse. The four-family stable-slice viewer uses the endpoint coordinate c = 1/q; that coordinate differs from the canonically normalized coefficient c0. The family reference records the normalization, including the additional phase of a reversing coefficient.
Named slices within the direct locus
| Slice | Reciprocal coefficients | Contraction weight |
|---|---|---|
| Collinear / common multiplier | (c, c) | w = 1/2 |
| Mirror-symmetric | (c, c̄) | w = 1/2 |
| Quarter-turn | (c, ic) | w = 1/2 |
| Fibonacci | (c, c²) | w = |c|/(|c| + 1) |
Here |c| > 1. Mirror symmetry describes conjugate coefficients in two direct maps. It is separate from a reversing generator. These slices impose parameter equations; they do not by themselves assert marked stability.
Explore the mirror-symmetric slice, three-dimensional sections, and contact families and certified stable patches in their dedicated viewers.
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 proves that the embedded-curve locus in the direct family λ− + λ+ = −1 is disconnected. This is the direct zipper family with traversal signature (−,−); traversal reversal is separate from planar orientation. Its attractors are connected throughout the contraction lens; embeddedness is the additional condition studied there. The atlas places this family alongside the other marked zipper surfaces in common four-dimensional charts.
| 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
The optional 3D address layer in the linked viewers places finite prefix centres at a height given by their contraction scale. It exposes branching and nested addresses. This scale height is a drawing convention; the planar attractor still lies in two dimensions, and the scaffold is not a contact graph.
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.
Exact structure within the larger atlas
The following results are documented in Bernat Espigulé’s working manuscript and are being consolidated in the main article and research companion. Both are in preparation; their PDFs await author review and approval. References in this table identify the 19 September working draft, so they will be updated when the reviewed manuscripts are released. The research ledger records the hypotheses and scope.
| Structure | Result in the manuscript | 19 September draft reference |
|---|---|---|
| Universal bounds | The square-sum criterion gives a connected core; connectedness requires the sum of contraction ratios to be at least one. Real coefficients give the exact outer slice. | Proposition 3.2; Theorem 4.3 |
| Exact phase families | Product and parallelogram models resolve complete slices, including a discontinuous radial frontier. A reversing triangle reaches the connected-core bound at every weight. | Corollary 5.2; Theorems 5.3 and 5.5 |
| Contact fibres | Fixed-address contact fibres are closed and recovered by nested prefix loci, with uniform distance bounds on compact contraction windows. | Theorem 7.1; Proposition 7.2 |
| Structural stability | A prescribed family’s stable parameters retain its forced coding quotient. Removing excess identifications gives a relative Gδ set. | Proposition 6.6; Proposition 7.3 |
| Twelve marked zippers | Endpoint traversal and plane orientation give twelve marked surface presentations. Three reversing signatures have exactly classified stable disks. | Theorem 8.2; Proposition 8.5; Theorem 8.6 |
| Finite stability certificates | Seven endpoint-return criteria reduce embeddedness to finitely certified separations. Outward bounds can establish a whole closed parameter cell. | Proposition 8.7 |
| Open separation regions | Complete finite prefix covers establish disconnection. A positive gap supplies an explicit neighbourhood of disconnected parameters. | Proposition 9.1; Lemma 9.2; Appendix A |
For the direct chart, the sufficient condition |λ−|² + |λ+|² ≥ 1 for connectedness also follows from Aoki–Fujimura–Taniguchi, Theorem 1.1: their normalized holomorphic dust-likeness locus lies in the strict square-sum region. The binary equivalence between dust-likeness and disconnectedness is recorded separately in their Remark 1.1. The manuscript’s Proposition 3.2 gives the bound for all three orientation charts.
Four stable slices, twelve curve examples
Open the four-family stable atlas to rotate the projected sheets and follow the twelve linked curve examples included in that numerical study.
The dated four-family study compares two direct and two mixed zipper presentations (legacy keys DD(1,0), DD(1,1), DO(1,0) and DO(1,1)). These are two-real-dimensional zipper surfaces inside four-real-dimensional orientation charts. Each coloured stable region is a finite inner cover made of certified parameter cells. The reflected direct presentation (legacy key DD(0,1)) remains in the full dictionary; displaying it again would duplicate that comparison.
The four-family viewer uses the canonical multiplier phases θ− = arg λ−, θ+ = arg λ+ and the following display map:
(4 + ρ cos θ−) sin θ+, ρ sin θ−).
For connected parameters, ρ ≤ 2. The distance from Φ to the display’s vertical axis is therefore 4 + ρ cos θ− ≥ 4 − ρ ≥ 2, which creates the central opening. Colour records the remaining coordinate, the contraction weight w. Patches that cross in this projection can have different weights and be disjoint in four dimensions.
For a coding map πp, let Ep = {(ω,ν) : πp(ω) = πp(ν)}. A marked stable family keeps this complete relation constant on its stated domain. Its attractors are canonically homeomorphic by matching the same addresses, with the two labelled generators intertwined.
A selected address equality guarantees a persistent contact. Stability requires control of every additional identification; a finite observed pattern does not establish the full relation. A stable patch, its algebraic contact carrier, and its ambient-boundary status are separate objects. The dated study above retains the scope of its certified cells.
Open questions include local contact geometry near exact critical examples, intersections of address fibres, stability in other prescribed families, and finite capture with independently varying multipliers.
Read the evidence beside the image
| Label | What it establishes |
|---|---|
| Established literature | A published result of the cited authors, applied with its stated family, orientation and hypotheses. |
| Published work by Espigulé and collaborators | Results such as Collinear Fractals and Bandt’s Conjecture, within the specified collinear families and ranges of arity. |
| Current manuscript result | A proof in the dated author manuscript, with its reference and hypotheses. The results table above belongs to this category. |
| Exact family theorem | A conclusion under the displayed mathematical hypotheses. Coefficients on screen are floating-point evaluations of the stated formula. |
| Certified stability cell | Outward bounds verify all required endpoint-return separation inequalities throughout the recorded parameter cell. |
| 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. |
| Computational observation | A pattern in the recorded samples, depths and parameter window; further computation or proof is needed to establish a general statement. |
| Conjecture / open question | A proposed or unresolved mathematical statement, identified explicitly as such. |
| Explorer functionality | A calculation, view or navigation control supplied by the interface, with its own sampling and resolution limits. |
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.
What the colours retain
| View | Colour meaning |
|---|---|
| Attractor Kp | Red and blue identify the actual two first-level pieces. |
| Contact difference Dp | The selected specimen colour links it to its parameter. Opacity represents the specified finite empirical density. |
| Radial atlas | A finite symbolic fingerprint, such as a selected surviving pair-address code. A constant hue does not define a topological equivalence class. |
| Four-family manuscript projection | Contraction weight w, the fourth real coordinate omitted from the three-dimensional position. |
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 direct-similarity laboratory’s original domain is α ∈ [0,π/2], γ ∈ [0,π] and w ∈ [0.35,0.65]. Reflections used in its display do not supply observations outside that sector. In particular, exchanging branches also exchanges w with 1−w; it cannot be silently used as a same-weight symmetry.
The direct-similarity 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.
The primary linked viewers use stable finite samples and recompute them when the parameter changes. Their point clouds, finite hulls and density rasters have different error and resolution controls. The reported prefix depth yields an analytic tail estimate in exact arithmetic; that estimate does not by itself bound every floating-point or raster error. A density image preserves address multiplicity and reports its finite sampling convention.
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.
Take a result back to its data
A useful figure should identify its parameter, family, arithmetic, finite budgets and evidence. Save its computation record together with the image. Camera settings and pixel dimensions belong to the figure; the mathematical parameter and proof status remain separate.
| Resource | Contents |
|---|---|
| Publication catalogue | Titles, individual authors, dated statuses, source links and citations. |
| Research ledger | Named claims, source references, hypotheses and interpretation. |
| Dataset catalogue | Domains, grids, finite-search settings and immutable source identities. |
| Publication checksums | Exact byte sizes and SHA-256 digests of the submitted poster and its web preview. |
| Parameter schema · Example record | A small portable mathematical record, independent of camera and UI session formats. |
| Agent entry point · Technical guide | Conventions, data discovery and evidence rules for scripts and AI-assisted research. |
| Maintenance guide | How to add a family, replace a publication with a dated revision, and validate a release. |
The historical laboratory payload remains 5,913,659 bytes. Its checksum is retained in the source manifest. A finer display mesh leaves that data record unchanged. A new survey belongs in a separately versioned dataset.
Sources and attribution
Mathematical research, complex-tree families and original interactive explorations by Bernat Espigulé. The programme connects the author’s earlier complex-tree work with the collinear doctoral research and the present four-dimensional atlas. Related papers retain their actual coauthors.
The manuscript announcements provide citations for Connectedness Loci for Pairs of Planar Similarities and its research companion. Both titles are provisional and both works are in preparation. Citations are available as BibTeX, LaTeX bibitems and CITATION.cff.
- 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.
- Bernat Espigulé, David Juher and Joan Saldaña, Collinear Fractals and Bandt’s Conjecture, Fractal and Fractional 8(12) (2024), 725. Results for the specified collinear n-ary families.
- 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. The binary direct slice with a common complex multiplier.
- 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. Boundary dynamics for the binary direct family with a common multiplier.
Individual family entries identify their source and scope. This guide documents the software and records the scope of manuscript results pending their reviewed release. Public availability does not change the authorship or licensing of the source publications.