⋈Complex TreesConnectedness AtlasFamily guide ↓

Beyond two branches

Ternary complex trees

Three multipliers. An entire landscape of self-similar shapes.

Choose a family

Re zIm z

Real partImaginary part

Shaded region: strict contraction · drag to explore

Rendering

Finite prefix depth

Every prefix at the reported depth is drawn. Greater depth resolves smaller pieces; it does not certify their topology.

Source-defined family

Fern family

1×

Preparing finite prefixes…

Similarity dimension—Σ |λⱼ|ˢ = 1 · not automatically dim_H

Contraction ratios—Colors identify first-level pieces.

Tipset maps

Read the geometry correctly

A tree and its tipset are different objects.

A complex tree starts at 1. Following letters c₁, c₂, … gives nodes 1 + c₁ + c₁c₂ + ⋯. The infinite limits form its tipset, the invariant set of the maps fⱼ(x)=1+cⱼx. The path view draws a finite collection of branches. The tipset view uses finite-prefix centers, or composed seed triangles for the canonical triangular family, to approximate the limiting self-similar set.

About this family

How this extends the binary atlas

The main atlas has four real parameters for a normalized pair of planar similarities. The named ternary families here have one complex parameter, hence two real parameters. Three independent planar similarities need a larger space: the editor exposes twelve real coefficients, before removing the four real degrees of freedom of a generic similarity normalization.

The displayed similarity dimension solves Σ|λⱼ|ˢ=1. It equals the Hausdorff dimension under the open set condition; no such equality is inferred from a rendering.