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So just to start off with a+ib is  polynomial of degree 1 in i.

This makes perfect sense NOW, especially as i realised that they arose as the solution to polynomials that otherwise had no real solution. HISTORICALLY mathematicians had been used to giving solutions in surd form so a+b√-1 had the right form but no real decimal name in the namespace. However i now see that these are polynomial solutions to polynomial equations. I will call them polynomial numerals. Later I will term them polynomial unary operators or polynomial rotations.

I have to acknowledge the influence of kujonai and timgolden  in formulating the basis of unary operators by exploring the notion of sign. My interest in the operators of the set FS under iteration have led me to reconfigure foundational notions of math. Unary operators came from understanding the action of i and j. This was then extended by understanding sign, a unary operator mod(2). Some of the language will have to be decided, for example polynomials of degree or power or signal 2!

Kujonai also introduced the mod(n) categorisation which means that the logarithmic additions are mod(n) clock arithmetic. This means that the actions of the unary operators on themselves are added mod(n) and thus we can think of them as acting in a kind of multiplicative way. So a unary operator acting on another unary operator is a product mod(n).
For example sign acts on the real numeral 2: sign⁰2 =+2 and sign¹2=-2
sign¹sign⁰2=sign¹⁺⁰2=sign¹2=-2
sign¹sign¹2=sign¹⁺¹2=sign⁰2=+2.

The indices to sign i have called signals. They look and act like powers mod(2). In polynomials they may also get referred to as degree. These notational references speak of the history of notation more than anything else, but it is important to be clear from the outset that these indices refer to the action of an operator on an appropriate operator not to numeral manipulation,numeral products or numeral multiplication. Therefore i retain the term signal. This means that if sign is taken as the nomial polynomials in sign will all reduce to signal 1 polynomials mod(2)   
        sign⁰2+sign¹2+ sign²2+sign³2
      =2sign⁰2+2sign¹2

signal 0 is defined as the identity signal and as we have seen the identity signal on sign means that a positive sign is symbolically attached.

Now i can use a consistent geometrical representation of a unary operator as long as i clearly define what that is. For this discussion i am going to define unary operators as acting in a plane only. I am implying that i can define them as acting in geometrical space but it is a bit of a tort to do so without establishing the definition in the plane first.

To have any geometry we need a specific orientation first. This orientation is essentially not definable by any geometrical reference frame i construct with it. So to be literal it is ground 0. Because of this every reference frame is relative to the observer, and it is only by agreement we construct a common reference frame .This orientation is the axiomatic orientation. However it helps me to see that a general property of space is orientation. i have a special capacity to fixate on a specific orientation and reference from that orientation.

Another fundamental property of geometrical space is extension . Orientation and extension are logically and practically inseperable and a sensory synaesthesia. The third fundamental of geometrical space for our discussion is rotation, and again this is inseperable from orientation and extension. However, customarily we ignore the sensory synaesthesia because we are not reference free, we live in a gravitational geometrical space and that determines our orientation fundamentally and frictional forces determine our rotation.

Nevertheless we need three fundamentals to establish a geometrical space : orientation, rotation, and extension. Of the three orientation is the ground 0 and cannot be defined,and extension can then be defined as being in a certain orientation (direction) and rotation defined as around a certain orientation (axis). I am going to use the notion of axis and axes to refer to an orientation different to and from  the axiomatic orientation, and to give axes therefore there own extension and rotation. Because of this an axis will have a direction in the plane referenced from the  axiomatic orientation.

I will define as a scale  a division iteration such as: divide into 10 parts a unit length. This iterated will produce the real numeral scale along the axes. From the axiomatic orientation i define unary operator cycle mod(1) to be a rotation about any axis that returns to the axiomatic orientation . I define the unit length as sweeping out a circle radius 1 circumference 2π and the axis of rotation as being always right to its direction which in a mod(4) unary operator scheme will be identified as π/2, and the axes under cycle mod(1) i define as sweeping out a plane, which makes a normal to a plane (the axis of rotation under cycle mod(1)) the definition of that plane. AS the mod(n) n increases i can define a scale by division iteration: divide 2π into n parts. However we normally use: divide the unit (radius) into 10 parts divide the cicumference into parts that are equal to or fractions of the unit (radius). By this more complex iteration we construct a radian measure of rotation.

It is worth noting that geometrical space has all these attributes but we only distinguish them under some operation,and the vector notion ought really to include rotation as axiomatic and within its definition, thus a vector has magnitude direction and rotation.

 I am going to use 3 mod (4) unary operators to construct a reference frame for geometrical space. As you will see this means i am going to construct polynomials of signal 3. Signal 0 will be as defined above the identity or axiomatic orientation for that rotation, ground 0.

The unary operators are i,j,v. They operate on the numeral line.

i only operates on i,j on j, v on v.

The operators act on the real numeral line which is the basis of the axes and to which rotation is imparted orthogonal to the direction of the axis, that is to say that an orthogonal axis to the axis in focus rotatates around that axis. However this implies and is made explicit that there is always an axis orthogonal to both that defines the plane the first two axes in focus are in. So i have our 3 requirements for the vectors orientation and an axiomatic orientation which is to the right of the screen for the positive numeral line (by the way sign is preserved in an even mod unary operator scheme), extension along the numerate axes, and rotation defined by i,j,v.

The reference frame is constructed. Thus the following rules describe the constructed scheme.

i,j,v are unary operators in a specified plane.

Using the real numeral line as the axiomatic orientation i can define i as rotating in a plane that goes into the page away from this orientation, j as rotating in the plane of the page away from this  orientation and v as being aligned so that its axiomatic orientation goes into the page and it rotates in a plane coming out of the page orthogonal to the i and j planes. This makes all the normals to the 3 planes orthogonal to each other, and if i take 0 as the "origin" i have the familiar octant division of geometrical space.

i⁰=1 i₀ which is the direction of the real number line.
j⁰=1 j₀ which is the direction of the real number line by construction.
v⁰= 1 v₀ which is the direction into the screen achieved by rotating i⁰₀ by  i¹.

i⁰=1i₀ the identity rotation with extension 1, i¹=ii₀ a π/2 rotation of the orientation, i²=i²i₀ a π rotation of the orientation, i³=i³i₀ a 3π/2 rotation of the orientation, i⁴=i⁰ a 2π rotation of the orientation.
j⁰=1j₀ the identity rotation with extension 1, j¹=j j₀a π/2 rotation of the orientation, j²=j²j₀ a π rotation of the orientation, j³=j³j₀ a 3π/2 rotation of the orientation, j⁴=j⁰ a 2π rotation of the orientation. The plane of this rotation is orthogonal to the plane of the i rotation.
v⁰=v₀ the identity rotation of v with extension 1i=1ii₀, v¹rotates v₀ to j¹₀=v¹₀ a π/2 rotation of the orientation, v²=v²v₀  =i³₀ - v²₀ a π rotation of the orientation of v, v³=v³v₀   =j³₀= v³₀ a 3π/2 rotation of the orientation of v, v⁴=v⁴v₀=i¹₀=v₀ a 2π rotation of the orientation. The plane of this rotation is orthogonal to the planes of the i and j rotations.

These rotations are quantized, which is a result of the definition as mod(4) unary operators. I point out that all complex math is so quantized and the amount of quanta is some power of 4.



 

So referring back to my vector definition , the axes form a set of orthogonal vectors each being the norm to a plane and each importantly having rotation included by definition. I will write the tensor (i₀,A₁,A₂,A₃)

where A₂,A₃ refer to the orthogonal axes with extension and rotation inherent, and i₀ refers to the orientation.

I can now rewrite this tensor as (i₀,(ei₀,v),(ev₀,j),(ej₀,i))

 where ei ej ev are extensions in the directions i₀,j₀,v₀ and i,j,v are the unary operators providing quantized rotation.

Because of the extension on each axis i can use scalar multiplication on each axis, but this operates as a unary operator only on the extension not the rotation.

Focussing on the rotations i can write the position of a point  as a polynomial of rotation from the orientation thus

(x,y,z)= xi⁰+yi¹+zj¹


Where x is the extension along the axiomatic orientation,y the extension along the axis formed by the rotation i¹ and z the extension along the axis formed by the rotation j¹.
To clarify matters i define i₀,j₀,v₀ as being of unit extension, these then become scalar products as desired.

An important note is that by design i₀ and j₀ are interchangeable as are i²₀ and j²₀ and v₀ and i¹₀ and v¹₀ and j¹₀, v²₀ and i³₀ and v³₀ and j³ ₀.They are only references to the same orientation (as are i⁰ ₀ and j⁰₀) so you cannot rotate v⁰ in the planes for i and j!  thus i*i≠v⁰*v⁰

Another interesting point to me is that setting the scalar to 0 does not remove the rotation. it simply gathers that rotation to the origin where it remains as an infinite potential, still quantized and still seperate from any other rotation that has been gathered there by setting the scalar to zero!

While i can now refer to any point in the octants by a rotational polynomial, i have not accounted for all the rotations by using the common form of notation.

writing (x,y,z)= xi⁰+yv⁰+zj¹
 
is better but means that rotations are independent or dependent , for example i cannot do v⁰ from the axiomatic orientation without doing an i rotation first. I will cover this in the next post.

{September 2013, i learned that rotations act on the cylinders around an axis! so this restriction i laid on the topic was why it was so hard to grasp what was going on!}

   
i₀ is the orientation from which all other rotations are measured anticlockwise. By construction j₀ is interchangeable with i₀ and j²₀ is interchangeable with i²₀.

The unary operators i,j,v are rotations by π/2 in a plane, each plane constructed as orthogonal to the other two. v⁰ is thus the orientation for v rotations and is not interchangeable with a j or an i rotation. However v⁰ is referenced by an i rotation and not by a j rotation.

Geometrically the system can be represented by rotations of a vector which has magnitude, orientation and rotation inherent, and graphically we can represent this by a scaled line with spin orthogonal to its extension. This scaled line we define as an axis and the i,j,v as rotatations of a unit extension on the axes. The rotation v⁰ is dependent on i and j in this system, but only as ways of referencing the outcome of each of 4 unary rotations under v. Thus the rotations are independent but rule bound as shown in the table above.

i cannot act on j. So ij =ji is defined by the orientation acted on.
i²₀ and j²₀ being interchangeable allows ij² to become ii² which can be resolved as i³ but only when acting on an i rotation.
Where i and j are symbolically written as acting on each other the action is defined only by the orientation acted on .

This system allows me to write a polynomial that represents a cartesian triplex or a complete polynomial that represents 8 cartesian points at once.

  (x,y,z)= xi⁰+yi+zj. This represents a point in the first octant obtained by rotating the extensions from i⁰/j⁰. The extensions are x,y,z and are scalars.

Writing them in the ordered triplex form can now be seen as a matrix of coefficients. The nomial matrix (i⁰,i,j) has i highest signal of 1

I allow other nomial matrices of unary rotations such as (i⁰,v⁰,j) or (j⁰,v⁰,j³) etc.

(x,y,z,x,y,z)= xi⁰+yi+zj+xi²+yi³+zj³ is the highest signal polynomial i can write and is the basis for the table above.

As i see it the axes cover all the octants and so this polynomial represents the 8 points of an object centred on the origin.
Thus (1,1,1,1,1,1) are the coefficients for a cube length 2

{September 2013. I was very confused in this post and in fact misleading about the actions. The axes and the rotation operators are confused in this post. I correct this later on but it was a struggle!}

(x,y,z)²=(xi⁰+yi+zj)²=

	xi0	yi	zj
xi0	x2i0	xyi	xzi0j
yi	yxi	y2i2	yzij
zj	zxj	zyji	z2j2

 

But (x,y,z)²=(xi⁰+yv⁰+zj)²=

	xi0	yv0	zj
xi0	x2i0	xyi0v0	xzi0j
yv0	yxv0i0	y2v0	yzv0j
zj	zxj	zyjv0	z2j2

 
                          
 Both tables express the squaring of the polynomial rotations for (x,y,z) but the first is with reference to the axiomatic orientation i₀ , the second is with respect to 2 orientations i₀ , v₀ .

To plot this i need to show what happens to the orientations under the new rotations and from that what happens to the coefficients . This then gives me the  position the pont is moved to.        

Now sign is the other unary operator which acts in the plane as a π rotation.

Thus sign⁰v⁰=v⁰,sign¹v⁰ =-v⁰=v² a π rotation from v⁰
so sign⁰i⁰=i⁰,sign¹i⁰ =-i⁰=i² a π rotation from i⁰
sign⁰j⁰=j⁰,sign¹j⁰ =-j⁰=j² a π rotation from j⁰.

Similarly   sign⁰v¹=v¹,sign¹v¹ =-v¹=v³ a π rotation from v¹
so sign ⁰i=i,sign¹i¹ =-i=i³ a π rotation from i¹
sign⁰j=j,sign¹j =-j=j³ a π rotation from j¹.

So the rules, just to be rigotous are:



It is clear to me that these associations are not commutative or associative in general. The X indicates actions that have no physical, geometrical or logical action. The simple rule is a rotation can only affect axes in its plane of rotation.However the orientations are non rotations so they have no action on any rotation. They serve to apply the extension to the axes being acted on in the association.

A couple of interesting notions that occur to me is that sign although a rotation of π in a plane is also a reflection through the centre of rotation. So rotation has a special transform called reflection involved with an axes . This property is a special combination of the rotation and extension and orientation already defined in the vector called an axis and does not need to redefine a vector. Reflection occurs only when one axis is rotated and only when that axis is in the direction of the reflection and the π rotation is in the vertical plane of that direction or is a π rotation of the horizontal axis in that direction. A rotation in the horizontal plane in that direction is a simple rotation.

A π rotation of 2 axes produces a reflection through the origin of rotation, which is the same as a rotation to that position. A π rotation of 3 axes gives a reflection again.

{September 2013 This is where i realise that rotation operators are different to orientations! But more subtly the orientations are labels and the labels remind us how the orientations transform when operated on by the rotations . My notation makes explicit what Hamilton was thinking, but loses the fluidity of his presentation}

There is an important distinction i am making between rotation and orientation . Rotation acts on orientation and results in an orientation. Defining i⁰₀ as the initial orientation and iⁿ₀ as orientations resulting from iⁿ rotations  i can represent the distinction between the i and j and v rotations. 

As i defined the iⁿ₀ are all orientations in a plane.jⁿ₀ are in a different plane and vⁿ₀ are in another plane. i,j,v are the unary operators for those planes, so no matter what i associating with j cannot be a rotation in a plane unless j rotates jⁿ₀ onto a iⁿ₀.

   it is therefore necessary to know which orientations are being rotated and by how many times. Where the resultant orientation falls determines whether the association can act or whether it fails to have an effect, Thus the table of associations above are based on rotations from the orientations of the system in the planes of the system. Thus i² is the rotation acting on i⁰₀ resulting in orientation i²₀.

ij are the rotations acting on j⁰₀ or i⁰₀, Whichever i choose the association fails. j rotates j⁰₀ to j¹₀, but now i cannot rotate the resultant orientation. Similarly j rotates i⁰₀ to j¹₀.

Now by construction j⁰₀ and i⁰₀ ,  j²₀ and i²₀ are orientations in the same plane so i orj  can rotate them but only as an i or j rotation, thus giving an i or j orientation.

Also by consruction j¹₀ , i¹₀ , j³₀ and i³₀  are orientations that lie in the v rotation plane. Thus v⁰₀ is i¹₀.

Now i³ acting on j only works if j rotates j¹₀ or j³₀ onto the i plane of rotation. It does not work if j rotates i⁰₀ or i²₀. So it is necessary to know which orientation is being rotated.

Where the X are is where i have made the observation that the first rotation acting has rotated the orientation into a plane the second rotation cannot act on. in this scheme 2 important orientations are rotated v⁰₀ or i⁰₀. In one instance i have had to emphasise that iii=ii=i=v⁰ v⁰ v⁰ = v⁰v⁰ = v⁰. These associations of i have to be mapped to the rotation that defines v⁰₀ very carefully.

(x,y,z)= (xi⁰₀,yv⁰₀,zj¹₀). This can be written as a polynomial rotation
(x,y,z)= (xi⁰i⁰₀+yii⁰₀+zji⁰₀) or
(x,y,z)= (xj⁰j⁰₀+yij⁰₀+zjj⁰₀)

I can then "factor out" the orientation to isolate the rotations

(x,y,z)= (xi⁰+yi+zj)i⁰₀

This then puts the table of actions in context and the table of associations  can be used to determine a resultant orientation that is valid.

I did think about using 0 in place of X but this is not just about neutering senseless associations but about reading the signals of what rotations are happening that would be missed.

I think that a lathed mandelbrot is sculpted because the quaternion math accentuates a v rotation under iteration unless missing terms are replaced.

I am going to call the reference frame i am constructing an initial reference frame. Whether it is the same as an inertial reference frame is a question for later. Using this initial reference frame as a basis i can define the position of origin of any relative reference frame and its orientation using the tensor (i⁰,ei⁰₀,v,ev⁰₀,j,ej¹₀,i).

 It is clear the quantized rotatations i,j,v are not subtle enough to describe any orientation in geometrical space,but again a later post will consider this. If i refer to this relative frame as XYZ, then it is important to observe that the association ji and ij have a effect on XYZ: they spin XYZ around the v⁰₀ or the j⁰₀ orientations. This means that far from j acting on i having no sensible meaning it in fact refers to an action on the rotation attribute of each axis. That is to say that if it were possible to quantify the spin in an axis (v⁰₀ in this case) j would increase the spin. Alternatively it would rotate the quantized j rotation in v⁰₀ by π/2.

This could thus be interpreted as a spin rate increase or a rotation of the other two orthogonal axes (that is the plane in which they lie) around the axis. j acting on i would then be not one axis rotation,( which i have discussed as producing reflection in the case of a π rotation) , but a plane rotation preserving the relationships on the plane but changing the orientations in the plane in the initial system. Since in XYZ nothing appears to have changed i define this sort or rotation as a relativistic rotation, or more simply a rotation in the initial reference frame.

So briefly looked at hamiltons quaternions and clifford algebras and some vector algebras and think that these are different to polynomial rotations based on unary operators, and constructed reference frames. I was also pleased to note that although widely described as dimensional this in no way necessitates an alternative to geometrical space. Hamilton i might add is quoted as inventing the term space-time even though it did not have the wide meaning it has today.

The unary operators in a plane, can  construct orientation vectors to span geometrical space from an initial orientation. This initial set of orientation vectors is the initial basis, but action takes place through the unary operators sign,i,j,v which are rotations,scalar multiplication and addition et al. This means that rotation translation magnification and reflection are all attributes of this constructed system.
 The term linear combination is the nearest equivalent term to my polynomial numeral. Polynomials have a long and important history in mathematical thought and i mean to refer to that in the naming so that one can easily search in the field of polynomials for solutions to particular problems.

So i have made some alterations to previous posts in the light of greater clarity. It is unfortunately easy to confuse what is being resolved by polynomial rotations as the custom and practice seems to be a bit confused with regard to rotations and orientations or directons.

I now have in place notation for orientation and notation for rotators and quantized rotations. It is also clear that a polynomial rotation should have an orientation as a resultant. It is also clear that associations of rotations cannot always be resolved without reference to the orientation being rotated.

Finally to be able to plot these polynomial rotations it is important to know what the resultant orientations are not the association of rotations. As the tables so far have been about association of rotations i will produce a table of resultant orientations and see how that turns out.

  A thought that needs development but just noted here.

Each axis has extension and rotation inherent with orientation coming from the axiomatic orientation i₀ and rotations i,j,v in planes with orthogonal norms j₀,v₀,i₀. Each norms rotation therefore acts on two orthogonal axes in a plane. Objects in this plane with the same norm will rotate about each others  norms depending on the spin ratios: the faster spins dragging the slower spins around repulsively, and if the norms are oriented in opposite directions then attractively.

 If the norm  of the objects in this plane are orthogonal they will be unaffected as to spin and thus the object will be neutral neither attracted or repelled except by objects with the same norm to their spin plane. these objects with orthogonal norms will tend to have there spin rate unaltered but there rotation around the spin plane norm will be at the norms quantized rate.

Would this provide a basis for action at a distance, that is through spin plane coupling on a resonance type model?


{September 2013 I discover through Norman that Euler has thought this problem through very clearly. However, Norman is the only professor who presents it correctly. Euler rotations as "vectors" double the angles of rotation! You have to use half Euler angles to get the desired angle rotation. The angles here are in fact radian "vectors" or arcs of great circles.}

http://www.youtube.com/watch?v=0_XoZc-A1HU 

i⁰₀=i₀
j⁰₀=j₀=i⁰₀
v⁰₀=v₀=i¹₀

I have adopted a convention that a rotation  produces its resultant orientation when that lies in its plane of operation, but an association of rotations is replaced  by its rotational equivalent in the plane.

under construction

Without confusion it would seem i can rewrite the orientatobsas

i¹₀=i₁
i²₀=i₂
i³₀=i₃

with similar transformations for j orientations and v orientations.

Thus v₀=i₁.

Although i think currently that polynomial rotations are different from quaternions polynomial numerals are not. I can write any quaternion as a polynomial numeral of signal 1 if i introduce a fourth rotation k which can be in any plane orthogonal or not to the three planes already defined. In geometrical space there is no fourth orthogonal plane and so the geometrical analogy breaks down,but the symbolic workings continue in much the same way. What i have to do is set the extension of the fourth rotation to zero, that is only applying to the origin. This gives potential at the origin for infinite variations on a theme all happening at the same time but only being realised when the rotation is given extension in place of another being set to zero. This is very much a quantum superposition, and a case of schroedingers cat!

However,orthogonality is not a prerequisite for spanning a space, and so we can have non orthogonal orientations of any number set up in a spanning system for which the underlying symbolic manipulation will be identical,but the geometric representation will show transformation modifiers as in a tensor description.

Using Terry Gintz Quasz programme which is bases entirely on hypercomplex math using quad variables. which is to say that each polynomial numeral is a quaternion i have been able to construct polynomial rotations in which the extensions are , and then by degrees i have learned how to modify the extensions into elements of and .

As i progress in exploring these variaions i will post some image results here.

I have adjusted the strucure of my inital table which very much was a sketch of what was in my head at the time. The table is misleading and so i will correct it based on what it is actually showing, but for developments sake i note here that the ideas for it have developed and clarified over time, and this is how all maths proceeds, from draft to draft until the symbology is faultless. Benoit Mandelbrot explains it well in his interview: maths must not get up its own arse and fly away from the senses. That may be a little"rough" on some people but hey that is what fractal originally meant! :embarrass:

Alice, angry now at the strange turn of events, leaves the Duchess's house and wanders into the Mad Hatter's tea party. This, Bayley surmises, explores the work of the Irish mathematician William Rowan Hamilton, who died in 1865, just after Alice was published. Hamilton's discovery of quaternions in 1843 was hailed as an important milestone in abstract algebra, since they allowed rotations to be calculated algebraically.

Just as complex numbers work with two terms, quaternions belong to a number system based on four terms. Hamilton spent years working with three terms - one for each dimension of space - but could only make them rotate in a plane. When he added the fourth, he got the three-dimensional rotation he was looking for, but he had trouble conceptualizing what this extra term meant. Like most Victorians, he assumed this term had to mean something, so in the preface to his Lectures on Quaternions of 1853 he added a footnote: "It seemed (and still seems) to me natural to connect this extra-spatial unit with the conception of time."

As Bayley points out, the parallels between Hamilton's mathematics and the Mad Hatter's tea party are uncanny. Alice is now at a table with three strange characters: the Hatter, the March Hare and the Dormouse. The character Time, who has fallen out with the Hatter, is absent, and out of pique he won't let the Hatter move the clocks past six.

Reading this scene with Hamilton's ideas in mind, the members of the Hatter's tea party represent three terms of a quaternion, in which the all-important fourth term, time, is missing. Without Time, we are told, the characters are stuck at the tea table, constantly moving round to find clean cups and saucers.

Their movement around the table is reminiscent of Hamilton's early attempts to calculate motion, which was limited to rotatations in a plane before he added time to the mix. Even when Alice joins the party, she can't stop the Hatter, the Hare and the Dormouse shuffling round the table, because she's not an extra-spatial unit like Time.

The Hatter's nonsensical riddle in this scene - "Why is a raven like a writing desk?" - may more specifically target the theory of pure time. In the realm of pure time, Hamilton claimed, cause and effect are no longer linked, and the madness of the Hatter's unanswerable question may reflect this.

Alice's ensuing attempt to solve the riddle pokes fun at another aspect of quaternions that Dodgson would have found absurd: their multiplication is non-commutative. Alice's answers are equally non-commutative. When the Hare tells her to "say what she means", she replies that she does, "at least I mean what I say - that's the same thing". "Not the same thing a bit!" says the Hatter. "Why, you might just as well say that 'I see what I eat' is the same thing as 'I eat what I see'!"

When the scene ends, the Hatter and the Hare are trying to put the Dormouse into the teapot. This could be their route to freedom. If they could only lose him, they could exist independently, as a complex number with two terms. Still mad, according to Dodgson, but free from an endless rotation around the table.

	xi0	yi	zj
xi0	x2i0	xyi	xzi0j
yi	yxi	y2i2	yzij
zj	zxj	zyji	z2j2

 


so (x,y,z)²=x²i⁰+y²i²+z²j²+2xyi+xz(j+i⁰j)+yz(ji+ij)

Acting on i₀

gives  x²i⁰i₀+y²i²i₀+z²j²i₀+2xyii₀+xz(j+i⁰j)i₀+yz(ji+ij)i₀

giving  x²i₀+y²i²₀+z²i²₀+2xyi₁+xz(j₁+i⁰j₁)+yz(ji₁+ij₁)

giving  x²i₀+y²i²₀+z²i²₀+2xyi₁+yzji₁+2xzj₁+yzij₁
This is the julia which is noticeably sparser than a quaternionic one
(http://www.fractalforums.com/gallery/2/410_19_05_10_11_55_12_0.png)
This is the mandy which again is less round and the bum cheeks are evident.
(http://www.fractalforums.com/gallery/2/410_19_05_10_11_55_12_2.png)

ji₁ is interpreted as a j rotation around the i₁ axis, that is the whole basis rotates around the i₁ axis.

Similarly ij₁ is a rotation of the basis around the j₁ axis.

In my initial exploration  of the origin of the notions of axes and there use in a new definition of vector i did not fully appreciate the distinctions this intimates. So firstly the unary operators do not act on scalars commutatively but on the axes which have scalar multiplication of a unit  extension as intrinsic. At the moment the operation of the unary operator has been defined as pre operative that is the operator is positioned prior to the axis it operates on. to make it commutative i need to define the post operative forms as being identical to the pre operative forms and there is no problem in doing that. I have utilised distributivity with no problem, but have shown that associativity cannot be assumed among the unary  operators.

Thus the tables explore and clarify the associative rules for the operators with reference to the axiomatic orientation and indeed any orientation  in {i₀,j₁,v₀}.

what i did not see until now is the effect of the rotation on the rotational elements of the vector definition. In my exploration i did not distinguish whether the rotation for each axes is 'forced' by external input or inherent by axiom.

Inherent rotation would be a constantly spinning axis which would be quantized  only in the sense of a snapshot or a "still frame". It would mean that by any iterative scheme of measurement or description i only "see" the resultant of the  spin motion on an orthogonal axis at that iteration point. In particular all three axes would be spinning and this should result in any point in one iteration stop being "mangled" by the next iteration stop! Regions referenced by fixed axial extensions would undergo this constant motile transformation of reflection rotation being squeezed into a point or a line or a plane and back into geometrical space!.

I leave it at that not to condemn it , as it may be that it holds an explanation for quantum fluctuation, but because it highlights that my construction was clearly not intended to be of this kind.

Forced rotation then has implications for the working out of resultants to rotations of axes. I particularly concentrated on orthogonal axes as this was the design goal, and set out an axes as having two variable elements once a common orientation reference is "factored out" of the notion. That is to say that any axis has three necessary variables orientation, rotation, and extension but by constructing a reference frame from a axiomatic orientation i can reduce this to two provided that the axes are in a vector system.

The implied effect of a rotation then is to force the rotation variable of an axis to impinge on all axes orthogonal to it ; so as to rotate the axes with it. The implication is a coupling between orthogonality of the axes and rotation around an axis. However the axes themselves do not need to be orthogonal to the rotating axis but need to be related to the axis by some metric that involves orthogonality currently. By this i mean that metrics as i am discovering in spaciometry do not need to be constructed of euclidian elements, and any fixed relation may do.

So it becomes clear that a rotation acts on an orthogonal axis without affecting the rotation element of that axis definition, which would  force a rotation around it. This is the case for non orthogonal axes also. However any regions referenced by the axes being rotated have to rotate in step with the rotation from the forcing axis. Thus regions spin on an axis parallel to the forcing axis.

Thus it seems clear that a rotation creates parallel spin throughout its region of influence but does not interact with the orthogonal rotation elements.

So now focusing on ji₁ i have a forcing rotation j forcing the j rotation variable in the axis definition of i₁ the only axis along with i₃ that it can force! The question is should this be understood as forcing a rotation of the two orthogonal axes?

I have a choice. From he above exploration the natural choice would be to couple the rotation of the orthogonal axes to any forced rotation of an axis. This leads to the notion of initial frames of reference and relativistic frames.

The other choice is to understand the rotation as torsion so the orthogonal axes do not rotate but the forced axis is twisted or spun. This makes the rotation of the orthogonal axes  independent of the axis and external to the axis and thus i have no notion of where this rotating transform comes from, or how it acts differentially on axes depending on orientation. To Avoid this, torsion would have to act on regions the same way no matter which  non orthogonal axis referenced it, thus regions would not rotate in step but would "torse" in step with the forced axis. in both cases the definitions deal with the rotation of the orthogonal axes by direct operation.

From a mechanical/fluid dynamic point of view the two choice could be related through viscosity of medium, the first choice being a viscous reference material and the second a very low viscous material.

Each of the choices (and some in between as it turns out) has an effect on the form sculpted by quasz both in the julia and mandelbrot boundary conditions

When i look at the tensor definition of the Axes system  A₁ has a unit extension  extension i₀ and a quantized rotation v. This means that v rotations force A₁ to rotate taking v₀/i₁ to v₁ or j₁ . So v acts on A1 and effects A2 and A3.

I can thus write vi₀ to mmean v is acting on axis A1 but it is more clear to write vA1. This difference in meaning is consistent with the tensor definition, but not consistent with the unary operator scheme outlined above, in which v i₀ has no action!

Now ii₀ is defined to have an action, but the Tensor definition would be iA₁ and i cannot force A₁ to rotate, it acts on A₃ and thus i A₃ would be written as i j₁.

To avoid confusion in the tensor form it would be better to standardise around the tensor form and thus to see the rotations as acing on their axes of rotation not on the orientation. This makes a helluva difference as the axis of rotation affects ostensibly the 2 orthogonal axes not just the one as commonly posited.

Thus a point can be referred to by its extensions along the orthogonal orientations but not by orthogonal rotations.
Orthogonal rotations represent a dynamic transformation of the space in which the point is referenced and thus indicate all points that are similar under the transformation. picking out orbits of points referenced by the orientation axes.

The scalar will directly affect the scalar  of the unit orientation.

Thus xi on yj₁=xyij₁. I then have to look at what happened to i₀ and i₁ under the action: i₀ goes to i₁ and i₁ goes to i₂

I then have to apply the next part of the transformation to the new arrangement

So i apologise once again for making something that is intuitively simple look horrendously complex. In my defense these are the outpourings of a "dysfunctional" brain, a "crippled" mind, enslaved and girded about by pedagogical half truths and grappling with historical misrepresentation, idiosyncratic predilection, and a false sense of subject boundary.

A diffracted mind shattered by the tortuous hoops we are made to jump through to receive feedback that is approval, but is no guide as to relevance or direction in one's own personal experiential continuum.

I have tried to explore space spaciometrically, refrerence frames relativistically, and transformation transformatively, without the key foundational insight: proprioception.

This, the notation the tensors the transformations are all proprioception, and that makes it simple and that gives it sense.

So i will complete my exploration of proprioception before i progress this further, but initial and inertial frames are important and linked through "tools" that reference the transformation process between the different encapsulating reference frames. Tools like the magnetic compass, the pole star, the sun shadow lengths , radials and boundaries. The notions of fixed and reference points derive from these proprioceptively, and the freedom of my imagination allows me to soar above the mundane and fly to celestial vantage points and view the whole boundaries from there.

What a perspective ! And it is iterative, and therefore produces fractal outcomes, rough approximations to the set notFS.

The set FS is a model in need of revision, but the fundamental revision is that it is Fractal. A rough model of approximations arrived at iteratively. It is Fractal. It will always be fractal and it is endlessly fractal and that is the beauty of it and its endless fascination.

I mentioned a possible connection between spin and action at a distance. Apparently the spin hall (http://www.newscientist.com/article/dn19601-allelectric-spintronic-semiconductor-devices-created.html) effect may be an illustration of this through magnetism.

A good book for beginners is:
Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace and Virtual Reality.
Jack B. Kuipers

Thanks Hermann!  I will definately check out that book.  Just off topic a bit..I just purchased "Chaos" by James Gleick.  Another question... Probably Kindergarden for most of you.. buuut What your referring to here are "polynomial rotations" looks like what I would call a quaternion/julia rotation.  Clarify please!

With the help of this book it is easy to understand the concept of rotation which is also the basic to understand polynomial rotations.
The book starts at the basics but you should be familiar with trigonometric functions like sin, cos, tan etc.
And you should now some basic algebra. (How to bring x on the other side of an equation.)

With Quaternions it is possible to produces very fine fractals:

(http://www.wackerart.de/quaternionen/quaternion17.jpg)

(http://www.wackerart.de/quaternionen/quaternion6.jpg)

A note to myself.

I will need to relook at this topic in the light of Bombelli (http://i-is-no-longer-imaginary.gmxhome.de/), Euler and Gauss (http://docs.google.com/viewer?a=v&q=cache:kI5VliWMtjoJ:www.inflibnet.ac.in/ojs/index.php/AUJSAT/article/view/81/78+Euler+gauss+quadruples&hl=en&pid=bl&srcid=ADGEESi6pFnAf761qcqthU6sOonTl0mb9GB1vO-v_0vPYwa38UwwCbyH6rcUkKDv6FxdD04KnqLf1JsFoLTZK1DxLGJ-pBYv8FtA0pap8IjYJYFyJZNTTDdhxeCVM23tqYBy95NwOGLm&sig=AHIEtbS0f3GzjPOVXBsXNg2QCHcOAzJX9A) and neusis.

Although attribution and partisanship play a major role in any human endeavour, allowing me to moderate my intemperate outbursts at Gauss for his treatment of Riemann, and alas to point out the same treatment of peer bullying was meeted out to Hamilton, especially in America, it ought not to diminish the contribution of individuals to human "advancement" in utility of any idea.

This practice of peer bullying is not a new thing and is a kind of social coercive method of achieving conformity and cohesion. These are and always have been goals and methods of power elites whether military, religious, academic or,political.

In an Economic analysis these may be called market forces, if laissez faire management is allowed(capatalism) but directed or planned economy(former despotic, and communist ideology) is more usual to imperial expansion. So in the round we cannot escape these tendencies and influences, but should strive to mitigate them wherever possible or desirable.

From an evolutionary viewpoint this dominant subdominant rotation is lways going to occur and the fact of the matter is that ideas will always appear before their time and have to compete for dominance. Whichever way you look at it recording and maintaining these ideas no matter how trivial may be the way to enhance the subsequent dominance of something found to be of utility.

Thus Fontano and Cardano prepared the ground but Bombelli planted the seed which Euler an Gauss played husbandmen to. All are necessary and all are contributors (http://math.fullerton.edu/mathews/n2003/ComplexNumberOrigin.html)

For me the confusion caused by calling these enities numbers has been lifelong, and it is only through these researches that i understand the reason, the confusion in mathematicians minds and the semantic synesthesia and distinction necessary to resolve the issue, in my mind.

At the risk of repetition to death! The Logos Response originates among other things a Geometry, but more than that a Topology. I have coined a spaciometry to cover all past present and future manifestations of topologies or abstract geometries. The history of polynomials is the histoy of Equations for mainly geometric concerns, but also for proportional concerns. Where number comes into it is as an abstract naming game with gematrial and astrological significance. Measurement has and always will be the apriori interaction of the animates with notFS.

Measurement and spaciometry are co interactions so that it is inevitable that tensors should arise, and their delay if you like has been due to lack of utility, and the dominance of number and numbering,

Why would we replace spatial freedom with the restriction of a sequence?

I pose the question as currently i have no answer.

It is recognised now that we do not have to count to evaluate anything from a quantity of object (prototensors) to the area or volume of forms. So why did we start to do it? I am afraid that our old friend the accountant may ultimately be the cause!

So the equations of Cardano were geometrical symbolic forms based on an algebra derived from geometry.
My contension is that all algebras are derived from a spaciometry.

Given that this was the case the polynomials represented manipulations of the geometry. There was one exception in symbolic geometry that was taken for granted in actual geometry: rotation!

There was no rotation operator in polynomials or equations up to the time of Cardano. This is an oversight in algebra not in geometry. It hardly seemed to matter as most solutions were done by inspection of the problem and drafting bespoke proportions according to the need. These bespoke proportions by inspection formed the basis of the equations of proportions, that is proportioning. Thus proportioning is the fundamental ground of equations and the equal sign is a mute testament to the extending of the proportion sign to this use of equivalence: that is proportionally the same.

Therefore mathematicians would freely move, rotate, measure against, expand , contract and even weigh geometrical constructions to find solutions by proportion.

I think it may even have been a practice to stretch and twist shapes or forms out of fixed aspect ratios as we do in differential geometry, but apart from a chinese principle adopted in europe i have still to explore that.

Bombelli made sense of Cardano's discovery of the absolute utility of these strange surds/radicals by re invoking a common practice in the past that had fallen into obsolescence: neusis.

Simply put neusis is slipping and rotating a straight edge around a fixed point in order to match up a fixed length with two curves. We could trace the axiom of paralleism in Euclid back to his practice for example, and the significance of that is gausianRiemannian geometry.

For the present purposes the point to be drawn is that this action is precisely caught by the algebra of adjugate numbers as Bombelli called them. Therefore although i is properly a rotation around a point n*i is a translation and rotation around a point as in neusis.

The consequence of this overtime was to develop first through generalising to polynomials from equations the neusis necessary to solve these equations firmly linking number to these adjugate forms, and then to abstract these forms as did Euler and link them through Napiers work to trigonometric and logarithmic forms. Despite the names i hope it is clear that this is still just slipping and sliding and rotating a ruler around a fixed point geometrically.

Of course with Descartes came an even more stringent emphasis on algebraic Geometry and the usefulness of these "numbers" was for a time superseded until the work by wallis and Euler Cauchy and Gauss. Not one of them it seems could make out what Bombelli meant by adjugate numbers requiring an algorithm, a method of use and calculation, because apart from Jakob Steiner all were held in the thrall of cartesian coordinates.

It really did take Hamilton to break free from the cartesian stranglehold on Geometry, which of course meant that these ideas developed in different ways. Rodrigues is known to have used trigonometric forms to derive the quaternion algebra, which i hope you can see is entirely consistent and is still just slipping and sliding around a fixed point.

I can point to a gradual development of rotational operators in geometry starting with Grassman and culminating in Rodrigues which leads to the development of vectors and matrices and tensors, all firmly cartesian. Where Hamilton stands out is in his search for a symbolic algebra that dealt with rotation and slipping and sliding!

Despite the fact that Rodrigues published first Hamilton first published on couples and then spent aprrox 10 years trying to eork out the extension to 3d. During this time Rodrigues published a cartesian style solution.
Because of his position and respect Hamilton was able to promote his ideas quite far and wide, but apparently upset some mathematicians and physicists in doing so. So at the first real opportunity they ditched Quaternions in favour of Cartesian style vectors, matrices etc. Although non commutativity was a major objection to people like Maxwell, and Lewis Carrol who thought it unnatural and ungodly, it did not seem to raise its head when they replaced it with equally non commutative matrices. so as they say :go figure!

Bombelli's adjugae numbers i have called polynomial numerals and i have also distinguished them from polynomial rotations. In my eyes they are not numbers, but geometric operators. And in fact i think the proper notion should be measures, so they are adjugate measures in the class of operators which are in the class of Algorithms which i place in the class of transforms. These transforms transform space not number, they transform measures, where we measure and in some cases how we measure and frequently in what sequence we measure.

Because they are not numbers at all but symbolically expressed algorithns, i reflect back onto their place of origin, that is polynomials and specifically in quartic and cubic equations. And as i have pointed out these are themelves algorithms of how to solve the problem of finding a measurement in a geometrucal situation. I can then go further and say that though the symbolic form have wide applicability, they are necessarily a symbolic representation of geometrical transformations.

I believe i will find when i investigate Steiner's work evidence of this assertion, but suffice it to say that Albrecht Duerer collated and demonstrated the 14th to 15th century evidence of this geometrical sophistication and facility with slipping and sliding and rotating round a fixed point.

Hamilton started ith the notion of time when he began his algebra on couples.

As he continues he begins to pair moments in time, and then naturally references these couples by a straight line. In doing so he references the the geometry of a straight line. From this point on, no matter what else he is thinking he is homomorphically equivalent to a displacement in space, that is a vector as we now call them.

Vectors are geometrical entities studied we know as far back as ancient Greece. Hamilton's couples he realised could be made to do a kind of arithmetic, which naturally led him to explore if they could do an algebra. It was Bombelli who codified the rules of an Algebra in his lifes work "Algebra the major part of Arithmetic". Up until then algebra was a final chapter in most books on the subject of Arithmetic a kind of "Art of Calculation" showing how to solve specific problems of calculation mostly geometrically and by proportioning geometric forms and by iterative techniques like repeated fractioning,again a geometrical notion.

Algebra was a kind of indicator that a general method could be found for some types of problems requirig a method of calculating called an algorismo in old italian from the arabic  Muhammad ibn Mūsā al-Khwārizmī.

What Bombelli did was to synthesise the form of modern algebra from these earlier works and insights, revealing that behind most of Arithmetic or specific calculations, there is a collection of "rules"(a geometric notion) on how to proceed and what to expect. In his time the new notion in the west was negative numbers, an far eastern (http://en.wikipedia.org/wiki/Nine_Chapters_on_the_Mathematical_Art) if not entirely Indian idea of accounting and geometric representation.

Bombelli in everyday language, like Duerer, explained what algebra was about and what he knew so far, which was a lot. He went on to initiate new notation for previously inscrutable or hard to grasp operations and denotations including piu di mene and men di mene the positive and negative √-1, which Euler later denoted by i.

Because of Bombelli the field of polynomials gradually developed as really the first Algebra of the field of arithmetic. Mathematicians in the west went on tentatively to explore the field proprties of Arithmetic in polynomial form, knowing that weird results like minus 1 and √-1 had a use!

So Hamilton had by using a line as a referent for his couples of momemts in time reverted back to an essential geometrical description which underlies all of mathematics Eastern or Western, and using Bombelli's "rules" Hamilto explored an algebra different to polynomial algebra, but related by his use of + to form Bombelli's adjugate numbers, essentially and especially when he is exploring "multiplication" of couples.

Step by step Hamilton goes through each of Bombelli's rules and brings in whatever current mathematical ideas he neede to achieve them.So one finds that Hamilton, Like Euler and Napier uses the trig functions to develop his idea on the multiplication of couples. So his grand opening on time has slowly and surely become a work of geometry, that is space. Time and space were thus fused in Hamilton's mind as were rotation and translation, neusis.

Hamilton always strived to break free from a cartesian analysis of what he was about, probably because he was struggling to form a concept of time not space, so he intuitively recognised that he was supporting his argument and reasoning by geometric analogies, but was adamantly striving for something that was essence itself to him, pure time.

However it did not escape his attention that rotation and translation were in his notation easy things to manipulate, which is where Rodrigues found difficulty, because like Hamilton he applied trig to three dimensional issues specifically on rotation, but not in the form of an Algebra. Hamilton was unique in that one thing, and after him and his work Algebra was able to move out o the underskirt of Arithmetic and stand on its own feet as a utilitarian branch of mathematics.

To me the struggle that Hamilton had in developing an algebra of time is due to the fact that his notion of time is not an entity. It is a cultural construct which is in fact a homomorphism on motion. This is why Hamilton's time couples morphed into displacements, and became the basis of he modern notion of vector.

This topic since Hamilton and Gauss has sometimes been presented as quadruples and triples, based on the cartesian ordered pair, and triple. There is no cartesian quadruple, so for Gauss to be thinking of Quadruples needs some explanation. It is simple to understand if like Hamilton and Rodrigues you think of the ordered bracket as not containing numbers but measures in a specified direction, that is as axis "vectors". The problem then becomes how do you relate these vectors in a way that makes sense geometrically and preserves the field properties of Arithmetic?

This is of course not always possible as Gauss showed and Hamilton demonstrated . But what Hamilton like Bombelli showed was that the Algebra was still useful! The Dominating idea of Arithmetic, both in India and the west was commercial applicability. This was because most patrons of mathematics were merchants. Similarly geometry was dominated by "Rules" and "rulers'" wishes to construct and quantify securely.

In a very real sense this may be a reason for the dominance of sequencing in numbering and counting: secure knowledge of exactitude!.

In passing the geometrical basis of mathematics and symbolic representation drew attention to properties of space not really distinguished before. Polynomials and their graphical representation began to highlight "smoothness", and scale and continuity and discontinuity. In addition polynomials of higher than signal 1 have always  graphically demonstrated the folding of space near a point and expansion away from that point.

When polynomials of more than one parameter were explored it became possible to represent surfaces and volumes directly graphically and to visualise thes deformations. This is a recent thing but of course has its roots in the study of polynomials of more than one parameter.

Where Benoit Mandelbrot fits in this development is in the confluence of computers, geometry and Algebra, and mathematics of scale and error approximation and statistics.

The Raven is like a writing desk because they both have blue-black Quills!

Grassmann's Ausdehnungslehre is more fundamental to this topic than i initially realised. Much of what i have explored here Grassmann has explored in depth and with greater consistency.

In the study of space i have to distinguish 2 iterative and complementary processes. The first is the action of motion on regions however small the scale. The second is the post processing of the resultants of the action by my computational perceptual output processes.

This duality exists fundamentally throughout the whole of my experiential continuum. There is a reality i cannot know except by a model which i experience as knowing. Thus the action of motion on regions  in space in their exactitiude are not knowable, but an approximate rendering of surfaces and perspectives is possible by taking a pragmatic approach to the actual exact data. Thus a processing schemen may use latest data point, average datapoint, median data point, statistical average data point, moving average data point, the variance etc in premising its processing model, thus determining its output.

These methods act as filters to raw data which is already heavily filtered by the data collection process. thus i arrive at a statisitical description of the raw data , thus requiring my model to be a statistical model, and my appreciation of the experiential continuum to be  a statistical process producing a statistical output from statistical data. My knowledge of reality is therefore purely statisitical, deriving from iterative processes within me reacting to iteraive processes of motion without me acting on regions of space in a way i can only sample statistically.

Things therefore are never quite what they seem. In this sense spaciometry or geometry is my best "average" of the whole scheme of things.

the precision i can describe here with mathematical language, has to be rendered in perceptual language to give the actual experience. The fractal generator enables this to happen in the most remarkable way, and consequently gives real insight into the spaciometry of my reality. For those not familiar with vectors or tensors, the fractal generator provides a hands on , sandpit way to grasp the essentials of these notions, for what it is worth. However even a simple geometrical knowledge will produce the most exquisite results.

Those who have worked on the formulae and those who have worked on the application have produced our most fundamental representative model of space as we experience it.

motions in space at all scales with surface computations at all scales is about the nub of it. A complex vector algebra, quaternion algebra, clifford algeba and Grassmann algebra are all the different ways the app can be used to describe relations of motion sequents in space. However, we can see, hear or feel nothing without the surface processing algorithms that output surfaces, the auditory processing algorithms that output sound intensity,and the haptic processing algorithms that output force intensity.

The possibilities are truly infinite.

I have long entertained the notion of a rotation vector, but that was because i did not know the origins of the word vector coined by Hamilton, and pinched by all and sundry but particularly Gibbs, who used it to deliver a glancing blow to Quaternions! Oh the gentlemanly misconduct of Hamilton's times! :dink:
In any case, my increasing understanding leads me toward the notion of "path" or "arc". Very traditional terms i know, but many times the old is better! :D

I can define a "rotating vector". but the notion of a rotation vector does not seem to fit that tool. And so i recognise the path of the tip of such a rotating vector is what i am after. Similarly, tangential vectors may envelop a path their envelope tracing out the path.  The path around the unit circle is of particular significance: i can use it to measure rotation in the plane. Thus if U is the radian arc of a unit circle, i can define a general path c*U which takes me around the circle c radians. I rotate c radians around the unit circle path. However, if i leave the unit circle , the path P of the radian arc becomes a length in whatever units the radius is measured, c*P still takes me round the larger circle, and c now scales an "angle" ratio, but the Arc length has to be remembered to calculate circular distance traveled in units, not radians. This is made simple by the system, because P is always = r the radius of the circle in question so that the ratio is always reducible to 1:1. Thus the scaled angle ratio * radius gives us the distance around the circumference of a circle in the units the radius is measured in.

I can then define a path P for a circle as being an arc equal in length to the radius of a circle, and a general circle path length  ΩP being of magnitude Ω radians *r the radius of the circle in its units. Thus a Path for a circle is the same as a radian only in the unit circle, but of course is in a ratio to radians of the unit circle.

Now i can add a path to a vector v so that v+ΩP= w where the magnitude of |w|= |v|

Clearly there is a vector x such that v+x = e*w=W where e is some scalar, and i can now divide |W| by |w| to give e, in a straightforward  fashion.
|W/w|=e= |(v+x)/(v+ΩP|)
giving in this case
|w|*|(v+x)|= |W|*|(v+ΩP)|: we can consider vector magnitude products as a vector magitude producted with a common vector and a circular path. There is no sense in going any further, as this is not a value enumeration, but a note to include the rotation effect in producting vectors.

One other thing to note is that the scalar e scales the vector v and the path ΩP, as one would expect. The effect is to change P to e*P leaving Ω the same.

Producting is derived from combinatorics, and is the consideration of how to relate and place one form against or with another. Thus the * and the plus operations need to be defined before assuming any meaning. I have not done so here to illustrate that "operations" often cause confusion because we think addition and multiplication are number bonds! They are not number bonds they are "combinatorial bonds" and define combinations.

The insights gained from the above manipulation will need to be expressed rhetorically until a good notation is devised to express it symbolically.

I learned a lot through the difficulties i explored in this thread, especially the difference between the rotation and the axis, but i did not understand vectors or modern vector calculus when i started. That is not a bad thing, because i think maths is poorly taught, as a set of skills not as a philosophical life style. Mathematicians today are just that, mathematicians, whereas in all of the early history of maodern maths mathematics was done by people who were more rounded : Philosophers, classical scholars, Administrators and Educators with a broad curriculum view. Mathematics, i know has lost its way, chiefly by being stripped of its true nature that is natural philosophy within the context of theosophy and theurgical practice.

Hamilton, when he formed the notion of vector, and then fell upon the idea of quaternions did so in the hopes of serving his god's great purpose. It mau be remarked that he was not unique in doing this, as the same trait is found in Grassmann, Euclid , Pythagoras, Gauss and of course Newton, to whom the laws of gravity were given "as if he were god's messenger, god's angel to mankind."

Whatever the inspiration, i have studied sufficiently to know that mathematics has been stripped of its essential nature, and its purveyors are feeding us drivel and stones instead of milk and bread and yes honey too.

http://docs.google.com/viewer?a=v&q=cache:x4zsghc8SoIJ:www.lrcphysics.com/storage/documents/Hamilton%2520Rodrigues%2520and%2520Quaternion%2520Scandle.pdf+quote+james+Maxwell+on+quaternions&hl=en&pid=bl&srcid=ADGEESjlbAVlGbzNNu8exy3jO1io5Py5XQwyZXZacV0WSkmPIVeaHzVZWG34Q17QvasPYTUs1r6MpMM6ZmX3549Gjb1c-oaNPAECpNNpNXes8GmFFrBiPzgCHNpZJ0mhAW9MApQgPwcl&sig=AHIEtbRnj3jj4Ddqjc5EX2axH6KuuB3RCA.

I hope you enjoy this link, which sets out what i had hoped to do in this thread in its hitorical context, and shows the remarkable and clear insight that our fathers had, which has been denied us in our mathematical education.

I lament, a bitter lamentation, that when i asked why fractions and ratios in our school text books that no one could tell me the fundamental nature of just these 2 things in the development of the vector algebras of today, nor the absolute and generative insight that Newton derived from just these 2 in grounding his calculus of fluxions, the differential calculus. The riches of just these 2 things go on and on and into a lifestyle of just proportionality and requisite fairness as advanced by Pyhagoras.

It is therefore with the greatest admiration that Benoit Mandelbrot can be seen to have reversed the trend by concentrating on these 2 fractions, from which he derived fractal, and ratios from which he derived "almost self similar". His work naturally sits amongs a great field of vectors and other algebras and provided entrance for computers into the very stuffy heart of mathematicians, reinvigorating the beauty and dropping from their eyes the scales by whiich they declared beauty "monstrous", repetiton and iteration overwhelmingly "boring", and instant public communication "dumbing down."

Polynomial rotations may be a backwater issue in the grand scheme of things, but i am glad i attempted to do it, inspired by the great energy exhibited on the forum for ways to understand space so we could create the Mamdelbulb. The insights here have helped me to grasp the more difficult presentations in unfamiliar notation elsewhere, and also revealed where bombast has taken over from common sense!

The combinatorics here are but one of many possible combinatoric models, but no combinatoric can be of any sensible use without the form and the sequences to which it refers. The forms are simple, the sequences natural, the combinations understandable, and the many iterations do-able. They do not need to be obscured in gibberish.

The triples or triplets which i sought to explore in this thread, having a burning question as to why had it not been fully explored, but specifically would it hold the searched for holy grail of the Mandelbulb, i have at last come upon a lucid account of Hamilton's researches into the very same.

Usually and annoyingly these points are presented as if incontestable fact, and alien to human sentiment. Yet the human scientific endeavour, hope brimmed , is much more informative and affective, revealing at once the doubts and uncertainties and triumphs that necessarily precede development of ideas. And indeed Hamilton was never satisfied with the revelation so far of his topic called quaternion.

http://encyclopedia.jrank.org/PYR_RAY/QUATERNIONS.html Follow the link and enjoy.

My own ideas here are related but different and are a treatment of the subject in analogy. Although some results may attain the general treatment is far more restricted than Hamilton's or Grassmanns and illustrates the assumptions that have to be made , in the light of having a fractal generator app to test results, in order to achieve calculable and manageable actions in space.

What i learned here was the difficulty that notation presents, and the falsity of many taught assumptions and expectations, but the joy of being able to create notation that actually worked in part.

Now, because i have a better apprehension of a tensor/vector algebra not as some abstract principles, but as everyday apprehensions and interactons of and with space, i recognise this as an attempt to form a complex vector algebra in assumed continuous space.I also recognise all these expressions as arising out of notation to describe relationships within a form in space, and as such a direct and immediate descendant of Euclid's Stoikeioon. I make the bold claim that in fact Euclid both in his Stoikeioon and in his other Books had substantially worked out the laws of spatial relationships and actions, and that Appolonius to Archimedes , with Ptolemy and Eudoxus sees a refinement fit to purpose.

What we have been forced to do is to reinvent the wheel! However, that is not a bad thing. What is bad is to think higher of ourselves than we ought, because it is evidently demonstrated that our forefathers "knew a thing or two"!

If i think of a Quaternion as a space attached to an external origin, that is a reference frame for a space with a position vector for the reference frame in some external reference frame, then the quaternion is a natural candidate for relative comparisons.

Now this is not "relativity" in the sense of the relative speeds. this is relativity in terms of frames of reference, each quantity in the quaternion being a "vector " in an external frame. This gives a complete local reference frame description in an external reference frame. As an example. if i am the origin of the external reference frame, then i give a loal reference frame to every object in my reference frame. Keeping it simple: when i observe a box of tissues i can immediately give the box its edges as its local reference frame, only taking the edge vectors from one corner, and distinguishing that corner by a position vector from me to it.

Now of course the constraints on the box have to be specified, and so i immediately get a lagrangian description of the relative situation. This kind of Lagrangian is of course called a Hamiltonian.

Clearly everything is specified except the orientation of the box in my reference frame, and to do that i need at least one more position vector from me to another corner of the box. However, if i do not do that then the box is free to rotate in my refernce frame.

Suppose now i have two such quaternions , two boxes in my space, what does quaternion addition mean? What does quarernion "multiplication " mean?

The underlying e;ements of this quaternion description are vectors, not so called scalar quamtities, so addition in this case means vector "addition". and so multiplication  means what?

This nicely illustrates the point that addition and multiplication are the "wrong" notions we get taught from early childhood. We only ever are talking about synthesis and antisynthesis, combinations of forms with regard to their relative orientations , positioning and quality of fit. So i do not want to know about multiplication, i want to know about factorisation: wha components fit where to make the whole and what are there relative positions. Thus the factor forms tha come out of any "multiplication" actually are an invitation to work ou what forms they could possibly be part of.

We make life easy on ourselves generally by chosing a form with high symmetry like a cube or a square or a sphere, and then we do ot need to worry about the form, and generally we do not need to worry about the orientation, because of symmetry, and so we use a cube or a square as a standard form.

However, in this case we cannot ignore these aspects of the details of a combinatorial expansion. for they do not necesarily refer to a cube, or if they do the cube is not necessarily uniform, in the sense that a cubes faces are all pointing in the correct orientation!

Doug Sweetster gives the current notation for quaternion product which involves the cross and dot products in its expansion
 http://www.theworld.com/~sweetser/quaternions/intro/multiplying/multiplying.html

When i interpret this way of notating a quaternion i see straight away that it produces a magnitude in my space, lets say a new position vector, three vectors , lets say for the vectors of the edges of the new volume and a cross product which signifies a rotation axisfor elements inthe new volume. The size of the volume can be determined by the wedge product of the edge evectors

So in general the product of thes two boxes in my reference frame is a box in my reference frame that has a volume, edges and an axis of rotation which is a local axis for its elements, and a new position vector in my reference frame.

This position vector is derived from the 2 position vectors of the generating quaternions and for simpliity we denote that direction as e hat, so the position always is in e hat, but what if it were not?

Well i havenot got to that yet! However, keeping e hat the same makes this version of quaternion product ideal for spacetime descriptions also. However, you will note "time" is dependent in some complex way on space variables.

I did say I was pleased I tackled this topic, and now I have found some clarity in Hamilton's own work. The problem I had was the confusion between orientation and the unwary operator idea I started with. As it turns out the unwary operator idea is a way of trying to understand some simple combinatorial construction rules which few understand, despite running through them as properties of groups and rings etc.
Suffice it to say that Hamilton was not unclear about what were operators and what they were operating on, and I was and always has been a Hamiltonian vector in the quaternion group structure, and also in the couples group or field structure as Hamilton defined it. Thus the famous i² =-1 was a statement about a hidden operation which was rotation. The confusion in calling it multiplication arises from its past use and a less than careful distinction between multiplication and algebraic multipling. A distinction Hamilton took care to make.
Algebraic multiplying in couples was making effective steps , but in quaternions it is effective rotations. The effective steps in the plane involved a step a rotation through pi/2 and then another effective step in the new direction. In 3d using quaternions the effective step is a step then a rotstion through pi.

Due to De Moivre and the Cotes Euler identity any direction can be moved in, any orientation accessed by the quaternions.
Dropping back to couples, it is commutativity which makes them appear different to quaternions, but in fact whatever operation applies to quaternions also applies to couples, that is complex numbers. So any rotstion is effected by a pre and post multiplication operation, when this is done , rotstion of the couples is also through pi. In general, pre and post " multipling"  doubles the angle of rotation, so half angles should be used.

This was my best shot, and I was getting closer to discovering Hamilton's triples, work he did before finally realising he needed a fourth calculation axis.

At this point I had nearly everything in place, I was just confused by my use of the term associativity. This was based on but not the same as the group associativity axiom, the combinatorial rules for group elements.

In my blog I carefully outline the rotation actions in the quaternions that I identified here as I,j,v. I clearly need to correct some of the tables in light of this.

 http://jehovajah.wordpress.com/jehovajah/blog/quaternion-8-clarifying-the-rotation-action (http://jehovajah.wordpress.com/jehovajah/blog/quaternion-8-clarifying-the-rotation-action)

This also helps in programming the results for Quasz.

At last! Hamiltons bubbles burst in iridescent hues in my feeble brain!
http://jehovajah.wordpress.com/jehovajah/blog/2012/08/28/hamiltons-bubbles
Polynomial rotations have got it wrong but so nearly right! :embarrass:

The conception of polynomial rotations was born out of an idea of unitary operators. I have long since abandoned that idea as I studied Hamilton and then later Grassmann.
Despite my protestations I have been dragged unwillingly into the ring and group theory group of mathematicians, the so called abstract algebraists.

Well as you can imagine I am not going to drown in obscure languages and jargon. I am not going to change a thing in this thread, because it serves to show how heuristics works, and how bashing your brains out with algebra which literally means in Arabic "the twisting" referring to the Indian method of reconciling things like -1 *-1=+1, which is not now they considered it at all! But it gets the point across I hope

Well this brain twisting actually does some good if you stick with it. I can actually trace back to this thread some ideas I have in regard to fluid dynamics and vortices!

Hi,

I worked too on the same subject as Tim Golden and Kujonai, and I have some threads with Tim.
I have proposed the absolien numbers last year on that site, wich are équivalent to polysign numbers (as I interstood later), but with à matrix formalism more simple I think : vectors with only positive values, it can be seen as generalization of signs, or the end of signs !

I believe that absoliens are isomorph with quotien ring algebras, and so are very related to polynoms and multicomplexes MCn.
I am interested with your opinion. My site is:     https://sites.google.com/site/yannispicart/   (See page absolien numbers).

Hi Yannis.
Tim and Kujonai what a blast.
I have looked over your site  and tried to follow your presentation.
The detail is a bit mind blowing in full mathematical garb, but it is essentially what I expected. The Matrix notation I mentioned to Kujonai in passing, in the form of a question, but I was too bogged down to pursue his work in depth.
I did a bit on the Quaternion 8  group and identified where Hamilton had made a "mistake" iGmetryeoms of brute force making Quaternions work, but apart from a better understanding of reference frames and solving the Hamilton triples I have not done a lot since.

I am mulling over the Vortex 9 and vortex 18 groups very slowly, but I have no reference frame to represent them..

Kujonai gave a Kujonai 27 group which I had a go at, but it too lacks proper reference frame to represent it.

The prior art to all of this is found in the work of Sir Roger Cotes and Augustus De Moivre students of the great Newton. The notion of the roots of unity are absolutely the bedrock of these types of attempts, including Musean numbers etc. for this reason I abandoned the interesting poly sign work and concentrated on the roots of unity.

However my puzzle really was to grasp why was so mysterious. It turned out for me that the whole concept of numbers was misleading, and the whole foundation of mathematics was bent out of shape.

The absoliens the roots of unity the Quaternions etc all turned out to be special cases of a more general group or ring algebra. The way of working to this conclusion, in modern times was pioneered by the Grassmanns,

This does not detract from your work. The detail you have explored is the necessary research to build this model. .

The thing that interests me is of course removing the contra sign. This was always possible but it is a pyrrhic victory when the sign is replaced by a more complex symbol that does the same job.

Contra and subtraction are similar but contra is a directional status while subtraction is a dismembering process.

The homemorphism to the cyclic groups is also well studied.

The trick is to build a tool to display n dimensions. Using simplexes for 4 dimensions  is also an explored topic..

I am going to take my time looking at your work to help me with the V9 group.

The anti ontology is a topic most do not consider. Basically it means if A exist then notA also exists, in a universal set diagram that means the complement of a exists if A exists.. I have also called this conjugation.

While I think a quotient ring for Quaternions exists, that is a division algebra, I also think commutative Quaternions exist also in a division ring.. The non commutative Quaternions arise because Hamilton did not recognise the conjugate of k in his famous rules!

The thread here is a fundamental assessment of the geometric algebra that underpins the notion of rotation.
http://www.fractalforums.com/new-theories-and-research/geometric-algebra-geometric-calculus/msg61550/#msg61550

There is a lot more to come.

The restrictions i put on these Twistors in the early days were misguided, and the reason why i could not then achieve the vision i had. Since then i have solved the issue through Newtonian Triples, which is a model built from a much broader Method Developed by Hermann Grassmann.

Funny how things turn out! :D

Yes, there is life in this old topic yet!

Just updated a few posts in the past with some clarifying remarks and highlighted Norman Wildbergers treatment of Euler Rotations. Pay attention to the doubling effect.

Just found an application for some of the ideas in this thread on my thread in Magneticuniverse.com !