{"id":2856,"date":"2026-11-01T07:00:26","date_gmt":"2026-11-01T12:00:26","guid":{"rendered":"https:\/\/mathvoices.ams.org\/featurecolumn\/?p=2856"},"modified":"2026-10-31T15:02:32","modified_gmt":"2026-10-31T19:02:32","slug":"what-is-a-foliation","status":"publish","type":"post","link":"https:\/\/mathvoices.ams.org\/featurecolumn\/2026\/11\/01\/what-is-a-foliation\/","title":{"rendered":"What is a Foliation?"},"content":{"rendered":"

This was not just art: It showed a beautiful, infinite, winding pattern inspired by real foliations…<\/em><\/span><\/p>\n

What is a Foliation?<\/h1>\n

Hui Zhao<\/b>
\nThe Lab of Computational Discrete Global Geometric Structures<\/b><\/p>\n

Introduction<\/h2>\n<\/p>\n

The mathematical concept of foliation is one kind of global geometric
\nstructure. A foliation is a way of cutting a smooth shape (like a 3D
\nspace or a curved surface) into many thin, parallel, non-intersecting
\n\u201csheets\u201d or \u201clayers,\u201d just like the pages in a book, or the layers of
\nplywood, or the grain in wood. These layers are called leaves. They never
\ncross each other, as shown in the figure below. They fill the whole space
\nsmoothly, without gaps or messy overlaps. In geometry and topology,
\nfoliations help us understand the intrinsic and global geometric
\nstructure of complicated curved spaces called manifolds. Foliations can be computed and visualized,
\nallowing for an intuitive understanding of this abstract concept.<\/p>\n

\n\"A
A foliation on a surface of genus 2\uff0cgenerated by code platform: Geometric.<\/figcaption><\/figure>\n

Artifacts<\/h2>\n<\/p>\n

Ancient peoples around the world have observed the forms of
\nfoliations and depicted them in various artifacts, as exemplified by
\nthe ancient Peruvian artifact shown below.<\/p>\n

\n\"Museum
Peruvian artifact. Photographed by Dr. Hui Zhao at the Harvard
\nMuseum, 2015.
\n<\/figcaption><\/figure>\n
\n\"A
Cypriot barrel jug artifact. Photographed by Dr. Hui Zhao at the
\nBritish Museum, 2024.
\n<\/figcaption><\/figure>\n

Drawing by Thurston and Sullivan<\/h2>\n<\/p>\n

William P. Thurston, a Fields Medalist, was one of the greatest
\ngeometers of the 20th century. In the early 1970s, while he was at UC
\nBerkeley, he and Dennis Sullivan painted a large foliation pattern
\ndirectly on the wall of the Evans Hall math building. This was not
\njust art: It showed a beautiful, infinite, winding pattern inspired by
\nreal foliations he was studying. The curves represented the leaves of a
\nfoliation on a special surface of in the thrice-punctured plane. It was
\nhis way of turning abstract, hard-to-imagine geometry into something
\nvisible and beautiful. The painting became famous: It appeared on the
\ncover of the AMS Notices<\/em><\/a>. It showed how deeply Thurston
\nthought in pictures and shapes, not just formulas. It became a symbol of
\ngeometry, intuition, and the beauty of mathematics.<\/p>\n

Thurston\u2019s hand-drawn foliation sketch remained on the wall for more
\nthan 40 years, inspiring generations of students. Sadly, Thurston\u2019s
\niconic foliation mural in Evans Hall was covered over and lost during
\nthe building\u2019s renovation between 2014 and 2015.<\/p>\n

Foliations on surfaces<\/h2>\n<\/p>\n

By hand, Thurston could only draw foliations on simple spaces in the 2D
\nplane. Using a computer algorithm, we are able to construct and visualize foliations on
\nvarious complex surfaces. Our algorithm places constraints on the possibilities using ribbon graphs embedded in the surfaces.<\/p>\n

\"A
A ribbon graph.<\/figcaption><\/figure>\n
\"The
The circles are leaves of a foliation; the lines connecting balls
\nbelong to a ribbon graph.<\/figcaption><\/figure>\n

A closed<\/strong> leaf of a foliation is a leaf that forms a
\nsimple, smooth loop without any endpoints, meaning it starts and ends at
\nthe same point and never intersects itself, unlike ordinary leaves that
\nmay extend infinitely or wind densely around the surface. In intuitive
\nterms, it is like a closed curve or a circle that is part of the layered
\nstructure of the foliation, and nearby leaves either run parallel to it,
\nspiral toward it, or spiral away from it, making closed leaves
\nespecially important in understanding the global behavior and the
\nsingular structure of foliations on surfaces.<\/p>\n

The zeros<\/strong> of a foliation on a surface are the
\nisolated singular points where the local, regular structure of the
\nfoliation breaks down. Away from these points, the foliation consists of
\nsmooth, non-intersecting curves called leaves that flow uniformly in
\nparallel directions, but the zero of a foliation is where the leaves
\nstop being smooth, collapse, break, or converge. These singular points are essential to the
\nglobal topology of the foliation, and their total number, counted with
\nappropriate indices, is determined by the genus of the underlying
\nsurface. For a closed oriented surface of genus g, the total number of
\nzeros of a generic foliation is given by the formula $4g\u22124$. This
\ninvariant reveals a deep connection between the local singular behavior
\nof the foliation and the global shape of the surface.<\/p>\n

\n\"The
A zero of index 2. <\/figcaption><\/figure>\n
\n\"Two
Two zeros of index 1.<\/figcaption><\/figure>\n

Foliations also exist on surfaces with punctures, in addition to
\nclosed surfaces. A pole<\/strong> of a foliation on a surface is
\na special type of singular point, often regarded as a kind of \u201csource\u201d
\nor \u201csink\u201d for the leaves of the foliation, near which all nearby leaves
\nspiral around the point and either wind infinitely toward it or unwind
\naway from it; unlike a generic zero where leaves may form saddles or
\nmore complicated local patterns, a pole has a much simpler rotational
\nstructure, resembling the way water swirls around a drain or the way
\nlines radiate or converge around a central point, and such poles appear
\nas isolated singularities that contribute to the total count of singular
\npoints of the foliation, linking the local rotational behavior near the
\npole to the global topological invariants of the foliation and the
\nunderlying surface.<\/p>\n

\"A
A foliation on a surface of genus 2 with a pole of degree 2.<\/figcaption><\/figure>\n
\"Two
A foliation on a surface of genus 2 with two poles of degree 1.<\/figcaption><\/figure>\n

Whitehead equivalence<\/strong> of foliations is simply a
\nflexible way to say two different foliations are essentially the same in
\na topological sense\u2014meaning you can smoothly deform, stretch, bend, and
\nadjust one foliation into the other without ever cutting, tearing, or
\ngluing the layers together, just like how you can reshape a crumpled
\npiece of paper with lines drawn on it back into a flat sheet without
\nbreaking those lines. All foliations that can be transformed into
\neach other through such smooth deformations belong to the same
\nWhitehead equivalence class<\/strong>, which groups foliations
\nnot by their exact, rigid shape but by their core, unchangeable
\nstructure, letting mathematicians classify and compare foliations by
\ntheir fundamental topological behavior rather than minor geometric
\ndifferences. Different perspectives on foliations belonging to different
\nWhitehead classes can be observed in our videos<\/a>.<\/p>\n

\"A
\n\"A
Two foliations inside two different Whitehead classes.<\/figcaption><\/figure>\n

Foliations can also be defined on surfaces with boundary and with
\npunctures as shown in below.<\/p>\n

\"Foliations
Two foliations on a surface of one boundary and one puncture.<\/figcaption><\/figure>\n
\"Foliations
Foliations on a surface of three boundaries.<\/figcaption><\/figure>\n

Harmonic foliations<\/h2>\n<\/p>\n

A harmonic<\/strong> foliation is a particularly nice and
\n\u201cbalanced\u201d kind of foliation on a surface, where the leaves\u2014the curves
\nthat make up the foliation\u2014are arranged in a way that is geometrically
\nsmooth and tension-free, much like the most natural flow lines you would
\nsee on a curved surface, without unnecessary bending or distortion. In
\nsimple terms, these leaves behave like minimal, relaxed curves that
\nminimize local curvature and follow the intrinsic geometry of the
\nsurface, making harmonic foliations both analytically clean and
\ntopologically well-behaved. They play an important role in
\nconnecting the geometry of the surface with its global topological
\nstructure, such as the number of singular zeros and the genus of the
\nsurface.<\/p>\n

\n\"The<\/p>
A non-harmonic foliation.<\/figcaption><\/figure>\n

A partial<\/strong> foliation on a surface is a foliation that
\nonly covers part of the surface. A partial foliation can be several
\nclosed loops as narrow as desired.<\/p>\n

\n\"The<\/p>
A partial foliation.<\/figcaption><\/figure>\n

A partial foliation can be gradually transformed into a harmonic
\nfoliation within the same Whitehead class by reducing the Dirichlet
\nenergy. Each Whitehead class contains exactly one harmonic
\nfoliation. The morphing procedure can be viewed in our videos<\/a>.<\/p>\n

\n\"Developing
A partial foliation morphs into a harmonic foliation.<\/figcaption><\/figure>\n
\n\"Developing
A partial foliation morphs into a harmonic foliation.<\/figcaption><\/figure>\n

A horizontal foliation and a vertical foliation, which are mutually
\nperpendicular, together form a new global geometric structure known as a
\nholomorphic quadratic differential and a meromorphic quadratic
\ndifferential.<\/p>\n

\n

<\/p>

A holomorphic quadratic differential consists of a red horizontal
\nfoliation and a green vertical foliation.<\/figcaption><\/figure>\n
\"The<\/p>
A harmonic foliation and its corresponding meromorphic quadratic
\ndifferential.<\/figcaption><\/figure>\n

Applications<\/h2>\n<\/p>\n

Some special meromorphic quadratic differentials divide a surface
\ninto a grid, that is, a square\u2011tiling of the entire surface. By
\ncontrolling the Whitehead class of the harmonic foliation, one can
\nobtain different square\u2011tiling patterns of the surface lattice. Such a
\ncontrollably square\u2011tiled grid partition is called a \u201cSuper Structured
\nQuad Mesh\u201d, and the problem of constructing or realizing a Super
\nStructured Quad Mesh is referred to as the \u201cLingshan Problem\u201d. Super
\nStructured Quad Meshes play an important role in CAD\/CAE industrial
\nsoftware.<\/p>\n

\"checkerboard
A square tiling.<\/figcaption><\/figure>\n
\"checkerboard
Another square tiling.<\/figcaption><\/figure>\n
\"foliations
Two foliations and their corresponding grids.<\/figcaption><\/figure>\n

Further Reading<\/h2>\n<\/p>\n
    \n
  1. \n

    Hui Zhao. A Global Touring Art Exhibition of Computational Discrete Global Geometric Structures. Project website. https:\/\/www.globalgeometry.cn\/<\/a>. 2025.<\/p>\n<\/li>\n

  2. \n

    Gabai, David, and Steven P. Kerckhoff. \u201cWilliam P. Thurston,
    \n1946\u20132012.\u201d Notices of the American Mathematical Society 62, no. 11
    \n(2015): 1318\u20131332.<\/p>\n<\/li>\n

  3. \n

    Hui Zhao. A Geometry and
    \nTopology Code Platform: Geometric<\/a>. Unpublished manuscript. Personal
    \nwebsite.     https:\/\/meshdgp.github.io\/.<\/a>
    \n2025.<\/p>\n<\/li>\n

  4. \n

    Hui Zhao, Shaodong Wang, and Wencheng Wang. \u201cGlobal Conformal
    \nParameterization via an Implementation of Holomorphic Quadratic
    \nDifferentials.\u201dIEEE Transactions on Visualization and Computer Graphics,
    \nvol. 28, no. 3, pp. 1529\u20131544, 2022.<\/p>\n<\/li>\n

  5. \n

    Hui Zhao, Global Geometric Structures: From Abstraction to Construction, Visualization, and Applications<\/a>, Notices of the American Mathematical Society<\/em>, Volume 73, Number 9, October 2026, pp. 733\u2013739, Feature Article, DOI: 10.1090\/noti3384.<\/p>\n<\/li>\n<\/ol>\n

    Acknowledgments<\/h2>\n

    This work is part of the Computational Discrete Global Geometric Structures\u201d (CDGGS) program<\/a>, 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.<\/p>\n

    Dedication<\/h2>\n

    Dedicated to my daughter Yudu Zhao and my wife Xiaoling Wang, with
    \nall my love and gratitude.<\/p>\n","protected":false},"excerpt":{"rendered":"

    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 cuttingRead More →<\/a><\/span><\/p>\n","protected":false},"author":2,"featured_media":1599,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"advanced_seo_description":"","jetpack_seo_html_title":"","jetpack_seo_noindex":false,"jetpack_seo_schema_type":"","_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[213,114,242,134],"tags":[243,244],"class_list":["entry","author-uwhitcher","post-2856","post","type-post","status-publish","format-standard","has-post-thumbnail","category-213","category-geometry-and-topology","category-hui-zhao","category-math-and-arts-humanities","tag-foliations","tag-william-thurston"],"jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"jetpack_featured_media_url":"https:\/\/i0.wp.com\/mathvoices.ams.org\/featurecolumn\/wp-content\/uploads\/sites\/2\/2023\/03\/mathvoices-banner-feat-col.png?fit=2760%2C580&ssl=1","_links":{"self":[{"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/posts\/2856","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/comments?post=2856"}],"version-history":[{"count":13,"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/posts\/2856\/revisions"}],"predecessor-version":[{"id":2892,"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/posts\/2856\/revisions\/2892"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/media\/1599"}],"wp:attachment":[{"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/media?parent=2856"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/categories?post=2856"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathvoices.ams.org\/featurecolumn\/wp-json\/wp\/v2\/tags?post=2856"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}