15 Beautiful Examples of Mathematics in Nature

15 – Snowflakes,

• You can’t go past the tiny but miraculous snowflake as an example of symmetry in nature. Snowflakes exhibit six-fold radial symmetry, with elaborate, identical patterns on each arm. Researchers already struggle to rationalise why symmetry exists in plant life, and in the animal kingdom, so the fact that the phenomenon appears in inanimate objects totally infuriates them.
• Snowflakes form because water molecules naturally arrange when they solidify. It’s complicated but, basically, when they crystallise, water molecules form weak hydrogen bonds with each other. These bonds align in an order which maximises attractive forces and reduces repulsive ones. This is what causes the snowflake’s distinct hexagonal shape.
• As you know, though, no two snowflakes are alike, so how can a snowflake be completely symmetrical within itself, but not match the shape of any other snowflake?
• Well, when each snowflake falls from the sky, it experiences unique atmospheric conditions, like wind and humidity, and these affect how the crystals on the flake form. Each arm of the flake goes through the same conditions, so consequently crystallises in the same way. Each arm is an exact copy of the other.

14 – Sunflowers,

• Bright, bold and beloved by bees, sunflowers boast radial symmetry and a type of numerical symmetry known as the Fibonacci sequence, which is a sequence where each number is determined by adding together the two numbers that preceded it. For example: 1, 2, 3, 5, 8, 13, 21, 24, 55, and so forth.
• Scientists and flower enthusiasts who have taken the time to count the seed spirals in a sunflower have determined that the amount of spirals adds up to a Fibonacci number. This is not uncommon; many plants produce leaves, petals and seeds in the Fibonacci sequence. It’s actually the reason it’s so hard to find four-leaf clovers.
• So, why do sunflowers and other plants abide by mathematical rules? Scientists theorise that it’s a matter of efficiency. In simple terms, sunflowers can pack in the maximum number of seeds if each seed is separated by an irrational-numbered angle.
• The most irrational number is known as the golden ratio, or Phi. Coincidentally, dividing any Fibonacci number by the preceding number in the sequence will garner a number very close to Phi. So, with any plant following the Fibonacci sequence, there will be an angle corresponding to Phi (or ‘the golden angle’) between each seed, leaf, petal, or branch.

13 – Uteruses,

• According to a gynaecologist at the University Hospital Leuven in Belgium, doctors can tell whether a uterus looks normal and healthy based on its relative dimensions – dimensions that approximate the golden ratio.
• Over a few months, Dr Verguts took ultrasounds of 5,000 women’s uteruses and compared the average ratio of a uterus’s length to its width among different age brackets. The data revealed a ratio that is about two at birth. This steadily decreases through a woman’s life until reaching 1.46 during old age.
• Dr Verguts discovered that, between the ages of sixteen and twenty, when women are at their most fertile, the ratio uterus length to width is 1.6. This is a very good approximation of the golden ratio.

12 – Nautilus Shell,

• A nautilus is a cephalopod mollusk with a spiral shell and numerous short tentacles around its mouth.
• Although more common in plants, some animals, like the nautilus, showcase Fibonacci numbers. A nautilus shell is grown in a Fibonacci spiral. The spiral occurs as the shell grows outwards and tries to maintain its proportional shape.
• Unlike humans and other animals, whose bodies change proportion as they age, the nautilus’s growth pattern allows it to maintain its shape throughout its entire life. Imagine never outgrowing your clothes or shoes. You could still be rocking those overalls your mum put you in when you were four years old.
• Not every nautilus shell makes a Fibonacci spiral, though they all adhere to some type of logarithmic spiral. Nautilus aren’t consciously aware of the way their shells grow; they are simply benefiting from an advanced evolutionary design.

11 – Romanesco Broccoli,

• Our next example can be found in the produce section of the humble grocery story. Romanesco broccoli has an unusual appearance, and many assume it’s another food that’s fallen victim to genetic modification. However, it’s actually one of many instances of fractal symmetry in nature.
• In geometric terms, fractals are complex patterns where each individual component has the same pattern as the whole object. In the case of romanseco broccoli, each floret is a miniaturised version of the whole head’s logarithmic spiral. This means the entire veggie is one big spiral composed of smaller, cone-like mini-spirals.
• Although it’s related to broccoli, romanescos taste and feel more like a cauliflower. It’s, of course, rich in vitamins, which is probably why kids hate eating it. Or it could be they subconsciously realise romanescos involve mathematics, and therefore share an association with school.
• 10 – Pinecones,
  
  • Pinecones have seed pods that arrange in a spiral pattern. They consist of a pair of spirals, each one twisting upwards in opposing directions.
  • The number of steps will almost always match a pair of consecutive Fibonacci numbers. For example, a three–to–five cone meets at the back after three steps along the left spiral and five steps along the right.
  • This spiralling Fibonacci pattern also occurs in pineapples and artichokes.
  9 – Honeycombs,
  
  • Bees are renowned as first-rate honey producers, but they’re also adept at geometry. For centuries, mankind has marvelled at the incredible hexagonal figures in honeycombs. This is a shape bees can instinctively create; to reproduce it, humans need the assistance of a compass and ruler!
  • Honeycombs are an example of wallpaper symmetry. This is where a pattern is repeated until it covers a plane. Other examples include mosaics and tiled floors.
  • Mathematicians believe bees build these hexagonal constructions because it is the shape most efficient for storing the largest possible amount of honey while using the least amount of wax. Shapes like circles would leave gaps between the cells because they don’t fit perfectly together.
  • Some party-poopers think the hexagonal shape of honeycombs is an accident. They refuse to believe that bees are architectural masterminds, and reason the hexagonal shape occurs when the wax around the bees’ circular cells collapses it into a hexagon. Either way, it’s a fascinating example of symmetry in nature.
  8 – Tree Branches,
  
  • The Fibonacci sequence is so widespread in nature that it can also be seen in the way tree branches form and split.
  • The main trunk of a tree will grow until it produces a branch, which creates two growth points. One of the new stems will then branch into two, while the other lies dormant. This branching pattern repeats for each of the new stems.
  • A good example is the sneezewort, a Eurasian plant of the daisy family whose dry leaves induce sneezing.
  • This pattern is also exhibited by root systems and even algae.
  7 – Milky Way Galaxy,
  
  • Symmetry and mathematical patterns seem to exist everywhere on Earth – but are these laws of nature native to our planet alone? Research suggests not.
  • Recently, a new section on the edges of the Milky Way Galaxy was discovered, and, by studying this, astronomers now believe the galaxy is a near-perfect mirror image of itself.
  • Using this new information, scientists have become more confident in their theory that the galaxy has only two major arms: the Scutum-Centaurus and the Perseus.
  • As well as having mirror symmetry, the Milky Way has another amazing design. Like nautilus shells and sunflowers, each ‘arm’ of the galaxy symbolises a logarithmic spiral that begins at the galaxy’s centre and expands outwards. Trippy dippy.
  6 – Faces,
  
  • Humans possess bilateral symmetry, and research suggests a person’s symmetry is of paramount importance when determining physical attraction.
  • Faces, both human and otherwise, are rife with examples of the Golden Ratio. People with lopsided faces would need to be really rich, really funny, or, um, really well-endowed to compensate for this perceived flaw.
  • Studies have shown that mouths and noses are positioned at golden sections of the distance between the eyes and the bottom of the chin. Comparable proportions can be seen from the side, and even the eye and ear itself, which follows along a spiral.
  • Of course, everybody’s different, but averages across populations lean towards phi. The closer our proportions adhere to phi, the more attractive those traits are perceived. For example, the most beautiful smiles are those in which central incisors are 1.618 wider than the lateral incisors, which are 1.618 wider than canines, and so on.
  • From an evo-psych perspective, it’s possible that we are primed to like physical forms that adhere to the golden ratio, as this is a potential indicator of reproductive health.
  • 5 – Orb Web Spiders,
    
    • There are approximately 5,000 types of orb web spiders. All of them create near-perfect circular webs that have near-equal-distanced radial supports coming out of the middle and a spiral that is woven to catch prey.
    • It’s not clear why orb spiders are so geometrically inclined. Tests have shown that orbed webs are no better at catching prey than irregularly shaped webs.
    • Some scientists theorise that orb webs are built for strength, with radial symmetry helping to evenly distribute the force of impact when a spider’s prey makes contact with the web. This would mean there’d be less rips in the thread.
    • But if this is a better web design, why aren’t all spiders utilising it? Some spiders have the capacity to produce orb webs, but don’t seem bothered. It’s a case that requires further study.
    4 – Crop Circles,
    
    • Some string, a board and the cover of night are the only conditions some pranksters need to create perfect symmetrical crop circles. Some of them are so intricate that many people refuse to believe their self-confessed human creators; they insist that only aliens could be capable of such a feat.
    • It’s possible alien-made crop circles exist on Earth; however, the fact the circles are getting more complicated suggests most are man-made. It’s counterintuitive to think aliens trying to make contact would create increasingly complicated messages that are near impossible to decipher. It’s more likely people are learning from each other through example.
    • No matter whether they come from aliens or some futuristic lawn-mowing service, crop circles are a sight to behold because they’re so geometrically impressive.
    • A study conducted by physicist Richard Taylor revealed that, somewhere in the world, a new crop circle is created every night, and that most designs demonstrate a wide variety of symmetry and mathematical patterns, including Fibonacci spirals and fractals.
    3 – Starfish,
    
    • Starfish or sea stars belong to a phylum of marine creatures called echinoderm. Other notable echinoderm include sea urchins, brittle stars, sea cucumbers and sand dollars.
    • The larvae of echinoderms have bilateral symmetry, meaning the organism’s left and ride side form a mirror image. However, during metamorphosis, this is replaced with a superficial radial symmetry, where the organism can be divided into similar halves by passing a plane at any angle along a central axis.
    • Sea stars or starfish are invertebrates that typically have five or more ‘arms’. These radiate from an indistinct disk and form something known as pentaradial symmetry.
    • Their evolutionary ancestors are believed to have had bilateral symmetry, and sea stars do exhibit some superficial remnant of this body structure.
    2 – Peacocks,
    
    • Most animals have bilateral symmetry, which means drawing an even centre line would create two matching halves.
    • The peacock takes the earlier principle of using symmetry to attract a mate to the nth degree. In fact, Charles Darwin, who famously conceived the survival of the fittest theory, detested peacocks. In an 1860 letter, he wrote ‘The sight of a feather in a peacock’s tail, whenever I gaze at it, makes me sick!’
    • Darwin thought the peacock’s tail was a burden that made no evolutionary sense. He remained furious until coming up with the theory of sexual selection, which asserts that animals develop certain features to increase their chances of mating. Male peacocks utilise their variety of adaptations to seduce sultry peahens. These include bright colours, a large size, a symmetrical body shape and repeated patterns in their feathers.
    1 – Sun-Moon Symmetry,
    
    • The sun has a diameter of 1.4 million kilometres, while his sister, the Moon, has a meagre diameter of 3,474 kilometres. With these figures, it seems near impossible that the moon can block the sun’s light and give us around five solar eclipses every two years.
    • So what causes these solar eclipses? By sheer coincidence, the sun’s width is roughly four hundred times larger than that of the moon, while the sun is about four hundred times further away. The symmetry in this ratio causes the moon and sun to appear almost the same size when seen from Earth, and, therefore, it becomes possible for the moon to block the sun when the two align.
    • Earth’s distance from the sun can increase during its orbit. If an eclipse occurs during this time, we see what’s known as an annular or ‘ring’ eclipse. This is because the sun isn’t completely hidden.
    • Every one to two years, though, the sun and moon become perfectly aligned, and we can witness a rare event called a total solar eclipse. You know that Madonna song, Total Eclipse of the Heart? This is like that, but with astrology.
    • Astronomers don’t know how common this symmetry is between other planets, suns, and moons, but theorise that it’s quite rare. Every year, though, our moon drifts roughly four centimetres further from Earth. This means that, billions of years ago, every solar eclipse would have been a total eclipse.
    • If things continue as they are, total eclipses will eventually cease entirely – as will annular eclipses, assuming the planet lasts that long. With this in mind, it’s easy to conclude that we’re simply in the right place at the right time to witness this phenomenon. Some have theorised that this sun-moon symmetry is the special factor which makes life on Earth possible.