Making Math

Catenary Arches

The catenary curve is interesting because there are many examples of it in the world around us.  The best way to visualize a catenary curve is to imagine the shape of a hanging chain.  (The word comes from the Latin word catena meaning "chain.")  Catenaries are used in engineering and architecture, for example in the shape of hanging bridges, or when inverted, in the shape of some arches.  One of the most impressive examples is the St. Louis Gateway Arch.  Catenaries can also be found in nature, for example in the curve of a spider web.

At first glance, catenaries might look like parabolas, but they have a completely different formula.  The formula gives a shape that has a special structural property when used as an arch.  When the chain shape is inverted into an arch and divided into building blocks, the blocks can support each other by gravity alone.  In these workshops, students will build catenary arches and physically experience their special properties.

To fully understand how catenaries differ from parabolas and why chains take the shape of catenary curves, students would need some calculus background. We will not go into those details in these workshops as pre-university students generally are not ready for this material.  Rather, we hope this activity inspires students to keep building their mathematics knowledge, leading towards integral calculus (and beyond).

Paper Catenary Arch.  In this activity, students build paper modules that assemble into an elegant arch 18 inches (48 cm) tall.   This construction is a fun dexterity challenge that can be adapted to different skill levels.

Cardboard Catenary Arch.  Students will build a five-foot tall version of the catenary arch using cardboard modules.  This activity provides a group exercise in collaborative construction.

Wood Catenary Arch.  This seven-foot tall arch is made of 1/8-inch thick laser-cut plywood.  It is a more permanent, architectural sculpture, that exemplifies the beauty of mathematics.