“Those who wish to destroy existing and future enemies should construct a fire altar in the form of a rhombus,” declare the Sulbasûtras. These geometry manuals from the Vedic period deal with religious sacrifice, but the language is very cryptic. For example, the above rite could easily refer to a rhombus-shaped military formation. According to Kim Plofker, a historian of mathematics, different theories lead us to infer that “geometrical figures symbolised religious ideas, and the need to manipulate them led to the relevant mathematics.” Other geometric shapes are also suggested — the falcon altar leads one to heaven, the tortoise shape to attainment of Brahman and so on.

Many Sulbasûtra constructions are about the transformation of one shape into another, while preserving the area, or keeping the number of bricks constant. Squares are transformed into rectangles, triangles, trapezia, and rhomboids. The problem of making a square equal in area to a given circle was also known to the Greeks (“the circlesquarers”).

One may connect this content of the Sulbasûtras to modern ideas of symmetry. If you rotate a square by 90 degrees, it looks the same. If you rotate an equilateral triangle by 120 degrees, it looks the same. Transformations that leave an object unchanged are called symmetries of an object. In the case of fire altars, the priests were trying to keep the area or number of bricks unchanged. In modern physics, the falcon and other shapes would be like molecules that remain unchanged if the constituent atoms (or bricks) are interchanged. Or several shuffles of a deck of cards that leave the original sequence unchanged.

According to Richard Feynman’s famous lectures on physics: “Nature often exhibits certain kinds of symmetry in the objects we find in the world around us. Perhaps the most symmetrical object imaginable is a sphere, and nature is full of spheres — stars, planets, water droplets in clouds. The crystals found in rocks exhibit many different kinds of symmetry, the study of which tells us some important things about the structure of solids. Even the animal and vegetable worlds show some degree of symmetry, although the symmetry of a flower or of a bee is not as perfect or as fundamental as is that of a crystal.” A crystal is made up of the same basic unit, repeated across space. Much like the Vedic altars of fire repeated bricks and the priests chanted mantras, repeating syllables across time.

In the late 1800s, Georges-Pierre Seurat, an artist, started using tiny coloured points to make large compositions on canvas. His technique foreshadowed the invention of computer pixels by over a century. It also emphasised the growing conviction that all matter in the Universe is composed of tiny identical particles, or atoms; and the birth of Marxism, wherein all individuals in a society should be considered on an equal footing, creating a dynamic ensemble or commune of indistinguishable entities.

As a result, the typical Seurat painting is highly granulated, giving the impression of coloured particles in restricted vibratory motion like a gas in a transparent chamber. Painted over two years on a canvas 2x3 metres in size, his major work — Un dimanche après-midi à l’île de la Grande Jatte ( A Sunday Afternoon on the Island of la Grand Jatte ) — is composed of over three million dots. The scientific implications of what Seurat was merely hinting at started becoming manifest by the turn of the century in 1900, when the architecture of the quantum world started emerging like a castle from the mists of history. Reality itself appeared to be made up of discrete, quantised dots. And all the smooth continuity of nature, merely an illusion.

By the faculties of the human mind, a Seurat painting transcends the disparate points it is composed of, and what we see is a constructed mirage. The obvious question to ask yourself is whether a slightly different arrangement of the points on Seurat’s canvas could create the exact same illusion? For example, we can analyse Jatte’s painting very closely, and try to shuffle some points around while making sure that the overall appearance of the painting remains the same. Mathematically, these new arrangements that leave the painting invariant under transformations can be called the symmetries of a Seurat painting.

The history of science is the history of symmetry. Everything that we see is made up of atoms and molecules and these, in turn, of smaller elementary particles. Even while nothing seems to change, everything is changing. The countless elementary particles are being exchanged in such a way — at the speed of light or, perhaps, faster — that the overall view remains the same. The falcon remains a falcon, the pond remains a pond, and the tortoise remains a tortoise.

Interesting symmetries happen when space and time are interchanged, when history is replaced by fiction. Perhaps quantum physics is not inspired by the Vedas, but the Vedas have been inspired by quantum physics. How would you destroy a future enemy 3,000 years before his appearance? The rhombus is a key figure in the theory of relativity, which describes the architecture of time.

(Rohit Gupta explores the history of science as Compasswallah @fadesingh)

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