It had rained heavily that morning of October 16, 1943, so when it cleared Sir William Rowan Hamilton He finished his whiskey (no whiskey, that’s why it was Irish) and asked his fragile wife if they would go out for a walk around Dublin. He had been working on a mathematical problem involving complex numbers for years without success, and decided it was a good idea to get some air.
That’s what he did. They walked, talked about their children’s future and as they crossed Broom Bridge Sir William’s light bulb suddenly went on. “Helen!” he exclaimed, “I don’t need to multiply triplets: I can use quadruples!” Helen didn’t know anything, of course, but at that moment quaternions were born, an extension of real numbers that more than a century or a half later are critical for NASA’s space missions and also for the video game industry. Good for Sir William.
Worthy successor to Sir Isaac Newton
Sir William Rowan Hamilton (Dublin, 1805-1865) stood out since childhood. At the age of thirteen he already spoke several European languages, but also Persian, Arabic, Sanskrit and Malay. When he was 8 years old, his fame was already notable, and the tour of the American calculus prodigy, Zerah Colburn, gave him the opportunity to prove his brilliance. That 9-year-old American boy crushed him on a mental arithmetic test, and that showed little Hamilton the way. He would continue studying languages, but what he wanted to do was dedicate himself to mathematics.
In 1823 that young man achieved first place among 100 candidates in the Trinity College exams. The prestigious Irish university soon discovered the brilliance of Hamilton, who already in his student days wrote part of his treatise on optics, the well-known “Theory of Ray Systems“.
That was key so that in 1827 he ended up occupying the position of Royal Astronomer of Ireland, a well-paid chair that was unheard of for it to end up in the hands of an undergraduate. Not only that: it gave Hamilton the opportunity to research freely, something he would not have been able to do in a hypothetical position as a professor at Trinity College.
His work in the field of optics would end up mixing with that of dynamics and algebra in the 1830s. His work with several colleagues led him to pursue a very special objective: to try to generalize complex numbers in order to represent rotations and movements of vectors in space. three-dimensional. If he succeeded, he would have a very powerful tool to formulate the basic laws of physics and describe the movement of rigid bodies in space.
In 1833 he presented a paper to the Royal Irish Academy in which he defined addition and maltiplication operations on pairs of real numbers. He was the first mathematician to treat complex numbers as ordered pairs (Gauss had done so before, but without publishing his discoveries) and his vision was closely related to physics.
To try to advance in that field, Hamilton tried to study what he called the “Triplet Theory”, hypercomplex numbers referred to three-dimensional space in the same way that complex numbers referred to two-dimensional space.
It was that that led him to the discovery of the quaternions. The triplets did not have the common properties of complex numbers when trying to multiply them and his obsession with the problem was such that even his children ended up asking him the same thing every morning: “Well dad, can you multiply triplets now?”, to which he replied: “no, for now I can only add and subtract them.”
And then came that ride. Hamilton would describe that happy moment of sudden discovery in a letter to one of his sons fifteen years after it occurred:
“Tomorrow will be the fifteenth birthday of the quaternions. They came into life, or into the light, fully grown, on October 16, 1843, when I was walking with Mrs. Hamilton towards Dublin, and we arrived at Broughman’s Bridge. That is, then and there, I closed the galvanic circuit of thought and the sparks that fell were the fundamental equations between i, j, k; exactly as I have used them ever since.
I took out, at that moment, a pocket notebook, which still exists, and made a note, on which, at that very moment, I felt that it would possibly be valuable to extend my work for at least the ten (or it could be fifteen) years to come. It is fair to say that this happened because I felt, at that moment, that a problem had been solved, an intellectual desire relieved, a desire that had haunted me for at least the previous fifteen years. I could not resist the impulse to take my knife and engrave on a stone of Brougham Bridge the fundamental formula with the symbols i, j, k:
i2=j2=k2=ijk=−1
which contained the solution to the Problem, which has since survived as an inscription.
Hamilton called a quadruple with those multiplication rules a quaternion, and he dedicated the rest of his life to studying them, developing them, and teaching them to students and academics.
Quaternions in space, quaternions in video games
The study of quaternions has led to many other mathematical discoveries, but their application has been amazing more than a century and a half after that walk. In fact, quaternions are used in flight computers or in simulation studies in which large changes in angle are involved when monitoring the altitude of the spacecraft.
The use of quaternions eliminate problems like Euler singularity and allows the use of only four parameters, in addition to being ideal for digital error control.
In fact the so-called unitary quaternions allow counting with a mathematical notation to represent the orientations and rotations of objects in three dimensions, and therefore are widely used in robotics or navigation satellite orbital mechanics and are used in NASA missions for decades.
That same ability to represent rotations in space is key for the development of 3D video games and even animation: several engines make use of these systems to represent those rotations and bring them to the virtual world with precision. We insist. Good for Sir William.
Image | Unsplash
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