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A physical model of the Reeb foliation

Gianluca Bande, Gregorio Franzoni

TL;DR

The paper builds a tangible bridge between abstract foliation theory and physical visualization by constructing the first full 3D-printed model of the Reeb foliation on $S^3$. It starts from foundational foliation notions and proceeds to visualize the Reeb foliation via stereographic projection into $\mathbb{R}^3$, before detailing a complete fabrication pipeline using Mathematica and Rhinoceros to produce watertight STL meshes. The authors present multiple models capturing both Reeb components and the Clifford torus, demonstrate their production with an FDM printer, and discuss educational and pedagogical considerations. They also outline future work to make the assembly fully detachable and self-contained, enhancing portability and teaching utility.

Abstract

In this paper, after explaining some basic aspects of the modern theory of foliations with the aim of describing the celebrated Reeb foliation, we propose the first construction of a comprehensive physical model of it. The construction of the model is obtained by an implementation of geometric methods for 3D printing.

A physical model of the Reeb foliation

TL;DR

The paper builds a tangible bridge between abstract foliation theory and physical visualization by constructing the first full 3D-printed model of the Reeb foliation on . It starts from foundational foliation notions and proceeds to visualize the Reeb foliation via stereographic projection into , before detailing a complete fabrication pipeline using Mathematica and Rhinoceros to produce watertight STL meshes. The authors present multiple models capturing both Reeb components and the Clifford torus, demonstrate their production with an FDM printer, and discuss educational and pedagogical considerations. They also outline future work to make the assembly fully detachable and self-contained, enhancing portability and teaching utility.

Abstract

In this paper, after explaining some basic aspects of the modern theory of foliations with the aim of describing the celebrated Reeb foliation, we propose the first construction of a comprehensive physical model of it. The construction of the model is obtained by an implementation of geometric methods for 3D printing.

Paper Structure

This paper contains 13 sections, 1 theorem, 22 equations, 13 figures.

Table of Contents

  1. Preliminaries
  2. Some examples
  3. The tangent bundle of a foliation and the Frobenius Theorem
  4. Applications of the Frobenius Theorem
  5. The Reeb Foliation
  6. Rendering and physical models
  7. Visualization in $\mathbb{R}^3$ via Mathematica®
  8. Physical realization by 3D printing
  9. About McNeel Rhinoceros 3D
  10. Hardware and software
  11. Existing physical models
  12. The models
  13. Further developments

Key Result

Theorem 1.3

Let $M$ be a smooth manifold, and let $\mathcal{D}$ be a $k$-dimensional smooth distribution on $M$, that is $\mathcal{D} \subset TM$ assigns to each point $p \in M$ a $k$-dimensional subspace $\mathcal{D}_p \subset T_pM$. The distribution $\mathcal{D}$ is completely integrable (integrable for short

Figures (13)

  • Figure 1: Two rendered images of the Reeb foliation in $\mathbb{R}^3$, showing four leaves belonging to the "external" Reeb component (the blu ones). The white surface is the compact leaf.
  • Figure 1.1: Transition maps.
  • Figure 1.2: Distinguished chart with plaques.
  • Figure 1.3: Level sets of the function \ref{['level sets']}, for $n=1$.
  • Figure 1.4: Level sets of the function \ref{['level sets']}, for $n=2$.
  • ...and 8 more figures

Theorems & Definitions (4)

  • Definition 1.1
  • Example 1.2
  • Theorem 1.3: Frobenius
  • Remark 2.1