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

Abstract

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.