The electromagneto gravitic equation or force formula produces some interesting Fractal patterns, showing the deformation of space.
The description of space is essentially related to these: an associated region, associated to a semion, an orientation to a subjective processing centre, a relative orientation to the indicated orientation that is a local notion of orientation a subjective volume of space associated with this region with associated parallax and perspectives, an associated set of behaviours such as deformation of space, rotation in space and lineal translation in space, and a uniform transformation of space called scaling or magnification.
Now we have stumbled across a vector algebra, called a complex vector algebra, by me. that fits my idea of a compass multivector network. They were for a long time presented by the "unlearned " as numbers when in fact they were known to be magnitudes, but poorly understood, and not well trusted for any results they might give.
They are not numbers, Wessel called them directed numbers or measurements, Gauss established them as linear combinations of magnitudes , Grassmann and Hamilton started with this idea and went on to derive extensions and quaternions. Hamilton did the full work in founding algebraically linear combinations of 2 magnitudes , and looked for a useful version of 3 magnitudes, which he discounted because he wanted to find a method of algebra for rotations. IN the mean time Grassmann generalises the linear combination to n magnitudes while Hamilton is going cock s hoop over the quaternions that is 4 magnitudes. Hamilton coined most of the words we use today especially vector, The complex linear combinations of 2 magnitudes were vectors and the quaternions were a development that contained vectors of 4 magnitudes in combination . Meanwhile Grassmann coined all the other words we use in a set theoretic description of linear algebra, except matrix, which was already in use as Tableau,
AT the same time Dedekind was kicking up a fuss about numbers and ended up classifying numbers, and ontologising numbers, and unfortunately defined the complex vectors as numbers.
Nobody bar Grassmann and Hamilton seemed able to grasp the enormity of their work on algebraic vectors and algebraic geometry. Meanwhile, Gibbs who saw a chance took Grassmann's ideas and called them his own, masterminded getting Hamilton's quaternions out of the way and promoted his own vector algebra of mechanics. He wanted the word vector defined his way. So the complex vector algebra was sidelined to quaint numbers, and quaternions to queer numbers.
It is hard for us to imagine what a social furore Hamilton's Quaternions created, but Lewis Carrol was deeply upset by them,and so it seemed were a lot of other individuals in positions of influence.
Both vector algebras allow us to position translate magnify and rotate and reflect in a centre of rotation. With differential fractions we can describe deformations, and we can effect deformations by using them in an iterative context and forming a multiple combination.
They enjoy widespread use in Physical and applied sciences but only now has the vector use of them been venerated .
I guess, Bombelli knew what he was dealing with, neusis. That is rotation and extension. His carpenters square was the first vector used to solve algebraic problems. Newton was the next person to use vectors to solve algebraic problems and De Moivre and Cotes learned a thing or two from him. One important person is John Napier. His logarithms combined the rotating radius of a circle to the proportions of the side of a right angled triangle called the sine. Thus the variable "Bombelli" vector was related to the trig ratios and a fixed radius.Due to Napier a great calculation was launched to derive the logarithm in table form and for differnt bases.
This calculation and the calculation of the sine tables was an underlying computation that led to the differential calculus. The calculus of interpolations involving difference formulations led to a great awareness of the relationship between difference forms for calculating smaller and smaller intervals accurately. This plus the compoungin interest method for aggregating growth were on Newtons mind when Wallis announced his involvement with the calculation of "Napiers logarithms" calculation.
The ratios of the right angled triangle and the unit circle was a particular interest of Newton's, thus the logarithm calculation highlighted the rotation in the unit circle. The rotation was a quarter circle. The quarter rotation was inherent in Newtons mind, and so was the tangent vector. Bombelli's rules for the surd of a negative magnitude involved a quarter turn. Euclid's method for finding the geometric mean involves a quarter rotation. Without formally expressing it i believe Newton understood that the imaginary magnitude involved a magnitude of rotation.
De `Moivre certainly developed this notion and worked out the versine and coversine relationship, and eventually the theorem with Cotes regarding the roots of unity.
Cotes really advanced the link between rotation and the trig ratios and the imaginary magnitude and the quarter turn.
Cotes did an amazing thing. He took a piece of cord and tethered it and drew a circle arc. He then used the same cord and laid it along the arc. This measure he called a radian. Then he attached the cord to the circle arc and pulled the cord along the line of the diameter extended outside the circle. As it is pulled the circle on which the arc sits rotates that distance. The arc represents a rotation and the horizontal distance can be used to represent a rotation.
Studying Napier's exposition of logarithm, Cotes realised that instead of using Arithmoi he could use arc lengths of the rotating radial. And using his new measure made it even more obvious: he could place the arc in the ratio name and link the length along an axis to the logarithm of a sine along the same axis. Using De Moivres relationship of the coversine Cotes was able to set out the relationship between the logarithm of the factors of 1 and the arc lengths
ix =log(cosx+i sinx). This only works because the factors are factors of 1 in the unit circle, and the geometric powers become the same as the arguments of the ratios, Demoivre Cotes theorem.
The astonishing thing is that Cotes used the surd form to represent different magnitudes ,disposed along different axes, axes at right angles. Thus he surd form represented a rotation of a quarter turn.
Newton had a fascination with spinning tops. They appear as a fundamental bridge axiom between the motion on the planet and the celestial motion in his Principia. De Moivre had been shown by Newton some of the properties of the unit circle in relation to the surd of a negative magnitude. Wallis had previously speculated that this surd would represent some point in Descartes plane. De Moivre was able to show that the power of a factor of 1 was equal to a combination of the cos and the sin of angles with multiples equal to that power. Cotes therfore felt he had found an invariant relation that liked all forms of measurement in harmony. He began to write his Harmonium Mensurarum with De Moivre. He died without completion.
The Euler Cotes formula links rotation in a quarter turn to the factors of unity and to a length along the diameter of a rotating circle through Napiers logarithm :It also manages to include the new surd measure. It offered Newton and Cotes a new way a spinning top way to look at the rotation of planets.
Euler's form eix = cosx + isinx links an exp vector function to a rotating unit vector and at the same time defines a rotation vector along a radian arc vector.
This can be extended from the plane to 3d. in which case the exp quaternion function is linked to a rotating unit quaternion and at the same time a 3d rotation vector is defined over a radian triangular surface vector which i think hamilton called a versor.
In general therefore Vectors and Versors were envisaged as dealing with rotation and expanson of space with translation and deformation, compex vectors for the plane and Quaternions for 4d which Hamilton hoped would include his notion of space time.eLM = cosM + LsinM
Usually the 3 quarter turns in a quaternion are defined ortho normally but in reality there is no such restriction on them inherently. The restrictions reflect one choice in an infinite set of choices. Thus we can define quaternions on generalised coordinate systems.
In a similar way the quarter turn in the plane does not inherently have to be a quarter turn and we can choose generalised coordinate systems for that. We can use any vector orientation as long as we define the relationships between the reference frame vectors. The quarter turn has historical preeminence because of pythagoras Theorem and the trig ratios. If we define tables of relationships for other triangular forms, or spiral forms we can use theese to describe sequences and relationships in space.