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Author Topic: Fractal Foundations of mathematics: Axioms notions and the set FS as a model  (Read 107964 times)
Description: All ideas welcome.Needed to revise mathematical thinking and exploration
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« Reply #360 on: January 05, 2011, 03:24:20 PM »

Of the continuing strangeness of imaginary magnitudes,but unnecessarily so.

So when Brahmagupta invented misfortunate numbers after meditating on the I ching he set in motion a wave or revulsion and confusion that exists till this day.

The greeks had drilled down to the core of magnitude, looking for unity, the indivisible unit of reality they called an atom.
For them, and for all cultures there was always something, never "nothing", and whatever this something was it was dynamic.

So Brahmagupta was traduced in translation! and then he was rejected through his unfortunate choice of examplars: misfortune and fortune!

Shunaya is the finest of tilths, the lightest of mists, the essence of perfume! but it was reduced to The cold blackness of misunderstanding.

Brahmagupa's operator, Bombelli's operator speak of yoked magnitudes, and the yoke that distinguishes them is the yoke of roots of unity, the yoke of rotation, the yoke of dynamic equilibrium.

The greeks , From Thales to Plato with their plethorate arithmoi, and their proportional relationships, developed the notion of scalar. The indians with their love of rhythm and vedic  sutras and verse developed the notion of compact aggregational structures.Both had a dynamic understanding of a motile reality, and responded to it dynamically.

So the concept of numbers has arisen as an amalgam of many notions but non more injurious than the conception first introduced by spherical geometrists and gradually developed in the west into the number line by Wallis and eventually Dedekind. The dynamic magnitude was being turned into a static symbol, a shadow of itself, and keeping all the bad associations without a hope of redeeming all the good dynamic qualities.

The dynamic qualities were spread abroad the landscape as one argument after another rigorously defined a number, leaving out this or that attribute. As the attributes of dynamic magnitudes  were discarded others came along collecting them into new entities called vectors, matrices, tensors and eventually a strange collection of measures were put in place of the dynamic magnitude of the indians and the Greeks.

So none treated dynamic magnitudes properly, and thus  weird notions proliferated and the weirdest was the notion of the "negative number" along with the notion of "nothing".

Brahmagupta understood from the I ching that unities emanated from shunaya in yoked pairs: the yoked pairs  swirled around like spokes in a wheel, in equilibrium, dynamic and or static. They were balanced and yoked and from Shunaya, the womb of all things, thus they were always the same as shunaya when aggregated, and they had their own rhythm.

These yoked pairs were balanced and therefore equal, but moving in a rotation and therefore constantly changing their orientation relative to "the world", but not to themselves. Brahmagupta mistakenly chose to distinguish them by fortune and misfortune, but we do not have to make the same mistake. I will distinguish them by their angular difference in radians 0 and π. or i could use π and 2π.

So now i can use 0 to distinguish all magnitudes which are yoked to π magnitudes.

If i use a boundary i can write these yoked magnitudes on the boundary with either a 0 or a π  and place them diametrically opposite. I could even use 0,π to further distinguish the yoke and let the magnitude and direction of the yoke be represented by a line. All lines cross at an intersection which i will call shunaya, the centre the origin, the balance point.

If as Brahmagupta advises all the magnitudes are unity we have 10---1π

And these aggregated are shunaya, the centre from which they came and around which they spin.


We have to recognise the rhetoric of former times. Thus we did not use a confusion of symbols to discuss these things as they were not in use then. Instead i attempted to describe, distinguish and demonstrate the notions to which i draw your attention, just as ancient scientists did, and consequently their rhetoric was misunderstood and lost in translation.

The Sutras allow us to apply Brahmagupta's advice  in this way and we see that a dynamic magnitude which emanates from shunaya is balance exactly by a dynamic magnitude which emanates precisely π radians rotated from it. Thus shunaya is Equilibrium of all things, also.

Every dynamic magnitude has a radial direction and a relative rotational magnitude to its balance magnitude. In a yoked pair this relative rotational magnitude is π radians, but as we now know there are more relationships than yoked pairs emanating from shunaya as roots of unity!

The important thing is that a dynamic magnitude has a radial direction and a relative rotation associated with it, and the vector notion is thus shown to be inadequate.

This was the conception allowed by Brahmagupta's attribution of yoked pairs to shunaya. Thi is a conception that Brahmagupta placed before us, but though original it is derived from preceding notions abour space, or shunaya from other traditions and cultures.

When Brahmagupta applied it to astronomy, he added additional rotational attributes to the yoked magnitudes, namely radial axis spin in seemingly opposite clockface  rotations. He also constructed a table of sines to help with his astronomical calculations and measurements, which i think like the reek was related to right angled triangles and chords, but additionally to rotating circle radials by Brahmagupta.
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« Reply #361 on: January 07, 2011, 10:26:44 PM »

Standard trochoid Use this web app to meditate.

Lazarus is not very communicative, but he is brilliant.

The boxes represent circles that are linked to one another, they are related, relative and yoked.

In the box the top control measures the radius rotation against the grey background. It measures it in revolutions. 0 means no rotation of the radius, 1 is the unit revolution, 2 is two revolutions per unit revolution etc.This measure affects the radius, and the radius is made visible by clicking one of the little squares in the top left hand corner.

The third control in the box, the lowest one is the circle radius measure.The radius of each circle is yoked to the centre of the next.The linking is supposed to be orthogonal, but of course they are actually generalised relationships, a type of Lagrangian set of constraints.

The yoke between the centres ha magnitude the sum of the two radii measures. The top control sets the rate these radii rotate against the background in units of per evolution, the app draws a rotating radius, or a rotating paraleleogram or a rotating hypercuboid  etc.

These dynamic forms are dynamic magnitudes that we otherwise call "vector sums", but this is an unnatural term which is best replaced by the dynamic magnitudes on screen.

The yoked relations are worth meditating on.

The very first new thing is that the proprioceptive mesh has a decision to make: which of all the sensory data does it take as the referential base?

Proprioception "draws" my geometry for me, it puts in place all the relations and connections in my immediately surrounding space, and ties all the other sensory mesh responses to this basic platform. Thus relativity is a choice of reference frame made by my proprioception. Once chosen the data is "snapped" to it to make a "sensible", ordered arrangement of space from that reference frame. As each reference frame is chosen the data is reorganised immediately relative it. Therefore i have to face a choice, a moment  of proprioceptive indecision. The organisation of the sensory data actually is changed by which reference frame i proprioceptively adopt!

I can see a spiral, a straight line, a complex rotating system, all depending on which proprioceptive response i take.
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« Reply #362 on: January 08, 2011, 11:55:09 AM »

Proprioception works like this: suppose I take a picture of a scene. If I take my picture, on my retinas as my basis I then proprioceptively step into the seen and attribute behaviours and properties to every object in the scene. I do this continuously through an iterative process of adjustment requiring me to compute scene changes and relations on a masive scale every moment. If I change my focus or attention, then proprioception takes that into account and provides valid scenarios for the new focus.

Sometimes the proprioceptively map does not quite fit the sensory data, and that is when I have paradoxes, illusions or misrepresentation of what is in the scene.

I may provide a perception of a scene proprioceptively which is lacking detail, but sufficient for purpose of navigation, situational awareness, echo location etc. But for detailed examination it is insufficient, and that is when I will focus on a small range of not FS a small region of space and enhance the data signal and develop a more detailed perception by iteration of the perceptual output . This new set of perceptions are organised into levels of percption so that I/ we can choose appropriately how much processing time we need to devote to recognising something.

Recognition then is perceiving at various levels of processing. Thus a perceptual process iterates through various levels adding more detail to the recognition and testinfpg the validity. Should the validity test fail recognition becomes confusion and should prompt reexamination at a slower and more detailed rate.

The rate of recognition is dependent on the numerosity of the perceptions and these are determined by the experience of the perceived, the fluidity of the processing state, and the decision/ control power the perceived has at the time. Certainly one can be more decisive after a rest, and with a clear head. So flooding the process with perceptions derived from sensory and proprioceptively meshes will not bring about quick recognition, but it may lead to the correct recognition with sifting of the data/ perceptual outputs or it may lead to new relations andn creative links.

There is a kind of homeostasis in our processing system, best expressed by occurs razor: the simplest recognition is the one we tend to use first. It may not be very good, but if it saves my life it is good enough!

Into this mesh of activity and perceptual behaviour comes the notion of method. Thus we develop individually a method of processing or managing all this computation that is unique to us. This method or collection of methods is then re iterates into the mesh and used in a self referential way to define our self identity.

Thus a particular perception behaviour distinguishes an individual, and the common perceptual behaviours distinguish groups of like minded individuals.

Thus scientists perceive in a particular iterative pattern, mathematicians in a distinct but similar pattern and so on for all the so called areas of study.

At the end of the day each one of us is a unique combination of perceptual traits that focus our thinking and behaviour in ways that are adaptive or maladaptive useful or dysfunctional, depending on the prevailing situation.

As my son is fond of remarking: if ever we get attacked by zombies he knows exactly how to save us all!


So relativity is firmly based on the proprioceptively perceptual mesh I use to apprehend a circumstance, and when I meditate on lazurus Plath's trochoids in motion I see this very clearly and intriguingly.
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« Reply #363 on: January 08, 2011, 12:43:25 PM »

In Lazarus trochoid browser app, the box contains from top to bottom

Radius rotator control
Radial ratio control
Radius control.


The radial ratio control is the largest figure in the box and it is the ratio of the arm link to the point drawing the trochoid curve to the radius of the circle rotating the arm link from it's centre.

This may make sense to you but not to me, when I look at Lazarus app. What I see is a White point with a grey trochoid path running through it which the White poit always stays on and travels round. But for some settings the point does not travel the trochoid curve does. So what is going on?

Well, so far I think Lazarus has used several output layers and combined them into one screen. Each output layer shows the result of different aspects of the calculation. Thus the trochoid is an out put of a normal java trochoid app; the radii are a drawing app that are timed to sync with the trochoid app, so that for each radial plot a trochoid is drawn going through that White point. The White point geometrically is very straight forward and is always the displacements from the pixel map of the radii. This draws a parallelogram which by the way is an arithmoi and in particular the aritmoi we use to define a vector sum.

So how does Lazarus get the trochoid to stick with the White point no matter what the trochoid is doing- and sometimes it is rotating relative to the background as well as the radii!

And the effect of purporting different aspects to different masks is that it is easier to see the absolutely wonderful relative symmetry of the whole system.

In this app the radial ratio determines the shape of the trochoid but not it's geometry apparently, and that is the strangeness. The definition I started out with does not seem to explain this. If the arm ratio increases should that not lead to a change in sizes/ magnitudes for a given radius? In short smaller or larger circles should be evident, shouldn't they? Or is there an alternative explanation of this ratio in terms of rates of rotation?
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« Reply #364 on: January 08, 2011, 03:41:27 PM »

Limits and continuity are related through ratio, similarity and magnification, in the following way: Proprioception and other mesh sensory systems compute a spaciometry. The system attempts to compute this efficiently and quickly and uses fast transforms of any changing parameters in a scene.

This is very similar to video encoding algorithms, particularly the VP* encoders by Google/ON2 , a brilliant codec developer swallowed up inside google somewhere!

Thus frame, frame rates, keyframes etc are all appropriate and proportionally applied to the discussion of computing a proprioceptive spaciometry of notFS.

So the limits that we inately expect are iteration or computation limits. This means that the mesh expects to compute a frame within so many iterations. When this is done the surface computed is sufficent for the purpose it was computed, containing detail or artifact def pending on the intentional teleos of the whole system.

 therefore like Albert we may at first see a lion, but then on a second look see a different form all together. (In fact this is the poem i remember a snippet from from my youth! I never knew until today!)

The limit of the perceptual computation is linked to magnification by unit size. thus the unit size allows the same computational limit to compute detail for a smaller area/volume.

As the unit size is gradually shrunk as in magnification the copputation smoothly adjusts, giving the ene of continuity and contiguity. However speed of change soon dispels this illusion, For very fast change we will experience a blur or even no resolved image at all. This shows that the actual event from which the mesh is calculating a response to is changing faster than it can compute , or there is no contiguity and/or continuity at that region of sensory data, or some intense convolution exists there.

Thus at all times our/my notion of limit and continuousness is a computational artifact of my proprioceptive and other sensory meshes.

I can reasonably expect my sensory meshes through the CNS and PNS to compute fractal scenes at any magnification, or unit size as long as the signal input variation is there for them to process.  
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« Reply #365 on: January 09, 2011, 12:34:51 PM »

Lazaus trochoid java app has at last revealed some answers.

In the top left hand corner above the circle boxes are some small squares which are controls for various settings. The first reveals a text box containing parameters and settings which i do not understand and cannot seem to change anyway. The next bu one reveals the radii and the geometriv magnitudes and forms, the next but one reveals the masks and some other aspects of the app, possibly surface shading? the next but 1 generates instances of circle boxes, that is you can place as many circles on screen as you like.

Now the masks reveal that the first control which i called Radius rotation actually generates a trochoid in black. The second larger control which i called radial ratio actually generates a trochoid in green and the 2 trochoids rotate relative to the background. There are some trochoid meshes related to the green trochoid which vary according to the diagonal measure of dynamic form produced by the radii(vector sum) and the white dot traverses both trochoids simultaneously!

Lazarus chooses he green trochoid to represent on the main screen/mask.

Looking closely at the trochoid mesh there are some unexplained short trochoid paths cutting across.

This still does not explain the arm ratio idea, and in fact it seems to support a geometric explanation of the path of the trochoids bases solely on rotation ratios and the dynamic diagonal magnitude of the geometric form.
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« Reply #366 on: January 09, 2011, 02:09:11 PM »

There is a curious attribute of rotatio of a form we take in our stride, but nevertheless is very interesting. As an object rotates it defines an axis of rotation. The notion is hardly explainable without a demonstration, and is an example of a recursive definition, a convolution or a tautology!

It seems everything to do with rotation is circular,but this notion needs to be avoided: we are dealing with closed loops or near closed loops of any path, we could include spirals, but we have to recognise the dependence of their definition on radials and closed loops. The circle is only one form of closed loop, and a very special one.

So we have an axis of rotation whnever we rotate that is apprehend a closed loop, and with that notion of axis is an unbreakable link to a radius or a radial line if this line is not constant in length.

So a closed loop, an axis of rotation and a radial line are tautologically linked to any definition of rotation. We simultaneously observe and appreciate measure and compute all 3 when we perceive rotation.

This simultaneity develops an innate sense of synchronicity. This relate Chronos to rotation. Their are other forms of simultaneity but unless they loop back they go out of observable range, therefore closed loop simultaneity is best for Chronos measurement.
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« Reply #367 on: January 09, 2011, 05:23:41 PM »

Chronos measurement requires a unit as does all measurement. In this case i can use a unit rotation around any closed loop. We rhen reference any closed loop rotations against this.

This of course produces scalar multiples, but leads to part rotation issues, and synchronicity issues. There are also issues with  regard to rotational quickness and constancy of quickness.

We apprehend quickness innately through our fundamental sensory mesh processing cycle frequency. Thus our basic iteration cycle frequency is an inate measure of quickness of change, and therefore all motion, including rotation.

Rotation cannot be measured by smaller closed loops, it gives the same result, and rotation frequency requires a unit rotation to establish, so the only thing left to measure rotationally is part rotations.

Part rotations lead to the notion of arc, not necessarily circular, which leads to the notion of angle between radials.

The precision way to divide angles into smaller and smaller parts has taken centuries to perfect. and is based on the circular arc.

With the angle measure established we can talk about angular magnitude as defining to rotational magnitude
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« Reply #368 on: January 10, 2011, 02:50:54 AM »

Angular magnitude can e aggregated as i have discussed in the previous posts on aggregation rhythms, but because it relates to a closed loop measurement of a dynamic rotational motion the mod() arithmetic is particularly appopriate.

The Babylonians used a mod(60) clock arithmetic and a mod(60)60n aggregation structure which include n=-1,-2 for minutes and seconds of arc.Thus there whole aggregation rhythm was tied to this rotational scheme by setting unit rotation=360.
Using mod(360) as a base it was easy to switch between everyday aggregation needs for commerce and astronomical aggregation needs. These astronomical needs related to yearly cycles of the sun, stars and moon and provided a strong astrological link to everyday commerce. and the notion of lucky numbers and good days and bad days on which to venture.

Today we tend to restrict our use of angle measure to a range of (-π,π] but there is no reason why we cannot establish an aggregation rhythm mod(π) and an aggregation structure mod(π)πn and use mod(2π) to factor out the rotations.

If 0 \leq \theta_n <\pi we can write \theta_n \pi^n for the terms.
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« Reply #369 on: January 10, 2011, 03:25:30 AM »

i have concentrated on the rotational and angular magnitude of the rotational motion, but have yet to deal with the synchronicity.

I can use a unit rotation as a frequency measure. One aspect of measuring rotation is to establish a radial as a marker an some reference external to the rotating radial and a counter to record the radial marker lining up with the external reference. This can be counted, enumerated, or experienced as a duration.

Duration is an experience which is proprioceptively induced. The normal mesh clock rhythm for processing data is attached to some simultaneously moving object by a process of reference. Certain proprioceptive responses require severs processing cycles to complete others are instantaneous. The sense of duration is therefore an artifact of the processing processes, which is associated simultaneously with some moving or changing object..

The final strange thing is the circumference of a circle. If as we do we have a radial arrangement of a disk of material and that rotates, it is hard to believe that it will stay together due to the increased circumference regions have to travel to remain in synchronicity.
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« Reply #370 on: January 10, 2011, 09:18:17 AM »

Lazarus Plath's circle box controls  use rotational frequency ratios to determine the shape of the trochoid. This relates precisely to roots of unity!
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« Reply #371 on: January 10, 2011, 09:59:39 AM »

This is not very clear, or thought through properly, but i sense that it may be possible to define synchronicity by a lagrangian set of constraints  for a rotating body made up of regions.

 the constraints would take equilibrium as the fundamental notion and the regions would rotate relativistically around  local centres of rotation, as well as around a centre of rotation for the whole collection or regions.

 a rule of contiguity would define the maximum displacement difference between a region and its local rotational centre before which rotational constraints could be applied to determine the local relative position, and after which the rotation around the body centre determines position of the region, or rather what i mean is the position of a region is determined by the dynamic magnitude formed by the body central rotation and the local centre rotation, with a cut off where the local centre no longer affects the position of the region because the body centre rotation creates to big a displacement, At that time the region is no longer conected to the body, as this is achieved through local equilibrium conditions, and therefor the body no longer has a rotational influence on the region.

At this radius the body simply breaks up into separate regions which may clump together hrough local rotational centre effects, independent of the main body.

This is a kind of gravitational system transmitting equilibrium and synchronicity through local rotational centres, with Lagrangian constraints.

I can see it but i have not fully described it.

The question is begged: what happens when  a region "breaks free" from a local rotational centre?  This till leaves the bodies rotational centre as the centre of rotation for the region, but without the constraint of synchronicity. This means that a region in this state is free to rotate around the body centre, but is notsynchronized to the centres rotation, so it has an extra set of degrres of freedom, precisely the "broken" lagrangian constraints.

However why would it rotate around the centre of the body? I cannot answer that as it is a mystical, divination question which is equally answered by: "equilibrium constraints" or "god would have it so!"

Newton chose a combination of the two and called it "action at a distance" and described how it works.

The Equilibrium constraint relates again to Shunaya, yoked roots of unity and rotational ratios, and therefore will be logarithmic in nature. Newton's estimate of the proportions was an inverse square "law"(presumably of god) of the radius yoke, but it is in fact a negative logarithm law of the roots of unity using radius rotation ratios of some form. Depending on the frame of reference the paths of motion of the "broken free" regions would describe trochoids .
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« Reply #372 on: January 10, 2011, 12:42:13 PM »

OMG! Lazarus has organized the control boxes in the circle box so they use the same radii. The rotational ratios are varied and that defines the trochoids for each of the 2 control boxes, but only one of the trochoids is displayed on the main screen and only one of the rotational radii frequencies is shown or picked out; the common one between the circle boxes not within the circle box. However the relative motion between the two trochoids is displayed, so the main trochoid rotates against the background, but by a function of its relative position to the undisplayed one.

Then Lazarus does something amazing. When a third circle is brought into play he calculates the coordinate or rather vector position of the centres relative to each other and calculates their common trochoidal path in 3d or rather in generalised coordinate representation.

It is brilliant!
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« Reply #373 on: January 11, 2011, 02:30:03 PM »

I aggregate to measure.

Thus all my aggregation, manipulation, summation and calculation is to measure.

My algorithms, mathesis , ruminations and procedural rituals are so i can measure.

All my tools, preparations and conventions, directed and agreed practices, points of view and perspectives are simply so i can measure.

What and how i measure then is not solely up to me. and carries with it a ritual a convention a history and a mathesis. It has adherents, proponents antagonists and pugilists, as well as heroes and sages, lords and serfs, in short it is a thoroughly human enterprise, all for te goal of measuring, and through that process manipulating what is going on around and among us.

It is to develop a proprioceptive basis to our understanding of our existential experience.It is to measure the mystical and mysterious and make it mundane.

But in our measuring we make one condition: our unit whatever it is must be fixed, not slippery, still not dynamic, stable not changing. And that condition, understandable as it is , fails us and dooms us to failure and approximation; or rather, more cheerfully, provies us a launching pad to iteration, and an ever changing reality of lights.

When we aggregat we aggregate visually, and that gives us spaciometric magnitude but only in part. The other spaciometric magnitudes are bi-aural aggregations, bi-olfactory aggregations , and if we would let them grow a bi-whisker aggregation.

The close circumspect circumflex-kinesthetic aggregation  contributes to a grounding of all the other aggregates of intensity- magnitude. So magnitude that we measure has form and/or intensity or unending  unbounded space, which we can explore and investigate forever!
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« Reply #374 on: January 12, 2011, 05:16:46 AM »

b              a*(a-b)/a
2b            a*((a-b)/a)^2
3b            a*((a-b)/a)^3
4b            a*((a-b)/a)^4
5b            a*((a-b)/a)^5    
 ....................
nb            a*((a-b)/a)^n

arithmoi          logos

The arithmoi is a geometrical form. In this case it is a rectangular cuboid, very thin, but increasing in length. It serves to record a regular plethoration of a unit 1b

The logos is the proportion written as a ratio.

Napier simply observes that a regular plethoration is in step with an iterative proportional decrease. The logos therefore had an associated arithmoi, it was the logos's arithmoi which melded to logarithm.

He had to draw attention to the arithmoi, because he noted that adding the arithmoi mutiplied the logos.

The idea of function is a later notion, but in everyway a table is the archetypical function definition, and Napier was developing a table. becaus of this he was able to develop a table look up notation that would define a function relationship in later eras.


Look up a certain value in the table and trace its logarithm:
 we write log(a*((a-b)/a)^n) gives n, and eventually over time it becomes =n


\log_{\frac{a-b}a}((\frac{a-b}a)^n) = n

In this case a-b/a= sin(π/2-\epsilon) \epsilon being a very small radian value.

\log_{sin(\pi/2-\epsilon)}(sin(\pi/2-\epsilon)^n) = n

\log_{1}(sin(\pi/2-\epsilon)^n) = n

sin(\pi/2-\epsilon)^n is used to work out a table of values for each n.

b              a*(a+b)/a
2b            a*((a+b)/a)^2
3b            a*((a+b)/a)^3
4b            a*((a+b)/a)^4
5b            a*((a+b)/a)^5    
 ....................
nb            a*((a+b)/a)^n
 Allows us to derive logarithms for any base =(a+b)/a .Thus for base 10 a=1 b=9
\log_{10}(10^n) = 9n

\log_{cos(0+\epsilon)}(cos(0+\epsilon)^n) = n

\log_{1}(cos(0+\epsilon)^n) = n
cos(0+\epsilon)^n is used to work out a table of values for each n.

\log_{cos(\pi/2+\epsilon)}(cos(\pi/2+\epsilon)^n) = n

\log_{0}(cos(\pi/2+\epsilon)^n) = n

cos(\pi/2+\epsilon)^n is used to work out a table of values for each n.

\log_{sin(0-\epsilon)}(sin(0-\epsilon)^n) = n

\log_{0}(sin(0-\epsilon)^n) = n
sin(0-\epsilon)^n is used to work out a table of values for each n.


\log_{i}(i)^n) = n would imply a-b/a=+&-1=i
and therefore b=a(1-i)

For  an arithmoi +&-1 means that the form is a unit square with one side +1 and an orthogonal side-1 this gives a unit area of -1 which is the yoked unit to +1.

Therefore using b= +&-1=i

gives (a-b)/a=(a-(+&-1))/a = a(1-(+&-1)/a)

If i set a=1

1-(+&-1) becomes the base

The basic algebraic rule separates different kinds of aggregated arithmoi

\log_{1-i}(1-i)^n) = n(i)

also for
(a+b)/a=(a+(+&-1))/a = a(1+(+&-1)/a)

If i set a=1

1+(+&-1) becomes the base

The basic algebraic rule separates different kinds of aggregated arithmoi

\log_{1+i}(1+i)^n) = n(i)
« Last Edit: January 12, 2011, 11:48:52 AM by jehovajah » Logged

May a trochoid of 去逸 iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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