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Continuous Flattening and Reversing of Convex Polyhedral Linkages

Erik D. Demaine, Martin L. Demaine, Markus Hecher, Rebecca Lin, Victor H. Luo, Chie Nara

Abstract

We prove two results about transforming any convex polyhedron, modeled as a linkage L of its edges. First, if we subdivide each edge of L in half, then L can be continuously flattened into a plane. Second, if L is equilateral and we again subdivide each edge in half, then L can be reversed, i.e., turned inside-out. A linear number of subdivisions is optimal up to constant factors, as we show (nonequilateral) examples that require a linear number of subdivisions. For nonequilateral linkages, we show that more subdivisions can be required: even a tetrahedron can require an arbitrary number of subdivisions to reverse. For nonequilateral tetrahedra, we provide an algorithm that matches this lower bound up to constant factors: logarithmic in the aspect ratio.

Continuous Flattening and Reversing of Convex Polyhedral Linkages

Abstract

We prove two results about transforming any convex polyhedron, modeled as a linkage L of its edges. First, if we subdivide each edge of L in half, then L can be continuously flattened into a plane. Second, if L is equilateral and we again subdivide each edge in half, then L can be reversed, i.e., turned inside-out. A linear number of subdivisions is optimal up to constant factors, as we show (nonequilateral) examples that require a linear number of subdivisions. For nonequilateral linkages, we show that more subdivisions can be required: even a tetrahedron can require an arbitrary number of subdivisions to reverse. For nonequilateral tetrahedra, we provide an algorithm that matches this lower bound up to constant factors: logarithmic in the aspect ratio.

Paper Structure

This paper contains 13 sections, 7 theorems, 3 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Our Results
  3. Definitions
  4. Lower Bound on Subdivision
  5. Flattening Convex Polyhedral Linkages
  6. Turning Equilateral Linkages Inside-Out
  7. Tetrahedral Linkages
  8. Triangular Prisms
  9. General Algorithm
  10. Example: Regular Icosahedron
  11. Proof of Theorem \ref{['th:i-o']}
  12. Reversing Nonequilateral Linkages
  13. Open Problems

Key Result

Theorem 1

There is an infinite family of convex polyhedral linkages that requires $\Omega(|E(G)|)$ subdivisions to flatten or reverse, where $G$ is the graph of the linkage.

Figures (11)

  • Figure 1: Two distinct flat-folded states of a linkage where edges $AC$ and $BD$ are touching. In (a), $AC$ lies above $BD$ (i.e., $AC \succ BD$), while in (b), $BD$ lies above $AC$ (i.e., $BD \succ AC$). Transitioning between these states requires a $2\pi$ rotation about edge $CD$.
  • Figure 2: An example of continuously folding a convex polygonal linkage into a spiky ball by subdividing all edges in half: (a) the original pentagonal linkage $ABCDE$ with a fixed point $O$, the midpoint $M$ of the edge $AB$, and its intermediate folding with vertices $A', B', C', D', E'$ and $M'$ corresponding to $A, B, C, D, E$ and $M$, respectively; and (b) the spiky ball with center $O$ reached by $A', B', C', D'$ and $E'$.
  • Figure 3: An example of flattening a tetrahedral linkage in two ways: (a) subdividing all edges in half, and (b) subdividing only one edge to obtain Figure \ref{['fig:cross']} (a) as a subset.
  • Figure 4: (a) An example of a tetrahedral linkage with an equilateral base $BCD$ and the peak $A$ such that $|AB|=|AC|=|AD| < 2 |BC|/\sqrt3$; (b) Motions of $A$ and midpoints of $AB, AC, AD$ in the beginning; (c) The figure of the linkage when midpoints reach on the cylinder of radius $|AB|/2$ with the $AA'$-axis (where $A'$ is the target location of $A$); (d) The final figure of $L$ which is the mirror image of the original. .75
  • Figure 5: (a) A linkage of an equilateral triangular prism with three stacks, where number $i$ stands for the vertex $v_i$; (b) The top stack transformed to a near tetrahedral linkage; (c) The near tetrahedral linkage transformed to the inside-out figure: (d) The second top stack transformed to a near tetrahedral linkage; (e) The near tetrahedral linkage transformed to the inside-out figure; (f) Edges folded in half moved in the outside.
  • ...and 6 more figures

Theorems & Definitions (13)

  • Theorem 1
  • proof
  • Theorem 2
  • proof : Proof
  • Theorem 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 4
  • ...and 3 more