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Constructing flexible polyhedra by twinning

Elvar Atlason, Simon Guest

TL;DR

The paper addresses the scarcity of flexible polyhedra by generalizing Raoul Bricard's 1897 construction through a twinning mechanism that creates infinite families of self-intersecting flexible polyhedra. It then shows how Connelly’s embedding methods can remove self-intersections to yield embedded flexible models, and it provides explicit Type I and II twinning theorems plus concrete examples including a novel crinkle and a flexible foxtrot polyhedron. The approach expands the catalog of flexible structures and offers practical templates (networks and tents) for embedding flexible polyhedra with controlled motion, with potential applications in deployable design and engineering. Crucially, the volume remains constant during flexion (Bellows theorem), validating the physical plausibility of these motions while enabling new, non-self-intersecting realizations.

Abstract

Polyhedra are generically rigid, but can be made to flex under certain symmetry conditions. We generalise Raoul Bricard's 1897 method for making flexible octahedra to construct an infinite family of flexible polyhedra with self-intersections. Removing an edge from any of these models gives a crinkle, and these can be used to create flexible polyhedra without self-interesection. We show this in a particular example, giving a flexible embedded polyhedron with a large range of motion. We also discuss a novel crinkle.

Constructing flexible polyhedra by twinning

TL;DR

The paper addresses the scarcity of flexible polyhedra by generalizing Raoul Bricard's 1897 construction through a twinning mechanism that creates infinite families of self-intersecting flexible polyhedra. It then shows how Connelly’s embedding methods can remove self-intersections to yield embedded flexible models, and it provides explicit Type I and II twinning theorems plus concrete examples including a novel crinkle and a flexible foxtrot polyhedron. The approach expands the catalog of flexible structures and offers practical templates (networks and tents) for embedding flexible polyhedra with controlled motion, with potential applications in deployable design and engineering. Crucially, the volume remains constant during flexion (Bellows theorem), validating the physical plausibility of these motions while enabling new, non-self-intersecting realizations.

Abstract

Polyhedra are generically rigid, but can be made to flex under certain symmetry conditions. We generalise Raoul Bricard's 1897 method for making flexible octahedra to construct an infinite family of flexible polyhedra with self-intersections. Removing an edge from any of these models gives a crinkle, and these can be used to create flexible polyhedra without self-interesection. We show this in a particular example, giving a flexible embedded polyhedron with a large range of motion. We also discuss a novel crinkle.

Paper Structure

This paper contains 8 sections, 5 theorems, 6 equations, 8 figures.

Table of Contents

  1. Rigidity
  2. Twinning
  3. Type I
  4. Type II
  5. Examples of twinned polyhedra
  6. A new crinkle
  7. Methods of removing self-intersections
  8. A novel flexible polyhedron

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 (8)

  • Figure 1: An embedded flexible polyhedron, using a pentagonal crinkle arising from a twinned digonal anticupola. A net is given in figure \ref{['net']}, and a model is available online at https://www.geogebra.org/m/fq3vdnmy.
  • Figure 2: The quadrilateral $ABA'B'$ has rotational symmetry.
  • Figure 3: The Bricard octahedron, or twinned pyramid. Two faces have been omitted to remove the self-inersections, creating a crinkle.
  • Figure 4: A twinned digonal anticupola and its twinned version, obtained by removing a diagonal of the square base with blue boundary and rotating a copy around its axis of symmetry. The equator is depicted in blue. Below, see how the equator may also be regarded as the red hexagonal boundary of a twinned hexagonal cone.
  • Figure 5: The quadrilateral $ABA'B'$ has reflective symmetry. $B'$ lies behind the gray plane $\pi$.
  • ...and 3 more figures

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Theorem 2
  • Corollary
  • Conjecture : Generic case
  • Theorem 3
  • proof
  • Theorem 4