I had thought to look at an aggregation rhythm based on the logarithms to different bases but realised that these make little sense as aggregates because they are not a rhythmical depiction og plethora or aggregation. They are a signal of a stage in a calculation algorithm
The Greek notion of plethora which taken together with unity forms our basic notion of scalar and scale is a useful one. It covers dynamic increase in the units whether by multiplication, scaling the rhythm of the musical bar while counting, or by division of unit size, setting up sub scales of rhythm within a larger initial unit. Thus a plethora is a dynamic description of unit motion whether observed in a living organism with cell division or a mass production or replication of the unit through some specified agency.
Replication of the unit is the macro notion of plethora, whereas plethora through a division process of unit size reduction is the micro notion of plethora, but it is always dynamic.
Aggregation is the apprehension of gathering items together, both physically and psychologically, mentally. An aggregation is a more general structure than form, and is an appreciation of gatheredness. It is not as dynamic and self organising as plethora, but of course could be made of many plethora!
The factorial number system, or as I would draw attention to it, the factorial rhythm aggregation has been explored in relation to probability distributions, and a link to logarithmic aggregate rhythms has been shown, but I would observe the underlying structure of any aggregation purporting to use or relate to logarithms would have to be an antilogarithm aggregate scheme, whereby the base of the contiguos unity in the arrangement will increase by one:
+ mod(n)*n^logarithm+ mod(n-1)*(n-1)^logarithm+……+ mod(2)*2^logarithm+ mod(1)*1^logarithm+mod(0)*0^logarithm
The + gate is not the +mod() gate we have encountered earlier. The factorial rhythm has a +mod(n!/n-1!)
+ gate. Because of this I cannot make an aggregate rhythm and therefore this type of aggregation is different.
Since this represents an aggregation of rhythmical aggregations each strand of the aggregation is a complete system by itself,so there is no need to move to a different strand to change scale, the logarithm does that , so this aggregation is a ranking or quality aggregation,allowing qualities or ranks to be aggregated.
The +gate thus is a quality yoke and does not allow summation except on an arbitrary imputed assignment. The transform between strands is a ratio of logarithms
When that ratio exists between the strand members those members can be summed at the logarithm level and we know that represents a product at the antilog level.
Apart from rank or quality measurement there may be other uses for such an aggregation mesh of rhythms.
The +gate has become an equivalence relation.
If asked to represent in any artform or mixed media the notion of curved motion, how many would use stereophonic soundscapes, tactile intensity maps, or gustatory intensity maps?
I have mentioned before the fundamental nature of the Logos Response to apprehending notFS, particularly in relation to proprioception maps. However this is only one aspect of the synaesthesia we use in apprehending shunya, notFS.
Originally posted by author:
Spin field effect transistor announced, using the effect of a spin helix field.
And i realise thar polar equations make spirals easy and plot [tex]r= theta^2+2*theta+1[/tex] in spacetime.us.
Jakob Steiner is my next post.
Originally posted by author:
Of the mesh of logarithmic aggregates and the dance rhythms of street dancers and Break dancing.
Of the factorial rhythm aggregate and the factorially fractal structure of infinite possibility space, as it condenses stochastic probability clouds in regional spaces , containing statistically significant propensities to computational rearrangement: some of which condense into dense regions of relativistic motion space.
Of discontinuous attributable space, to which regionalised properties are attributed subjectively, and which regions are dependent on the measuring instrument adopted subjectively.
of a dynamic density function that regionalises space into condensing and evaporating potentials, and which is dynamically in equilibrium
Originally posted by author:
3d polar coordinate geometry is quite hard to find a very general treatment of it which is not fully wrapped in tensor or vector notation.
Take a unit pole and a variable rod, and attach the rod o the poke so that it is joined in a corner which is free to rotate. This measring device will be the basis of some measurements of regions, and works by pointing the pole at one point on the boundary of the region and moving the rod around the region boundary noting the scalar length of the rod as it moves round and the radian angle measure between the rod and the pole as it moves in this way. I thus record [tex](r, theta)[/tex] but clearly not the rotation of the polar axis( the pole) as the rod traces round the region boundary. For this purpose i have an orthogonal unit that is attached to the pole, thus the pole becomes a right triangle, a Bombelli vector, and i have a third marker of unit length orthogonal to the pole but free to rotate in a plane orthogonal the pole . This marker is also fixed on a point on the boundary of the region if possible or some point relative to which the region is fixed.
Now i can measure pole's axial rotation in radians. This is exactly like using a pair of compasses with the region being traced out by the pencil. Thus we record [tex]r, theta,alpha[/tex] .
We can then write equations that have to distinguish [tex]theta[/tex] and [tex]alpha[/tex] and therefore these require +gates and the cos and other trig functions.
When we look at the role of the roots of unity we find that they do one thing, they rotate the unit magnitude in space. So + and – are π rotations of the unit magnitude. The functions that control this rotation and its rate and nature are the trig functions. The + and the – are +gate modifiers, like mod() and control how units are aggregated. and what quantity.
[tex]r=theta^2+theta*cosalpha+1[/tex] for example.
Standing in the middle of the bank holiday sale motions of people , discerning the aggregation rhythms of their motions, the dance of their hurried predilections.
Suddenly the logarithmic rhythm of their dance gradually emerges.
I think of disaggregation and am immediately surprised by the difference in rhythm. The modulo minus gate -mod() starts with a bang! The rule that you only sum units that are the same in any aggregation applies to disaggregation, but the dissolution of a unit into a smaller nested unit is a shattering surprise! One sees a whole unit dissolve before ones eyes into smaller units that disaggregate and disperse in unexpected ways!
Really the imposed order of aggregation lulls one into a false sense of an orderly build up. But the unfamiliarity of disaggregation means that anything is possible!
So A unit kind of dissolves into smaller units enabling smaller units to be disaggregated. Whole units can obviously be moved away as a whole, or roll away in blocks, etc, but smaller units require unit to dissolve or shatter into smaller units.
Therefore the shattering of a plate when it drops on the floor, the shearing off of a precipice of ice from a glacier, a disintegration of a piece of wood into shards and splinters are all examples of the logarithmic rhythms of disaggregation.
So now the aggregation of things has a moment of unification of a larger unit which is somehow less dramatic than the shattering of a unit, but in fact it is more interesting than that because we see these unifications in everyday situations. Traffic jams, where a collection of rhythms forms solid looking block, that dissipates as the rhythms play out. The formation of planets where a cloud of smaller units surrounds and moves with the larger units. Standing observing a crowd one sees clumping and dispersing of groups of people, larger groups forming and splitting , moving and whirling away and together, finally splitting into single units.
The formation of waves in a dynamic ocean as mounds of water surrounded by smaller dissipating ripples, amonst the coagulating mounds.
This is a great cycle of motion in a motion field experiencing aggregation and disaggregation at the same time, leading to a density distribution that is dynamic and polarised around intense density and extreme rarefaction.
Thus it became a syncopated thought that the logarithmic rhythms of aggregation and disaggregation play their part in the dance of cosmogeny.
Originally posted by author:
What we attribute is what we tend to find.
Aggregation and disintegration images
Here's a thought.
When meditating on units dissolving into smaller and smaller units do I observe each unit dissolving into nothing or rather into the finest of tilths deservedly called shunya?
The possibilities are indeed infinitesimal!
It maybe that the factorial aggregation rhythm allows us to deal with the very very large and the very very small in a sensible way, and that the very very small written in this way are Newton's fluxions, his infinitesimal quantities that apparently "vanish".
Shunya the alpha and the omega of all, and the mother of all unity if we be the father of it. Man is indeed the measure of all his conceptions, and all our children turn out as we do.
I saw a man in a dug out canoe with a rolled out kite dangling a spider's web over the dynamic waters and thereby catching the largest of fish.
It seemed that the strength in the finest of things when combined with man and dynamic space produces miraculous results, and that an open learning brain may "mind" a new way to do a necessary action for survival. That spider's have a thing or two to teach us about what it is possible to catch with an n dimensional web and that Klein surfaces may just relate to spider webs and roll up into a 3d form on contact with a "seed" form.
And so the wisdom of the trochoids in all their spiral forms web together all the roots of unity into the finest, the silkiest of tilths called shunya.
And there are webs within webs…..
So to create a spiral trigonometry first pick your spiral.
Then pick your radial hypotenuse.
Then record the ratios as your hypotenuse meanders gracefully along the spiral curve.
Such a table of ratios may have some worth, I think.
The notion of an angle is a "planar" notion having a long history of derivation. However we seem to get scared when it comes to generalising the notion into 3d, because of the overwhelming choice we have for a "corner" or "wedge" notion. Such is life, full of infinite choices!
However I suspect that the most useful generalisation of angle would involve "trig circles" on a unit sphere.
I call them trig circles because the radii would of course be trig function values.
The cones and other polyhedral within these arrangements would form a fractal arrangement of decompositions of 3d space, and would link the roots of unity to such a decomposition.
We often confuse our measures with "reality". This of course is a common mistake. They are not reality after all, they are only our measures.
So the question is: can we ever know "reality" without our measures?
The answer is: i do not know. I suspect not as we are fundamentally a network of measuring sensors, and does it really, truthfully matter?
In fact if it is so, the imputation is that the possibilities are not infinite and there is an absolute unchanging "perfection" from which every measure derives its norm.
As i say, i do not know, but i do not think so.
If, however you wish to think in this way, be my guest. I have no argument with you. Nor do we frankly care. Not said or writ in arrogance, or haughty high mindedness, but in expression of present concerns.
When i speak in higher mathematics to my cats
Or hieroglyphics to the greeks
It's not that i pontificate!
But rather i communicate
Are pertinent to what one seeks!
Let us suppose,with some foundation, that our understanding of the development of mathematical ideas has been refracted through the predilections and national interests of certain individuals who having found a use for a certain praxis or mathesis propose It as an adherent with some historical or situational persuasiveness in eloquent language of influence.
That such ideas and texts being of utility are redacted or written up into courses of study and learning and promulgated through educational opportunities to eager minds of the young. What may we deduce?
Mainly that nothing presented to us as children is as it seems.
When we become adult, providing we have not become crippled in mind by the process we ought to thoroughly take time to review what we have been taught from our mother's knee to our most revered and wise tutors of education, retaining only that which appears to be useful, and supported by the discoverable data of our research. We may then construct if we so wish our own version of how things came to be, with out the imposition of censure or coercion from long dead philosophers or scientists or other sources of influence.
As there is no basis for unity, I say; that is it is my opinion that this is a notion to be ventured , and a rule of thumb i use in my meanderings, there is no basis for a beginning of this or that idea or a preeminence of this or that notion other than that some wag or wit or respected person has declared it so.
So before Descartes the Greeks and indeed the Indians and earlier cultures managed perfectly well without Cartesian coordinate systems. In fact Descartes managed perfectly well without them and would be a little bemused by the extent to which they have been put to use! For Descartes the praxis was the thing, the mathesis of ordering the thinking and empirical process to come by degree to a solution that is built of smaller mor obvious steps. No crazy notion of mathematical proof existed in his time, just a Greek and gentleman's agreement that if it was obviously so then it was so. The burden then was to demonstrate that it was obviously so.
Of course what is obvious to me is not obvious to you thus some time for meditation is necessary to assist communication of a notion. However in a school or a gymnasium, such time was not given by your competitors and so an element of coercion has always existed in the forced education of the young. Being thus brainwashed was par for the course, but left many educated but dead to innovation and insight. So only those peculiar few who escaped this harsh environment or were resilient to it's grinding conformity ever got to play around with the basic notions of their craft or art.
Men of renown could earn good coin by teaching redolent youth their methods of production, their praxis and thus it is of no surprise that students received harsh treatment for their mistakes, a man's reputation being at stake, and thus his economic survival. The economics of renown therefore are as important in deciphering the development of mathematical notions as the enthusiasm of student or adherent, that is in common parlance the fanaticism of fans!
However it is put or whichever way it is glorified the Cartesian coordinate system did not just magically occur to Descartes . The precursor was trigonometry of the Greeks and Indians and Chinese the utilisation of the ratios of the right angled triangle, the use of shadows cast by a gnomon to measure immeasurable distances by proportional equations.
In his praxis Descartes set himself the task of starting from the simplest of components, and for him this appeared to be the measurement of lengths of a right angled triangle. By Pythagoras theorem and trigonometry all manner of things could be measured and calculated.
Descartes praxis was to show how Euclid can be derived from these simpler concepts of the right angled triangle with as few additional tools as possible. Thus as and when needed and only then he added additional tools in his demonstrations. To think that Descartes imagined a great and rambling use for his crtesian coordinates is to impute to much. Descartes barely acknowledged that they could be used in 3 dimensions, confining himself to weightier matters of philosophy and analytical reductionism.
This is not to say that Descartes was a haughty man. Far from it . His work is engaging and inviting and inclusive and left things for students to do and explore., but his insight was limited even though influential. His jibe at Bombelli's " imaginary numbers" failed to kill off the topic but rather created a reference to it. So he would definitely be surprised at the use of his Cartesian coordinates in the " explanation" of these "impossible" quantities.
And so Descartes coordinates are a much trumpeted version of trigonometry of the right angled triangle, an idea well understood by the Greeks and Indians, but apparently available only to classical scholars in the Western world in dribs and drabs and by catholic educational seminaries.
Despite the seats of learning in the east and far east western scientists were restricted by religious strictures to a received notion of the world and it's origin and it's measurement. Thus Descartes praxis was a way to deduce from foundational ideas the knowledge already deduced by the Greeks. But more importantly, it was a means of spreading to a wider western audience the geometry of the Greeks. What Bombelli had to travel to find out in Rome and other places, Descartes presented dished up on a plate using his own method of deduction and demonstration.
Given that Cartesian coordinates are the systematisation of the right angled triangle geometries of the Greeks we may now proceed to a different understanding of differential and integral calculus.
These topics I venture are applied trigonometry.
Thus the analysis of the tangent to a curve is the application of the tangent ratio of a right angled triangle to a curve or line. Because Greek geometry is dynamic this is the application of neusis to the dynamic situation. The limit process as it is now called initially was simply the tangent at a point being the tangent ratio at that point, found easily enough by similar triangles, that is an expansion to a larger triangle with a hypotenuse parallel to the tangent to the curve!. Thus a table of tangents is sufficient to provide the differential to any curve having drawn the tangents to the curve.
The derivation of formulae for differentials appears normal to the topic, but in fact this applies interpolation formulae to known ponta to find out from table the in between results. Thus the logarithm of sines and cosines derived by Napier were of extreme importance in the development of differentials enabling the full exploration of methods of calculating sines and their interpolations and cosines and from them tangents.
Napier's logarithms were also fundamental to the notion of compound interest, a co related notion to differential calculus and the basis of integration.
As the prevailing interest was in calculating area the area under a curve becomes a simple summation of the areas under the right angled triangles to a curve based on the tangents to a curve. This areas is therefore related to right angled triangle area formulae and proportionately to the cosine of the requisite right angled triangle with the given tangent.
In particular the area of the triangle can be shown to be proportional to some aggregating product of tangent ratio, thus revealing that integration is a inverse procedure to differentiation which forms the tangent ratios by disaggregation( division).
The details are of interest but do not need to detain us here. Suffice it to say that logarithms sines cosines and tangents are intimately related in the calculus and are all applications of the Greek theory of proportions and right angled triangle trigonometry, or rather trigonometry of whatever flavour Greek Indian or Chinese or Arabic.
The link through spherical trigonometry to astronomy and astrology etc has to be mentioned but more importantly the notion of a dynamic magnitude that measures geometrically dynamic situations is an old idea too, which we have disguised once again under the notion of vector, admittedly a notion advanced by Hamilton but not a new one, rather a restating in modern clothes of an old ancient view of the dynamic nature of shunya.
Originally posted by author:
It occurs to me that my measure with a little modification becomes recogniseable as a surveyors theodolite. This essentially renders surveyors as potentially supreme spherical coordinate producers..
I wonder how they record their data?
Finally found what i was looking for ! Geodesics for Dummies! Nice!
For symmetrical forms a spiral or vorticular path is the most efficient space filling "tiling" path, as the inherent rottion is disguised by the symmetry.
For all other forms the most efficient path splits into efficient paths of rotated forms, and these too are spiral or vorticular . In fact all paths are trochoidal in nature and relation, trochoids of spirals/vortex. These trochoids support either an inward spiral/vortex. an outward spira/vortexl or a closed circuit such as an ellipse/ ellipsoid. In special cases a circular/spherical closed circuit is observed.
We may derive trigonometric tables for any boundary, thus the special case of circular or spherical rotation is not the only trig database we could develop. Each of these databases would be a fingerprint of a form up to similarity, no matter what orientation, but corrections for parallax and perspective would need to be included.
We can even derive trig tables for open or dynamically varying forms, if they vary systematically by some trochoidal path.
The number concept makes fractions difficult to comprehend. Proportions are the natural way to understand the scalar nature of spatial measure. Units and nity are natural constructs for the human brain, the only difference is the scale sizes of these unities, the ratios and proportions.
Fractions seem alien and are indeed are alien to our natural way of measuring. Our neural networks and meshes work from the smallest common unit signal.
Now the modification to the measure discussed earlier is based on 2 Bombelli vectors. One is a half unit square cut diagonally (π/4 set square) and the other a Bombelli vector with a unit side while the other 2 are variable, but always forming a right angled triangle.
Hinging these 2 vectors at the right angled corner allows free movement in a unit sphere which can be used to determine radian relative angle motion. The semi unit square Bombelli vector i will denote as the pole and the variable Bombelli vector i will denote as the rod, and the angles of rotation will be described by spherical triangles on the unit sphere.
This will be called a quaternion measure.
Units are dynamic magnitudes. There appear to be some basic dynamisms: growing and shrinking, rotating relative to a pole, radial translation from a centre region, shearing and twisting morphisms relative to a plane, rarefaction and condensing.
For growing , shrinking and radial translation i can use units that "plethorate" that i can scale by replicating and aggregating or disaggregating using +gate algorithms.
For rotating and twisting i need units that "modulate", They reflect the cycle of a rotatin object. This distinctive behaviour requires modul or clock enumeration systems. and aggregation requires +mod() gates
For shear we require units that reflect conservation of area and/or volume but not perimeter or boundary magnitudes. A curious unit which i have yet to explore.
For rarefaction and condensing we need units that conserve kinaesthetic sensations of pressure but again not boundary magnitudes, so proprioception of geometry is different despite kinaesthetic pressure being the same. Sensitivity to this type of unit will depend on sensor density, so that the average over a given number of sensors will be different as the unit boundary magnitudes grow smaller, eventually resulting in spike like signals like Diracs integral for impulse force distributions.
How these last units are aggregated i do not know yet, but it seems clear that some regional limitation will apply, requiring an iterative approach to aggregation, therefore a fractal arrangement of some description is still the outcome. I suspect that nested unities will be involved with some weighted function controlling the +gates, the functions possibly using boundary parameters. Thus for shear the bigger he boundary the less impact on the overall aggregate, whereas for density or rarefaction, the bigger the boundary the greater the impact.
For units of rotation i can distinguish four +gates of renown: +, +&-, – ,-&+. Bombelli's rules apply.
+&- tends to be written a i when it is a rotation, when i is an axis it is the same as the y axis and not a rotation.
-&+ would then be i^3 or equivalently -i.
The confusion between axes and rotations i have worked through in polynomial rotations which is unnecessarily complicated because of it!
When i think about the +mod() gate for rotations i see that though the base is 4 the aggregation is not a logarithmic rhythm. Thus i venture that roots of unity are not aggregations that can be set out in an obvious logarithmic form. They have to be written as cosines and sines to do that, which is where Cotes formula reveals something about rotational +gates: they are in fact yoked roots of unity.
Aggregation algorithms are therefore really interesting operators, and require some tabulation of examples to establish. This is as close as one gets pragmatically to axiomatic algebras: a detailed model of the parameters and their interactions has to be set forth initially before any general arithmetic can be done. Thus Brahmagupta, Napier, Bombelli and Hamilton and numerous others laid the foundation by detailing these relationships, or bonds.
Originally posted by author:
Some basic sequence definitions
A sensible explanation of recursive / convolution
so iteration can be understood as going round the convolution/recursion again with the new value.
In algorithmic terms the recursion/ convolution is a systematic sequence of operations on a start point resulting in an end point or teleos, that is a goal, which is necessary and sufficient for the next iteration of the recursion, where iteration means using this goal result in the recursion as a new start point.
As complicated as it reads most activities in our lives are recursive/convoluted, and we get to where we want to be, or where we end up 9depending on how one perceives it) by iterating the convolutions in our lives r being iterated though th convolutions in our experiences.
Why functions are recipes
And Lazurus Plath gets round to explaining a bit of how his applications work