This was not just art: It showed a beautiful, infinite, winding pattern inspired by real foliations...

What is a Foliation?

Hui Zhao
The Lab of Computational Discrete Global Geometric Structures

Introduction

The mathematical concept of foliation is one kind of global geometric structure. A foliation is a way of cutting a smooth shape (like a 3D space or a curved surface) into many thin, parallel, non-intersecting “sheets” or “layers,” just like the pages in a book, or the layers of plywood, or the grain in wood. These layers are called leaves. They never cross each other, as shown in the figure below. They fill the whole space smoothly, without gaps or messy overlaps. In geometry and topology, foliations help us understand the intrinsic and global geometric structure of complicated curved spaces called manifolds. Foliations can be computed and visualized, allowing for an intuitive understanding of this abstract concept.

Artifacts

Ancient peoples around the world have observed the forms of foliations and depicted them in various artifacts, as exemplified by the ancient Peruvian artifact shown below.

Drawing by Thurston and Sullivan

William P. Thurston, a Fields Medalist, was one of the greatest geometers of the 20th century. In the early 1970s, while he was at UC Berkeley, he and Dennis Sullivan painted a large foliation pattern directly on the wall of the Evans Hall math building. This was not just art: It showed a beautiful, infinite, winding pattern inspired by real foliations he was studying. The curves represented the leaves of a foliation on a special surface of in the thrice-punctured plane. It was his way of turning abstract, hard-to-imagine geometry into something visible and beautiful. The painting became famous: It appeared on the cover of the AMS Notices. It showed how deeply Thurston thought in pictures and shapes, not just formulas. It became a symbol of geometry, intuition, and the beauty of mathematics.

Thurston’s hand-drawn foliation sketch remained on the wall for more than 40 years, inspiring generations of students. Sadly, Thurston’s iconic foliation mural in Evans Hall was covered over and lost during the building’s renovation between 2014 and 2015.

Foliations on surfaces

By hand, Thurston could only draw foliations on simple spaces in the 2D plane. Using a computer algorithm, we are able to construct and visualize foliations on various complex surfaces. Our algorithm places constraints on the possibilities using ribbon graphs embedded in the surfaces.

A closed leaf of a foliation is a leaf that forms a simple, smooth loop without any endpoints, meaning it starts and ends at the same point and never intersects itself, unlike ordinary leaves that may extend infinitely or wind densely around the surface. In intuitive terms, it is like a closed curve or a circle that is part of the layered structure of the foliation, and nearby leaves either run parallel to it, spiral toward it, or spiral away from it, making closed leaves especially important in understanding the global behavior and the singular structure of foliations on surfaces.

The zeros of a foliation on a surface are the isolated singular points where the local, regular structure of the foliation breaks down. Away from these points, the foliation consists of smooth, non-intersecting curves called leaves that flow uniformly in parallel directions, but the zero of a foliation is where the leaves stop being smooth, collapse, break, or converge. These singular points are essential to the global topology of the foliation, and their total number, counted with appropriate indices, is determined by the genus of the underlying surface. For a closed oriented surface of genus g, the total number of zeros of a generic foliation is given by the formula $4g−4$. This invariant reveals a deep connection between the local singular behavior of the foliation and the global shape of the surface.

Foliations also exist on surfaces with punctures, in addition to closed surfaces. A pole of a foliation on a surface is a special type of singular point, often regarded as a kind of “source” or “sink” for the leaves of the foliation, near which all nearby leaves spiral around the point and either wind infinitely toward it or unwind away from it; unlike a generic zero where leaves may form saddles or more complicated local patterns, a pole has a much simpler rotational structure, resembling the way water swirls around a drain or the way lines radiate or converge around a central point, and such poles appear as isolated singularities that contribute to the total count of singular points of the foliation, linking the local rotational behavior near the pole to the global topological invariants of the foliation and the underlying surface.

Whitehead equivalence of foliations is simply a flexible way to say two different foliations are essentially the same in a topological sense—meaning you can smoothly deform, stretch, bend, and adjust one foliation into the other without ever cutting, tearing, or gluing the layers together, just like how you can reshape a crumpled piece of paper with lines drawn on it back into a flat sheet without breaking those lines. All foliations that can be transformed into each other through such smooth deformations belong to the same Whitehead equivalence class, which groups foliations not by their exact, rigid shape but by their core, unchangeable structure, letting mathematicians classify and compare foliations by their fundamental topological behavior rather than minor geometric differences. Different perspectives on foliations belonging to different Whitehead classes can be observed in our videos.

Foliations can also be defined on surfaces with boundary and with punctures as shown in below.

Harmonic foliations

A harmonic foliation is a particularly nice and “balanced” kind of foliation on a surface, where the leaves—the curves that make up the foliation—are arranged in a way that is geometrically smooth and tension-free, much like the most natural flow lines you would see on a curved surface, without unnecessary bending or distortion. In simple terms, these leaves behave like minimal, relaxed curves that minimize local curvature and follow the intrinsic geometry of the surface, making harmonic foliations both analytically clean and topologically well-behaved. They play an important role in connecting the geometry of the surface with its global topological structure, such as the number of singular zeros and the genus of the surface.

A partial foliation on a surface is a foliation that only covers part of the surface. A partial foliation can be several closed loops as narrow as desired.

A partial foliation can be gradually transformed into a harmonic foliation within the same Whitehead class by reducing the Dirichlet energy. Each Whitehead class contains exactly one harmonic foliation. The morphing procedure can be viewed in our videos.

A horizontal foliation and a vertical foliation, which are mutually perpendicular, together form a new global geometric structure known as a holomorphic quadratic differential and a meromorphic quadratic differential.

Applications

Some special meromorphic quadratic differentials divide a surface into a grid, that is, a square‑tiling of the entire surface. By controlling the Whitehead class of the harmonic foliation, one can obtain different square‑tiling patterns of the surface lattice. Such a controllably square‑tiled grid partition is called a “Super Structured Quad Mesh”, and the problem of constructing or realizing a Super Structured Quad Mesh is referred to as the “Lingshan Problem”. Super Structured Quad Meshes play an important role in CAD/CAE industrial software.

Further Reading

1. Hui Zhao. A Global Touring Art Exhibition of Computational Discrete Global Geometric Structures. Project website. https://www.globalgeometry.cn/. 2025.

2. Gabai, David, and Steven P. Kerckhoff. “William P. Thurston, 1946–2012.” Notices of the American Mathematical Society 62, no. 11 (2015): 1318–1332.

3. Hui Zhao. A Geometry and Topology Code Platform: Geometric. Unpublished manuscript. Personal website. https://meshdgp.github.io/. 2025.

4. Hui Zhao, Shaodong Wang, and Wencheng Wang. “Global Conformal Parameterization via an Implementation of Holomorphic Quadratic Differentials.”IEEE Transactions on Visualization and Computer Graphics, vol. 28, no. 3, pp. 1529–1544, 2022.

5. Hui Zhao, Global Geometric Structures: From Abstraction to Construction, Visualization, and Applications, Notices of the American Mathematical Society, Volume 73, Number 9, October 2026, pp. 733–739, Feature Article, DOI: 10.1090/noti3384.

Acknowledgments

This work is part of the Computational Discrete Global Geometric Structures” (CDGGS) program, which develops abstract global geometric structures into algorithmic constructions, numerical realizations via code, visualizations, and applications, and conversely introduces algorithmic thinking and tools into the mathematical study of global geometric structures.

Dedication

Dedicated to my daughter Yudu Zhao and my wife Xiaoling Wang, with all my love and gratitude.