How a Tagged Equation Changed Mathematical History

On October 16, 1843, Irish mathematician William Rowan Hamilton had an epiphany while walking along the Royal Canal in Dublin. He was so excited that he took out his penknife and carved his discovery on the spot on Broome Bridge.

This is the most famous graffiti in the history of mathematics, but it seems rather modest:

i² = j² = k² = ijk = –1

Yet Hamilton’s revelation changed the way mathematicians represent information. And this, in turn, has simplified a multitude of technical applications, from calculating forces when designing a bridge, an MRI machine or a wind turbine, to programming search engines and the orientation of a rover on Mars. So, what does this famous graffiti mean?

Rotating objects

The mathematical problem Hamilton was trying to solve was how to represent the relationship between different directions in three-dimensional space. Direction is important in describing forces and speeds, but Hamilton was also interested in 3D rotations.

Mathematicians already knew how to represent the position of an object with coordinates such as x, Yes And zbut understanding what happened to those coordinates when you rotated the object required complicated spherical geometry. Hamilton wanted a simpler method.

He was inspired by a remarkable way of representing two-dimensional rotations. The trick was to use what are called “complex numbers,” which have a “real” part and an “imaginary” part. The imaginary part is a multiple of the number I“the square root of negative one”, defined by the equation I ² = –1.

By the early 1800s, several mathematicians, including Jean Argand and John Warren, had discovered that a complex number could be represented by a point on a plane. Warren had also shown that it was mathematically quite simple to rotate a line 90° in this new complex plane, like turning the hand of a clock from 12:15 to noon. Because that’s what happens when you multiply a number by I.

Hamilton was very impressed by this connection between complex numbers and geometry and set about trying to make it in three dimensions. He imagined a complex 3D plan, with a second imaginary axis in the direction of a second imaginary number jperpendicular to the other two axes.

It took him many arduous months to realize that if he wanted to extend the magic of 2D rotational multiplication into I he needed four-dimensional complex numbers, with a third imaginary number, k.

In this 4D mathematical space, the k-the axis would be perpendicular to the other three. Not only k be defined by k ² = –1, its definition is also necessary k = I = –ji. (By combining these two equations to k given ijk = –1.)

Putting it all together gives I ² = j ² = k ² = ijk = –1, the revelation that struck Hamilton like a bolt of lightning at Broome Bridge.

Quaternions and vectors

Hamilton called his 4D numbers “quaternions” and he used them to calculate geometric rotations in 3D space. This is the type of rotation used today to move a robot, for example, or orient a satellite.

But most practical magic comes into play when we consider only the imaginary part of a quaternion. Because this is what Hamilton called a “vector”.

A vector encodes two types of information at once, including the magnitude and direction of a spatial quantity such as force, speed, or relative position. For example, to represent the position of an object (x, Yes, z) relative to the “origin” (the zero point of the position axes), Hamilton visualized an arrow pointing from the origin to the location of the object. The arrow represents the “position vector” x I + Yes j + z k.

The “components” of this vector are the numbers x, Yes And z-the distance traveled by the arrow along each of the three axes. (Other vectors would have different components, depending on their magnitudes and units.)

Half a century later, the eccentric English telegrapher Oliver Heaviside helped usher in modern vector analysis by replacing Hamilton’s imaginary framework. I, j, k with real unit vectors, I, j, k. But in all cases, the components of the vector remain the same, and therefore the arrow and the basic rules for multiplying vectors also remain the same.

Hamilton defined two ways of multiplying vectors together. One produces a number (called today a scalar or dot product) and the other produces a vector (called a vector or cross product). These multiplications appear today in a multitude of applications, such as the formula for the electromagnetic force that underlies all our electronic devices.

A single mathematical object

Unbeknownst to Hamilton, the French mathematician Olinde Rodrigues had proposed a version of these products three years earlier, in his own work on rotations. But calling Rodrigues multiplications products of vectors is a step backwards. It was Hamilton who connected the separate components into a single quantity, the vector.

Everyone from Isaac Newton to Rodrigues had no idea of ​​a single mathematical object unifying the components of a position or force. (In fact, there was one person who had a similar idea: a self-taught German mathematician named Hermann Grassmann, who independently invented a less transparent vector system at the same time as Hamilton.)

Hamilton also developed compact notation to make his equations concise and elegant. He used a Greek letter to denote a quaternion or vector, but today, following Heaviside, it is common to use a bold Latin letter.

This compact notation has changed the way mathematicians represent physical quantities in 3D space.

Consider, for example, one of Maxwell’s equations relating electric and magnetic fields:

∇ × E = –∂B/∂t

With just a handful of symbols (we won’t go into the physical meanings of ∂/∂t and ∇ ×), this shows how an electric field vector (E) propagates through space in response to changes in a magnetic field vector (B).

Without vector notation, this would be written as three separate equations (one for each component of B And E)—each a tangle of coordinates, multiplications, and subtractions.

The power of perseverance

I chose one of Maxwell’s equations as an example because the eccentric Scotsman James Clerk Maxwell was the first great physicist to recognize the power of compact vector symbolism. Unfortunately, Hamilton did not live to see Maxwell’s approval. But he never abandoned his faith in his new way of representing physical quantities.

Hamilton’s perseverance in the face of mainstream rejection really moved me when I was researching my book on vectors. He hoped that one day – “no matter when” – he might be thanked for his discovery, but this was not vanity. The possible applications he envisioned were exciting.

He would be delighted that vectors are so widely used today and that they can represent digital as well as physical information. But it would be particularly fortunate if, in programming rotations, quaternions often remain the best choice, as NASA programmers and computer graphics designers know.

In recognition of Hamilton’s achievements, math enthusiasts retrace his famous walk every October 16 to celebrate Hamilton Day. But we all use the technological fruits of this unassuming graffiti every day.

This article is republished from The Conversation under a Creative Commons license. Read the original article.