On October 16, 1843, Sir William Rowan Hamilton was crossing the Royal Canal, Dublin, with his wife, on his way to presiding over a meeting of the Royal Irish Academy. While crossing Brougham Bridge, he had a moment of inspiration, and discovered the quaternions, in the process carving their fundamental equation i2 = j2 = k2 = ijk = -1 into the stone of this bridge. Since 1989, an annual ‘pilgrimage’ called the Hamilton Walk is held from Dunsink Observatory to this Royal Canal bridge, in which scientists and mathematicians remember this inspirational moment.
The year 1843 was around the time that mathematicians were attempting to put the concept of complex numbers, a two-dimensional system based on the equation i2 = -1, on a more solid, mathematical footing. Hamilton was trying, unsuccessfully, to obtain a three-dimensional system of numbers akin to the two-dimensional complex numbers.
For some reason, his focus must have shifted momentarily to a four-dimensional system, perhaps due to the futility of his three-dimensional endeavours. While crossing this bridge, this idea of a system of numbers based on the four numbers 1, i, j and k that satisfy the above equations came to him, and the rest is history.
Hamilton was so proud of his quaternions that he demanded that they are to be studied in school. Soon, they became a compulsory examination topic in Dublin. He also founded a ‘school of quaternionists’ to continue to popularise the subject. One reason why Hamilton was so obsessed with quaternions was the discovery that they can describe what we nowadays call vectors – a word coined by Hamilton himself – and the operations which are referred to as the scalar and vector products.
Unfortunately, in the 1880s, today’s vector techniques, spearheaded by Gibbs and Heaviside, began taking over, partly because the way Hamilton described quaternions in his posthumous book Elements of Quaternions, containing no fewer than 762 pages, was difficult to read. Physicists started to favour Gibbs’ and Heaviside’s approach to vectors, which was seen to be simpler.
The story, however, does have a happy ending. At around the start of this century, quaternions were brought back to life, finding applications in computer vision, quantum physics, robotics and in the fast implementation of three-dimensional computer graphics. This is because quaternions can describe spatial rotations in a more compact way than using other methods such as matrices, allowing such rotations to be implemented in a more efficient manner. Thus, almost 200 years later, Hamilton can finally have a contented sigh in his grave.
Alexander Farrugia is a senior lecturer at the University of Malta Junior College with a PhD in mathematics and a top writer on Quora.
Did you know?
• A quaternion is of the form p + qi + rj + sk, where p, q, r and s are numbers and i, j and k are vectors pointing towards each of the three spatial dimensions (breadth, height and depth). For example, 1 + 2i - 3j + 4k and 0.5 - 3.7i + 4.6j + 9.1k are both quaternions.
• In the quaternion 1 + 2i - 3j + 4k, for example, the number 1 is called the scalar part, while the rest of the quaternion 2i - 3j + 4k is called the vector part. This terminology was coined by Hamilton himself. Nowadays, we still use the terms ‘scalar’ and ‘vector’ in mathematics.
• Normally, when we multiply two numbers by each other, the order is not important. For example, 2 x 3 and 3 x 2 are both equal to 6. However, this is not true for quaternions; multiplying two quaternions by each other may yield a different result depending on whether we multiply the first quaternion by the second or vice-versa.
• The initial motivation for Hamilton to introduce quaternions was to be able to divide three-dimensional vectors by each other. In fact, Hamilton’s very first definition of a quaternion, in his book Elements of Quaternions, was ‘the quotient (division) of two vectors’. Thus, a vector is transformed into another vector via multiplication by a suitable quaternion.
For more trivia see: www.um.edu.mt/think
Sound bites
• Quaternions have recently turned up in a new kind of image processing system called the Fractional Quaternion Cosine Transform (FrQCT). In the paper Chen, B., Yu, M., Su, Q. et al. ‘Fractional quaternion cosine transform and its application in colour image copy-move forgery detection. Multimed. Tools Appl. (2018)’, the authors are suggesting using this new algorithm to detect image forgeries. They demonstrated that FrQCT takes only half the computational time of other methods, and the proposed FrQCT-based forgery detection algorithm achieved a better performance than other state-of-the-art algorithms.
• During this decade, quaternions have also been used as values in artificial neural networks. Such networks are called Quaternion-Valued Neural Networks (QVNN), which are an extension of the Complex-Valued Neural Networks (CVNN), themselves an extension of the real-valued neural networks. Recently in the paper A. B. Greenblatt and S. S. Agaian. ‘Introducing Quaternion Multi-Valued Neural Networks with Numerical Examples. Information Sciences (2017)’, artificial neural networks based on QVNNs were used in two applications: chaotic time series prediction and image illumination classification. The model used by the authors of this paper significantly outperformed a variety of other neural network structures.
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