An Eggcellent shape

On the bewildering variety of egg shapes, and the search for a universal formula.

How would you describe a roundish-but-elongated-on-one-side figure? The word “egg shaped” probably springs to mind. Eggs are so familiar to us — and have been since ancient times, as evidenced by the Roman phrase ab ova ad mala or “from eggs to apples”. Eggs used to be served at the beginning of a meal and fruits at the end, so the phrase basically means “from the beginning to the end”.

If you move your thoughts away from food, though, eggs are a real wonder. They’re such an unfamiliar way of containing a life form: small enough to slip out of the body and yet it hosts the future living form in it — containing everything needed to stay alive. No wonder using egg-shaped capsules for space exploration feels so sensible.

Using egg shapes in this way calls for careful calculation — as indeed it does even in the case of bird studies,poultry farm design, and architecture. And for calculation, we need precise definitions. Which brings up the question: what, exactly, is the shape of an egg?

To understand the egg, one must first understand that they aren’t all ovoid and white. And to drive that home, let’s take an extreme example.

Fifteen-year-old Eragon is out hunting, when he comes across what looks like a large gemstone in the forest. He tries to sell it — only to discover that its value cannot be discerned, since it does not break on contact with a hammer. And yet, it resounds like a hollow object!

The “gemstone” is perfectly elliptical, a shape known for its symmetry, and which increases the surface area on which any force acts. Eragon’s find gets accidentally dropped several times, but not a crack appears on its surface — a testament to its strength. Although the material would have played a very important role in this, there is no denying that the shape itself was a massive help.

Eventually, the object does crack open. And out hatches…the dragon Saphira, who goes on to be Eragon’s constant companion.

Using the phrase “the shape of an egg” is, in some ways, as sensible as saying “the shape of a gem”. We all know that gems occur in various shapes and sizes, and eggs are no different.

Let’s look at another beloved franchise: How To Train Your Dragon. We’re introduced to a group of goofy, loveable dragons, but what’s more interesting is investigating their egg shapes and how they tied into what kind of dragon they grew up into.The Gronckle, a species known for its love for rocks and sociable behaviour, lays eggs that are spherical with a rock-like structure — with the added twist that they are bouncy, something that an elliptical egg could never hope to achieve! This egg’s  welcoming nature corresponds with the puppy-like behaviour of the Gronckle.

Contrast this with another dragon— the Deadly Nadder. The eggs of these beasts are pear-shaped, covered with predator-deterrent spikes. Deadly Nadders are known to be cautious, and their spiky eggs reflect that character.

One doesn’t have to go into fantasy to find strangely shaped eggs. Those of the guillemot, which nests right on the cliffside, are remarkably pointy. It was believed that this shape prevented them from rolling off, but newer research suggests two alternative reasons: having more eggshell in contact with the surface, to prevent against impacts, and having a shape better suited to keeping clean.

The guillemot is just a well-known example: wherever you look, you will find birds with very specific and unusually shaped eggs.

To bring some order to the chaos, various indices have been used to classify eggs into four shapes: sphere, ellipsoid, ovoid, and pyriform. The range from the perfect ball laid by owls to guillemot-level elongated pointy-on-one-side eggs.

But while this certainly helps in studying eggs, there is no universal formula that can be tweaked to apply to any egg. Such a formula would be a boon to science and help inspire new technology in the future. (Before that, of course, it would also be a help to the food and poultry industry).

Is there a single universal formula that can be applied to any egg? That’s what a recent mathematical study aimed to find out.

Researchers considered the eggs of four birds, each representing a different shape-class from round to elongated. Ural owl eggs stood in for perfect spheres and emu eggs for ellipsoids; song thrush and osprey eggs stood in for the ovoid shape, while pyriforms had the guillemot egg as their representative.

Each shape is this list is considered a more general form of the previous shape. Thus, every sphere is an ellipsoid; every ellipsoid is an ovoid — and not the other way around.

Simple geometrical shapes such as triangle, circle, square, cube, rectangle are aesthetically pleasing and have attracted the human eye since antiquity as evidenced in its deployment in classical Greco-Roman architecture. They’re also pretty easy to make calculations with.

Round things are a bit more difficult.

If you want to define a cube, all you have to do is set up lines of a certain length, at right angles to each other. For a sphere, the formula is more like this:

Take a random point in space and a particular radius, r. Now, choose all the points that are r distance away from the random point. That’s your sphere — a 3D version of this circle from school.

If you open it and play around with the formula, you will see it growing larger and smaller as you tweak it — but it doesn’t really change shape.

This is, incidentally, also the shape of an Ural owl egg.

Owls like to nest in nooks and crannies, or the hollows of trees, so there’s not much chance of rolling out. This means owls can afford to lay near-perfect balls, with all the advantages it brings such as compactness and being less of a hassle to find space for in the nest — after all, one needs to fit in between one and thirteen eggs in a relatively small space.

While Greco-Roman architecture liked simple geometric shapes, the egg hasn’t been lying low either. In modern times, the egg has inspired modern buildings such as the roof of London’s City Hall and the Gherkin, London’s most recognisable tower.

As it turns out, this shape is better for the environment — at least, in terms of energy efficiency. Because it exposes 25% less surface area to the sun than a cubic building of the same volume, an egg-shaped building does not overheat in summer or lose too much heat in winter. A principle that one speculates must apply to actual eggs as well.

Chicken eggs are incredibly common — since the time of the Romans, apparently —but other eggs can get very expensive. About ten years ago, emus had become quite the craze in India, with emu eggs selling for thousands of rupees. Unfortunately, people realised that one couldn’t really do anything with emu eggs except sell them to someone else…until the country ran out of people to sell to.

Emu eggs are the next simplest thing to a sphere: in 2D, they’d be elliptical. They live in grasslands, where, again, the eggs don’t roll. Unlike Ural owl eggs, though, these ones are elongated one way probably so they can squeeze more baby bird in.

In mathematical life, there’s an easy way to make an ellipse: take a cone, and slice its top off. If you slice it perfectly straight you’ll get a circle, but if you slice it at an angle what you’ll have is an ellipse! In fact the word “ellipse” comes from the Latis ellipsis meaning “falling short” or “deficit”. The shape you have just falls short of a cone.

This formula has two parameters, which you can adjust independently: this is where you start seeing various shapes emerge.

Egg shapes can be subtle. Even the ordinary ones you eat for breakfast aren’t perfectly elliptical, as you’d know if you follow the advice of storing your eggs narrower-side-down to give them a longer shelf-life. We’re conditioned to think the rounded side is the bottom and the pointier side the top, but it’s actually the other way round. Every egg has a naturally occurring air bubble there, which helps keep the yolk centred and lets your eggs stay fresh for longer — if the bubble isn’t displaced.

Now that you’ve been made acutely aware of the difference in egg sides, you can see our mathematical model doesn’t really capture the subtleties.

For this, we need to turn to Hügelschäffer’s formula. Essentially, the idea is to multiply one of the elliptical directions by a function such that the width of the ellipse is different at different points. If you look at the forumla, you’ll find three parameters, which can be adjusted to match the shapes of thrush and osprey eggs.

Last but not the least is the pyriform egg. These are conical eggs, the ones laid by the cliff-laying guillemots being an extreme example. The pyriform shape generally refers to a pear-like structure, with a blunt, round bottom and a slightly pointed top. Pyriform eggs are generally laid by birds that tend to nest on cliffs and slopes, the guillemot being the obvious example.

Formulae for all these shapes already existed. The question was: how to combine them into one giant formula; a formula that can work for every egg in the world?

The result turned out to be this monster of a formula that took me quite some time to type into Desmos, my online mathematical calculator.

This general formula has four parameters: egg length L, maximum breadth B, shift of the vertical axis W, and the diameter at one quarter of the egg length DL4. Here’s a fun challenge for you: open the link, and try adjusting the four parameters so that you can obtain all of the four shapes mentioned above for different sets of parameters.

The mathematicians admit their formula might not fit every last contour of an egg, but that their work has expanded the area of mathematics to give another shape to describe the majority of  real-world eggs.

Of course, that still doesn’t cover the Gronckle.