Connectedness Loci for Pairs of Planar Similarities
Connectedness loci for pairs of planar similarities: the parameter space, its topology, contact criteria and exact families.
In preparation · PDF not yet availableCite this manuscript →Manuscripts in preparation
Connectedness Loci for Pairs of Planar Similarities
Two planar similarities determine a self-similar set. Their two complex coefficients give four real parameters in which to study connectedness, symbolic contacts and distinguished families.
The main article and research companion are in preparation. Their titles are provisional. PDFs will be made available after the author has reviewed and approved the release versions.
Forthcoming manuscripts
Connectedness Loci for Pairs of Planar Similarities: Contact Families, Sections and Landmarks
Contact families, exact landmarks and two-dimensional slices, with their placement in three-dimensional sections and links to the computational atlas.
In preparation · PDF not yet availableCite the companion →Barcelona Mathematical Days 2026
Four-Dimensional Mandelbrot Sets for Binary Self-Similar Sets of the Plane
The submitted poster connects the equal-contraction atlas to four selected families: the collinear, mirror-symmetric, Fibonacci–Mandelbrot and quarter-turn slices. It includes the exact S and U boundary landmarks.
Interactive resources
Four-dimensional explorer
Vary the contraction weight and orientation type. Inspect a parameter together with its attractor and marked difference set.
Open the explorer →Linked atlas
Place probes on the numerical parameter survey and follow the corresponding attractors.
Open the atlas →Four-family stable slices
Inspect the direct and mixed zipper families, each with two traversal choices, in their coefficient-phase projection. Colour records contraction weight; the coloured cells give finite certified inner covers in their specified families.
Open the stable-slice viewer →Mirror-symmetric trees
The poster’s mirror slice, the historical fan gallery, addressed contact curves and their stable domains.
Open the poster QR destination →Planar slices and 3D cutaways
Move a point in a 2D slice and inspect its attractor and both chart lifts in a clipped 3D section.
Explore linked cutaways →Companion contact families
Fifteen marked stable families, with contact addresses, certificate scope and real and imaginary carrier cuts.
Browse the family plates →Methods and data
Read the coordinate conventions, calculation budgets and distinctions between exact results, validated certificates and numerical approximations.
Read the mathematical guide →Citations
The manuscript citations below identify work in preparation and preserve that status. The titles may change before release.
Download BibTeX · Download LaTeX bibitems · CITATION.cff
Main article
\bibitem{Espigule2026PlanarSimilarities}
B. Espigul\'e,
\emph{Connectedness Loci for Pairs of Planar Similarities},
Manuscript in preparation (provisional title), 2026.
\url{https://complextrees.com/4D/publication/#flagship}.
Research companion
\bibitem{Espigule2026ContactFamilies}
B. Espigul\'e,
\emph{Connectedness Loci for Pairs of Planar Similarities: Contact Families, Sections and Landmarks},
Research companion in preparation (provisional title), 2026.
\url{https://complextrees.com/4D/publication/#companion}.