Mathematics · computation · visual exploration

Bernat
Espigulé

Mathematical researcher · self-similar sets · computational exploration

I study self-similar sets and the spaces of parameters that shape them. My work connects proofs and computation with interactive maps, mathematical images, and sculpture.

Complex Trees is a research and mathematical-visualization project by Bernat Espigulé.

PhD candidate · Universitat de Girona · Catalonia

Explore the work View CV

The Beauty of Roots: a gold and blue map of polynomial roots with fine branching detail and marked algebraic landmarks. — The Beauty of Roots · 2019Mathematical cartography ↗

September 2026 Thesis deposited at the Universitat de Girona. Defence pending.The doctoral thesis

From a parameter to a world

Explore the mathematics.

All projects & materials ↗

Detailed angular survey of the 4D atlas, with coloured bands of finite numerical signatures.

Interactive atlas · Research preview

Connectedness in four dimensions

Explore four real parameters through linked attractors, contacts and angular surveys. Open the Tip Atlas or inspect a chosen parameter locally.

Explore ↗Tip Atlas Guide

Four coloured first-level pieces of a collinear self-similar set, captured from the current explorer.

Research software · 2026

Collinear Fractals Explorer

Link a complex parameter to its self-similar set, compare first-level pieces, and investigate finite capture and restricted polynomial roots.

Explore ↗Source

Six yellow, blue and red 3D-printed complex trees in the 2019 Arboretum.

Geometry · sculpture · animation

Mathematical forms

Explore 3D-printed complex trees, geometric sculpture, polynomial-root images and animations, with exhibition records for each work.

Explore ↗Works & records

Recent research

Publications & preprints.

Browse all works ↗

2024

Collinear Fractals and Bandt's Conjecture

Journal article

Bernat Espigule, David Juher, Joan SaldañaFractal and Fractional 8(12), 725

DOI ↗arXiv ↗PDF ↗

Cite

@article{espigulejuhersaldana2024bandt,
  author = {Espigule, Bernat and Juher, David and Salda{\~n}a, Joan},
  title = {Collinear Fractals and {Bandt}'s Conjecture},
  year = {2024},
  journal = {Fractal and Fractional},
  volume = {8},
  number = {12},
  pages = {725},
  doi = {10.3390/fractalfract8120725},
  url = {https://doi.org/10.3390/fractalfract8120725}
}

2026

Finite capture and the closure of roots of restricted polynomials

Preprint

Bernat Espigule, David JuherarXiv:2603.07397

arXiv ↗PDF ↗Explorer ↗

Cite

@misc{espigulejuher2026capture,
  author = {Espigule, Bernat and Juher, David},
  title = {Finite capture and the closure of roots of restricted polynomials},
  year = {2026},
  eprint = {2603.07397},
  archivePrefix = {arXiv},
  primaryClass = {math.DS},
  note = {Preprint},
  doi = {10.48550/arXiv.2603.07397},
  url = {https://arxiv.org/abs/2603.07397}
}

2026

Infinitely many holes in connectedness loci for collinear affine iterated function systems

Preprint

Bernat EspigulearXiv:2606.01467

arXiv ↗PDF ↗

Cite

@misc{espigule2026holes,
  author = {Espigule, Bernat},
  title = {Infinitely many holes in connectedness loci for collinear affine iterated function systems},
  year = {2026},
  eprint = {2606.01467},
  archivePrefix = {arXiv},
  primaryClass = {math.DS},
  note = {Preprint},
  doi = {10.48550/arXiv.2606.01467},
  url = {https://arxiv.org/abs/2606.01467}
}

2026

Explicit interpolations among the Sierpiński, Rauzy, and Apollonian gaskets

Preprint

Bernat EspigulearXiv:2606.23278

arXiv ↗PDF ↗Explorer ↗

Cite

@misc{espigule2026gaskets,
  author = {Espigule, Bernat},
  title = {Explicit interpolations among the {Sierpi\'nski}, {Rauzy}, and {Apollonian} gaskets},
  year = {2026},
  eprint = {2606.23278},
  archivePrefix = {arXiv},
  primaryClass = {math.DS},
  note = {Preprint},
  doi = {10.48550/arXiv.2606.23278},
  url = {https://arxiv.org/abs/2606.23278}
}

Doctoral thesis · 2026

Collinear Fractals
and Connectedness Loci

Topology, Finite Capture,
and Restricted Polynomial Roots

Bernat Espigulé PonsUdG

The doctoral thesis

Topology, finite capture,
and polynomial roots.

My doctoral research brings together the geometry of collinear self-similar sets, the topology of their connectedness loci, and roots of polynomials with restricted integer coefficients.

Supervised by David Juher and Joan Saldaña.
Corrected circulation edition · 15 September 2026.

Read about the thesis ↗Cite the thesis

Research & experience

A path through mathematics.

Trained in physics and mathematics, I have worked in research, scientific computing, museum content, and teaching.

Full academic & professional CV ↗

2022–present: Doctoral research · Universitat de GironaSelf-similar sets, connectedness loci, and restricted polynomial roots. Thesis deposited; defence pending.
2021: MSc Advanced Mathematics · Universitat de BarcelonaMaster’s thesis completed in 2018; degree awarded in 2021.
2013–2017 2020–2021: Consultant · Wolfram ResearchMathematical computation, interactive content, and technical communication.
2012: BSc Physics · Universitat Autònoma de BarcelonaInternational study at UC Santa Barbara and Heidelberg.