Sierpiński--Apollonian stack

Top (Z = 2.0): Apollonian Gasket limit.

Base (Z = 0.0): Sierpiński triangle limit.

Iteration Level 5

Low Detail High Detail

Transformation Progress (Z-Axis)

Base (Sierpiński) Top (Apollonian)

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 3D topological deformation mapping the affine Sierpiński triangle into the conformal Apollonian Gasket.

This is achieved via a complex parameterized Iterated Function System (IFS) that morphs linear transformations seamlessly into non-linear Möbius transformations, bending the straight edges into perfectly mutually tangent circular holes.

Companion links

This web application accompanies the public preprint linked below. Its canonical public URL is complextrees.com/sierpinski-apollonian-stack. The bend slider is a nonliteral display option and does not change the underlying mathematical family.

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".
Apollonian Gasket: Apollonius of Perga (c. 200 BC), Mumford, D., Series, C., & Wright, D. (2002). Indra's Pearls: The Vision of Felix Klein.

Citation

@misc{EspiguleApollonian2026,
  author = {Bernat Espigul\'{e}},
  title  = {A {M\"obius} deformation from the {Sierpi\'nski} gasket to the equilateral {Apollonian} gasket},
  year   = {2026},
  note   = {Preprint}
}

@misc{EspiguleApollonianCompanion2026,
  author       = {Bernat Espigul\'{e}},
  title        = {Companion interactive demonstration for the {Sierpi\'nski}--{Apollonian} deformation},
  howpublished = {\url{https://complextrees.com/sierpinski-apollonian-stack}},
  year         = {2026},
  note         = {Web application}
}

A Möbius deformation from the Sierpiński gasket to the equilateral Apollonian gasket

Preprint, 2026.

Open preprint PDF Open LaTeX source Open bibliography Open sister app

Abstract

This application accompanies the paper on an explicit one-parameter family of Möbius maps deforming the equilateral Sierpiński iterated function system into the parabolic generators of the equilateral Apollonian gasket. The paper proves disk invariance and contraction for every t < 1, continuity on each truncated interval [0, 1 − ε], and an exact variational formula for the Hausdorff dimension of every hyperbolic fiber.

Application notes

The default stack orientation follows the deformation direction emphasized in the paper: base = Sierpiński and top = Apollonian. 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 family defined in the manuscript.

Suggested citation

Bernat Espigulé, A Möbius deformation from the Sierpiński gasket to the equilateral Apollonian gasket, preprint, 2026. Companion application: Bernat Espigulé, Companion interactive demonstration for the Sierpiński--Apollonian deformation, web application, complextrees.com/sierpinski-apollonian-stack.