How to Fold a Triangle: The Connection between the Grashof Fourbar Linkage and Pythagorean Triangles

Summary

This paper presents two different methods for folding a triangle onto a flat line by converting it into a Grashof Special Case fourbar linkage. Foldable triangles have many applications, ranging from space structures to collapsible furniture. In the first method, a generic triangle with positive real side lengths is shown to be foldable inward and outward by adding a pin joint at a specific location to one of the sides. The second method shows how to create a collapsible linkage using a Pythagorean triangle; we demonstrate that all four links in the Pythagorean fourbar have integer length and can be easily built using rods with uniformly spaced holes (e.g., LEGO bricks, Unistrut, etc.). Next, the formulas for finding the interior angles of the Pythagorean fourbar are presented, for the purpose of plotting collapsible structures with mathematical software. We conclude with a demonstration of a sample application of Pythagorean fourbars to collapsible arch or tower structures and a surprising proof that some types of collapsible arch structures are unrealizable with Pythagorean fourbars.

Additional information

Notes on contributors

Eric Constans

Eric Constans ([email protected]) is a professor of Mechanical Engineering at the Rose-Hulman Institute of Technology. He is the author of a textbook on computational kinematics and mechanical design.

Nicola Golfari

Nicola Golfari ([email protected]) is an architect and designer who graduated from the Politecnico di Milano. Born in Italy in 1969, he began thinking about design by observing nature and dismantling toys and devices. In 2010 he founded his own studio, working on architecture and product design (www.nga.archi). With the design collective Recession Design he is coauthor of two books on Do It Yourself design thematics (published by Rizzoli). In 2018 he began his research of folding frame mechanisms, focused on new typologies of deployable structures.