Rauzy--Sierpiński stack

Top (Z = 2.0): Sierpiński gasket limit.

Base (Z = 0.0): Rauzy gasket limit.

Iteration Level 5

Low Detail High Detail

Transformation Progress (Z-Axis)

Base (Rauzy) Top (Sierpiński)

Illustrative bend (display only) 0%

Literal stack Stylized bend

Auto Animate

Solid Mesh Mode

Auto Rotation

Reverse stack orientation

Custom Color

Aspect Ratio (Height) 2.0

Flat Tall

About the Geometry

A continuous three-dimensional stack interpolation between the projective Rauzy gasket and the affine Sierpiński gasket.

The original idea to illustrate the topological equivalence between the Rauzy gasket and the Sierpiński triangle was communicated to the author (Bernat Espigulé) by Pierre Arnoux at ICERM in 2019.

Companion links

This web application accompanies the public preprint linked below. Its canonical public URL is complextrees.com/rauzy-sierpinski-stack. The default stack orientation matches the paper: base = Rauzy and top = Sierpiński. The bend slider is a nonliteral display option.

Open preprint PDF · Open sister paper · Open sister app

Seminal References

Sierpiński triangle: Wacław Sierpiński (1915). "Sur une courbe dont tout point est un point de ramification". Comptes rendus hebdomadaires des séances de l'Académie des sciences, 160: 302-305.
Rauzy Gasket: Pierre Arnoux & Štěpán Starosta (2013). "The Rauzy Gasket". In Further Developments in Fractals and Related Fields (pp. 1-23). Springer, Boston, MA.

Citation

@misc{EspiguleArnoux2026,
  author = {Bernat Espigul\'e and Pierre Arnoux},
  title  = {An explicit affine-projective interpolation between the {Sierpi\'nski} gasket and the {Rauzy} gasket},
  year   = {2026},
  note   = {Preprint}
}

@misc{EspiguleCompanion2026,
  author       = {Bernat Espigul\'{e}},
  title        = {Companion interactive demonstration for the {Rauzy}--{Sierpi\'nski} interpolation},
  howpublished = {\url{https://complextrees.com/rauzy-sierpinski-stack}},
  year         = {2026},
  note         = {Web application}
}

An explicit affine-projective interpolation between the Sierpiński gasket and the Rauzy gasket

Preprint, 2026.

Open preprint PDF Open LaTeX source Open bibliography Open sister app

Abstract

We study an explicit one-parameter family of projective maps on the standard simplex that interpolates between the affine Sierpiński system at t = 0 and the Rauzy system at t = 1. The paper proves parameter-independent first-level geometry, uniform contraction for every t < 1, stability of the symbolic quotient up to the Rauzy endpoint, a product structure for the three-dimensional stack, explicit constant-address asymptotics, and exact Hausdorff dimension for every intermediate fiber.

Application notes

The default stack orientation in this application matches theorem 7.1 of the paper: the base face is the Rauzy fiber and the top face is the Sierpiński fiber. The reverse-orientation toggle is provided only as an inspection aid.

The bend slider is an intentionally nonliteral display mode. It is useful for visual exploration, but it is not part of the mathematical object defined in the manuscript.

Suggested citation

Bernat Espigulé and Pierre Arnoux, An explicit affine-projective interpolation between the Sierpiński gasket and the Rauzy gasket, preprint, 2026. Companion application: Bernat Espigulé, Companion interactive demonstration for the Rauzy--Sierpiński interpolation, web application, complextrees.com/rauzy-sierpinski-stack.