The Hidden Maths of Nature - How symmetry and mathematical transformations appear in the natural world

Beneath the surface of general perception, mathematics is the architect of all nature. While mathematics is largely thought of as being confined to textbooks and classrooms, the principles of maths today are seen in all elements of our world: the perfect symmetry of a snowflake; faultless polyhedra of a beehive; and the ever-continuing parallel lines of sand dunes. Natural forms are molded on mathematical concepts, and often go unnoticed.

Transformations in Nature

Mathematical transformations such as rotations, translations and scaling are abundant in nature. The Fibonacci Sequence is a mathematical sequence consisting of the Fibonacci numbers (1,1,2,3,5 etc) which create a series of squares with lengths equal to these numbers. When a line is traced through the diagonals of each square, a Fibonacci spiral is formed. Examples of this spiral are observed throughout nature. Notice the similarities when looking at the chambers of a nautilus shell, the centre of a sunflower and shape of a galaxy - all examples where Fibonacci’s sequence appears. Transformations can also be seen in wave patterns, both visibly in water and through the vibration of particles in sound waves. Physics has allowed us to understand how the oscillations and distribution of energy creates all types of wave transformation, such as reflection and refraction, to become a vital mathematical component of our natural world. 

Symmetry in Nature

Nature is overflowing with symmetry. A harmonious and distinct quality of being made up of exactly similar parts facing each other or around an axis, symmetry can be found in many different forms. Bilateral symmetry refers to an object which has two sides that are mirror images of each other; most animals have this type of symmetry. Moths and butterflies are a prime example as they have a single line of symmetry down the middle of their body to shape identical patterns on each wing. The replication of patterns is theoretically always the same because patterns are hard-wired in the genome and the genetic code is uniform for all cells on both wings. As proven, the symmetry of their wings is a crucial part in their mating displays and camouflage. Other examples of bilateral symmetry include the human body, leaves, worms, cats, dogs, clams and snails. 

Radial symmetry, in which a centre point with numerous lines of symmetry can be drawn, is also present in many aspects of the natural world. A starfish, for example, exhibits dihedral symmetry (the group of symmetries of a regular pentagon); there are five axes of reflection at each vertex of a starfish allowing us to see the same image when rotated by 72 degrees. The beautiful and complex structures of snowflakes, formed from frozen water vapour in the atmosphere, represent hexagonal (six-sided) symmetry, as they produce the same image each time they are rotated by 60 degrees. Geometry is key in the growth of snowflakes, and all of them exhibit the same patterns of symmetry. When looking into the science of snowflakes, this symmetry is created by water molecules which have tetrahedral shapes forming hexagonal rings when bonded with each other. The rings then stack in a hexagonal lattice - the fundamental unit of these natural wonders. As proven, nature is full of perfection due to symmetry. 

The natural world is filled with patterns which can be explained by mathematical principles -  these patterns not only show us the beauty in our physical world, but also provide us with insight into its workings. Maths has existed in the Universe before humans have, and now helps us explain the natural phenomena of our planet and beyond. Although hidden to the ordinary mind, it is undeniably key in blueprinting the smallest snowflakes to the largest galaxies. Ultimately, nature continues to remind us that mathematics is influential in every aspect, shaping the world in ways both remarkable and beautiful.