Later, Leonardo da Vinci tried to find the golden ratio in his famous drawing of the human figure. It was clear to him: “Human ingenuity… will never invent anything that is more beautiful, simpler or more precise than what nature achieves; for their inventions lack nothing and have nothing superfluous about them.”
About butterflies, starfish and mirror axes
But what is the secret of this omnipresent design language? For physicists and mathematicians, symmetry is, in principle, nothing more than the ability to survive certain operations unchanged. All symmetries are based on concrete mathematical rules and principles. But we often recognize instinctively whether a shape is symmetrical or not.
A butterfly, for example, is clearly symmetrical. Its two halves are like mirror images. If you hold a mirror in the middle of the animal, the half complements itself to form a whole. Such a “right-left” similarity is called bilateral symmetry. It is typical for most animals – including humans. But a starfish or a snowflake can also be reflected along certain axes. However, with them there is not just one, but several levels of symmetry. In the starfish with its five arms, there are five ways to complement it into a whole through reflection, so it has a five-fold mirror symmetry.
The more such planes an object has, the more symmetrical it is. There are infinite levels of symmetry in a sphere, which is why it was considered the epitome of beauty and a divine principle in the Middle Ages. In 1543, Nicholas Copernicus wrote in his book “On the Circular Motions of the World’s Bodies”: “The spherical form is the most perfect of all… …partly because it forms the most spacious form, which is most adapted to contain and preserve everything…or because everything tends to limit itself in this form…”
Wind turbines, spiral staircases and DNA
Many flowers or, for example, wind turbines also appear to be mirror-symmetrical at first glance. But upon closer inspection, it becomes apparent that there are no two completely equal halves. However, many flowers are also symmetrical. Their form of symmetry is called rotational symmetry. The crucial operation here is rotation: If you rotate such a flower or, for example, a wind turbine around its axis, after a certain angle you reach a position that can no longer be distinguished from the starting position.
But there are other, more hidden forms of symmetry: what about spiral staircases, the spiraling horns of a mountain sheep, or the fronds of a palm tree? Although not obvious at first glance, they are also symmetrical: if they are rotated through a certain angle and then moved along the central axis, they also remain unchanged.
The underlying mathematical operation here is a combination of rotation and translation. If this operation is repeated always with the same amount, as in the spiral staircase or the double helix structure of DNA, it is a helix. If the amount of rotation and displacement changes with each operation, it is a spiral.
But it’s even more complex…
January 9, 2026 – Nadja Podbregar