A living parameter atlas · Direct–direct family
Follow the shape of connection
Move a coloured locator. Watch two fractal pieces and their difference change together.
Turn the surface, find a ridge, and travel into its fine structure.
Preparing the live atlas…
Kp
Two first-level pieces
Dp
λ₋K − λ₊K Locator A
Independent finite addresses, with multiplicities retained. Opacity uses the same density scale at every locator.
Adapts to keep movement responsive; refines at rest.
Advanced view layersStructure, sampling & evidence
Explore the construction
Tree height is the cylinder scale |λu|. Address branches descend towards the planar attractor.
Numerical detail
The contact question is 2 ∈ Dp. The surface, hulls and density are finite numerical views. Their apparent overlaps do not decide limiting contact.
Read the geometry
A parameter space you can move through
The two similarities are f₋(z) = −1 + λ₋z and f₊(z) = 1 + λ₊z, with λ₋ = (2w/ρ) exp(i(α−γ)) and λ₊ = (2(1−w)/ρ) exp(−i(α+γ)). Weight changes contraction lengths; addresses are sampled with equal symbol probabilities.
The torus displays the full angular coordinates. Its radius is the outermost transition detected by a finite radial survey, and its colour records a selected finite pair-address code. The torus hole belongs to the display. Colour is not a proved classification of topological type.
Controls, density & numerical limits
Choose A–F to revisit your six locators. Drag a locator on the visible surface, or focus it and use arrow keys to change its angular coordinates. Drag the background to orbit. Double-click a visible point to move the active locator there along a smooth angular path. A new gesture interrupts travel. Reduced-motion preferences make travel immediate.
K shows sampled finite addressed centres. D uses independent pairs of finite words, retaining multiplicities, with a mass-preserving smoothed histogram. Its opacity is a shared logarithmic display of probability per complex-plane unit². Sampling grid, bandwidth and word depth are reported above; neither a limiting density nor a dimension estimate is asserted.
The convex hulls approximate finite prefix hulls, with numerical tail bounds. The optional tree is an address scaffold: the node for u is (fu(0), |λu|). It does not identify which branches meet in the limiting set. The original high-resolution study retains its exact image provenance and downloads.