Title: The simplicity of the unity of Dynamic Directed magnitudes
Post by: jehovajah on February 18, 2011, 09:15:41 AM
I would not want anyone to go away with the notion that the complex numbers are inherently complex as they are not.
Firstly they are not numbers but trigonometric ratios written in a polynomial form.
Secondly they are not complex but they are applied to geometrical situations that look simple but turn out to be complex!
In fact what we have in the past termed complex is better described by thr notion "fractal" as commonly used in these times.
I know some would like a strict definition of fractal, but the genie is out of the bag and normal language development of meaning now applies.That more often than not is a good thing as it indicates spreading of the idea to the general consciousness, and providing confusion is headed off at the pass can make communication simpler.
Most of us are familiar with negative numbers but maybe not understanding that their proper name is directed numbers, as they have a geometrical application beyond double entry book keeping. As a consequence we may be familiar with the rotation of the number line to describe negative values. We however are misdirected precisely at this point in our understanding.
We may then come across the trigonometric ratios and the importance of direction and quadrant without ever making the connection. We may then be introduced to the weird and wonderful world of complex arithmetic through the imaginary "route", exploring briefly and confusingly the roots of x^2+1=0. Again no understanding or explanation that makes sense is usually given, and the relation to trigonometry is totally void.
Finally vectors may be thrust upon you with the trigonometric conversions and no understanding of where you are, where you have just been, and where this is all leading you toward!
I hope that fairly explains an initial interaction with the so called complex numbers.
In my experience one never quite gets over the √-1!
So lets get over it together.
First of all directed numbers are a special case as they only address opposite directions! What if a directed number is not in an opposite direction, how do we notate that?
If you keep this question in mind then everything else becomes real simple.
We chose, confusingly i think, + and - for the opposite directions, but what did we choose for other directions?
Bombelli attempted to set up a notation for the other directions but was cunfused himself. He chose +√- (piu di meno) and -√- (meno di meno).
This was quite a complex sign which for a long while had no commonly accepted geometrical meaning, but which for Bombelli and Descartes always meant at right angles to!
Therefore as geometers they always looked and moved at right angles to the +- direction in their geometry.
When it became popular to use their system by fixing the ordinate and coordinate in the page then the right angles direction became known as y. Thus historically y came before i but they both represent the same thing, the orthogonal direction to the +- direction you define on the page.
To say this became obscured is to make an understatement.
Most of us have heard the rule of thumb: the square root of a number is that number you have to square to get the number you start with
Some are familiar with its origin in the geometric mean a*b=x^2 for a pair of crossed lines divided in the ratios a:b and x:x.
These in turn are magnitudes of similar triangles where a:x=x:b for corresponding sides. Thus the geometrical construction of finding the geometrical mean underlies the expression "to square root"
When directed magnitudes are introduced into the geometry what we find is that square rooting involves a rotation through π/2 from the positive direction, in either an anticlockwise or clockwise rotation.
Thus, very straightforwardly and without any artifice the square root of a magnitude in the negative direction involves rotating clockwise or anticlockwise by π/2
We simply have to set up some convention for these rotations and the simplicity is made secure.
So for example √-1 may be defined as a clockwise rotation by π/2 and -√-1 as an anticlockwise rotation or vice versa. However The Bombelli operator does the job perfectly and should be retained as the standard.
So the rule of thumb then becomes: the square root of a directed magnitude is the rotation of the direction by π/2 and the geometrical mean of the magnitude, under the rules of operation described by Bombelli.
We need only now to realise that √-1 is a trigonometric ratio called)
if anticlockwise rotation is defined or )
if clockwise.
The definition of √-1 is implicit in the trig identity:
geometric mean of)
=)
*)
From which we derive as a directed magnituce with direction radial 
in the unit circle and magnitude unity from the unit circle.
It is to be called the fourth root of unity, thereby recasting all numeral schemes as scalars of roots of unity. Our understanding of measure is to be derived from this basis, making -1 simply a measurement in the direction - of 1 unit magnitude, and allowing such measurements in an infinity of directions denoted by radial directions in the unit sphere, with a special subset in the unit circle.
In the unit sphere we define the eighth root of unity for the sphere as
In 3 dimensional spherical)
geometric mean of(geometric mean of(,0,0) *()
,0,
))
Where the magnitudes of the vectors/ tensors are used.
From this we see that a matrix or tensor format is desirable over a linear form, but a linear form can be written with the additional conventions that the axis are represented by orthogonal vectors, and the aggregation is an algorithm for defining a position in 3 space.
Firstly they are not numbers but trigonometric ratios written in a polynomial form.
Secondly they are not complex but they are applied to geometrical situations that look simple but turn out to be complex!
In fact what we have in the past termed complex is better described by thr notion "fractal" as commonly used in these times.
I know some would like a strict definition of fractal, but the genie is out of the bag and normal language development of meaning now applies.That more often than not is a good thing as it indicates spreading of the idea to the general consciousness, and providing confusion is headed off at the pass can make communication simpler.
Most of us are familiar with negative numbers but maybe not understanding that their proper name is directed numbers, as they have a geometrical application beyond double entry book keeping. As a consequence we may be familiar with the rotation of the number line to describe negative values. We however are misdirected precisely at this point in our understanding.
We may then come across the trigonometric ratios and the importance of direction and quadrant without ever making the connection. We may then be introduced to the weird and wonderful world of complex arithmetic through the imaginary "route", exploring briefly and confusingly the roots of x^2+1=0. Again no understanding or explanation that makes sense is usually given, and the relation to trigonometry is totally void.
Finally vectors may be thrust upon you with the trigonometric conversions and no understanding of where you are, where you have just been, and where this is all leading you toward!
I hope that fairly explains an initial interaction with the so called complex numbers.
In my experience one never quite gets over the √-1!
So lets get over it together.
First of all directed numbers are a special case as they only address opposite directions! What if a directed number is not in an opposite direction, how do we notate that?
If you keep this question in mind then everything else becomes real simple.
We chose, confusingly i think, + and - for the opposite directions, but what did we choose for other directions?
Bombelli attempted to set up a notation for the other directions but was cunfused himself. He chose +√- (piu di meno) and -√- (meno di meno).
This was quite a complex sign which for a long while had no commonly accepted geometrical meaning, but which for Bombelli and Descartes always meant at right angles to!
Therefore as geometers they always looked and moved at right angles to the +- direction in their geometry.
When it became popular to use their system by fixing the ordinate and coordinate in the page then the right angles direction became known as y. Thus historically y came before i but they both represent the same thing, the orthogonal direction to the +- direction you define on the page.
To say this became obscured is to make an understatement.
Most of us have heard the rule of thumb: the square root of a number is that number you have to square to get the number you start with
Some are familiar with its origin in the geometric mean a*b=x^2 for a pair of crossed lines divided in the ratios a:b and x:x.
These in turn are magnitudes of similar triangles where a:x=x:b for corresponding sides. Thus the geometrical construction of finding the geometrical mean underlies the expression "to square root"
When directed magnitudes are introduced into the geometry what we find is that square rooting involves a rotation through π/2 from the positive direction, in either an anticlockwise or clockwise rotation.
Thus, very straightforwardly and without any artifice the square root of a magnitude in the negative direction involves rotating clockwise or anticlockwise by π/2
We simply have to set up some convention for these rotations and the simplicity is made secure.
So for example √-1 may be defined as a clockwise rotation by π/2 and -√-1 as an anticlockwise rotation or vice versa. However The Bombelli operator does the job perfectly and should be retained as the standard.
So the rule of thumb then becomes: the square root of a directed magnitude is the rotation of the direction by π/2 and the geometrical mean of the magnitude, under the rules of operation described by Bombelli.
We need only now to realise that √-1 is a trigonometric ratio called
The definition of √-1 is implicit in the trig identity:
geometric mean of
From which we derive
It is to be called the fourth root of unity, thereby recasting all numeral schemes as scalars of roots of unity. Our understanding of measure is to be derived from this basis, making -1 simply a measurement in the direction - of 1 unit magnitude, and allowing such measurements in an infinity of directions denoted by radial directions in the unit sphere, with a special subset in the unit circle.
In the unit sphere we define the eighth root of unity for the sphere as
In 3 dimensional spherical
geometric mean of(geometric mean of(
Where the magnitudes of the vectors/ tensors are used.
From this we see that a matrix or tensor format is desirable over a linear form, but a linear form can be written with the additional conventions that the axis are represented by orthogonal vectors, and the aggregation is an algorithm for defining a position in 3 space.
Title: Re: The simplicity of the unity of Dynamic Directed magnitudes
Post by: jehovajah on February 28, 2011, 01:35:14 PM
The versine 
and the coversine 
reveal the general geometric mean definition of the square root . as it relates to the unit circle.
Thus the diameter of the unit circle is coversine +versine
and the area of the arithmoi or rectilinear forms that can be represented is the product coversie*versine
The geometric mean of the coversine*versine is the sine.
Thus we have a definition of the general square root identity for thr unit circle
)
*  )
=
This is of course relating the geometric mean to the right angled triangle in the unit circle, where pythagoras gives
= 
.
The analytical use of the geometric mean reveals that if directed magnitudes are applied rather than absolute magnitudes, then complex behaviour results dependent on the sign of the magnitudes.
Since the sign is just shorthand for opposing directions in general, the interplay of the opposing directions in directed magnitudes accounts for the complex behaviour found in the study of complex "numbers".
This reveals that recursive definitions or applications of just the directions which oppose each other have a powerful significance.
Each relation and /or set of relations chosen, then can form a seperate algebraic field, group or ring on the trig ratios which may or may not return useful applications for the attributions we initially gave them.
The relations between these fields etc are admittedly complex, and any arithmetic devised from them will have "complex" geometric associated forms. This does not mean they will look necessarily out of the ordinary, but rather the relation between parts of the form will exhibit complex dynamic relations. These may or may not be useful in describing, cataloguing or displaying physical phenomena in the experiential world.
All in all, i mean to say that the simplicity of the greek dynamic conception of geometry and their development of trigonometry has founded the study of reality on about as sound and useful a foundation as we can humanly devise!
The proper area of mathematical research turns out to be in the area of these trigonometric relations and identities, with of course the biggest trig identity innovation being Napier's Logarithms.
I have found and am finding a euphony of applicability to all measurement in utilising these simple but profound relations ratios and conventions derived from spherical and plane trigonometry and geometry. In addition the general naturalness of spherical and polar measurement systems over cartesian.
The unit sphere as applied by Eudoxian proportional theory found in Euclids elements is indeed the "bees knees", and allows for the development of relative reference frames and Napierian logarithms as the most applicable measure of physical reality and fractal design and demographic description,
For example it immediately becomes apparent that Newtons Laws of gravity are a second "order" approximation of an infinite logarithmic series description of gravitational behaviour!
Another of my favourites is that mass is the logarithmic count of charge particle quantity
)
where q is the charge vector of a single region with magnitude of charge, orientation of the region and 2 spin axes, one in the direction of the orientation and the other orthogonal to the direction of the orientation. I prefer to call them dynamic magnitudes or dynamic tensors.N={0,1,2,...,Avagadros number/constant}.A is advogadro's constant/number.
A vector sum is implied to denote the charge summations which may or may not equal the zero charge vector.
These are just a few random thoughts that arise from the application of the simplicity of the trigonometric identities and relations enhanced by Napierian Logarithmic trig identities and polar coordinate reference frames.
Thus the diameter of the unit circle is coversine +versine
and the area of the arithmoi or rectilinear forms that can be represented is the product coversie*versine
The geometric mean of the coversine*versine is the sine.
Thus we have a definition of the general square root identity for thr unit circle
This is of course relating the geometric mean to the right angled triangle in the unit circle, where pythagoras gives
The analytical use of the geometric mean reveals that if directed magnitudes are applied rather than absolute magnitudes, then complex behaviour results dependent on the sign of the magnitudes.
Since the sign is just shorthand for opposing directions in general, the interplay of the opposing directions in directed magnitudes accounts for the complex behaviour found in the study of complex "numbers".
This reveals that recursive definitions or applications of just the directions which oppose each other have a powerful significance.
Each relation and /or set of relations chosen, then can form a seperate algebraic field, group or ring on the trig ratios which may or may not return useful applications for the attributions we initially gave them.
The relations between these fields etc are admittedly complex, and any arithmetic devised from them will have "complex" geometric associated forms. This does not mean they will look necessarily out of the ordinary, but rather the relation between parts of the form will exhibit complex dynamic relations. These may or may not be useful in describing, cataloguing or displaying physical phenomena in the experiential world.
All in all, i mean to say that the simplicity of the greek dynamic conception of geometry and their development of trigonometry has founded the study of reality on about as sound and useful a foundation as we can humanly devise!
The proper area of mathematical research turns out to be in the area of these trigonometric relations and identities, with of course the biggest trig identity innovation being Napier's Logarithms.
I have found and am finding a euphony of applicability to all measurement in utilising these simple but profound relations ratios and conventions derived from spherical and plane trigonometry and geometry. In addition the general naturalness of spherical and polar measurement systems over cartesian.
The unit sphere as applied by Eudoxian proportional theory found in Euclids elements is indeed the "bees knees", and allows for the development of relative reference frames and Napierian logarithms as the most applicable measure of physical reality and fractal design and demographic description,
For example it immediately becomes apparent that Newtons Laws of gravity are a second "order" approximation of an infinite logarithmic series description of gravitational behaviour!
Another of my favourites is that mass is the logarithmic count of charge particle quantity
where q is the charge vector of a single region with magnitude of charge, orientation of the region and 2 spin axes, one in the direction of the orientation and the other orthogonal to the direction of the orientation. I prefer to call them dynamic magnitudes or dynamic tensors.N={0,1,2,...,Avagadros number/constant}.A is advogadro's constant/number.
A vector sum is implied to denote the charge summations which may or may not equal the zero charge vector.
These are just a few random thoughts that arise from the application of the simplicity of the trigonometric identities and relations enhanced by Napierian Logarithmic trig identities and polar coordinate reference frames.
Title: Re: The simplicity of the unity of Dynamic Directed magnitudes
Post by: jehovajah on April 03, 2011, 09:04:33 PM
Quote
Since the sign is just shorthand for opposing directions in general, the interplay of the opposing directions in directed magnitudes accounts for the complex behaviour found in the study of complex "numbers".
This reveals that recursive definitions or applications of just the directions which oppose each other have a powerful significance.
Each relation and /or set of relations chosen, then can form a seperate algebraic field, group or ring on the trig ratios which may or may not return useful applications for the attributions we initially gave them.
This reveals that recursive definitions or applications of just the directions which oppose each other have a powerful significance.
Each relation and /or set of relations chosen, then can form a seperate algebraic field, group or ring on the trig ratios which may or may not return useful applications for the attributions we initially gave them.
In my blog (http://my.opera.com/jehovajah/blog/2011/07/05/ferf-merf-sg-lg-elg-and-eg-and-smurf) i elaborate on this vague set of statements. The importance of mirror reflection is what i am discussing here without realising it clearly.
It is a thought that Bombelli came up with his operator rules while looking in a shaving mirror...I wonder? This observation applies equally to Brahmagupta,although a more meditative repose seems more appropriate.
The mirror reflection is why i elsewhere describe "complex" arithmetic as a fix, and why Lewis Carrol sought to use the paradigm of a looking glass to parody Hamiltonian "mathesis" of the imaginaries.
The mirror is for me a forgotten geometrical tool, but its significance is so much more than that!
Look at this:
general linear trigonometric form for a planar point/vector/polynomial rotation/directed magnitude
cos0 is the directed magnitude in the orientation 0,
Thus the trig ratios become or are used as directed magnitudes. This in and of itsellf is worthy of the designation "complex"
Now here comes the fix, the switch.
The versine coversine identity is written in magnitudes. We need to to rewrite it in directed magnitudes.
The versine becomes
This change results in the covert "mirror reflection" of the versine into the coversine.
Now at
And similarly
at
Thus through this covert mirror reflection we can apply a geometric mean(GM) identity without dropping down into the direct GM!
Thus we now get
Which gives a result in the general directed magnitude form
I wanted to show you this as it reveals the source of rotation (GM) , mirror reflection(the directed magnitudes) and the fundamental basis in trig ratios of the "complex numbers".
I do not suppose that we would have easily located these sets of identities and relations if we had not had Bombelli's operator and insight. As it is, even with his insight and the minds of some great mathematicians we still managed to not quite put it all together until now. 600 years of fumbling in the dark!
Of course this is only one of an infinite set of identities and relations amongst the trig ratios. This makes the trig ratios a rich vein to mine.
But it does not end there. The trig ratios are formed on a closed boundary. We can extend the trig ratios to an open boundary, specifically a spiral. The spiral i choose is Theodorus spiral. It probably adequately repesents a Hilbert space and links in to quaternions.
`i wonder if hyperbolic functions are doing this already?
Before i leave complex trig ratios in directed magnitudes there is an extension into 3d that follows the geometric mean to find the cube root.
Title: Re: The simplicity of the unity of Dynamic Directed magnitudes
Post by: jehovajah on July 06, 2011, 08:49:05 AM
I finally went on to the realisation that it is not mirror reflection, reflection in a plane that is fundamental, but rather reflection in a centre of rotation. The difference is subtle but profound.
Title: Re: The simplicity of the unity of Dynamic Directed magnitudes
Post by: jehovajah on May 13, 2012, 10:18:34 AM
My latest thoughts on this process.
http://my.opera.com/jehovajah/blog/2012/05/04/the-shunyasutras-and-1
Jehovajah the shunyasutras and at Wordpress.com.
Google it.
http://my.opera.com/jehovajah/blog/2012/05/04/the-shunyasutras-and-1
Jehovajah the shunyasutras and at Wordpress.com.
Google it.
Title: Re: The simplicity of the unity of Dynamic Directed magnitudes
Post by: jehovajah on August 24, 2012, 12:59:01 PM
As I am currently looking at Quaternions in some detail it is just a note to say that the master of the imaginsries, Sir William Rowan Hamilton, says the equivalent of what I have stated here.
He defined quaternions as the quotient of directed lines in 3d space. He later called these directed lines vectors. Thus quaternions are the quotient of 3d vectors.
Now a quotient is the correct term used for the resul from performing a fraction operation, and a fraction system is a ratio specifically defined as a number system. Thus we are looking at the ratios of vectors when considering quaternions. However this is a bit confusing until you realise Hamilton established the foundation of group theory, and what he is saying is that the quaternions form a group structure with 2 operations commensurate with addition and multiplication, and they have a division algebra. Thus the quaternions were the first group algebra with a ring structure which also supported division. They lack one property which is commutativity, but that in general is a special property of. Indirect reference.
Of course vectors derive from Hamilton himself, and they are a trigonometric combination structure. The gauss combination of vectors relate directly to this trigonometric form. The use of numbers and the Cartesian framework tends to obscure all this detail about the fundamental nature of vectors.
Now since vectors are lines, they cannot be operators. This clears up the confusion about I which is a vector not an operator in this group structure. The operator is and always has been the rotation motion which has never had a proper symbol to identify it. We have only ever identified the resultant of it. In polynomial rotations I dealt specifically with that issue without ever fully understanding it!
He defined quaternions as the quotient of directed lines in 3d space. He later called these directed lines vectors. Thus quaternions are the quotient of 3d vectors.
Now a quotient is the correct term used for the resul from performing a fraction operation, and a fraction system is a ratio specifically defined as a number system. Thus we are looking at the ratios of vectors when considering quaternions. However this is a bit confusing until you realise Hamilton established the foundation of group theory, and what he is saying is that the quaternions form a group structure with 2 operations commensurate with addition and multiplication, and they have a division algebra. Thus the quaternions were the first group algebra with a ring structure which also supported division. They lack one property which is commutativity, but that in general is a special property of. Indirect reference.
Of course vectors derive from Hamilton himself, and they are a trigonometric combination structure. The gauss combination of vectors relate directly to this trigonometric form. The use of numbers and the Cartesian framework tends to obscure all this detail about the fundamental nature of vectors.
Now since vectors are lines, they cannot be operators. This clears up the confusion about I which is a vector not an operator in this group structure. The operator is and always has been the rotation motion which has never had a proper symbol to identify it. We have only ever identified the resultant of it. In polynomial rotations I dealt specifically with that issue without ever fully understanding it!
Title: Re: The simplicity of the unity of Dynamic Directed magnitudes
Post by: jehovajah on July 16, 2013, 08:16:08 AM
The Stoikeioon book 5, in which Eudoxus introduces the Logos analogos concept is the root cause of this difficulty experienced by Cardano and Tartaglia. They had inherited analogos duality without realising it was different to isos duality. Bombelli, by sticking closely to the Stoikeioon, or rather the works of Proculus and Pappus , at least sensed the difference and created appropriate notation.
Brahmagupta was fully aware of the difference and used the logos analogos methods to develop his concepts of Shunya. Within this concept, the notion of yoked magnitudes is developed to explain fortune and misfortune, both from Shunya. Thus to conclude that Shunya does not mean everything is one of the grossest errors of translation ever made! Traduce trattore !
Simply, in isos duality quadrature naturally was based on neusis, that is on gonu or rotation around the circle. In book one Euclid demonstrated a construction that showed how to transform any rectilineal form into a square that was its isos! This naturally involved Pythagoras theorem, which naturally is in regard to an Ortho gonu, a rotation of a right angle!
However, when coded in the Logos Analogos duality, all neusis has to be done prior to the encoding. Thus it is hidden within the code. It only shows up due to the rule that different kinds cannot be compared. Thus a magnitude in a different direction or orientation is a different kind of magnitude to one in the standard direction.
This fact, because it is a subjective distinction , was appreciated by the Pythagorean school where Eudoxus learned of it and developed his full theory of logos analogos duality. The myth that
created a crisis among the Pythagorean leadership tells us more about the geopolitical situation in Italy than it does about the truth! Pythagoras faced down an attempt to wrest control of his commune by some politically motivated " devotees" who wanted to destroy his growing influence in the area where he gave Kairos!
In the Logos Analogos system, because of its rigour, rules or formal procedural statements are fundamentally necessary. It is by following these rules that agreement with the more natural, veinal, fibrous duality is obtained. The logos analogos duality is a subjective internal judgement deliberately linked to the mosaic patterning on the floor of the pythagorean mousaeion . Consequently it represented a generalisation of the Isos duality, which enabled Pythagoras and his school to link projective geometries together.
The 3 main projective geometries are the parallel projection, the point projection and the circular projection. These were all projections onto a line. Or in 3 dimensions( Mekos, plates, bathos) onto an epiphaneia, especially an epipedos( mosaic) os speripedos surface.
The introduction of negative numbers was some invention that although traceable back to Brahmagupta has a lot more to do with the misconceptions of later Mathematicians. It was heavily resisted and resented when first misconceived. The fact that Cardano and Tartaglia resorted to them was considered highly inauspicious!
The confusion they created was deliberate. It was the custom to claim intellectual primacy by sending coded messages to publishers, societies etc, and it was the custom to keep secret what one had discovered. Both Hooke and Newton are examples of this tradition.
Descartes thus never fully understood Bombelli's prior Art in his "ars del Algebra", declaring the complex roots as "imaginary" in a perjorative sense. He did tend to big up his own ideas in La Geometrie which is an algebraic text . This annoyed Newton, who had no problem with the
, because he understood the logos analogos duality, and utilised it throughout his Principia.
It is therefore less remarkable tht his student De Moivre , along with Cotes fully explored and explained the Cotes De Moivre theorem on the so called roots of Unity. These notions were heavily linked to the ongoing work on the tables of the Sines and the newer Napierian and Briggs logarithmic tables. All these works were the centuries long computation of the logos analogos ratios which underpin our notions of numbers and mathematics today, and our empirical metrics and data analysis and synthesis.
Brahmagupta was fully aware of the difference and used the logos analogos methods to develop his concepts of Shunya. Within this concept, the notion of yoked magnitudes is developed to explain fortune and misfortune, both from Shunya. Thus to conclude that Shunya does not mean everything is one of the grossest errors of translation ever made! Traduce trattore !
Simply, in isos duality quadrature naturally was based on neusis, that is on gonu or rotation around the circle. In book one Euclid demonstrated a construction that showed how to transform any rectilineal form into a square that was its isos! This naturally involved Pythagoras theorem, which naturally is in regard to an Ortho gonu, a rotation of a right angle!
However, when coded in the Logos Analogos duality, all neusis has to be done prior to the encoding. Thus it is hidden within the code. It only shows up due to the rule that different kinds cannot be compared. Thus a magnitude in a different direction or orientation is a different kind of magnitude to one in the standard direction.
This fact, because it is a subjective distinction , was appreciated by the Pythagorean school where Eudoxus learned of it and developed his full theory of logos analogos duality. The myth that
In the Logos Analogos system, because of its rigour, rules or formal procedural statements are fundamentally necessary. It is by following these rules that agreement with the more natural, veinal, fibrous duality is obtained. The logos analogos duality is a subjective internal judgement deliberately linked to the mosaic patterning on the floor of the pythagorean mousaeion . Consequently it represented a generalisation of the Isos duality, which enabled Pythagoras and his school to link projective geometries together.
The 3 main projective geometries are the parallel projection, the point projection and the circular projection. These were all projections onto a line. Or in 3 dimensions( Mekos, plates, bathos) onto an epiphaneia, especially an epipedos( mosaic) os speripedos surface.
The introduction of negative numbers was some invention that although traceable back to Brahmagupta has a lot more to do with the misconceptions of later Mathematicians. It was heavily resisted and resented when first misconceived. The fact that Cardano and Tartaglia resorted to them was considered highly inauspicious!
The confusion they created was deliberate. It was the custom to claim intellectual primacy by sending coded messages to publishers, societies etc, and it was the custom to keep secret what one had discovered. Both Hooke and Newton are examples of this tradition.
Descartes thus never fully understood Bombelli's prior Art in his "ars del Algebra", declaring the complex roots as "imaginary" in a perjorative sense. He did tend to big up his own ideas in La Geometrie which is an algebraic text . This annoyed Newton, who had no problem with the
It is therefore less remarkable tht his student De Moivre , along with Cotes fully explored and explained the Cotes De Moivre theorem on the so called roots of Unity. These notions were heavily linked to the ongoing work on the tables of the Sines and the newer Napierian and Briggs logarithmic tables. All these works were the centuries long computation of the logos analogos ratios which underpin our notions of numbers and mathematics today, and our empirical metrics and data analysis and synthesis.
Title: Re: The simplicity of the unity of Dynamic Directed magnitudes
Post by: jehovajah on August 06, 2013, 09:53:12 PM
I discuss how the Circle theorems in Stoikeioon 3 miror Apollonius' work on circles, and how it provides the framework for the whole of "modern mathematics", including the confusion about complex magnitudes. The problem arose due to a poor understanding of Eudoxus' theory of proportions.(Logos Analogos)
http://my.opera.com/jehovajah/blog/2013/08/05/factorisation-and-the-sphere
http://www.youtube.com/watch?v=qNMFNk6o_xY
http://jehovajah.wordpress.com/2013/08/02/eudoxus-on-logos/
http://www.youtube.com/watch?v=YAdEfQsIGt8
While Norman gives a modern overview, it is a modern interpretation of the greek in the Stoikeioon, so unfortunately it has to be ignored. I show it to highlight the confusion created by the misconception we call number.
http://my.opera.com/jehovajah/blog/2013/08/05/factorisation-and-the-sphere
http://www.youtube.com/watch?v=qNMFNk6o_xY
http://jehovajah.wordpress.com/2013/08/02/eudoxus-on-logos/
http://www.youtube.com/watch?v=YAdEfQsIGt8
While Norman gives a modern overview, it is a modern interpretation of the greek in the Stoikeioon, so unfortunately it has to be ignored. I show it to highlight the confusion created by the misconception we call number.
Title: Re: The simplicity of the unity of Dynamic Directed magnitudes
Post by: jehovajah on August 08, 2013, 04:20:37 AM
The concept of number arose in the Renaissance, as individuals rediscovered and translated ancient texts, principally Indo Greek texts as preserved in Arabic, Aramaic and Persian.
The introduction of negative magnitudes was an innovation That philosophers have never recovered from. They were resisted for the better part of their history, but were finally bludgeoned into our consciousness by determined Pedagogues. The consequence of numbers and then negative numbers leads inexorably to imaginary numbers.
While Descartes is often opinionated, his etiphet "imaginary" was apt and not at all derogatory initially. He did however deride those who attempted to make such magnitudes representative of "real" magnitudes in heaven or on earth! However, his argument was always on shaky ground , since he accepted the notion of numbers per se.
We do not need numbers. The concepts, algorithms, tricks and shortcuts they provide are based on sensory and kinaesthetic experiences which are all utilised in many other ways, and with many other meanings.
the ingenuity and inventiveness of the human mind is astonishing, but due to these ideas we overlook the similar and profound genius in those other animates we share this planet with.
At the level of cells we find an even greater similarity, and truly at the level of so called microbes, every so called uniquely human or animate or plant trait is evident in abundance. We marvel at the computational power of our processors while ignoring the astonishing computational power of the organisms we call enzymes, which in the end are proteomes.
Are crude molecular theories simplify and conceal an incredible complexity. while seeming to give us god like powers of transmutation. None of this is due to us or our ingenuity. We tap into an incredibly dynamic universe which exists without us, creates without us and destroys without us.
We are here for the ride!
The introduction of negative magnitudes was an innovation That philosophers have never recovered from. They were resisted for the better part of their history, but were finally bludgeoned into our consciousness by determined Pedagogues. The consequence of numbers and then negative numbers leads inexorably to imaginary numbers.
While Descartes is often opinionated, his etiphet "imaginary" was apt and not at all derogatory initially. He did however deride those who attempted to make such magnitudes representative of "real" magnitudes in heaven or on earth! However, his argument was always on shaky ground , since he accepted the notion of numbers per se.
We do not need numbers. The concepts, algorithms, tricks and shortcuts they provide are based on sensory and kinaesthetic experiences which are all utilised in many other ways, and with many other meanings.
the ingenuity and inventiveness of the human mind is astonishing, but due to these ideas we overlook the similar and profound genius in those other animates we share this planet with.
At the level of cells we find an even greater similarity, and truly at the level of so called microbes, every so called uniquely human or animate or plant trait is evident in abundance. We marvel at the computational power of our processors while ignoring the astonishing computational power of the organisms we call enzymes, which in the end are proteomes.
Are crude molecular theories simplify and conceal an incredible complexity. while seeming to give us god like powers of transmutation. None of this is due to us or our ingenuity. We tap into an incredibly dynamic universe which exists without us, creates without us and destroys without us.
We are here for the ride!
Title: Re: The simplicity of the unity of Dynamic Directed magnitudes
Post by: jehovajah on November 21, 2013, 11:44:15 AM
Sometime in the past 5 years, I do not remember when, but I distinctly remember where, I realised that mathematics had never defined the unit quadrant in the plane in the Wallis -Descartes-DeFermat coordinate system.. That is to say that Bombelli had defined it in terms of his carpenters set square , and tried to communicate it through his pui de meno poem. But others had ignored it. And Wallis got close to defining it when he said that Square root –1 must be in the plane of the coordinate system, based on his work on the circle and general Conics.
http://youtu.be/EnxV3_1oaOU
The problem was that geometers were happy to define the coordinates of a square in positive terms, but. Not in mixed terms. Thus a rotated square falls into 4 quadrants. By the sign rules the positive square exists, the negative square remained undefined, or considered a reflection, not a rotation..
The difference this makes is in the definition of the square root. Geometers were happy for it to be positive, less happy for it to be negative, but they conceded, but they would not have it mixed between the 2! Thus they would not define the other 2 roots of unit square. But by calling it i for for infinite Euler slipped it in as an imaginary quantity or magnitude with strange behaviours.
This was a strange algebraic number notion that was completely bemusing because no one would geometrically define the reflective quadrant as a rotation involving mixed axes.
Once we do allow mixed axes in the definition of the Square root we resolve the problem and return to a methodological consistency.
The method Norman shows can be used to demonstrate that the square root of -1 oscillates between 1 and –1, which is made sense of by defining the reflective square as the negative unit square which has sides which are measured positively in one axis and negatively in the other axis.
Switching to vectors / Strecke enable the rotational effect of the product to be highlighted over the reflection.
It was Cotes who decided that as a magnitude square root –1 was an arc , and this lead him to his famous equation decades before Euler. It was Euler who concluded that in the measurement of Arcs i as he defined it was the magnitude of a quarter turn or a quarter arc of a circle. The valuation 1 and –1 came at the intersection of the circle with the coordinate axes. His famous equation was the first vector treatment of this rotation. Cotes formula was the logarithm and also contained the vector cosø + isinø, but it was not until Argand, Cauchy and Gauss that this format came to be established. However it was Grassmann that completely reworked the theoretical base of all arithmetical measurements in general, necessary to set out the method soundly.
http://youtu.be/EnxV3_1oaOU
The problem was that geometers were happy to define the coordinates of a square in positive terms, but. Not in mixed terms. Thus a rotated square falls into 4 quadrants. By the sign rules the positive square exists, the negative square remained undefined, or considered a reflection, not a rotation..
The difference this makes is in the definition of the square root. Geometers were happy for it to be positive, less happy for it to be negative, but they conceded, but they would not have it mixed between the 2! Thus they would not define the other 2 roots of unit square. But by calling it i for for infinite Euler slipped it in as an imaginary quantity or magnitude with strange behaviours.
This was a strange algebraic number notion that was completely bemusing because no one would geometrically define the reflective quadrant as a rotation involving mixed axes.
Once we do allow mixed axes in the definition of the Square root we resolve the problem and return to a methodological consistency.
The method Norman shows can be used to demonstrate that the square root of -1 oscillates between 1 and –1, which is made sense of by defining the reflective square as the negative unit square which has sides which are measured positively in one axis and negatively in the other axis.
Switching to vectors / Strecke enable the rotational effect of the product to be highlighted over the reflection.
It was Cotes who decided that as a magnitude square root –1 was an arc , and this lead him to his famous equation decades before Euler. It was Euler who concluded that in the measurement of Arcs i as he defined it was the magnitude of a quarter turn or a quarter arc of a circle. The valuation 1 and –1 came at the intersection of the circle with the coordinate axes. His famous equation was the first vector treatment of this rotation. Cotes formula was the logarithm and also contained the vector cosø + isinø, but it was not until Argand, Cauchy and Gauss that this format came to be established. However it was Grassmann that completely reworked the theoretical base of all arithmetical measurements in general, necessary to set out the method soundly.
Title: Re: The simplicity of the unity of Dynamic Directed magnitudes
Post by: jehovajah on July 04, 2014, 07:00:05 AM
It occurred to me yesterday that while complex numbers is a totally misleading name for these processes, complex ordinals is quite apt!
Most of us know the ordinal numbers only in their lesser role, but in fact ordinals underpin every fundamental aspect of thought. We cannot count without ordinals and we certainly cannot measure without them.
Ordinals are thus suitable for encapsulating the notion of Arithmos, which is a standardised ordinal net . A complex ordinal is thus equivalent to an Arithmos, a net of standard square forms used to order , meaur And rotate in pace!
Most of us know the ordinal numbers only in their lesser role, but in fact ordinals underpin every fundamental aspect of thought. We cannot count without ordinals and we certainly cannot measure without them.
Ordinals are thus suitable for encapsulating the notion of Arithmos, which is a standardised ordinal net . A complex ordinal is thus equivalent to an Arithmos, a net of standard square forms used to order , meaur And rotate in pace!
Title: Re: The simplicity of the unity of Dynamic Directed magnitudes
Post by: jehovajah on April 20, 2016, 08:06:48 AM
It is a while since I posted here and some links have broken and google has abandoned some platforms!
Wessel gave Gauss the Confidence to publish on imaginary magnitudes, but the French Argand and Cauchy we're steaming ahead with applications and methods, albeit quite involved in layout.
Few really appreciated the work of Newton as expanded by DeMoivre and sir Robert Cotes, not even Bernoulli who handed of the problem to Euler his young protogé.
Like Cotes and DeMoivre Euler studied Newton and focused on the unit circle( that is rafius 1) . This powerful device can be found in Newtons scratch pad, and in the works of his 2 students. The one noteable difference that Euler introduced was the circle perimeter measure or arc segment. However it is reasonably clear that this was what Cotes was about to share with Newton before he died.
The symbol i took on many meanings in Eulers work on the calculus of arcs , but its commonly accepted meaning is)
.
However it is clear that Euler viewed this as a magnitude of arc! That we have not grasped this is due to an academic blindness to arcs in favour of trigonometric line segments! Newton himself showed how to resolve circular arcs into trig line segments.
The measurement of an arc length is philosophically problematic, but by defining the semi perimeter as pi Euler and Coates provided a consistent measure that was always approximate but always in true ratio to any approximation of the arc segment by a straight line measure .
We do not need a straight line measure, we need the diameter of the appropriate circle to construct the correct circular Arithmoi with which to measure .
The i thus becomes a symbol of arc measurement .
The linear combination of the 2 trig line segments defines i as a quarter arc!
DeMoivre showed that using that arc we can define a symbol for any arc segment in a proportion to the trig line segments .
It is and was a complex notation for the visually obvious arc segment of a circle.
Wessel gave Gauss the Confidence to publish on imaginary magnitudes, but the French Argand and Cauchy we're steaming ahead with applications and methods, albeit quite involved in layout.
Few really appreciated the work of Newton as expanded by DeMoivre and sir Robert Cotes, not even Bernoulli who handed of the problem to Euler his young protogé.
Like Cotes and DeMoivre Euler studied Newton and focused on the unit circle( that is rafius 1) . This powerful device can be found in Newtons scratch pad, and in the works of his 2 students. The one noteable difference that Euler introduced was the circle perimeter measure or arc segment. However it is reasonably clear that this was what Cotes was about to share with Newton before he died.
The symbol i took on many meanings in Eulers work on the calculus of arcs , but its commonly accepted meaning is
However it is clear that Euler viewed this as a magnitude of arc! That we have not grasped this is due to an academic blindness to arcs in favour of trigonometric line segments! Newton himself showed how to resolve circular arcs into trig line segments.
The measurement of an arc length is philosophically problematic, but by defining the semi perimeter as pi Euler and Coates provided a consistent measure that was always approximate but always in true ratio to any approximation of the arc segment by a straight line measure .
We do not need a straight line measure, we need the diameter of the appropriate circle to construct the correct circular Arithmoi with which to measure .
The i thus becomes a symbol of arc measurement .
The linear combination of the 2 trig line segments defines i as a quarter arc!
DeMoivre showed that using that arc we can define a symbol for any arc segment in a proportion to the trig line segments .
It is and was a complex notation for the visually obvious arc segment of a circle.
Title: Re: The simplicity of the unity of Dynamic Directed magnitudes
Post by: jehovajah on April 24, 2016, 10:00:55 AM
A simple line segment and a simple circular arc segment are Things that we call magnitudes. There are other magnitudes but typically we use one magnitude to represent another, usually a convenient one to stand for a more involved one .
Proprioception enables us to make these kind of links or mappings.
The exploration of our proprioceptive thought patterning as it applies to magnitude was initiated by the Pythagireans. But the Grassmanns took it on board to reexamine these proprioceptions.
Justus along with Abel and others of his era considered the fundamental combinatorial structure of perception and Natures complicity in that. So we perceive Combinatorially because Nature is itself combinatorial?
How can we know?
In point of fact we cannot express how we as individuals know but in general it is an acceptance of associations .
Sometimes it seems these associations "fit" Combinatorially naturally . But we can never know that is the case, because all we do at a sensor level is compare !
Assuming uniformity is therefore alwaysbourvstarting point. It is the Prime Mover in all our systems of reference or comparison..
But now we can apprehend that uniformity is fractal. That is we fragment topology into uniform tessellating forms of varying complexity at any scale.
The prime mover is itself not moved. That is just a way of saying everything is relative until we accept an absolute such as a Prime Mover, then everything is relative to that Absolute.
The way tjis works out in measurement principles is discussed by Newton in his preamble to The Principles for Astrologers (Mathematikoi). He therefore posits 2 absolutes: time and Space. These are fractal of a uniform structure against which all else is compared and motion is intrinsically revealed in all its forms.
Against these 2 prime Movers Newton reveals the general trochoidal motion in and of space as Tyme is set out as a record of spatial positions!
Thus a series of snapshots records Tyme and spatial position as an indissoluble link or complex. And these snapshots when themselves in motion past the observing Eye play back the dynamics of all they record!!
In this analysis and synthesis the circular arc segment has been greatly mystified because it too is a complex link of 2 magnitudes and dynamics. Nevertheless Newton and His Peers did not shrink from setting it at the foundation of their principles and setting out sound rules to transform between the 2 fundamental extensive magnitudes, the circular arc followed by the subsequent straight line that is dependent on that circular dynamic for definition and realisation!
So I struggle still to explain a line segment as in general a trochoidal line but in particular a combinatorial dynamic of a circular arc and a radial line segment both of which are free to vary independently .
This would seemingly be a recipe for " chaos" we're it not for the remarkable fact that for any given whole circle a right triangle sits upon its diameter with any point chosen on its perimeter!
Thus the unit diameter circle is our most fundamental measuring scheme, and from it we may construct all fractal measures, all Arithmoi.
But more still: all dynamical measures!
Both Cotes and Euler recognised this deep thought in Newtons scrapbook along with DeMoivre. But how to express what Even Newton could not express!
The magnitude i holds the symbolic expression of the quarter arc turn . The amazing identity between the exponential function as an infinite series and the complex sum or rather lineal combination of the sine and cosine infinite series beggars belief,
But when you sit back and realise these are computations of interpolations of the proportions of right triangles sitting on a diameter in a circle it becomes visually clear!
The line segment has no " length" that is intrinsic. It is a magnitude that requires a fractal scale to be designed and assigned to it . Such a scale is what we mean by length. But similarly and simply an arc requires a fractal scale to be designed and assigned to it! It then becomes clear that the scales are of a different character. They are Not homogenous!
The Pythagoreans stated this in the beginning, Eudoxus explains this in book 5 onwards and yet we still puzzle over the value of
!
And, i and the quarter turn are thus indissolubly linked.
So any system of arc and line segments is called a lineal algebra because of the work of Grassmann and Hamilton, but they drew on the works of Newton, Lagrange and Euler, Cotes, DeMoivre and the Pythagoreans to puzzle out the combinatorial patterns we impose though proprioception on our magnitude perception.
Proprioception enables us to make these kind of links or mappings.
The exploration of our proprioceptive thought patterning as it applies to magnitude was initiated by the Pythagireans. But the Grassmanns took it on board to reexamine these proprioceptions.
Justus along with Abel and others of his era considered the fundamental combinatorial structure of perception and Natures complicity in that. So we perceive Combinatorially because Nature is itself combinatorial?
How can we know?
In point of fact we cannot express how we as individuals know but in general it is an acceptance of associations .
Sometimes it seems these associations "fit" Combinatorially naturally . But we can never know that is the case, because all we do at a sensor level is compare !
Assuming uniformity is therefore alwaysbourvstarting point. It is the Prime Mover in all our systems of reference or comparison..
But now we can apprehend that uniformity is fractal. That is we fragment topology into uniform tessellating forms of varying complexity at any scale.
The prime mover is itself not moved. That is just a way of saying everything is relative until we accept an absolute such as a Prime Mover, then everything is relative to that Absolute.
The way tjis works out in measurement principles is discussed by Newton in his preamble to The Principles for Astrologers (Mathematikoi). He therefore posits 2 absolutes: time and Space. These are fractal of a uniform structure against which all else is compared and motion is intrinsically revealed in all its forms.
Against these 2 prime Movers Newton reveals the general trochoidal motion in and of space as Tyme is set out as a record of spatial positions!
Thus a series of snapshots records Tyme and spatial position as an indissoluble link or complex. And these snapshots when themselves in motion past the observing Eye play back the dynamics of all they record!!
In this analysis and synthesis the circular arc segment has been greatly mystified because it too is a complex link of 2 magnitudes and dynamics. Nevertheless Newton and His Peers did not shrink from setting it at the foundation of their principles and setting out sound rules to transform between the 2 fundamental extensive magnitudes, the circular arc followed by the subsequent straight line that is dependent on that circular dynamic for definition and realisation!
So I struggle still to explain a line segment as in general a trochoidal line but in particular a combinatorial dynamic of a circular arc and a radial line segment both of which are free to vary independently .
This would seemingly be a recipe for " chaos" we're it not for the remarkable fact that for any given whole circle a right triangle sits upon its diameter with any point chosen on its perimeter!
Thus the unit diameter circle is our most fundamental measuring scheme, and from it we may construct all fractal measures, all Arithmoi.
But more still: all dynamical measures!
Both Cotes and Euler recognised this deep thought in Newtons scrapbook along with DeMoivre. But how to express what Even Newton could not express!
The magnitude i holds the symbolic expression of the quarter arc turn . The amazing identity between the exponential function as an infinite series and the complex sum or rather lineal combination of the sine and cosine infinite series beggars belief,
But when you sit back and realise these are computations of interpolations of the proportions of right triangles sitting on a diameter in a circle it becomes visually clear!
The line segment has no " length" that is intrinsic. It is a magnitude that requires a fractal scale to be designed and assigned to it . Such a scale is what we mean by length. But similarly and simply an arc requires a fractal scale to be designed and assigned to it! It then becomes clear that the scales are of a different character. They are Not homogenous!
The Pythagoreans stated this in the beginning, Eudoxus explains this in book 5 onwards and yet we still puzzle over the value of
And
So any system of arc and line segments is called a lineal algebra because of the work of Grassmann and Hamilton, but they drew on the works of Newton, Lagrange and Euler, Cotes, DeMoivre and the Pythagoreans to puzzle out the combinatorial patterns we impose though proprioception on our magnitude perception.
Title: Re: The simplicity of the unity of Dynamic Directed magnitudes
Post by: jehovajah on April 24, 2016, 10:11:38 AM
Why is 2i different to i2?
It is a notational thing. For example 2*2 is different to 22 but they give the same resultant evaluation.
Is i+i therefore the same as -1?
No , because we are adding quarter arcs to get a half arc, but i2 is a different process it is a composition of dynamic motions of the plane, not solely an arc magnitude,
Here we come to grips with 2 uses of the label. One identifies an arc magnitude, the other identifies a rotational magnitude , we use the same symbol !
It is a notational thing. For example 2*2 is different to 22 but they give the same resultant evaluation.
Is i+i therefore the same as -1?
No , because we are adding quarter arcs to get a half arc, but i2 is a different process it is a composition of dynamic motions of the plane, not solely an arc magnitude,
Here we come to grips with 2 uses of the label. One identifies an arc magnitude, the other identifies a rotational magnitude , we use the same symbol !
Title: Re: The simplicity of the unity of Dynamic Directed magnitudes
Post by: jehovajah on April 24, 2016, 11:14:43 AM
Grassmann Hamilton Euler really define I as a planar magnitude, that is a circular arc segment. Thus they defone Planar rotation.
The composition rules for their product designs impose the dilation associated with the rotation. In addition the factorisation of magnitudes imposes constrictions on the way outputs may be geometrically represented .
Learning all these regulations enables an expert to methodically design processes that are notationally simple but procedurally complex!
The composition rules for their product designs impose the dilation associated with the rotation. In addition the factorisation of magnitudes imposes constrictions on the way outputs may be geometrically represented .
Learning all these regulations enables an expert to methodically design processes that are notationally simple but procedurally complex!
Title: Re: The simplicity of the unity of Dynamic Directed magnitudes
Post by: jehovajah on April 26, 2016, 12:58:45 AM
We use th product form for many purposes . It is a design form
So i+ i is a combinatorial form but i*i is a composition or a product form . What is composition?
It is a larger conception of arrangment, usually more complex and more artistically designed .
Combining objects or elements has a simple notion, that of gathering or collecting together . However if a patterned arrangement is introduced into this simple gathering then we may begin to apprehend composition.
The first notion of composition is in fact factorisation. This is dividing a form into a pattern of blocks or objects that can refit back together to recreate the original form. Thus the original form is represent able as as a multiple form . These sub foms are the factors of the original.
However we cn go evn further in our design, so thst the factors carry a pattern in 2 or more independent " directions" . It is these directions in the patterning that enable us to compose or recompose th form.
Hermann Grassmnn identified this notion of direction as being key to his system designs for products . This notion of direction is clearly spaciometric . Most of us will be stuck if asked to think of morevthan3 directions in which to pattern a structure, but fortunately Justus Grassmann found a mineralogy study at the same time as he was piecing out his combinatorial ideas. The patterning in a mineral , like a diamond actually shows how nature can pattern in more than 3 directions , and how faceted crystals are !
These multiple directions reveal a deeper decomposition of direction, into orientation and translation along a line segment or arc segment in that orientation.
So i gives us an orientation and a translation path. The orientation however is not that of a line segment but that off plane, and the translation is not solely along an arc but also around the centre of that arcs enveloping circle as a translation or a pure rotation.
The basic combination of arc segments will be a bigger arcs of a Circle or even whole perimeters and more. However if I want to rotate a pure arc around its centre that is a change in orientation, and in Grassmann product design any change in orientation requires a sign clearly indicating that.
So i is a quarter arc i*i is a change in orientation of that uarter arc by a quarter arc in the plane that i identifies.
We cn view this orientation changed in many ways: a plane rotates carrying the quarter arc, the quarter arc rotates in the plane around its centre , the observer rotates relative to th plane by a quarter arc.
So now 5*i is. Form with several possible definitionS: a big arc of more than one perimeter, a proportional arc segment on a circle with 5 times the radius
So i+ i is a combinatorial form but i*i is a composition or a product form . What is composition?
It is a larger conception of arrangment, usually more complex and more artistically designed .
Combining objects or elements has a simple notion, that of gathering or collecting together . However if a patterned arrangement is introduced into this simple gathering then we may begin to apprehend composition.
The first notion of composition is in fact factorisation. This is dividing a form into a pattern of blocks or objects that can refit back together to recreate the original form. Thus the original form is represent able as as a multiple form . These sub foms are the factors of the original.
However we cn go evn further in our design, so thst the factors carry a pattern in 2 or more independent " directions" . It is these directions in the patterning that enable us to compose or recompose th form.
Hermann Grassmnn identified this notion of direction as being key to his system designs for products . This notion of direction is clearly spaciometric . Most of us will be stuck if asked to think of morevthan3 directions in which to pattern a structure, but fortunately Justus Grassmann found a mineralogy study at the same time as he was piecing out his combinatorial ideas. The patterning in a mineral , like a diamond actually shows how nature can pattern in more than 3 directions , and how faceted crystals are !
These multiple directions reveal a deeper decomposition of direction, into orientation and translation along a line segment or arc segment in that orientation.
So i gives us an orientation and a translation path. The orientation however is not that of a line segment but that off plane, and the translation is not solely along an arc but also around the centre of that arcs enveloping circle as a translation or a pure rotation.
The basic combination of arc segments will be a bigger arcs of a Circle or even whole perimeters and more. However if I want to rotate a pure arc around its centre that is a change in orientation, and in Grassmann product design any change in orientation requires a sign clearly indicating that.
So i is a quarter arc i*i is a change in orientation of that uarter arc by a quarter arc in the plane that i identifies.
We cn view this orientation changed in many ways: a plane rotates carrying the quarter arc, the quarter arc rotates in the plane around its centre , the observer rotates relative to th plane by a quarter arc.
So now 5*i is. Form with several possible definitionS: a big arc of more than one perimeter, a proportional arc segment on a circle with 5 times the radius
Title: Re: The simplicity of the unity of Dynamic Directed magnitudes
Post by: jehovajah on April 26, 2016, 02:23:31 AM
N*i can therefore take on 2 other notions : phase difference and wavelength.
Ith in the same circle N*i gives an arc length that is a phase difference for each rotation of an object or plane.
Here N stands for any rational number. Thus N*i*i rotates each phase by a quarter arc.
However where N*i labels a proportional segmnt in a Circle with radius N compared to a unit radius then N is Leo a measure of rotationl length. It also is thereby connected in time varying systems to wavelength and frequency, as well as amplitude.
It is one of those amazing rotating disc facts that for a highly viscous substance the outer edge of a disc of the substance physicallybtraversesca greater path than the inner portions. This tends to highlight inertial behaviours which can lead to a shearing or shattering of the material.
The materialmatbthevedge is moving with a greater tangential velocity than any material inside it . A tangential velocity is compensated by a radial velocity or any circular velocity or angular velocity. That actually means for rotating ystms the longer wavelengths move at greater velocity to maintain angular phase .
If angular phase cannot be maintained in a rotating system then a spiral patterning emerges where the tangential velocity may be constant but the phase difference varies.
Thus for spiral systems the greater wavelength may travel with the same velocity but their transverse variations are so slow( but huge) that they are discounted and experienced as " low pitched" . They seem slw but in fact it is the immensity of there transverse motion that is what takes time.
Ith in the same circle N*i gives an arc length that is a phase difference for each rotation of an object or plane.
Here N stands for any rational number. Thus N*i*i rotates each phase by a quarter arc.
However where N*i labels a proportional segmnt in a Circle with radius N compared to a unit radius then N is Leo a measure of rotationl length. It also is thereby connected in time varying systems to wavelength and frequency, as well as amplitude.
It is one of those amazing rotating disc facts that for a highly viscous substance the outer edge of a disc of the substance physicallybtraversesca greater path than the inner portions. This tends to highlight inertial behaviours which can lead to a shearing or shattering of the material.
The materialmatbthevedge is moving with a greater tangential velocity than any material inside it . A tangential velocity is compensated by a radial velocity or any circular velocity or angular velocity. That actually means for rotating ystms the longer wavelengths move at greater velocity to maintain angular phase .
If angular phase cannot be maintained in a rotating system then a spiral patterning emerges where the tangential velocity may be constant but the phase difference varies.
Thus for spiral systems the greater wavelength may travel with the same velocity but their transverse variations are so slow( but huge) that they are discounted and experienced as " low pitched" . They seem slw but in fact it is the immensity of there transverse motion that is what takes time.
Title: Re: The simplicity of the unity of Dynamic Directed magnitudes
Post by: jehovajah on April 26, 2016, 09:09:25 AM
5*i is often defined as equal to i*5, that is that commutativity is defined as part of this composition. However Grassmann does not in general see commutativity as a fundamental characteristic of any product design. In his methodical analysis, commutativity has to be demonstrated. Associativity is the only general compositional form that appears to be fundamental .
Thus when we blithely write 5i*5i = 25*i*i we are often writing something that is physically unfounded!, or spaciometrically inane. This often comes about through a confusion about Number.
The concept of Number has changed over the centuries from Arithmos a mosaic of regular sub forms and thus a multiple form, to a pattern of Arabic numerals. Dedekind tried to ground number y a concept call the Number line which was a revision of Wallis's measuring or measurement line, which in turn was the pythagorean segmnted "good" line, often translated as straight, but surprisingly that is not necessarily its only meaning.
Thus Dedekind defined number as an abstract notion often called length or the magnitude of extension in one orientation.
Everyone of a philosophical mind knows length or extension does not exist in space on its own! In fact it only exists in our proprioceptive decomposition of our experiential continuum.
Arithmoi as mosaics do exist in space and have all manner of oriented extension. From these we may select one as primary or principal. When this is done then the spatial object imposes conditions on how all other orientations and their extensive quality may be combined! Justus in his careful combinatorial study of Minerals was forced into these realisations, and published little read pamphlets on the topic.
Fortunately his sons Hermann and Robert ingested his ideas.
While it does place so called Geometry at the heart of So called Mathematics alongside Arithmetic and symbolic Arithmetic ( Algebra) it highlighted logical assumptions that were pure arbitrary wishful thinking and not grounded in spatial constraints. The most innocuous of these was the concept of multiplication! Justus just could not give a logical ground for it, he could only point to a geometrical figure! This was because he held on to the oncept of commutativity.
It was Hermann who accepted that commutativity is not logically justified in general, and applies only to special situations in which processes of construction are truly independent of starting position, or thr symmetry of the product means the process is not uniquely definable !
So product design is crucially a thing of expertise in which every calcultivar process in the product has to be demonstrated as giving the designed outputs. Fortunately the method of induction or inductive reasoning means structures can be built in a logically defensible way and fundamental behaviours can be demonstrated as an outcome of a prior process.
Thus it is not always simple to say " let us assume " and then fail to demonstrate the outcomes forbthevprocesses based on those assumptions.
These demonstrations can be very tedious initially, so it is fortunate for those of us with Los boredom thresholds that ou forebears were not distracted by TV's radios and games, but often delighted in tedium for the small rewards it gave!
Evenso, the more romantic would rush ahead with great visions of what was to come hoping that those who would plod through thr detail would justify their exuberant forecasts.
Hermann presented himself as a plodder providing a " ringside" or stadium seat for all those interested in the drama but unable or unwilling to engage in the necessary arduous trainings! His 1844 print revels in this metaphor, whilst at the same time profusely apologising for any error he might commit in the process due to the part time nature of his studies!
For all it's imperfections it is and was a masterpiece, and recognised as such by Hmilton, perhaps the only other Natural Philosopher naturally inclined to that way of thinking! We may dismiss St Vainant as a plagiarist attempting to claim primacy for France, even though he may very well have been studying the same ideas from the greats of his time like LaGrange,Euler, the Bernoulli's and of course Newton. The tedium is what distinguishes Hermanns work from any other. He has lived through these boring demonstrations and somehow managed to develop a huge enthusiasm based on them! Only a very few would thereby be drawn to sit at his feet as a student and Hamilton was one of those ! Bill Clifford was another, but few others! Most would take his ideas and fly their own jets with them!
So what does the composition i*5 depict, or refer to or lable?
Thus when we blithely write 5i*5i = 25*i*i we are often writing something that is physically unfounded!, or spaciometrically inane. This often comes about through a confusion about Number.
The concept of Number has changed over the centuries from Arithmos a mosaic of regular sub forms and thus a multiple form, to a pattern of Arabic numerals. Dedekind tried to ground number y a concept call the Number line which was a revision of Wallis's measuring or measurement line, which in turn was the pythagorean segmnted "good" line, often translated as straight, but surprisingly that is not necessarily its only meaning.
Thus Dedekind defined number as an abstract notion often called length or the magnitude of extension in one orientation.
Everyone of a philosophical mind knows length or extension does not exist in space on its own! In fact it only exists in our proprioceptive decomposition of our experiential continuum.
Arithmoi as mosaics do exist in space and have all manner of oriented extension. From these we may select one as primary or principal. When this is done then the spatial object imposes conditions on how all other orientations and their extensive quality may be combined! Justus in his careful combinatorial study of Minerals was forced into these realisations, and published little read pamphlets on the topic.
Fortunately his sons Hermann and Robert ingested his ideas.
While it does place so called Geometry at the heart of So called Mathematics alongside Arithmetic and symbolic Arithmetic ( Algebra) it highlighted logical assumptions that were pure arbitrary wishful thinking and not grounded in spatial constraints. The most innocuous of these was the concept of multiplication! Justus just could not give a logical ground for it, he could only point to a geometrical figure! This was because he held on to the oncept of commutativity.
It was Hermann who accepted that commutativity is not logically justified in general, and applies only to special situations in which processes of construction are truly independent of starting position, or thr symmetry of the product means the process is not uniquely definable !
So product design is crucially a thing of expertise in which every calcultivar process in the product has to be demonstrated as giving the designed outputs. Fortunately the method of induction or inductive reasoning means structures can be built in a logically defensible way and fundamental behaviours can be demonstrated as an outcome of a prior process.
Thus it is not always simple to say " let us assume " and then fail to demonstrate the outcomes forbthevprocesses based on those assumptions.
These demonstrations can be very tedious initially, so it is fortunate for those of us with Los boredom thresholds that ou forebears were not distracted by TV's radios and games, but often delighted in tedium for the small rewards it gave!
Evenso, the more romantic would rush ahead with great visions of what was to come hoping that those who would plod through thr detail would justify their exuberant forecasts.
Hermann presented himself as a plodder providing a " ringside" or stadium seat for all those interested in the drama but unable or unwilling to engage in the necessary arduous trainings! His 1844 print revels in this metaphor, whilst at the same time profusely apologising for any error he might commit in the process due to the part time nature of his studies!
For all it's imperfections it is and was a masterpiece, and recognised as such by Hmilton, perhaps the only other Natural Philosopher naturally inclined to that way of thinking! We may dismiss St Vainant as a plagiarist attempting to claim primacy for France, even though he may very well have been studying the same ideas from the greats of his time like LaGrange,Euler, the Bernoulli's and of course Newton. The tedium is what distinguishes Hermanns work from any other. He has lived through these boring demonstrations and somehow managed to develop a huge enthusiasm based on them! Only a very few would thereby be drawn to sit at his feet as a student and Hamilton was one of those ! Bill Clifford was another, but few others! Most would take his ideas and fly their own jets with them!
So what does the composition i*5 depict, or refer to or lable?
Title: Re: The simplicity of the unity of Dynamic Directed magnitudes
Post by: jehovajah on April 28, 2016, 01:10:13 AM
i*5 has a meaning encoded in the form.
The structure a*b is a compositon fotm . It is a combinatorial structure encoding a process of combination of specified elements . The elements cn be magnitudes, quantities, counts, dimnsional, forms objects etc etc.
In 5*i I specified i as a magnitude of arc and 5 as a count of that magnitude. The implication is magnitude is counted to the left of the * symbol. So puttin i to the left of 5 makes no sense , and 5 is undefined.
If we define 5 asv count then we alo have to define i*5 . The simplest is i with the count written to the right.
We can not justify this as commutativity because it is an identity by definition.
However if I count in bundles, and then count the contents of a bundle, those two counts do commute. The spatial arrangement can be topologically commuted.
Suppose we define 5 to be a line segmented into 5 concurrent segments . Then i*5 has to have a sensible topological expression. Normal geometric expression is in terms of rectangles. . We could replace rectangles by curved Shunyasutras.
Alternatively we could define it to describe a process of rotating the line segment as a whole into a position about one end that is a quarter arc rotation to the start position. whatever we take or define it to mean is context bound and that is why symbolic combinatorial structures are not truths in and of themselves.
The structure a*b is a compositon fotm . It is a combinatorial structure encoding a process of combination of specified elements . The elements cn be magnitudes, quantities, counts, dimnsional, forms objects etc etc.
In 5*i I specified i as a magnitude of arc and 5 as a count of that magnitude. The implication is magnitude is counted to the left of the * symbol. So puttin i to the left of 5 makes no sense , and 5 is undefined.
If we define 5 asv count then we alo have to define i*5 . The simplest is i with the count written to the right.
We can not justify this as commutativity because it is an identity by definition.
However if I count in bundles, and then count the contents of a bundle, those two counts do commute. The spatial arrangement can be topologically commuted.
Suppose we define 5 to be a line segmented into 5 concurrent segments . Then i*5 has to have a sensible topological expression. Normal geometric expression is in terms of rectangles. . We could replace rectangles by curved Shunyasutras.
Alternatively we could define it to describe a process of rotating the line segment as a whole into a position about one end that is a quarter arc rotation to the start position. whatever we take or define it to mean is context bound and that is why symbolic combinatorial structures are not truths in and of themselves.
Title: Re: The simplicity of the unity of Dynamic Directed magnitudes
Post by: jehovajah on April 28, 2016, 03:46:58 AM
http://bpeers.com/blog/?itemid=1001
The power series
The power series
Title: Re: The simplicity of the unity of Dynamic Directed magnitudes
Post by: jehovajah on April 30, 2016, 04:19:39 AM
Whenever i is implemented on a computing platform it has to be implemented as /2 . The i s not a computable magnitude whereas /2 can be approimsted.
This is because Eulers form of Cotes Euler formula introduces it as notation in an identity that acts as an arc indicator like the ° mark but also as a radian indicator as well as reminder
It most general role is as a quarter circular arc in the plane. Thus each pone has an i equivalent inherent.
I,j,k belong to 3 orthogonal planes.in any lineal combination( arc segment ) they represent a spiral on a spherical surface that intersects all 3 planes.
This is because Eulers form of Cotes Euler formula introduces it as notation in an identity that acts as an arc indicator like the ° mark but also as a radian indicator as well as
It most general role is as a quarter circular arc in the plane. Thus each pone has an i equivalent inherent.
I,j,k belong to 3 orthogonal planes.in any lineal combination( arc segment ) they represent a spiral on a spherical surface that intersects all 3 planes.
Title: Re: The simplicity of the unity of Dynamic Directed magnitudes
Post by: jehovajah on May 01, 2016, 11:34:33 AM
http://bpeers.com/blog/?itemid=1001
I chose W00t because he goes through the expected Demonstration of the quaternion as an exponential and logarithmic form.
This is Mathemythics!
The Thought Pattern is: there must be a way of showing that this is consistent with standard arithmetical processes especially differentiation ones.
The student and the teacher are left in awe snd somewhat bemused for lack of understanding.
The whole demonstration begins with product deign! Thus it is not a natural consequence of everyday arithmetic, it is a designed use of standard arithmetical forms but the forms are analogies. They are analogies by design. And it is because they are analogies that we call the process symbolic arithmetic aka Algebra!
In designing these analogies the combinatorial structures are important , but the internal semantics are different. So we replace the arithmetical operations by combinatorial and compositional ones.
Algebra since Hamilton Boole and Grassman et al has pushed this formal analogy to the limit.
The label for the exponential form is just that. eix is a label or function form . Somewhat we put in its argument is what we express as defined.
The design ofnthebquatenions product rules and addition rules ( composition and combination) are dealtbwithnextensivly by Grassmann in his paper on " where to find Hamiltons Quaternions in the Ausdehnungslehre"! It is a very sophisticated read of a very sophisticated thinker ( Hamilton) . Hermann dispels the Mathmythics by clearly demonstrating the constructed nature of these combinatorial forms.
The geometrical implementation/ motivation/ interpretation is then rather obvious . Where the obscurantism lies is in conforming the processes to the expectations and intuitions of arithmetical mathematicians!
For most Mathematicians it is precisely this " mind fluff!" of symbols that confuse 95% of the rest of humanity that is such a drug! We love to tortuously work through the twist and turns of any demonstration to get a result! But it has to be the "correct" result! It has to Bethesda right answer.
Hermann and other Algebraists of his time recognised the constructive psychology of mathematical behaviour and gave an exposition of it.
Math,sticians I'd not like this, it was too revealing of what they were actually doing! But philosophers were saying: don't be Luddites! You can simplify and improve and extend your expertise by understanding what you are actually doing!!!
The exponential label came after the logarithmic process was clarified by Napier and Briggs and Bruges(?). Mercator worked out he Logarithmic series which Newton Wallis et al confirmed. These were difference processes for interpolation something Mathematicians had been engaged in for centuries in calculating the Sine or Chord tables for different arc segmnts/ angles. The establishing of a serviceable labelling convention was part of the so called Function revolution. Mathematicians got so bogged down in detail they lost sight of what they were doing psychologically and philosophically, and to a certain extent theologically.
If you read Newton, Euler and Grassmann you find a clear philosophicl,and psychological adjustment to what they were about. As Berkely pointed out it was as nebulous as any religious discusdions or definition or debate of their time! Those who pursued this kind of constructive thought patterning usually had some expertise to apply it with. Thus a master craftsman might develop these kinds of notions over years of studying nd applying his/ her trade.
Newtons trade was Mechanics Astrology and Alchemy, all of which he applied throughout his lifetime.
So dear reader do not get the impression that quaternions are strange otherworldly entities, as Hmilton poetically inclined to believe. Realise that Grassmann explained it as a constructed analogous method or system for labelling and expressing ordinary spatial experince of dynmic entities like moving and shaping a block of wood !
Of course they have more sophisticated applications beyond spatial even temporal experinces, but you need to grasp Grassmanns analysis of extensive and intensive magnitudes to appreciate how a line segment and a step rise can encode those distinctions.
There is one point about commutativity that is prominent in W00ts explanation. The issue is not about arithmetic or counting numbers. These are compositional expressions and in general commutativity of compositions has to be defined or constructed . Nothing can be assumed .
If we define the Composition of two exponential quaternions we have to be aware of th conjugate something Hamilton was not initially clear about in his original presentation of the rules of composition of ijk. Indeed he was not clear about a lot of things regarding i j k ! For that matter Grassmann struggled in his early work on rotation trying to grasp the way to cope with the immense freedom line segments or rays had in the Raum!
The solution is that i,j,k specify planes of circular arc that are orthogonal. This was a olution based on Justus Grassmans work on specifying a system for points planes etc in the Raum.
I chose W00t because he goes through the expected Demonstration of the quaternion as an exponential and logarithmic form.
This is Mathemythics!
The Thought Pattern is: there must be a way of showing that this is consistent with standard arithmetical processes especially differentiation ones.
The student and the teacher are left in awe snd somewhat bemused for lack of understanding.
The whole demonstration begins with product deign! Thus it is not a natural consequence of everyday arithmetic, it is a designed use of standard arithmetical forms but the forms are analogies. They are analogies by design. And it is because they are analogies that we call the process symbolic arithmetic aka Algebra!
In designing these analogies the combinatorial structures are important , but the internal semantics are different. So we replace the arithmetical operations by combinatorial and compositional ones.
Algebra since Hamilton Boole and Grassman et al has pushed this formal analogy to the limit.
The label for the exponential form is just that. eix is a label or function form . Somewhat we put in its argument is what we express as defined.
The design ofnthebquatenions product rules and addition rules ( composition and combination) are dealtbwithnextensivly by Grassmann in his paper on " where to find Hamiltons Quaternions in the Ausdehnungslehre"! It is a very sophisticated read of a very sophisticated thinker ( Hamilton) . Hermann dispels the Mathmythics by clearly demonstrating the constructed nature of these combinatorial forms.
The geometrical implementation/ motivation/ interpretation is then rather obvious . Where the obscurantism lies is in conforming the processes to the expectations and intuitions of arithmetical mathematicians!
For most Mathematicians it is precisely this " mind fluff!" of symbols that confuse 95% of the rest of humanity that is such a drug! We love to tortuously work through the twist and turns of any demonstration to get a result! But it has to be the "correct" result! It has to Bethesda right answer.
Hermann and other Algebraists of his time recognised the constructive psychology of mathematical behaviour and gave an exposition of it.
Math,sticians I'd not like this, it was too revealing of what they were actually doing! But philosophers were saying: don't be Luddites! You can simplify and improve and extend your expertise by understanding what you are actually doing!!!
The exponential label came after the logarithmic process was clarified by Napier and Briggs and Bruges(?). Mercator worked out he Logarithmic series which Newton Wallis et al confirmed. These were difference processes for interpolation something Mathematicians had been engaged in for centuries in calculating the Sine or Chord tables for different arc segmnts/ angles. The establishing of a serviceable labelling convention was part of the so called Function revolution. Mathematicians got so bogged down in detail they lost sight of what they were doing psychologically and philosophically, and to a certain extent theologically.
If you read Newton, Euler and Grassmann you find a clear philosophicl,and psychological adjustment to what they were about. As Berkely pointed out it was as nebulous as any religious discusdions or definition or debate of their time! Those who pursued this kind of constructive thought patterning usually had some expertise to apply it with. Thus a master craftsman might develop these kinds of notions over years of studying nd applying his/ her trade.
Newtons trade was Mechanics Astrology and Alchemy, all of which he applied throughout his lifetime.
So dear reader do not get the impression that quaternions are strange otherworldly entities, as Hmilton poetically inclined to believe. Realise that Grassmann explained it as a constructed analogous method or system for labelling and expressing ordinary spatial experince of dynmic entities like moving and shaping a block of wood !
Of course they have more sophisticated applications beyond spatial even temporal experinces, but you need to grasp Grassmanns analysis of extensive and intensive magnitudes to appreciate how a line segment and a step rise can encode those distinctions.
There is one point about commutativity that is prominent in W00ts explanation. The issue is not about arithmetic or counting numbers. These are compositional expressions and in general commutativity of compositions has to be defined or constructed . Nothing can be assumed .
If we define the Composition of two exponential quaternions we have to be aware of th conjugate something Hamilton was not initially clear about in his original presentation of the rules of composition of ijk. Indeed he was not clear about a lot of things regarding i j k ! For that matter Grassmann struggled in his early work on rotation trying to grasp the way to cope with the immense freedom line segments or rays had in the Raum!
The solution is that i,j,k specify planes of circular arc that are orthogonal. This was a olution based on Justus Grassmans work on specifying a system for points planes etc in the Raum.