Kala · The yajña altar
Rope Geometry
Find east with a shadow. Stretch a rope, change a shape, and lay out a great altar. Twelve hands-on problems inspired by the Śulba Sūtras.
Why not just look at sunrise?
The Sun rises due east only around the equinoxes, in March and September. At other times of the year it rises a little north or south of east.
The two shadow marks, one in the morning and one in the afternoon, sit the same distance on either side of noon. Join them and you get an east–west line.
The part you can check yourself
Tomorrow, with a string
Fold a string into three equal layers, then fold it in half twice, and mark each fold: 12 equal parts. Tie it to one stick at 0 and to another at 4, bring the 12 back to the first stick, and pull it tight at 7. The angle at the second stick is a right angle. Check it against the corner of a book.
About the Śulba Sūtras
Śulba means rope. The Śulba Sūtras are manuals for building Vedic fire altars with a rope and sticks (śaṅku). Nine survive, from about 800 BCE to 400 CE, and Baudhāyana's is the oldest. They are the only mathematics we have from the Vedic period.
Altars came in set shapes: a round household fire, a square offering fire, a half-moon fire to the south, and for some rites a great falcon of brick, laid in five layers. Different shapes often had to hold the same area, so a square had to be doubled exactly, two squares added into one, and a square turned into a circle.
In 1975 Nambudiri priests in Panjal, Kerala, built the falcon altar during a twelve day Agnicayana, recorded by Frits Staal. The mathematician A. Seidenberg argued that geometry itself began with rituals like these, not with measuring fields. That idea is still debated.
- Kim Plofker, Mathematical Thought in Vedic India, pp. 19 and 22: Kātyāyana’s shadow and doubled-cord constructions; the rectangle-to-square procedure. The shadow model fixes the sun’s seasonal position; outdoor results are approximate. The diagrams and interactions here are modern teaching adaptations.
- Āpastamba Śulba Sūtra 5.1–5.2: the mahāvedi’s dimensions and 54-unit cord, marked at 12 and 15.
- Shulba Sutras and Vedi (altar), Wikipedia.
- Seidenberg, A. "The ritual origin of geometry." Archive for History of Exact Sciences 1, 1962.
- Staal, F. Agni: The Vedic Ritual of the Fire Altar. 1983. The 1975 performance was also filmed.
- "Cognition of the circle in ancient India." arXiv, 2017.