New marketing of video games focuses on making the gamer the creator of the games with a particular focus on children. The "Coding for Kids" education fad builds on the seductive attractiveness of the games by teaching the child how to become a game programmer / artist / designer. This tack, aimed at creating permanent consumers of alternate reality contesting for click bait on social media, now extends down to the pre-school level. But key enabling features in the video games themselves point the way to advances in all human knowledge if exploited by those who understand the irony.
In a surprising but lawful way, the search for the type of mathematics that proves to be the most useful tool for smoothly modeling such things as the rotational action of the animated avatars featured in video games, conjures up historical battles about the nature of the universe and the capacity of mathematics to describe it. This pearl, extracted from the mud of video culture, is now being explored in popular videos by Grant Sanderson.
Sanderson is the 28-year-old master of the "math visualization video,” with over four million YouTube subscribers and featured coverage in Popular Mechanics, on ABC News, and in Quantum Magazine. He views his channel as “primarily about discovery and creativity in math" and often talks about finding the "story" behind the math. Whether conscious of this or not, Sanderson has stumbled upon one of the most fundamental fonts of creativity in mathematical theory. To the extent these issues become fully conscious, it is actually and ironically possible to talk about a real value emerging from the focus on the smooth action of an avatar figure in video game animations.
As Lyndon LaRouche often said, the true nature of human creativity will thrust itself forward, at surprising times and in surprising places. This issue of video game programming is one of those surprises.
Real World Spatial Rotation
The ability to efficiently model 3D spatial rotation is not in any way a trivial matter in the hard sciences. Back in 1970, Jim Lovell and the other Apollo 13 astronauts nearly spun their spacecraft out into the far reaches of the universe because of the lack of such a capability.
When their spacecraft suffered an oxygen tank explosion, and they were experiencing uncontrolled gyrations, Lovell had to override certain computer guidance controls and manipulate the gyroscopic guidance manually. Here is an excerpt from the 1995 movie, Apollo 13, where, if you listen closely, you can hear in the first three minutes, as the astronauts are frantically struggling to regain control of their spacecraft, there are exclamations from both Lovell and from a Houston ground control engineer of something like, "Watch out, we're close to gimbal lock!".
To prevent any future catastrophe, NASA's computational arsenal was expanded by the hypercomplex number system called quaternions -- and, thereafter, the question of smoothly modeling a spacecraft's 3D rotational action without any possibility of glitches was resolved.
The theory of a quaternion number system, discovered by the Irish scientist William Rowan Hamilton in the mid-19th Century, and the story of the suppression of that discovery over the subsequent 150 years, forms the heart of our story. It is this theory the gaming industry is using to perfect the control mechanisms in their computer circuit board cards to enable smooth rotational action in their games.
Grant Sanderson has produced an excellent and popular video discussing quaternion theory, giving some attention to the controversy that surrounded the creation of the theory in 1840's Britain (involving what he humorously calls, some "old-timey trash-talk").
This video likely helped to spread the quaternion theory to a broader audience of scientists. Now, in just the last few months, there has been exciting news from research scientists who have been studying quaternions -- this time not from those involved in the space program, but from teams involved in research into fusion energy prototype reactors.
Two different fusion research teams, one at the University of Washington and one at The Japan Society of Plasma Science and Nuclear Fusion, have announced breakthroughs in their research due to the application of quaternions to their spatial control systems. The Japanese team has developed new controls for winding coils for plasma confinement, and the UW team has developed internal control mechanisms of vortex formation inside the plasma itself.
The Japanese team describes their breakthrough as follows:
"The quaternion, a four-dimensional hypercomplex number, is good at describing three-dimensional rotation and has been utilized in three-dimensional game graphics programming theory. Utilizing the quaternion capability, we have analyzed a matrix converter by three-dimensional rotation instead of transforming to two-dimensional rotation in alpha-beta coordinates…
"A matrix converter can produce an output voltage waveform of arbitrary frequency and phase angle, and can be utilized as a power supply for a resonant magnetic perturbation coil...which makes a fusion reactor more efficient…"
The UW campus newspaper announced their team's breakthrough simply as, "Gaming Graphics Card Allows Faster, More Precise Control of Fusion Energy Experiments".
W.R. Hamilton’s Invention
The 1700's fight over priority claims between Isaac Newton and Gottfried Leibniz for the invention of the calculus continued to roil British universities when H.R. Hamilton attended the University of Dublin. Although Newton had not actually published anything about his calculus system, the British Royal Society -- of which Isaac Newton was the president at the time -- declared the German, Gottfried Leibniz, to be a plagiarist when he dared to publish his own.
The Leibnizian approach was based on a metaphysical principle of the certainty of the "infinitesimal", and on the related least-action physical principles which underlie the perfect shape of the catenary curve -- i.e., the shape that is determined by the balance of forces at each infinitesimal point in any ordinary hanging chain -- a concept from which the idea of Leibniz's calculus emerges naturally.
Even though he distinguished himself early in the sciences, being named Irish Royal Astronomer at age 21, Hamilton came into the British scientific community as an outsider. His original aspiration was to become a poet, with the intention to live by his pen as his English contemporaries, the great poets, John Keats and Percy Shelley, were doing at the time. As a student, he was befriended by William Wordsworth, the mentor to Keats and Shelley; but Wordsworth famously encouraged him to follow where his talents lay, which was in the sciences, and not in poetry.
As a young professor, W.R. Hamilton rejected the stilted Newtonian version of things. It was not merely that Hamilton accepted and used the Leibnizian calculus system, but that he also made his own original discoveries in the physics of dynamical systems based on Leibniz's least-action principles.
In his scientific career, Hamilton preserved his bent for the creative life and persisted in upsetting the British scientific establishment with his pursuits based on the Leibnizian outlook. He escalated his “offense” when he built on the work of Carl Gauss and the German Gottingen school, by expanding upon the ideas of numbers in the complex domain.
The complex number system itself had been developed as a challenge to the lifeless domain of the Cartesian number grid. The domain of complex numbers was lively, allowing for a mapping of the real-world actions of stretching and rotating as if those actions were projected onto a two-dimensional plane. The breakthrough that Hamilton made with his quaternions was to extend that capability to mapping such actions into a three-dimensional space.
The attacks that flew at Hamilton, and especially his students and followers who supported the quaternion system after his death, were twofold. The first was a simple extension to Hamilton of the kind of attacks that had been suffered by Carl Gauss and others who had dared to insist on the integrity of numbers in the complex domain that had been labeled as "imaginary", or impossible, by such noted empiricist mathematicians such as Euler and D'Alembert. In fact the letter "i" is used to denote such an imaginary thing.
The Imaginary is Sometimes the Most Real
The most unique and the most useful number in the domain of the complex number system does not exist in the usual Cartesian grid of numbers. It is a product of a higher order of geometry which subsumes the mathematics allowable by Cartesians.
This number is denoted as the square root of -1. Now, the operation of finding a square root is supposed to mean find the number which, when multiplied by itself, gives you -1. Therefore, we cannot find such a number. In the domain of the logic of the lifeless compilation of number lines which makes up the Cartesian grid, the square root of -1 is meaningless.
But in the domain of a number system which relies on notation to describe action, the most action-filled number is that i. Contrast Leonhard Euler's lifeless pronouncement on the subject to Carl Gauss' action-filled explanation.
From Euler's 1770 "Elements of Algebra":
"Of such numbers we may truly assert that they are neither nothing, nor greater than nothing, nor less than nothing, which necessarily constitutes them imaginary or impossible.”
Gauss' response, 1832, "Second Treatise on Bi-Quadratic Residues":
"The difficulty, one has believed, that surrounds the theory of imaginary magnitudes, is based in large part to that not so appropriate designation (it has even had the discordant name impossible magnitude imposed on it). Had one started from the idea to present a manifold of two dimensions (which presents the conception of space with greater clarity), the positive magnitudes would have been called direct, the negative inverse, and the imaginary lateral, so there would be simplicity instead of confusion, clarity instead of darkness…."
Carl Gauss did not publicly posit that such a veil of darkness might have been deliberately thrown over the scientific community at that time. But Lyndon LaRouche, in several of his writings aimed at educating his youth movement in the early 2000's, placed the issue in just such a broader context. He emphasizes that such a deliberate mischaracterization of a very real and extremely critical concept in furthering mathematical physics is but one aspect of a broader historical battle of humanity against the forces of oligarchy.
In his "Special Report: Science and Infrastructure", published on September 27, 2002, LaRouche writes:
"Already, during the interval from Archytas and Plato, through the work of Eratosthenes and Archimedes, Plato and his associates had recognized that the physical differences among the notions of line, surface, and solid, were not consistent with a naive conception of linearly extended space and time; the difference among these species of physical existence represented the action of specific powers, as Plato emphasized in his Theaetetus dialogue. This notion of powers is that employed by Leibniz for defining a science of physical economy; it is the use of the notion of powers employed by Gauss in defining the fundamental theorem of algebra, the same notion Gauss employed in number theory, in defining the significance of residues.
"The appearance of the falsely named 'imaginary numbers' in number theory and geometry, is a reflection of the efficient existence of such physical powers for defining all mathematics suited for the practical requirements of physical science."
In his "Science in Its Essence: On the Subject of 'Insight,' " published on May 30, 2008, LaRouche extends this concept:
"This 'imaginary number' fraud by de Moivre, D'Alembert, Euler, et al., was not merely a reflection of their apparent ignorance of elementary principles of physical geometry known since no later than Archytas and Eratosthenes. It was to be seen as an echo of the 'Malthusian' oligarchical-model hoax expressed by the Olympian Zeus of Aeschylus' Prometheus Trilogy."
The Double Sin of Quaternions
For the nominalists and reductionists of the 1800's, who attacked and successfully buried the hypercomplex number system called quaternions, Hamilton's use of imaginary numbers was just the beginning of his crimes.
Hamilton was also accused of creating a great "evil" by hypothesizing a possible fourth dimension, which, when projected down to three dimensions produces four degrees of freedom for measuring the real action which takes place in our three-dimensional world.
The adherents of nominalism and empiricism, who might be comfortable with projecting our 3D world down to a 2D plane to do their computations, are yet horrified by what they consider to be any furtherance of "imaginary" concepts, even though those concepts might be much more coherent with efficient action in the real world than anything that their flatlander viewpoint can produce.
The British mathematicians, Oliver Heaviside, and Lord Kelvin, along with the American, Josiah Willard Gibbs, sliced and diced up Hamilton's quaternions, taking out the "imaginary" numbers, while leaving in certain "real" numbers that denote directionality -- which Hamilton had called vectors. They then announced to the world that they had a new system called "vector analysis". The system was still comfortably conformable to a Cartesian grid but had a capability to describe such things as electromagnetism. They hoped that everyone would just forget that Maxwell had earlier used quaternions to describe the same thing. And everyone did seem to forget, at least until recently.
Now, the whole lurid story of how quaternions were squelched is beginning to be exposed for all to see. Grant Sanderson spoke of the "old-timey trash talk" involved in the fight. Here are some examples.
Oliver Heaviside wrote in Nature magazine in 1883, "Vectors versus Quaternions":
"A vector is not a quaternion; it never was, and never will be, and its square is not negative; the supposed proofs are perfectly rotten at the core...
"It is to Prof. [Peter] Tait's devotion to his master [Hamilton] that we should look for the reason of the little progress made in the last 20 years in spreading vector-analysis… [Tait] thinks nothing of the inscrutable negativity of the square of a vector in Quaternions; here, again, is the root of the evil."
Lord Kelvin wrote in an 1892 letter to Robert Hayward:
"[Quaternions,] although beautifully ingenious, have been an unmixed evil to those who touched them, including Clerk Maxwell."
Oliver Heaviside continued his rants in his 1893 book, "Electromagnetic Theory"::
"I came later to see that, as far as the vector analysis I required was concerned, the quaternion was not only not required, but was a positive evil of no considerable magnitude; and that by its avoidance the establishment of vector analysis was made quite simple and its working also simplified, and that it could be conveniently harmonized with ordinary Cartesian work."
What insights into electromagnetic phenomena and innumerable other fields could have been generated if the truncated, crippled stepchild produced by the opponents of Hamilton had not replaced Hamilton's comprehensive system?
A similar operation was carried out against Hamilton's other area of research involving Leibniz's principles of least action. Hamilton had a very prescient understanding of the analogy between the least-action and least-time principles that govern the propagation of light, regardless of whether light is considered as a particle or as a wave. The fathers of modern quantum theory, Max Planck, and Erwin Schrodinger, both credit the rediscovery of Hamilton's unique interpretation of Leibniz's least-action concept, as being fundamental to the development of quantum physics.
Schrodinger began his seminal 1926 treatise (Part II), with a subheading, "The Hamiltonian Analogy between Mechanics and Optics". He lamented how Hamilton's rich concept had previously been presented to the world in only a stripped-down version:
"Unfortunately, this powerful and momentous conception of Hamilton is deprived, in most modern reproductions, of its beautiful raiment as a superfluous accessory, in favour of a more colourless representation of the analytical correspondence."
The same could be said for the rediscovery, today, of Hamilton's system of hypercomplex quaternions. The recovery of this concept opens the door to a rediscovery of its "beautiful raiment", which in this case, means at least the Leibnizian and Gaussian concepts.
Rather than merely lament the sad condition in which our young people are languishing today, let us pursue the openings which current events have given us to prod and poke into existence an upsurge in creative thinking on their part. It is always the youth who led in momentous change. When they realize how much fun and excitement there is in creating a Renaissance, there is nothing which will stop them.