The mathematics of gift wrapping 

From Christmas to Hanukkah to Kwanzaa, the gift-giving season is upon us. After we track down the perfect items for our favourite people, another task awaits us: gift wrapping. It’s not just an art—it’s math in disguise. 

We spoke to Dr. Adam Martens, UBC mathematics postdoctoral fellow and differential geometer about the best shapes to reduce waste—and why a donut-shaped object can be wrapped perfectly, but only if you work in four dimensions. 

What is a differential geometer? 

 A geometer is a specialist in geometry, or the study of points, lines, angles, surfaces, and solids. A differential mathematician studies smooth objects called ‘manifolds’, for example, a flat piece of paper or the surface of a ball. We also think about higher-dimensional objects, like the space-time of the universe. 

What is the easiest shape to wrap? 

No surprises here, but a box. The nice thing about wrapping a box is that each side is flat, and the flat edges meet at simple creases. Wrapping paper can be easily folded over the edges—mathematicians call this a manifold with corners. 

Wrapping paper is inherently flat and rigid. It can be folded, but from a mathematical point of view, it cannot be warped so that it lies flat on a curved surface. 

This means it’s mathematically impossible to wrap a sphere perfectly i.e. without any creases or folds. The only way to effectively wrap a ball is to put the ball in a box. 
A closely related theorem in calculus is the “hairy ball theorem,” which says you can’t comb a hairy ball flat without creating a cowlick or hair swirl. 

What is the most difficult shape to wrap? 

Technically, any shape that is not flat is equally difficult because they are all impossible. You cannot bend the wrapping paper to fit non-flat shapes. You could work around this by cutting and taping, but if any point is not flat, it’s impossible – at least not without creasing the wrapping paper. 

That being said, there are shapes that seem impossible to wrap but are actually technically doable. Take a donut shape, what we call a “torus” in math. This object sits inside four-dimensional space where, if you were a 4D creature, you could make a torus flat and wrap it— so potentially not very helpful for your holiday shopping since we’re 3D beings and can’t visualize what is going on.  

We can see this by taking a flat piece of paper. If you glued the long sides together, you would get a cylinder. You can’t do this in 3D because the paper would crinkle, but if you bend the paper and glue the short ends together, you’re able to take a flat piece of paper and bend it into a torus. 

What gift-container shape minimizes the amount of wrapping needed? 

In geometry, the isoperimetric inequality is a principle that tells us that a sphere is the most efficient shape for enclosing an item. An example of this is when we blow bubbles in a glass of water—the bubbles form as spheres because the air inside of them wants to take up as little space as possible due to the air pressure they face on the outside. In this sense, a sphere would be your most optimal shape for minimizing wrapping, except it wouldn’t really because, as we know, you can’t really wrap a sphere very well. 

The next best option would be a cube—not an arbitrary rectangular box—where all sides are equal in length. For a fixed volume, a cube minimizes the surface area that needs to be covered in wrapping paper. 

How about gift bags? 

It’s not always about optimization. As human beings, we tend to find things aesthetically pleasing when they’re not square. Gift bags, for example, are elongated in one direction. We like the look of this. A lot of it has to do with the golden ratio—1.618, also known as Phi—which we can find in nature, including in the radial spiral of pinecones or sunflower seeds, in art in the proportions of the Mona Lisa’s face and torso, and architecture, in the proportions of the Parthenon. I even have it tattooed on my arm. Many people think that some of these appearances in nature are just a coincidence or selection bias, but something about this ratio is very pleasing to the eye.