Polynomial rotations

[jehovajah]

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!

[jehovajah]

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.

[jehovajah]

	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

This is the mandy which again is less round and the bum cheeks are evident.


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.

[jehovajah]

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.

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

[jehovajah]

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

[jehovajah]

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.

[jehovajah]

I mentioned a possible connection between spin and action at a distance. Apparently the spin hall effect may be an illustration of this through magnetism.

[hermann]

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

[Bent-Winged Angel]

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!

[hermann]

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:

[jehovajah]

A note to myself.

I will need to relook at this topic in the light of Bombelli, Euler and Gauss 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

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.

[jehovajah]

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 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.

[jehovajah]

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.

[jehovajah]

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.

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

[jehovajah]

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.