A wrapped gift, as lovely as it may be, isn’t perfect without a colorful and springy ribbon. If you've ever wondered what causes a ribbon to change its shape in this way - here’s an explanation for the curliness.

The next time you wrap a gift for a friend, try this trick: Grab the wrapping ribbon and a pair of scissors. Hold the ribbon by one end, and carefully slide the scissors’ blade along it with your other hand (see the Wikihow website for a detailed guide). When done correctly, you’ll notice the previously straight ribbon transforming into a delightful curl. Ever wonder why this happens? Fortunately, physics can provide an explanation, which ties together geometry, elasticity, and other everyday phenomena you’ve surely come across in your daily life.

## What is elasticity?

To understand why the ribbon curls and other related phenomena that we’ll mention later on, we need to understand the concepts of elasticity and elastic range. According to the physical definition, elasticity is a property of a material that indicates the extent to which it can change its shape when influenced by the external forces that act on it (or cease to act on it). The elasticity of an object depends on its material composition as well as on other factors, such as its shape and the type and direction of the force applied to it. According to this definition, a sponge is very elastic upon compression, since it takes very little force to change its shape and compress it, while wood is not very elastic. “Elasticity” is therefore also the name of the field of physics and mathematics that describes how different objects respond to the forces applied to them. Specifically, when we study elasticity phenomena, we are particularly interested in changes in distance and in relative positions between different parts of the material. These changes are referred to as “deformations”.

The elastic range is a property that describes how much a material can be stretched or shrink with respect to its original length, and still return to its original form when we stop applying the force. If a material is stretched beyond its elastic range, its form will become permanent and it won’t return to its original state after we stop stretching it.

The 17th-century English scientist, Robert Hooke, founded the research field of elasticity. He used relatively simple mathematical approaches combined with experimental observations, and concluded that deformations to an elastic object are relative to the force applied to it. In its simplest form, this claim is called Hooke’s Law, and every physics student can tell you how this law describes the action of a spring: the spring will compact when we apply a force to it, and if we apply double the force, it will compact twice as much. According to Hooke’s approach, we can think about any material as a collection of springs linked to each other, and thus calculate how they will respond to different forces acting on them. In practice, it turns out that this claim is a very good approximation if the force acting on the object is relatively weak, but it’s less accurate when the force is stronger or with respect to materials with unique properties, such as very thin materials. In modern physics, elasticity is studied in the language of a more complex mathematical theory called differential geometry—a mathematical theory that describes surfaces and their properties, such as curvature. Through differential geometry, physicists can describe how different types of surfaces are deformed and change; this theory was even used by Einstein to describe time and space within the framework of the General Theory of Relativity.

A spring in three states within its elastic range: At the top - no force applied, in the middle—tensile force applied, and at the bottom - compressive force applied | Honourr, Shutterstock

## The Side That Stretches and The Side That Contracts

If so, let’s get to the point—how does physics explain why the ribbon curls? Every elastic object “tries” to change its shape in such a way that the internal forces acting within it balance the external forces acting on it. Roughly, two types of forces can act on any object: compressive (or tensile) forces that compact (or stretch) it and forces that induce bending. The thinner the object, the greater its tendency to bend and deform rather than stretch, and vice versa. For example, if you hold a metal ruler at both ends and try to bend it, it is likely to bend - since the ruler is very thin, it tends to bend rather than to contract - or in physical terms, bending is energetically preferred over contracting. In fact, thin objects made of various materials usually tend to bend rather than to stretch or contract. This explains, for instance, why it’s difficult to insert a page into a nylon storage pouch. We try to slide the page inside, but friction makes it difficult. Opposing forces act on the paper: on one hand we push it inside, and on the other hand the friction prevents it from entering. If we don’t open up the pouch’s mouth and allow the page to slide smoothly inside, it will wrinkle, in other words, it will bend instead of getting compressed.

How is this related to the curling of ribbons? When we glide scissors along the ribbon’s length, we stretch the fibers composing the ribbon in one direction. Given the scissor blade's acute edge and the way we perform the action, we apply more pressure to the outer side of the ribbon (the one facing away from us) than to the inner side, resulting in differential stretching. Consequently, the inner side of the ribbon remains shorter than its outer side. Since one ribbon cannot have two different lengths, the ribbon faces two alternatives: the first alternative is that the force that the long side exerts on the short side will cause it to stretch slightly, and vice versa - the force that the short side applies to the long side will cause it to shrink slightly. The result is that each side of the ribbon will achieve equilibrium between the external force we applied to it (stretching with the scissors) and the internal forces acting on it (the pull of the other side of the ribbon). Hence the two sides attain uniform length.

But the stretched ribbon faces another alternative: it can bend. This solution can be understood if we imagine a running track in an Olympic stadium: if the runners complete a full lap around the stadium they will start and finish at the same point, but the runner in the lane closest to the center will run a shorter track than those on the outer lanes. In the same way, when the ribbon bends and curls in the direction of the shorter, inner side, the two sides of the ribbon will be able to match each other without alterations in the length they reached during our initial stretching with the scissors. As mentioned earlier, thin objects usually tend to bend rather than stretch; so the result, in practice, is that the ribbon curls.

Using the scissors, we unevenly stretch the ribbon on both its sides. This causes the ribbon to favor curling as a means of reconciling the different lengths. Wrapped boxes decorated with curly ribbons | RoJo Images, Shutterstock