Copied from my thread on the Ausdehnungslehre at Fractalforums.com just google “jehovajah whatever”
While I cannot quite frame it yet I have the notion of the logarithm of rotations. By this I mean that rotations may be described as exponents of the exponential function, and in fact must therefore be complex or quaternionic logarithms. Whether these can be extended to ndimensional Tensors or reference frames I do not know.
The structure of Grassmanns algebras allows for sums and products of these exponential forms as well as quotients. While Grassmann was at a loss until he researched Hamiltons Qaternions( only then realising he had solved his “looseness” problem for swivelling arms in 3d without realising) he later set a task for himself to do the incumbent processing to continue his planned development. However he died before he could make much more headway
The tantalising snippet from his proposed “Schwenkunglehre” whets the appetite and the imagination for more detail. The logarithm of rotations is precisely his idea of the log of a quotient. There he considers it to be the point of intersection of 2 lines forming the angle of swing. My notion is more related to Napier’s logarithms using the sines.
By the way, it seems clear that this name is misleading. These proportions are found in the sine tables, but they are true geometric terms, the angle is insignificant to the calculation in fact the geometric mean between sin90 and sin30 is sin 45 not sin60.
The logarithms do not index angles uniformly, nor indeed the tangent line. The concordance however is fairly accurate up to tan45, by then over 4 million calculations of the proportion had been done. To get to tan60 so as to calculate sin30 2 million more calculations needed to be done.
It is not necessary to perform the calculations with these constraints as Brigg showed, but in doing so Napier reveals how logarithms can be shaped to any scale or form, as they are only indices not measures. Setting them out in a measured way allows a calculation process to trace out a form by the logarithms.
As an example, placing the logarithms uniformly on a circle drives a rotation around the circle by an infinite iteration,
Napier also placed the proportions on a line parallel to the logarithm line. He could have placed them on the radius of a circle, starting at the centre. As the sine decreased so the angle of the radius decreases and the point on the radius traces out a small semi ctcle, if it is rotated by the sine itself. But if it is rotated by the evenly spaced logarithms of the calculations it will trace out an inward spiral.
The relationship between the trig ratios is extremely convoluted. It requires the subjective process to account for orientation, direction and direction of rotation. It has multiple concordances between ratios and many surprising and convoluted algorithmic identities. Accordingly it is a rich field for meditative exploration, and a school for concepts of rotation, reflection , translation, rotational and reflective symmetries and computation.
It is the computation or arithmetic which is the odd one out! The introduction of quantity into a magnitude is simply so we can make a song and dance about it, literally. The experience of magnitude is entirely subjective, to communicate to an external other we have to specify and bound a region, this quantifies it and we can then communicate that specific region to smother. Depending on the conscious process of that other, they may understand the quantity on face value or as a label for the experience of magnitude .
The quantification of a magnitude always introduces a difficulty. The form or magnitude in a form is as is. The quantity we introduce is totally subjective and arbitrary. Thus as we compare,count , distinguish thus generating a logos or language model of the activity of comparison, the dynamics of it, we have no way of predicting if the comparison will be artios or perisos , perfect fit or approximate fit ( even or odd, which I hope you can see is now inane!) . Consequently we have subjectively moved from an indeterminate whole without anxiety to an indeterminate multiple form with anxiety! Will the quantity specified fit?
These specific quantities are called Metria a single ( Monas) one (en) is called a Metron. The idea of singling “one” out ( ekateros) is fundamental to book 7 of the Stoikeia of Euclid. This Monas becomes the standard monad or unit for a process of covering (sugkeime) which is done by placing the monad down(kata) onto the form/ magnitude to be measured/ quantified/ compared, and counting( Katametresee). This count is literally a cultural song and dance, by which we interact with and order space.
The form so covered by contiguous( edge joined) Metrons as monads are experienced as multiple forms( pollaplasios). But in fact they are also experienced as epipedoi or floor coverings. We came to call these things mosaics . Archeologists finding these patterns on the floors in Mousaion, houses for the Muses coined this term.
The mathematical significance of Mosaics is a fundamental and continuing nalysis of the Pythsgorean school of philosophy. Indeed no Pythagorean astrologer could qualify as an Astrologer( Mathematikos) without a deep muse inspired intuition of these forms.
These mosaics did not consist of standard Metrons, ie a cube tile or a hexagonal tile, but of a mixture of tile or block forms that continuously tiled or blocked the space being compared. Thus the spaces were Topologically described and counted. Area as a standard concept of counting only standard monads is a much later idea and of a different school of thought.
Mosaics were aesthetically designed to inspire, and thus often depicted scenes as well as just abstracted patterns. Such patterns were often traces of shadow dynamics throughout the yearly cycles.
By introducing standardised Metrons, a standardised approach to topology was introduced before we came to realise how limiting that was. Also anxiety was increased because one form as a Metron does not fit all!. The proclivity for perfect fitting forms drives aspects of mathematics today, but it is perisos or approximate fits that these mathematicians see as monsters! These standardised multiple forms are called Arithmoi. Thus all Arithmoi are mosaics but not all mosaics are Arithmoi. The counting of these standard forms eventually became confusingly modified into the notion of number.
Engineers and architects however love these perisoi! These approximate fits are pragmatically engineered to construct or sculpt real objects and structures. Pragmatics chooses the best approximation for the task, nd filling snd dmoothing gives the final fom. It is artisans and engineers who apply forms iteratively in construction projects which are grand mosaics! We live and have our conscious bring in these grand mosaical structures of our own hands and minds. And we continue to process the experiences around and in us in this way.
As much as this is formal and subjective it is also our experience that magnitude is regionalised. The very deepest mening of this we encapsulate in the perfected magnitude, a formal creation, called the sphere.
There are 2 other formal creations hich result from the deep processing subjectively of the sphere itself, these are the plane and the later straight line. Neither exist as magnitudes in our experience. We formally construct these notions from regions, that is from plane segments or line segments. In fact it is clear that the sphere is a formal construction from an iterative process of construction requiring infinitesimal regions.
The complexity of the notion has fascinated ever since it was first conceived nd continues to this day. The sphere encapsulates all our notions of analysis and synthesis, all our methods or processes of calculus both differential,integral and logarithmic. All our conceptions of topology and finally all our concept of spatial mosaics.
Because we quantify and thus introduce perisos anxiety it is not surprising, after do long a time of philosophising about it that we should find some counts of seemingly unit magnitudes should involve an endless process. In fact Zeno and Parmenides drew pointed attention to this. The pragmatist had no problem identifying the solution, as do engineers. You embrace approximation!
At some stage you simply decide enough is enough! This is essentially the principle of Exhaustion! Motivating such a principle is not only tiredness but also a notion of cyclical count. This count, as a record of planetary positions became known as Time and is dynamically measured, by dynamically cyclical objects in motion. Such measures are called Metronomes!
As you can see the Metron concept underpins all our measurement, including dynamic ones.
Dynamic measures answer the Zeno Parmenides conundrum. An infinite subjective process of analysis occurs like everything else within a dynamic cycle. Thus unless we actually extend the analytical process into infinite cycles, we can stop at any cycle by design or exhaustion! In particular. We can note that Dynmical systems traverse thes infinite process measurements in cyclically finite ways! That is to say I can count a number of cycles and while thus distracted a dynamic object would have traversed a magnitude I was unable to determine by an infinite process!
The issue therefore is pragmatic. Is the infinite process necessary ? The answer is no, but to be able to be as ” accurate ” as desired or needed is necessary. The use of the term accuracy and exact is misleading. Simply we can choose the form as the standard and then it is exactly and accurately 1 !
The underlying process is a comparison. The count is to determine a ratio. The ratio is to be reapplied pragmatically and iteratively in some construction or synthesis process. We only need to be using ratios that do not ” fall over”, crumble or shatter under stress and vibration! In addition, if we can we want to use ratios that are aesthetically pleasing. And we want to do most of this within the dynamic cycles of a lifetime! Pragmatics and aesthetics govern many of our most fundamental processes.
Before I finish, it is good to observe that iteration of the pragmatically generated ideal forms is fundamental to our experience of change. Having devised these forms and reapplied them to interacting with space we are reminded of why we had to devise them in the first place! Everything moves! Panta Rhei!! Our anxieties drive us to try to keep things still, but in so doing we lose contact with real life experience. We Leo often kill the thing that caught our interest and so inspired us in the first place. However a compromise is to abstract by analogy a form and then use it iteratively to identify the dynamic experience. This is precisely how are neural networks work!
A good example is found in film or video capture. Each frame captures an analogous form to the real life obje t. As the cycles continue the analogous foms change and we thereby capture change by iterative analogous forms synthesised into a contiguous mosaic.of frames.
Grassmann in his analysis and synthesis intuitively understood that these forms found our notions of everything, and their mosaic combinations are the stuff of our Musings. He therefore worked very hard to establish a labelling system that made this very clear, and rediscovered a deep and abiding connection to the philosophical enquiries, observations and formulations of the Pythagoreans!
The heuristic, mnemonic and whimsical approach is actually psychologically consistent with the way we subjectively process our interactions with space.
The modern number concepts, devoid of this rich association actually obscure the natural human processes involved in the logos analogos response: how we language our experience of real life , and thus synthesise a language model of our subjective Kosmos!