A Carved Equation: Hamilton's Breakthrough on Broome Bridge

On October 16, 1843, Irish mathematician William Rowan Hamilton had a moment of brilliance and  etched a remarkable equation into the stone of Broome Bridge in Dublin:

i² = j² = k² = ijk = -1

This seemingly simple inscription marked a pivotal moment in the history of mathematics, introducing the concept of quaternions.

Quaternions are a type of number system that extends the complex numbers. While complex numbers involve a real part and an imaginary part (represented by the imaginary unit i), quaternions introduce three additional imaginary units: i, j, and k. These units are defined by the following relationships:

• i² = j² = k² = -1
• ij = k, jk = i, ki = j
• ji = -k, kj = -i, ik = -j

Hamilton's discovery of quaternions was a breakthrough in algebra. It provided a new way to represent rotations and orientations in three-dimensional space, which had been a challenging problem for mathematicians.

Why did he carve it on a bridge? Hamilton was so excited by his discovery that he felt compelled to record it immediately. Carving the equation on the bridge was a way to commemorate the moment and share his breakthrough with the world. The inscription on Broome Bridge serves as a lasting reminder of Hamilton's ingenuity and the importance of his discovery.

Hamilton's quaternions have had a profound impact on mathematics, physics, and engineering. They are used in various fields. In computer graphics, quaternions  are used to represent rotations and orientations of objects in 3D space. In robotics, quaternions are used to control the movement of robots and robotic arms whereas in quantum mechanics, quaternions are used to describe the spin of particles in quantum mechanics. In navigation, quaternions are used in navigation systems to represent the orientation of a vehicle.

Hamilton's discovery on Broome Bridge continues to inspire mathematicians and scientists today, serving as a testament to the power of human ingenuity and the enduring beauty of mathematical ideas.  Let me know what you think, I'd love to hear.  Have a great weekend.