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Cutting along a symmetric quadrilateral to construct an embedded flexible dodecahedron

Elvar Atlason

TL;DR

The paper addresses the problem of constructing embedded flexible polyhedra with the smallest possible number of vertices. It introduces a general cut-and-glue framework along symmetric quadrilaterals, using a twist or a reflection to generate new flexible surfaces from known ones, and proves that these modifications preserve the original degrees of freedom. By applying this method to a Bricard type I octahedron, it constructs a flexible dodecahedron on eight vertices with a substantially larger range of motion and provides an alternative proof of the eight-vertex minimality, supported by a concrete net and parameter choices. The work broadens the toolkit for embedded flexible polyhedra and confirms that the simplest embedded flexible polyhedron has eight vertices, rather than the previously conjectured eight-vertex being tied to Steffen’s nine-vertex example, with practical realizations available via nets and interactive models.

Abstract

Until recently, the simplest known flexible polyhedron was Steffen's polyhedron on nine vertices. However, in 2024, an embedded flexible polyhedron on eight vertices was announced. It attains the known lower bound for the number of vertices, showing that the simplest embedded flexible polyhedron has eight vertices. We introduce a method for making new flexible polyhedral surfaces from old ones. This general method applies to the above minimal example, giving another proof of its flexibility. We also construct a different flexible dodecahedron on eight vertices. This improves both the range of motion and the simplicity of the exposition.

Cutting along a symmetric quadrilateral to construct an embedded flexible dodecahedron

TL;DR

The paper addresses the problem of constructing embedded flexible polyhedra with the smallest possible number of vertices. It introduces a general cut-and-glue framework along symmetric quadrilaterals, using a twist or a reflection to generate new flexible surfaces from known ones, and proves that these modifications preserve the original degrees of freedom. By applying this method to a Bricard type I octahedron, it constructs a flexible dodecahedron on eight vertices with a substantially larger range of motion and provides an alternative proof of the eight-vertex minimality, supported by a concrete net and parameter choices. The work broadens the toolkit for embedded flexible polyhedra and confirms that the simplest embedded flexible polyhedron has eight vertices, rather than the previously conjectured eight-vertex being tied to Steffen’s nine-vertex example, with practical realizations available via nets and interactive models.

Abstract

Until recently, the simplest known flexible polyhedron was Steffen's polyhedron on nine vertices. However, in 2024, an embedded flexible polyhedron on eight vertices was announced. It attains the known lower bound for the number of vertices, showing that the simplest embedded flexible polyhedron has eight vertices. We introduce a method for making new flexible polyhedral surfaces from old ones. This general method applies to the above minimal example, giving another proof of its flexibility. We also construct a different flexible dodecahedron on eight vertices. This improves both the range of motion and the simplicity of the exposition.

Paper Structure

This paper contains 5 sections, 4 theorems, 9 equations, 5 figures.

Table of Contents

  1. Symmetric quadrilaterals in $\mathbb{R}^3$
  2. Cutting along a quadrilateral
  3. General description
  4. A flexible dodecahedron
  5. Minimality

Key Result

Theorem 1

Let $A$, $B$, $A'$, $B'$ be four points in $\mathbb{R}^3$ such that $AB = A'B'$ and $AB' = A'B$. Then there exists a line $l$ in $\mathbb{R}^3$ such that a half-rotation in $l$ swaps $A$ with $A'$, and $B$ with $B'$.

Figures (5)

  • Figure 1: The quadrilateral $ABA'B'$ has rotational symmetry.
  • Figure 2: The quadrilateral $ABA'B'$ has reflective symmetry. $B'$ lies behind the gray plane $\pi$.
  • Figure 3: Cutting and reflecting a Bricard type I octahedron along the quadrilateral $DB'A'B$. The red and blue colours indicate segments of equal lengths, $AB' = A'B = BD$ and $AB = B'A' = B'D$.
  • Figure 4: A net for a flexible dodecahedron. Valley folds are given by dashed lines and mountain folds by solid lines. The dotted edge marked $y$ can go from a valley fold to a mountain fold during the flexing motion, so should be scored on both sides. Gluing instructions are indicated by the colour of the edges. An online model is available at https://www.geogebra.org/m/pb4nqczx.
  • Figure 5: Polyhedral graphs with triangular faces on seven vertices or fewer. Vertices of degree three, coloured red, can be removed without affecting flexibility. Thus, all but three graphs are rigid.

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • Theorem 4