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oomorphology and how to draw eggs

Eggs are tough to crack.

Not an egg. Bird eggs are, generally speaking, pretty easy to crack. Perhaps you know this because you yourself have dabbled in omelette making, or at least are familiar with the 1990s this is your brain on drugs PSA, in which Rachel Leigh Cook taught us 1) that eggs crack easily, and that 2) despite her name, she is a very bad cook.

rachel, you have severely misunderstood how to operate a frying pan} />

Yes, eggs plural are easy to crack, but The Egg presents a problem that has challenged Serious People for well over a century.

A while back, I sat down to code a simple egg simulator. It was supposed to be a quick, goofy distraction to send to my co-founder William and the founders in our current Restless Egg cohort, where you could tweak a few parameters and make a cute little eggmoji to use as an avi. Just a lil cutesy hehe an egg sim gag, a fun distraction for me, and a fun way to distract William from the fact that I was behind on actual work.

Like many seemingly straightforward endeavors, it quickly went from an afternoon procrastination to a deep rabbit hole that derailed my entire week.

The problem, I learned, is that The Egg is one of those forms that we think we know well just because we’re so familiar with it, like your own face, or the moon. It should be simple to quantify the shape of an egg, but it’s surprisingly evasive. Every mathematical formula has some caveat, every eggquation (sry) that produces an eleggant (i’m trying to stop) curve has some special eggception (i’m not well) where the curve fails to fit the egg of some pick-me freak shorebird.

In other words, much like my brain, an egg looks smooth and simple. But unlike my brain, its appearance belies its complexity.

This is the story of that pursuit: the search for a universal equation of the egg, a shape so familiar it disappears, so slippery that mathematicians, engineers, ornithologists, and poultry scientists have spent the better part of 200 years trying to pin it down.

This is the brief history of oomorphology: the scientific study of egg shapes.

Oomorphology! This is a real word, folks. And a real world, full of history, and full of obsessives who have devoted their lives to one absurdly specific question. These are eggfolk. My folk, apparently.

At first glance, an egg looks like an oval. And an oval seems like the type of thing the brain trust over at Math Inc. should have solved a long time ago. We have circles, we have ellipses. We have dropdown menus with those PLUS the much more edgy and seemingly complex arrow and speech bubble. The quotidian egg demands more?

The trouble is, a bird’s egg isn’t a symmetrical oval like the ones drawing tools make. One end is rounder and fatter, the other end narrower and pointier. It’s not simply elongated—it is biased. It has a point of view. Rotate an ellipse 180º and passersby are none the wiser. Rotate an egg 180º and everyone can tell. Classic OG shapes like the circle or ellipse are too well behaved, too symmetrical to capture this. In math terms, an egg is an asymmetric curve.

In biological terms, it’s a product of compromise, produced by the pressures of an oviduct, the needs of the embryo, the behavior of a nest, the muscles that make wings flap, the physics of rolling, the material constraints of calcium carbonate, and whatever evolutionary bargain a given bird has made with the world.

D’Arcy Wentworth Thompson, the eminent biologist and author of the foundational shape-of-stuff-in-nature tome On Growth and Form (1917), saw the egg as an ideal case for asking what shape is actually made of. He was, diplomatically, skeptical of the easy adaptation stories that naturalists liked to tell after the fact: the guillemot egg does not become pointed simply because a naturalist has noticed that pointed things roll in tighter circles and thus are less likely to roll off of cliffs. He ascribed this explanation to “very hasty logic…by dipshit naturalists who wouldn’t know a syrinx from a cloaca.” Paraphrasing a bit, but more or less the gist.

For Thompson, the egg was a mechanical consequence: an incompressible fluid enclosed in a deformable membrane, pressed by the muscular tube of the oviduct before the shell hardened. The blunt end, the pointed end, the fact that eggs are prolate rather than flattened—these were not separate curiosities, but outcomes of pressure, tension, curvature, friction, and timing. He even wrote an equilibrium relation between external pressure, membrane tension, internal fluid pressure, and the radii of curvature, then made the baller claim that, suitably generalized, it was “the equation of all eggs in the universe.”

This is both grander and less convenient than a formula you can drop into a simulator. Thompson’s conception of an egg was not a mathematical generalization of form so much as an argument that form is a diagram of force. He takes us, as he says, as far as one can safely go without actual quantitative measurements of the forces in each particular case. Beautiful, yes. Useful, debatably. But not as easily programmable as a tidy parametric curve that lets you slide Pointiness from 0.12 to 0.87 and summon a guillemot.

Still, Thompson’s mechanical account did not immediately settle the more practical problem of describing an egg’s visible outline. Even as engineers and mathematicians began proposing particular egg-like curves, some biologists remained skeptical that the actual contour of a bird egg could be captured in any useful general way. In 1949, the husband-and-wife ornithologist duo of Alexis and Anastasia Romanoff wrote in their canonical text, The Avian Egg, that egg contours “obviously cannot be expressed in mathematical terms.” Alexis was so devoted to eggs that when his wife suggested they buy a house, he replied that they could not, because it “Would take time from study of egg. Impossible, my dear. Egg is all.”

portrait of Alexis Romanoff from a wonderful 1953 New Yorker profile of the egg expert. Illustration by Abe Birnbaum.} />

This is a beautiful sentence, partly because it is wrong. But it is wrong in the way good wrong sentences often are wrong. It names the difficulty, allows a horizon beyond which the beauty of the natural world cannot be reduced. It says, here is where the wildness of nature slips through the net.

Which is extremely charming as a poetic position, and extremely annoying if you happen to be mathematically inclined and fixated on egg shapes (greetings, howdy, a pleasure).

So people kept trying.

Over the decades, a parade of enthusiasts, some driven by the mathematical curiosity, some by biological interest, some by the commercial need to understand shells, volumes, airflow, packaging, incubation, structural strength or other practical applied egg matters. Each privileged and offered differing philosophies of form: some valued elegant simplicity, others demanded accuracy and universality. Some wanted a model with inputs you could measure with calipers, rather than more complex means or abstractions.

The history of the egg equation is therefore also a history of what it means to describe anything real. How simple can a model be before it’s too broad to be representational? How complex can it be before it becomes too specific to be utile?

Our first stop is Paris, 1818, where we meet Gabriel Lamé, a French mathematician and engineer. Lamé was not, as far as I can tell, scheming on bird eggs. He was doing math. Nevertheless, he laid the groundwork for all future egg describers by generalizing the equation of an ellipse.

The standard ellipse is familiar enough. It is the circle asked to stuff itself into a suitcase. Lamé extended this into with a whole family of oval curves defined by an equation of the form:

|x/a|ⁿ + |y/b|ⁿ = 1

where a and b set the half length and half width, and n is an exponent that tunes the shape. If n = 2, you get a standard ellipse; if a and b are equal, it becomes a circle. Change n and the shape transforms. Higher n values make it more rectangular and architectural, and lower n values make it more pointed or diamond like. The result is what became known as a superellipse, or a Lamé curve.

This is an elegant idea: you get a nice sense of the character of ovals when you can adjust the transform via a knob. You’re basically controlling how rectangular the suitcase is.

Lamé’s discovery lived a mostly quiet life in math books until over a century later, when the Danish poet, inventor, and designer Piet Hein popularized the superellipse as an aesthetic form. Hein first used it in a design competition for a roundabout around a rectangular plaza, and it soon became an inexhaustible motif in his work, applying it in everything from an astronomical calendar, to book cover designs, to furniture. He even created an object called the “superegg” by rotating a superellipse around its long axis into three dimensions, which he manufactured and sold a miniature brass version of as a novelty toy for 1960s executives to place on their teak Danish desks, possibly between a decorative chrome table lighter and an 11:00AM tumbler of scotch.

The superegg’s schtick was that it could stand upright on a flat surface, which is the type of neat parlor trick that seems inconsequential until you remember that the rise of entire civilizations has been predicated on objects that can stand upright on flat surfaces.

The appeal of the superellipse to Hein makes sense when placed into the Danish modern design milieu he was working in. It’s neither hard and rectilinear, nor soft and bloblike, but something intentional and in between: modern, minimalist, human, pleasingly referential but not pastiche.

But as egg equations go, aye, therein lies the superllipse’s critical inadequacy: it is much too fair. Too balanced, too symmetrical. It appeals to a designer like Hein because it is eggy enough, but without the egg’s essential bias. A real egg does not stand upright ed note: this is a lie, it’s pretty trivial to balance chicken eggs on their fat end (egg balancing), but just rock with me for the sake of the piece.. It has a blunt end and a pointy end, for which it will never apologize.

Lamé gave us a very expressive oval. But he did not give us an egg.

China’s National Centre for the Performing Arts in Beijing, aka “The Boiled Egg” is a superellipse sliced on its long axis. Note the perfect symmetry illustrated by its reflection.} />

Fast forward to the 1940s, and the scene shifts to wartime Germany and Fritz Hügelschäffer, a German engineer whose name lives somewhat improbably in the annals of egg geometry.

I don’t have much biographical information on Fritz, which is probably fortunate for him, given the time, place, and aviation related context of his work. But we know from patent applications that his egg work was motivated by a practical application: aerodynamics. In 1944, in the midst of designing streamlined forms (sure, perhaps for Luftwaffe airplanes or bombs, but maybe just for fun!), Hügelschäffer introduced what he called an “oviform” curve—essentially an ellipse that’s been nudged off-center to make one end fatter than the other. This was maybe the first mathematical formula explicitly for an egg shape.

Hügelschäffer’s construction is v pleasingly simple in concept. Imagine two circles on a flat plane: one circle is centered at the origin, and the second circle is shifted a bit to the side (along the horizontal axis). If the two circles were on top of each other (concentric), we’d just get a regular ellipse by the classical paper-and-string method. But Hügelschäffer moved one circle slightly off the other. By following a particular geometric logic, he traced a curve that starts at one circle and ends on the other. The result is an asymmetric oval, wider on one side, just like an egg when you view it from the side.

2·w·x·y² + b²x² + (a² + w²)·y² − a²b² = 0

If you set w = 0 (no offset), the equation reduces to a plain ellipse. But with w > 0, the symmetry breaks, and the widest part of the curve shifts away from the middle.

There is something vaguely comic about the modesty of the move. The whole big egg problem, or at least one version of it, begins with a lil nudge. There is nothing flashy or complex here, just: what if the center were like…not quite in the center?

For me, this was the point in my egg simulator rabbit hole where the egg shape problem became interesting as an idea. The aha moment was less about ellipses themselves, and more realizing how many different agendas were hiding inside the shape. To the mathematician, it’s a question of symmetry; to the aerospace engineer, it’s a problem of drag; to the birder, it’s a signature of life. The value of going down rabbit holes like this is that it forces you to inhabit all of those perspectives at once, which is a skill worth honing even if there’s no obvious end goal.

The Hügelschäffer egg curve turned out to have some useful properties. It keeps a nice continuous curvature, which makes it attractive in applications involving airflow and other fluid movement. It’s found use in airplane fuselages and automotive design, but also civil engineering where egg shaped sewer pipes manage flow behavior better than circular ones.

london’s city hall, an egg in glass} /> an egg-shaped sewer pipe — better flow at low volumes than a circle} /> hügelschäffer’s construction in motion: two circles, one nudged off-center, trace the oviform} />

Hügelschäffer’s work proved that a pretty good egg shape could be captured by a fairly tidy equation. This was a big leap from the days of “it cannot be expressed in math.”

However, the Hügelschäffer egg, while more flexible than a superellipse, is still fairly limited in the scope of natural eggs it can fit. Essentially, it introduced one asymmetry parameter (the offset w). Nature’s eggs are far more morphologically diverse. Some are almost spherical, some quite elongated, some gently ovoid, and some extremely pointy (the very cool pyriform or pear-shaped eggs of shorebirds like the guillemot, which almost look like a different genre of object entirely).

the pyriform eggs of the guillemot. naturalists argue their shape is owed to evolutionary pressures, while thompson argued it was owed to pressures of big strong wing flapping muscles} />

With just three parameters (a, b, and w), Hügelschäffer’s model could handle many oval shapes, but the most extreme conical eggs aren’t well fit, and thus the hunt for the universal egg formula continued.

Across the ocean around the same time, and in the decade after WWII, eggs attracted the attention of a British-born scientist named Frank W. Preston. We’ve got quite a bit more biographical info on Preston, and he was a fascinating character.

His main work was in glass technology: he founded Preston Laboratories and became a leading expert on glass breakage, melting, and industrial glass processes.

In his spare time, he led a second life as an amateur ornithologist and ecologist, and here in the avian field was also quite prolific. Egg shapes weren’t even his primary interest, but he published a pioneering study “The Shapes of Birds’ Eggs” in 1953. In contrast to Hügelschäffer’s engineering angle, Preston was driven by biological curiosity: why are eggs shaped the way they are, and how might that shape relate to the birds that laid them?

Preston approached the problem like a true scientist. First, he acknowledged the biological stakes while throwing a bit of shade at pure maths dorks: “This investigation was not undertaken primarily as a mathematical amusement,” he wrote, hoping it would shed light on the biology and ecology of bird eggs. He speculated that the egg’s shape might reflect the forces in the oviduct as the eggshell forms, in line with earlier ideas by Mallock (1925) and D’Arcy Thompson. But before one can connect shape to biology, one needs a way to quantify shape. And so, Preston set about formulating a more generalizable egg curve.

His approach was essentially to start with an ellipse and then allow it to deform in a flexible way. In his 1953 paper, he described the egg profile in terms of what he called “circles of derivation”—an iterative geometric construction—that we can simplify like this: he came up with a parametric equation for the egg’s outline, with four parameters to tweak:

y = b · sin θ · ( 1 + c₁·cos θ + c₂·cos² θ + c₃·cos³ θ )

Here, θ is an angle parameter running from 0 to π (top to bottom of the egg). If you ignore the c₁, c₂, c₃ terms, x = b·cosθ and y = a·sinθ would just trace an ellipse. But Preston added those extra sinusoidal terms, each with a coefficient that can be tuned. These extra terms allow the curve to bulge out more on one side and taper more on the other, essentially capturing asymmetry and different degrees of “pointiness.”

The beauty of Preston’s formula was that it could fit almost any bird egg with remarkable accuracy. Fat round owl eggs? Hoo are you kidding, quick work. Sleek symmetrical emu eggs? Not emune to description. Off-center ovoid like a song thrush’s? Not even that could thrushtrate Preston. Super-pointy guillemot egg? Well well well, uhh…hmm…running out of bird puns, this is getting aukward [the crowd goes absolutely nuclear as I pull this one off].

Indeed, Preston demonstrated that with four judiciously chosen parameters, the entire spectrum of avian eggs could be described. This was essentially a universal mathematical model for eggs, at least in an empirical sense.

There was, however, a catch: the parameters in Preston’s model, while powerful, weren’t very user-friendly. They didn’t correspond to simple intuitive traits like “pointiness” or “skinniness” directly; instead they were coefficients in an equation that one usually had to obtain by fitting the formula to a traced outline of an egg.

Preston himself later defined some derived indices—elongation, asymmetry, and what he called “bicone” (referring to how tapered the ends are)—to help interpret egg shapes.

But those indices required measuring curvature at the egg’s tip with a special device and never caught on widely. Which is surprising to me, since I’m always looking for an excuse to bust out my “Optical Spherometer for Measuring Curvatures of the Ends of Relatively Small or Frail Birds’ Eggs.”

Nevertheless, Preston’s work was hugely influential. He showed definitively that egg shape is describable in mathematical terms, you just need the right parameters. For decades, his four-parameter equation was the gold standard for quantitative egg shape analysis. Later researchers, armed with computers, applied his formula to thousands of egg specimens, confirming that it was uncannily accurate. In fact, only very recently did anyone improve significantly on the precision of Preston’s model, and it’s still considered one of the best representations of an egg’s silhouette.

By the 2010s, then, the state of eggfairs was: we had a great mathematical description (Preston’s) that works, but is a bit unwieldy, and several ad hoc measures that are user-friendly but fail on some eggs. The holy grail remained a formula both universal and simple. And that’s what our final protagonist claimed to find in 2021.

In August 2021, news broke that a team of researchers from the University of Kent and colleagues in Ukraine had done it: a “universal equation for the shape of an egg.”

I would like to take a moment to talk about the lead author, Dr. Valeriy Narushin. From what I can gather, Narushin is the Lebron of the egg shape equation world. The man has published a heroic amount of work on egg shapes, egg volumes, egg densities, egg parameters, poultry science, and in 2016 he came back from a 3-1 deficit to lead the Cleveland Cavaliers to an NBA championship. I can only assume that when he steps into the poultry science conference, he is swarmed by hordes of rabid bird scientists holding up their eggs for autographs.

There are also a handful of videos on Youtube with him from the past few years, and he seems a charming man, both surprised and chuffed that anyone outside of the poultry science world cares about his oomorphology work.

Anyway, the 2021 paper received unusually broad attention because it made a plain, irresistible claim: after centuries of partial solutions, they had devised a formula that could describe all natural bird egg shapes.

Narushin’s team built upon the foundation of earlier work (they do cite the likes of Hügelschäffer and Preston in their paper). The key innovation was to incorporate the pyriform (pear-shaped) eggs into a single analytical formula.

Remember: the sphere, ellipsoid, and ovoid shapes already had formulas; it was those most difficult, markedly asymmetrical, conical shorebird eggs of guillemots and murres that lacked a simple equation.

Narushin solved this by taking an existing ovoid formula and introducing an extra term, essentially a function that adds a tapered tail to the oval. This produced a new curve that they described as “the last stage in the evolution of the sphere-ellipsoid,” completing the continuum of egg shapes.

Crucially, Narushin’s formula uses four parameters that are all easily measurable from an actual egg. In their formulation, those parameters are: the egg’s length (L, from pole to pole), its maximum breadth (B, the diameter at the widest part), the offset of the widest part from the midpoint (w), and the diameter at one-quarter of the egg’s length (D^¹ᐟ⁴, essentially how wide the egg is halfway between the tip and the middle).

If you’ve ever held an egg, you can intuit these: L and B are obvious; the shift tells you if the fat end is towards one side, and the quarter-diameter captures how quickly the egg tapers toward the pointy end. Despite obsolescing my Optical Spherometer for Measuring Curvatures of the Ends of Relatively Small or Frail Birds’ Eggs, I must admit there are practical advantages of the Narushin formula.

The actual equation is a bit long. The team even noted it looks intimidating—it took a full paragraph in their paper. But the point is, it’s one unified formula that you plug those four numbers into, and out comes a smooth curve. With different parameter sets, it can mimic a spherical egg, a normal chicken egg, or a pyriform egg, all by tuning L, B, w, D^¹ᐟ⁴. In tests, it fit real egg profiles extremely well.

The authors were justifiably excited. They called it the first truly universal egg shape formula, filling the gap left after all those years of pretty-good-but-not-universal-solutions. It’s elegant in concept because each parameter ties to a real physical dimension of the egg, so a farmer or biologist could, in theory, measure an egg with calipers and then use this formula to compute volume, surface area, or curvature.

But the formula has a stranger appeal that exceeds utility too. There is a certain philosophical satisfaction here. One of Narushin’s co-authors remarked on the “philosophical harmony between mathematics and biology” embodied in an egg.

It is hard not to feel that tug. The egg is one of the oldest human symbols: world egg, cosmic egg, the primordial Ur-thing from which difference emerges. Creation myths understood long before mathematics did that the egg is not merely a shape, but a proposition.

The proposition is this: form precedes emergence; a whole world can begin as a curve.

What makes the universal equation beautiful is not that it reveals to us the one perfect egg floating beyond space and time. It is almost the opposite. It suggests that eggedness is not a single fixed ideal, but a family resemblance, a grammar capacious enough to hold the sphere, the ellipse, the ovoid, and the pear. The perfect egg, if such a thing exists, is not the egg that excludes all the others. It is the form that can contain their differences.

That feels right, and extremely comforting to me. Also, importantly, it solves my egg simulator thing.

As my elliptical jaunt came to its end and back to its origin, my little egg simulator no longer felt like a hehe little joke. Or rather, it still felt like a joke (and one that sincerely distracted me from so much actual work that I’m realizing right now I still haven’t done, fuck, jesus christ, ahhhhhHHHHHH), but one that has opened me to new and valuable questions.

The egg had become a way for me to think about form itself: the struggle between simplicity and specificity, between a clean model and the weirdness of the actual thing. Nature so rarely gives us the Platonic object. Instead, it gives us the thing that’s a little screwy, a little random, a little too messy to simulate.

Like so much else organic and natural, the egg is a story of adaptations and trade offs: an egg must be big enough to nourish a chick, but not so big that the mama bird can’t lay it; it needs to be strong enough to survive incubation under a hen, yet fragile enough to be broken from the inside by a teeny beak; it needs to roll in a circle, not off a cliff. It is not designed for a single purpose, but negotiated among many.

No wonder its geometry is special.

The history of egg equations follows a similar logic. Mathematicians begin with an ideal, then approach the complexity by layering simple ideas: a circle (too simple) grows into an ellipse; an ellipse skewed by a circle gives an ovoid; an ovoid tweaked by one more function yields a pyriform. Each leap—Lamé, Hügelschäffer, Preston, Narushin—added a parameter or a term, making the formula less simple but more encompassing.

This has also had the surprising effect of complicating my operating bias toward minimalism. Sometimes the most elegant truth of a thing is not revealed by stripping it to its most spartan form, but by giving it just enough freedom to become itself.

The “perfect” mathematical egg turned out to require the messy collaboration of many minds over time, each building on the last, from pure math to engineering to biology and back.

Now, it’s your turn to enjoy the eggs of their labor. Below, you’ll get to design your own egg. Slide the parameters and see how each tweak channels one of these historical approaches, from a superellipse’s symmetry to a Hügelschäffer’s lopsidedness or a Narushin pear-shape. As you do, remember: you’re not just drawing an oval, you’re participating in a story that started with the cosmic egg, and egg by egg, continues to this day.

Happy egg shaping. May you find your own perfect egg, or at least enjoy the obsessive, absurd journey of looking for it.

✦ finis ✦

jk, please download to your heart’s content, share your beautiful eggs far and wide} />

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