[Paper Review] Rigid foldability of the augmented square twist

Thomas C. Hull, Michael T. Urbanski|arXiv (Cornell University)|Sep 13, 2018

Advanced Materials and Mechanics3 references3 citations

TL;DR

This paper proves that adding a diagonal crease to the classic square twist crease pattern enables rigid folding, transforming it into a 1-degree-of-freedom mechanism. It fully characterizes the configuration space, identifying exactly four non-degenerate rigid folding modes from the unfolded state, using kinematic analysis of degree-4 and degree-5 vertices via folding angle equations and loop constraints.

ABSTRACT

Define the augmented square twist origami crease pattern to be the classic square twist crease pattern with one crease added along a diagonal of the twisted square. In this paper we fully describe the rigid foldability of this new crease pattern. Specifically, the extra crease allows the square twist to rigidly fold in ways the original cannot. We prove that there are exactly four non-degenerate rigid foldings of this crease pattern from the unfolded state.

Motivation & Objective

• To determine whether adding a single diagonal crease to the classic square twist enables rigid foldability.
• To characterize the complete configuration space of the resulting augmented crease pattern.
• To identify all non-degenerate rigid folding modes connected to the unfolded state.
• To apply Balkcom’s method to degree-5 vertices to derive kinematic equations for rigid folding.

Proposed method

• Using folding angle equations for degree-4, flat-foldable vertices with α=45°, β=90°, and deriving mode-specific relations involving tan(ρ₁/2) and tan(ρ₂/2).
• Applying Balkcom’s procedure to degree-5 vertices by modeling 3D rotations via rotation matrices R_x(θ) and R_z(θ) to enforce geometric consistency.
• Deriving implicit equations for dependent folding angles (φ₁, ψ₁, φ₂, ψ₂) as functions of independent variables (u₁, ζ) at each degree-5 vertex.
• Enforcing loop closure at the central degree-4 vertex v₂ to reduce the system from 2 to 1 degree of freedom via the constraint φ₁ = ±2 arctan((√2−1) tan(φ₂/2)).
• Analyzing intersection curves of folding angle functions across different vertex modes to identify valid rigid folding paths.
• Using Mathematica to compute and visualize the configuration space curves, with validation through graphical analysis of folding angle plots.

Experimental results

Research questions

• RQ1 Can the classic square twist be made rigidly foldable by adding a single crease?
• RQ2 How many non-degenerate rigid folding modes exist for the augmented square twist from the unfolded state?
• RQ3 What is the kinematic behavior of the degree-5 vertices in the augmented square twist?
• RQ4 How do the folding modes of the degree-4 vertices (mode 1 vs. mode 2) affect the overall rigid foldability?
• RQ5 What is the configuration space structure of the augmented square twist, and how is it constrained by loop closure?

Key findings

• The augmented square twist is a 1-degree-of-freedom rigidly-foldable mechanism, with one degree of freedom removed by loop closure at the central vertex.
• There are exactly four non-degenerate rigid folding modes that connect to the unfolded state, each corresponding to distinct combinations of vertex folding modes.
• The folding angle curves for the degree-5 vertices are derived using rotation matrices and yield two solutions (±) for each dependent angle, forming the basis of the configuration space.
• The loop constraint at the central degree-4 vertex reduces the system’s degrees of freedom from 2 to 1, confirming the 1-DOF nature of the mechanism.
• Three of the four non-degenerate modes are traced by curves passing through the origin in the configuration space, with one mode following y = -2 arctan((√2/2) tan(ζ/2)).
• Degenerate cases (six total) lie along the ζ=0 axis and correspond to the non-augmented square twist, which is not rigidly foldable.