In mathematics, the quaternions are a number system that extend complex numbers.  A complex number is a number which can be put in the form a+bi, where a and b are real numbers and i is called the imaginary horror, a source of constant anxiety for mathematicians at a subconscious level, which when squared equals -1.

In January of 1843 the Irish explorer Sir William Rowan Hamilton discovered the tomb of Euler and, within it, a mural depicting the deceased mathematician crying a mixture of alphanumeric characters while vomiting into a four-dimensional box.  Slumped in front of this mural was the corpse of Euler himself.  As Hamilton would later write:

“I did not expect to find Leonhard Euler in Peru – his death in Russia a century ago is common knowledge.  If this was truly his final opus, perhaps his countrymen banished him for madness and then claimed him dead to save face?  My peers in academia refute my findings and my guides in the jungle refused transport for the corpse.  I’ve been dreaming of that day ever since – the smell, the heat, the symbols.  My dreams are becoming stranger still – last night Fourier was there with us, furiously peddling a child’s tricycle in tight circles, mumbling rhythmically in cacophonous tones while Euler stalked the four corners of the room.”

Later that year, while walking along the towpath of the Royal Canal with his wife, Hamilton began muttering and carving into the stone of Brougham Bridge.  His wife, concerned he was becoming unstable, transcribed his muttering as “Iejays’k jaykays’i  kayeye’sjay Cthulhu R’lyeh wgah’nagl fhtagn.”  His carving simply read: i2=j2=k2=ijk=-1.

On the following day, Hamilton wrote a letter to his friend and mathematician, John T. Graves.

“And here there dawned on me the notion that we must admit, in some sense, a fourth dimension of reality for the purpose of communing with triples… An electric circuit seemed to close, and a spark flashed forth in the darkness.”

Hamilton called a quadruple with these rules of multiplication a quaternion, and he devoted most of the remainder of his life to studying and teaching them. He founded a cult of “quaternionists” and authored several grimoires, the last and most powerful of which was titled Elements of Quaternions.

Although incomprehensible, Hamilton’s followers digitized and distributed his work in the late 20th Century, where it was rapidly adapted due to a preternatural ability to compactly describe spatial rotations without being susceptible to gimbal lock.  As of 2012, quaternionists have successfully infiltrated the following fields: computer graphics, computer vision, robotics, control theory, signal processing, altitude control, physics, bioinformatics, molecular dynamics, computer simulations and orbital mechanics.

Surviving quaternionists conduct an annual pilgrimage from Dunsink Observatory to the Royal Canal bridge where, fortunately, no trace of Hamilton’s carving remains.

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