Rectangle to Square Transformation
Pythagorean Triples
The Legend
The Sulba Sūtras (circa 800-200 BCE) are ancient Vedic texts describing precise geometric constructions for ritual fire altars (yajña). Different ceremonies required altars of specific shapes and areas—squares, rectangles, trapezia, circles. The texts encode sophisticated geometry including area-preserving transformations, the Pythagorean theorem (centuries before Pythagoras!), and approximations of √2 and π. These weren't abstract mathematics but practical instructions for sacred architecture.
The Mathematics
Area Preservation: Transform a rectangle into a square with equal area while maintaining ritual significance. If rectangle has dimensions w × h, the equivalent square has side √(w·h).
Construction Method (simplified):
1. Draw rectangle with width w and height h
2. Area = w × h
3. To find square side s where s² = w × h:
4. Use geometric mean: s = √(w·h)
5. Construct using compass and straightedge
Pythagorean Theorem: The Sulba Sūtras state: "The diagonal of a rectangle produces both areas which its length and breadth produce separately." This is equivalent to a² + b² = c².
Pythagorean Triples: Integer solutions (a, b, c) where a² + b² = c². Generated using formula:
• a = m² - n²
• b = 2mn
• c = m² + n²
for integers m > n > 0
Examples: (3,4,5), (5,12,13), (8,15,17), (7,24,25)...