How ancient Hindu temples have influenced the world of geometry

What do the structure of our DNA, stock market trends and the architecture of ancient Hindu temples have in common, you may ask. Well, they all contain repeating, self-similar patterns, which are also known as fractals. These patterns can be found in nature all around us, from the crystals on a snowflake to the spiral of a nautilus shell. Although the term fractal was first coined by the polymath Benoit Mandelbrot in the late 20th century, originating from the Latin term ‘fractus’, which refers to the irregularity and fragmented nature of these patterns, fractals were being used well before they were named. The use of geometry and fractals in the construction of ancient Hindu temples is widely believed to be one of the origins of the study of Mathematics in India, a process which would later lead to some of the most significant discoveries, which have paved the way for modern mathematics. Not only did these temples give us some of the earliest known examples of use of fractal geometry, but they also can be credited with some other considerable breakthroughs, such as the discovery of Pythagoras’ theorem, up to 300 years before Pythagoras himself discovered it.

Ancient Indians were fascinated by an area of study known as ‘rekha ganita’, translated as ‘line computation’, and used it repeatedly to help with construction of their temples. They used the idea of vastupurashamandala, which is the timeless and infinite square that represents the whole universe in the Hindu religion. This was used to construct temples as mandalas, which are large square grids, containing a square number of squares, usually 64 or 81, with each smaller square being an area to worship each individual God, as Hinduism is a polytheistic religion which worships many different Gods. The centre square was reserved for Brahman, the supreme being, hence the importance of using a square number of squares to construct the mandala.

The Sulva Sutras were a series of texts published from the 8th century BC to the 4th century BC, which set out ‘rule of the chord’, and this was used as the main guidance on how to construct the ancient temples and altars which were so important to them. Fire altars in particular formed an integral part of the Hindu religion at the time, and they were the centre for many important rituals, such as marriage or funeral rites. The shapes of these fire altars were not based simply of the whims of the constructor, but rather defined by a set of rules and instructions. For example, it was said that fire altars in the shape of falcons were fit for those who desired heaven, whilst people who wished ruin on their enemies should use rhombus shaped fire altars. The Sulva Sutras would have provided key information to builders in the Vedic period (1500 to 500 BC), through instructions on how to construct the specific shapes required.

Whilst those involved in building the temples just used the texts for the cause at hand, modern mathematicians have realised the significance of many discoveries which were made during this period, including many which were thought to only be first discovered much later in Europe. One key example of this is Pythagoras’ theorem, which was initially thought to have been discovered in around the 5th century BC. However, a statement from the Sulvasutras of Baudhavana, which was written in 800 BC, states that ‘In a Deerghchatursh (Rectangle) the Chetra (Square) of Rajju (hypotenuse) is equal to sum of squares of Parshvamani (base) and Triyangmani (perpendicular)’ (Saikia, 2011): . This is essentially Pythagoras’ theorem, and suggests that it was discovered around 300 years than was previously thought. The advanced list of Pythagorean triples also included in the text (3,4 and 5, 5, 12 and 13, 8,15 and 17, and 12, 35 and 37) would have enabled people to construct right angled triangles. Another Sulvasutras, written in 400 BC, gives a value of √2, which is now known to be accurate to 5 decimal places, an impressive feat given the limited tools available to mathematicians at the time.

These mathematical concepts, and many more can be seen all across Hindu and Jain temples, both ancient and modern. The Hindu belief that the cosmos is holonomic and self-similar, means that there are a lot of examples of fractal geometry being used in Hindu temples. The temples are seen as a smaller part of Hinduism as a whole, so the use of fractal geometry allows the whole religion to be reflected and honoured in each individual place of worship. For example, Virupashuka temple uses fractal geometry to reflect the patterns of nature and their repetitive tendencies. Meanwhile, the main shikhar (spire) of the Kandariya Mahadev temple in Khajuraho is surrounded by smaller, identical spires, which in turn are surrounded by smaller spires, all of which are replicas of the main shikhar.

The designs of these temples also sometimes served a practical use, as well as aesthetic purpose. The design of Konark Sun temple allows people to tell the time, as the 8 major spokes serve as a sundial, splitting 24 hours into sections of 3 hours, although time is told anticlockwise. The Modhera Sun temple uses geometry to represent the calendar year. 52 ornately carved pillars represent each week of the year, whilst the mandapa (the main pavilion) is divided into 7 sections, one for each day of the week. At the bottom of the mandapa, 365 elephants symbolise each day of the year.

Although the confines of their religion prevented many people from properly realising the significance of their mathematical discoveries, due to the fact that the intricacies of the construction of temples were viewed as sacred knowledge for high priests only, many of the discoveries made during this period paved the way for modern mathematics as we know it today.