The Catenary – Mathematics All Around Us.

[music plays] You may be aware that mathematics is all around us. It’s in the petals of a flower, it’s in the honeycomb of a bee hive, it’s in the spirals of a sea shell. So it’s all around us in the natural world, but what about closer to home? Here I have a piece of rope. If I hold it between my hands, it forms a curve. This is called a catenary curve, and it is not a parabola. It is its own type of curve, with its own type of mathematical description. [music plays] The word ‘catenary’ is derived from the Latin for ‘chain,’ because it’s the shape formed from any chain or rope hanging freely under its own weight. And you will see this simple and beautiful piece of mathematics everywhere, from the wires in an overhead power line, to the strands of a spider’s web. But why do we see this curve appearing again and again in the natural world? To answer that question, we’re going to turn to the world of bubbles. To minimize their surface tension, bubbles form perfect spheres. If we try to form a bubble between two rings, it cannot make a sphere, so instead makes a curved shape. This time, to minimize its surface tension, it makes the catenary. So a chain, like in the world of bubbles, wants to minimize its tension. So it forms the catenary, and it is not a parabola. The parabola is the arc of a ball flying through the air. [chalk board noise] Its mathematical description is y equals (ax) squared plus bx plus c. [y=ax^2+bx+c]. By eye alone, it looks like a catenary. Galileo realized that the curve of a free-hanging chain was very similar to a parabola, but something was amiss. The mathematical description of a catenary is actually quite different. It is y equals a cosh x over a. [y=a cosh(x/a)] That’s equal to a over 2 outside of e to the power x over a plus e to the power minus x over a.
[y=a/2(e^x/a + e^-x/a) The curve changes as the distance between the two ends and the length of the chain changes. But they can be described with the same formula, and are never parabolas. Man uses this natural form in the worlds of engineering and architecture. For example, the wires of a suspension bridge are parabolic due to the weight of the bridge it carries. But without this weight, it is more like a simple rope bridge, carrying the same weight along its length, and will again be a catenary curve. But if the catenary is the shape of a rope or a chain that minimizes forces in tension, then by turning this upside down, we form an arch, which minimizes the forces in compression. And it’s actually the strongest kind of arch– strong enough to support its own weight. For example, That’s what the master builders of the Middle Ages did to construct this, the ceiling of King’s College Chapel in Cambridge. So to make the strongest kind of arch, you can simply mark out the shape of a free-hanging chain, and turn that shape upside down. [music plays] So now that you know what a catenary curve is, you’ll be able to spot it everywhere. So the next time you see an archway, a bridge, or a spider’s web, remember: you’re seeing mathematics! [music plays]