END OF AN ERA, FRACTALFORUMS.COM IS CONTINUED ON FRACTALFORUMS.ORG

[f=0.1*z+i*(2*pi/3+0.0)
g=0.1*z+j*(2*pi/3+0.0)
h=0.1*z+k*(2*pi/3+0.0)
z=e^(f+g*2+h*3)+e^(f*2+g*3+h*1)+e^(f*3+g*1+h*2) +2*c

another crude implementation. Sculptures in julia and mandelbrot mode

Since my introduction to the Mandelbulb effort, when suddenly I had to decipher what this Iconic"set" was, and what  complex multiplication was, and what a fractal was, and hat a fractal generator was, I have always understood the sculptures S carvings of the quaternion block( inQuadz) and a kind of " negative" of the dynamics.. Here however is the most simplest and relevant conception of these sculptures: they are Chladni forms!

To me that is satisfyingly ignificant because it means the Mandelbrot set and the Mandelbulb definitely have an associated sound!

From the shape alone my guess is that it sounds like a gun shot explosion!

The other aspect of this relationship would be that they both have an atomic and or molecular analogue, some element or substance that they represent,

And finally they image the magneto Thermo sono electro complex which fractally distributed permeate what we call space!

Toward the physical Crystalnomics and  the geometrical doctrine of Combination
By Justus Günther Grassmann
First pamphlet,
 Stettin
By consideration of
Friedrich Heinrich Morin
1829

One of the striking patterns in the work and reasoning of Hermann Grassmann is the pattern of 3. This makes V9 a very interesting group to work with. In addition Hermann starts with the product and then identifies the fundamental elements that fulfill the product.

Quasz is set to make easy implementation of a tally mark( in this case one derived from a continuous measure rather than a discrete count) and an arbitrary  line segment , combined as a synthetic knitting .

This is often called a 4 dimensional object, as if it was outside our every day experience. In fact it is not . Think of a brolly and you essentially have a meaningful concept of a quaternion!

V9 can be thought of in a similar pattern . It would naturally sit in a 10 dimensional space, and exhibit complex vortex patterns.

Having read ahead I do not know if I want to commit the whole of the Vorrede to translation. In many ways it is similar to Hermanns Ausdehnungslehre, but admittedly a more plain verse style and accessible. In addition Justus is less intriguing than Hermann.

But experience dictates that if I miss something out i will only have to come back and do it again!

I have watched several mathematical presentations recently and find them very obscure and off putting. There is something about Normans presentations that are so appealing. Even when he is hurrying through pretty dire obscurity say in differential geometry I still learn something. Not so of many other presenters!

However with regard to combinatorics Norman has not done a full series on that topic so Fredrico is the best I have found, even if he is not so engaging a presenter.

It seems to me, and I have written on this topic in my blog, sequences and sequencing is fundamental to my interaction with space. Thus combinatorics of extensive forms is crucial groundwork to any topology and thus geometry and Astrology of space. On the flip side: these notions provide a framework for modelling intensive magnitude.

I already see the emergence of the Ausdehnung Größe in Justus development if Verbindungslehre.

Toward the physical Crystalnomics and  the geometrical doctrine of Combination 1829
-------------------------------------------------------------------
It itself opens a new Field of the spiritual developing,
 it yet not allows itself to overlook  extending magnitude and fruitable quality  of the Field,
but it  entitles to the most important expectations, that already now confront us in a concording neighbourhood of the same expectations:
 the representations of Spirit at the same moment as representations of Nature.

For this pure expertise,
which in general the space-like or geometrical combinatorial doctrine can come to be named,  
specifically,
are the Shapes,
which to us the Nature in the crystals proffers up,
only the resultant of the analysing of a singular free like comprehending personal exercise!

The  claims,
which in this declaration lie,
are all kinds of things
from thusly out-of-the-ordinary artform,
that only the material itself it   everyway  was pleasing to finish with rights.

But plainly this is also it,
 through which I myself hold everyway encumbered responsibility ,
they, howwholly with unified  striving against,
in their complete biting sharpness,
express out.

 I was believing to permit nothing to everyway give to the  material ,
 I was believing to have to bring to it itself my personal feeling
to the sacrificial offering to have to bring !
 In the flow  of writing of each artform,
which in un-overlookable crowds daily henceforward pressed themselves,
  the work-like new and fostered entity  
becomes  itself only  easy to overlook,
if it not
impudent and impolite ( cheeky and churlish) henceforward presses
and  to make itself "empowering" knows.

It processed itself here therefore around the handy entities,
and the valid quality of a new self standing mathematical expertise,
Whose goal it is,
simple and set together shapes to henceforward shout out.,
and their  melding together to show,
and which makes the claim,
all crystal shapes in a single personal exercise to comprehend!

If this comes to be publicly known,
thusly comes to be the criticism to the Author over his presentation,
and what he therein selected to fillet,
 indeed not like- valid, but yet from the most meagre  assigning to be.,
there no one can feel it any more as it itself,
how least it is until now slivered therefor,
and how  least of the least it  is
Here in the stand to give!

In which I overhere the criticism of the knower quietly expect ,
I do  yet hold
myself, in everyway encumbered with responsibility over
   the rooting and rising up and piecemeal  developing of my alongside aspect of
       the "here treated" content,
    thusly  how over :
     the contents table,
     Sharp-cut around and form of the before you lying content
     and the succeeding   booklet,
 myself in all shorthand to expound,
Themselves towards non finite times out of this   rooting and rising up
would like to output results

some unifying  not completely unimportant moments
 
to the appraisal of my endeavourings .

How I
through the involved working with the combinatorial doctrine
piecemeal to the super generator  ( of conviction) am come ,
     that  the until now notation of mathematic too strict  be,
     in which it rigorously taken the combinatorial doctrine has,
      which according to its  whole character  undoubtedly relates to the
        mathematic,
not with comprehending enfoldment !
have I
   partly at many places of the before you laying script (S.3,S.14-19)
   partly already for 12 years in the first volume of my space doctrine,
    which volume the uneveryway mixed geometrical combinations
    in the plane is  comprehendingly enfolding,
 expressed( S.10-12 the Foreword)

But Out of this outwardly widened label of mathematic outputs itself a completely other progression of Combinatorial doctrine.
Because if it the moment of the unlike quality Is, (inequality)
which a synthesis to a combinatorial entity makes,
thusly thereout follows, that also the companion ordering,
Distinguishing partitioning, nomenclature of the complexions in relating onto the same thing
 must happen/occur,  
and that thereby considering the tally mark of the elements
only in so far as keeping at the back of ones Mind
everyway serves
as it subordinates itself to that moment,
that brands, so far as through the tally mark of the elements
 a differing quality  comes to be henceforward brought.

This conducts then with necessary quality onto the developing concording to Forms,
and this shows itself as the completest forms,
yes singularly satisfying forms,
 thuslydirectly from complexions with unrestricted repetitions
 out of a finite crowd from elements
the discourse is.

Plainly this moment of unlike quality ( inequality) places the combinatorial doctrine in direct comparative situation against the arithmetic,
and travels it from the same in which, what to it innermost centring action property is,
completely not there(los),
thusly that it a centring activity property, from no other dependent expertise comes to be,
the plainly-of that comparative situation comes to be
but to wayfare
  a conscious parallelism with the Arithmetic comes to be claimed
and related by ancestry
synthetic entities become placed( cf.S.31 sq.)

Out of the outwardly widened label then follows further,
that also space-like contents,
which a differing quality allow to be  placed,
 for the combination  come to be needed,
but only in so far as can build a representation of a separated entity
or un everyway mixed entity  
with afar off self standing and centring action property part of the combinatorial doctrine ,
so far as this differing quality yet from that magnitude everyway arrangement independent is.

This remarking ,
that  Must itself allow to be created whole upon this way  ,
   the central like material of the geometry
    in lines-everyway-knitting entities and figures,
everyway bound with the  remarking,
guides towards the geometrical combinatorial doctrine,
how such a doctrine in its first initiating grounds in the above named writings
as a fore preparing ( a priori) to the geometry is presented,
and I have pulled myself over,
 siding to the entity many times ,
that only therethrough
can come to be   realised by work,
the henceforward stepping entity
    which in the geometry tracked lines-everyway-knitting entities and figures  
       the   seemingly arbitrary and surprisingly strange entities accepted,
        the stricken companion ordering of the geometrical propositions
                 as a natural and self necessary entity from the Initiator Apprehended,
and
  a complete settling of the claims of the everyway standing entity
alongside the System of the geometry .

Commentary on the crystalnomics foreword.

There is a lot of in these verses of plain song or plain verse. Justus has had 12 or more years to craft nearly every verse as his response to the lack of interest or understanding of his major work the Space Doctrine!

There are many literary and historical allusions and words have a double or deeper meditative meaning.

"Vene,vidi,vici" for example is alluded to in how the strict labelling for combinatorics has led to a major mis-comprehension of how combinatorics actually enfolds lines and lineal figures and forms in space.. Justus came, he saw the mis-relation and the labelling took the combinatorics captive! " gekommen,gehört, genommen".

There is also an allusion to the revolutionary times or the Moment in which he lived. The reference to the moment of Inequality draws on the spirit of the times, the French revolution and the Prussian Spring or Rennaisance. The Humboldt reforms if the Prussian Education system were in full Swing and Bismark was busy consolidating his power base and industrialisation programme to rebuild Prussian pride and dominance on the geopolitical stage. So an opportunity presented itself for alternative views to be looked at to rebuild the self actuating capability of Prussian children and scientists.


As a loyalist Justus did his part and waited patiently to be heard, but his work was overlooked! And so it remained for nearly half a century. If it were not for the loyalty of his sons Hermann and Robert it would have been buried in the historical turmoil.

We shall see how doggedly he pursued his conviction that combinatorics of the line segment enfolded in the plane was the key to setting geometry on the correct basis!

As a child I played with all manner of bricks, rods and sticks and mud. How I combined those materials was intuitively understood to be important to an apprehension of geometry. However nobody study the rules of combinations directly related to such materials. It is that aspect that Justus picked up on. Their is a centrally acting property to the combinatorics of spatial objects that is analogous to arithmetic! Indeed arithmetic derives its being and rules from this a priori property of combinations of space-like things. The trick is how one teases that out of our subjective processing! It was not easy!

Again at this stage I would remind the reader that the line segment should necessarily even principally include the circular arc segment. The predilection is and was to exclude the circle from these initial concepts, but in fact an examination of the Stoikeia does not justify this. Historically we may suspect that Apollonius ofbPerga criticised geometers for doing just this and redacted the texts accordingly.

The unending line and the unending circle are a fundamental analogy, by which all counting and measurement is ultimately deriveable.