Buzz
For centuries, symmetry has intrigued philosophers, astronomers, mathematicians, artists, architects, and physicists. The ancient Greeks were particularly captivated by it—and even today, we tend to embrace symmetry in everything from arranging our furniture to styling our hair.
The reasons behind its omnipresence are still a mystery, as is why the math behind it seems to influence so much of the world around us. However, the following ten examples undeniably showcase its presence.
Just a heads-up: once you start noticing it, you might find yourself unable to resist spotting symmetry everywhere you look.
10. Romanesco Broccoli
You might have walked past romanesco broccoli in the store and assumed, due to its peculiar look, that it was some sort of genetically engineered food. In reality, it’s just one of the many stunning examples of fractal symmetry in nature—though it’s certainly an eye-catching one.
In geometry, a fractal is a detailed pattern in which each smaller part mirrors the whole. For romanesco broccoli, each floret follows the same logarithmic spiral as the entire head (just on a smaller scale). In essence, the whole vegetable forms a massive spiral made up of tiny, cone-shaped buds that are themselves mini spirals.
Interestingly, romanesco is closely related to both broccoli and cauliflower, though it shares more in common with cauliflower in taste and texture. It’s also packed with carotenoids and vitamins C and K, making it a healthy and mathematically stunning addition to our plates.
9. Honeycomb
Bees aren’t just experts at producing honey—they also seem to have a natural gift for geometry. For thousands of years, humans have been amazed by the flawless hexagonal shapes in honeycombs and have wondered how bees can instinctively craft a shape that we can only reproduce using a ruler and compass. The honeycomb exemplifies wallpaper symmetry, where a repeating pattern covers an entire surface (like a tiled floor or mosaic).
So, why do bees seem to favor hexagons? Mathematicians argue that hexagons are the ideal shape for storing the maximum amount of honey while using the least amount of wax. Other shapes, such as circles, would leave gaps between the cells because they don’t fit together perfectly.
Some skeptics, who don’t fully trust the bees’ geometric brilliance, suggest that the hexagonal shape is simply a result of “accident.” They believe that bees initially create circular cells, and the wax naturally collapses into a hexagonal form. Regardless of the theory, it’s a phenomenon of nature—and undeniably impressive.
8. Sunflowers
Sunflowers showcase radial symmetry and an intriguing form of numerical symmetry known as the Fibonacci sequence. The Fibonacci sequence progresses as follows: 1, 2, 3, 5, 8, 13, 21, 24, 55, 89, 144, and so on, with each number being the sum of the two preceding ones.
If we were to count the seed spirals in a sunflower, we’d discover that their number corresponds to a Fibonacci number. In fact, numerous plants (including romanesco broccoli) organize their petals, leaves, and seeds according to the Fibonacci sequence, which explains why four-leaf clovers are so rare.
Counting spirals on sunflowers can be tricky, so if you’d like to test this idea yourself, try counting the spirals on larger objects such as pinecones, pineapples, or artichokes.
But why do sunflowers and other plants follow mathematical principles? Like the hexagonal patterns in beehives, it’s all about maximizing efficiency. To put it simply, a sunflower can fit the most seeds if each seed is positioned at an angle defined by an irrational number.
As it turns out, the most irrational number is known as the golden ratio, or Phi, and interestingly, when we divide any Fibonacci or Lucas number by the preceding number in the sequence, we get a value close to Phi (1.618033988749895 . . .). Therefore, for any plant following the Fibonacci sequence, there should be an angle known as the ‘golden angle’ that corresponds to Phi between each seed, leaf, petal, or branch.
7. Nautilus Shell
In addition to plants, certain animals, such as the nautilus, also display Fibonacci numbers. For example, the shell of a nautilus grows in what’s known as a ‘Fibonacci spiral.’ This spiral forms because the shell maintains the same proportional shape as it grows outward. For the nautilus, this growth pattern allows it to preserve its shape throughout its life, unlike humans, whose bodies change proportion as they age.
As is often the case, there are exceptions—so not every nautilus shell follows a Fibonacci spiral. However, they all follow some form of logarithmic spiral. And before you imagine that these cephalopods could have aced your math class, keep in mind they’re not consciously aware of how their shells grow. Instead, they simply benefit from an evolutionary design that allows the mollusk to grow without altering its shape.
6. Animals
Most animals exhibit bilateral symmetry, meaning they can be divided into two identical halves along a central axis. Humans are no exception, and some researchers argue that the symmetry of a person's face plays a key role in determining their physical attractiveness. So, if your face isn’t perfectly symmetrical, you might want to rely on other charms.
One animal that may have taken the concept of symmetry as a way to attract a mate a bit too far is the peacock. Darwin was genuinely displeased with this bird, famously stating in an 1860 letter that “The sight of a feather in a peacock’s tail, whenever I gaze at it, makes me sick!”
To Darwin, the peacock’s tail seemed like a burden and didn’t fit his theory of ‘survival of the fittest.’ He was frustrated until he formulated the theory of sexual selection, which suggests that certain traits evolve because they help animals secure mates. Clearly, peacocks have mastered sexual selection, sporting vibrant colors, large size, and a symmetry in both their bodies and the repeating patterns of their feathers to catch the attention of potential mates.
5. Spider Webs
With about 5,000 species of orb-web spiders, all of them create nearly flawless circular webs, featuring almost perfectly spaced radial lines radiating from the center, along with a spiral structure designed to capture prey. Scientists are still unsure why these spiders are so meticulous about geometry, as tests suggest that orb webs don't catch prey any better than those with irregular shapes.
Some scientists speculate that the circular shape of orb webs provides strength, with the radial symmetry helping to evenly distribute the impact when prey strikes the web, which reduces the likelihood of the threads tearing. But if this design truly offers an advantage, why do other spiders not adopt it? Some spiders that aren't orb-weavers seem capable of making such webs but choose not to.
For example, a newly discovered species of spider in Peru constructs each part of its web with identical size and length, showing its ability to ‘measure.’ However, it then assembles these uniform pieces into a chaotic, irregular web. Are these Peruvian spiders missing something that orb spiders know, or have they simply not realized the importance of symmetry?
4. Crop Circles
Give a few tricksters a board, some string, and a bit of darkness, and it turns out humans can create symmetrical patterns too. In fact, the extraordinary symmetry and intricate designs of crop circles have led many to believe that only extraterrestrial beings could be responsible, even though the creators have since come forward and demonstrated their own handiwork.
It's possible that a mix of human and alien-made crop circles exists—though one of the clearest signs that they are entirely man-made is the increasing complexity. It doesn’t make much sense to think that aliens would make their messages harder to understand, especially when we didn't even grasp the meaning of the earliest designs. More likely, people are learning from each other and progressively improving their circles.
Wherever they originate, crop circles are undeniably fascinating to observe, mainly due to their remarkable geometric precision. Physicist Richard Taylor conducted a study on crop circles and discovered—besides the fact that one is created on Earth every night—that most of the designs exhibit an impressive array of symmetries and mathematical structures, including fractals and Fibonacci spirals.
3. Sun-Moon Symmetry
Despite the fact that the sun has a diameter of 1.4 million kilometers and the moon measures only 3,474 kilometers, it seems almost unbelievable that the moon can completely block the sun’s light, giving us approximately five solar eclipses every two years.
How does this phenomenon occur? Remarkably, although the sun’s diameter is about four hundred times greater than the moon's, the sun is also about four hundred times farther away. This symmetry in the ratio makes the sun and the moon appear nearly identical in size when viewed from Earth, allowing the moon to fully obscure the sun during an alignment.
Of course, the Earth’s distance from the sun can vary throughout its orbit—and when an eclipse happens at this time, we observe an annular (or ring-shaped) eclipse, since the sun isn’t completely covered. But about once every one to two years, everything aligns perfectly, granting us the magnificent spectacle of a total solar eclipse.
Astronomers aren’t sure how widespread this type of symmetry is among other planets, stars, and moons, but they believe it’s quite rare. Even so, we shouldn’t get too full of ourselves, as it all seems to come down to chance. For instance, each year the moon moves about four centimeters farther from Earth, meaning that billions of years ago, every solar eclipse would have been a total one.
If current trends continue, total eclipses will eventually fade away, followed by the disappearance of annular eclipses (if the planet endures that long). It seems we’re simply in the right place at the right time to observe this occurrence. Or are we? Some suggest that this sun-moon symmetry could be the crucial factor that makes life on Earth possible.
2. Milky Way Galaxy
As we've discovered, symmetry and mathematical patterns are nearly everywhere—but do these natural laws apply only to our planet? It turns out they don’t. Recently, astronomers found a new section at the outer edges of the Milky Way Galaxy, leading them to believe that the galaxy itself is a nearly perfect mirror image of itself. This new data has strengthened the theory that the galaxy has only two major arms: the Perseus and the Scutum-Centaurus.
Beyond its mirror symmetry, the Milky Way also boasts another incredible feature—akin to the spirals of nautilus shells and sunflowers—where each “arm” of the galaxy forms a logarithmic spiral that begins at its center and stretches outward.
1. Snowflakes
Even something as minuscule as a snowflake follows the rules of order, as most snowflakes display six-fold radial symmetry, with intricate, identical patterns on each of its arms. Understanding why living organisms favor symmetry is challenging enough, but how do inanimate objects—like snowflakes—figure this out?
It seems that chemistry holds the answer; specifically, how water molecules arrange themselves as they freeze (crystallize). As water molecules solidify, they form weak hydrogen bonds with one another. These bonds align in an organized pattern that maximizes attractive forces and minimizes repulsion, ultimately forming the hexagonal shape of the snowflake. But, as we all know, no two snowflakes are the same—so how can a snowflake be perfectly symmetrical yet different from every other snowflake?
As a snowflake falls through the atmosphere, it encounters distinct conditions like humidity and temperature, which influence how the crystals on the flake “grow.” Since all the arms of the flake experience the same conditions, they crystallize similarly—each arm a perfect mirror of the others. However, no two snowflakes share the exact same journey down, which is why every flake appears slightly different from the next.
