I love it when plan comes together and bers fruit. I might say, plant, as it was while decomposing a plant into its elasson, its essential meros that it all came together!
Arithmos is that part of space used to fit to a form. The essential idea is to construct or manufacture a form from its constituent parts, its essential building blocks of space. Arithmoi are used to show how "it" all fits together. It answeres the question" how do i use all these forms and theorems in the first 6 books of the stoikeioon to build things in the places and spaces where i live? HOW DO I MANUFACTRE A COPY OF A PLANT? What is a plant a multiple form of? Which of the shapes i know and understand theoretically can i use to construct a multiple form from that will give me a plant form, and more importantly the attributes of a given plant form?
In book 7 Euclid addresses this question generally. Aris mezoos means to fit it all together into a harmonious form, and arith moi are hose parts used to do that. Ar comes from the idea of grasping , and thus joining together flows naturally from its etymological development. We see it in art in arthritis in articulate and in area. The connected, joined whole, emphasisng the manufacture, the manipulation of constituents into a joined up form.
The point is, Euclid gives a general method, or praxis or mathesis to synthesie any perceivable form. This method is a gematria of the arithmoi of which we have only been taught such part as our tutors understood or thought relevant. And yet the very method is as natural for a seamstress and a tailor as it is for the mosr erudite scientist to apply.
This Pythagorean derived method,as filtered through the mind of plato is the basis of the whole of this worlds technological achievement, the foundation of any research and development and manufacturing process, the basis of our discovery of and apping of the human genome, and the manufacture of genetic products.
So the Pythagorean Theurgy has had a breathtaking effect on this world, as the Pythagorean Philosophy has reached out to encompass the whole world. That is not to say that other philosophies have had no effect. It is to say that they have all had an abalogous effect, but without a doubt, the framework of the pythagorean is able to contain them all, but not without subtle modification to them. Other Philosophies serve to give deeper insight into the pythagorean, not to supplant it, while it eiher competes on equal terms or utterly supplants all others!
So for my plant i chose a monad that consists of a tube with a leaflike surface extending out all the way round, This leaflike surface is able to curl to form a type of tube in addition to the resident tube. Thus gy placing this basic monad in various rotate arrangments i am able to build up a leaf structure or the grain structure of a tree stem or trunk by folding the extra surface into another tube.
Now some may be aware of the iterative nature of certain plant structures, Thus a simple iterative algorithm may be used to create a leaf or a stem bundle which will develop into a tree structure with all its roots, trunks, branches twigs and leaves.
The simple significance is not in that this is not how a plant like a tree grows, but in the fact tha a model of tree growth can be developed using the Euclidean general method of the Arithmoi.
Now , in addition to what is in the books by Euclid, it has become necessary to notate certain actions otherwise taken for granted as commonly understood, or left out by translators who did not understand the significance of the distinction. For example the fundamental inportance of the gnomon for forming rectilinear and linear forms is subsumed under the notion of multiplication, but not properly or completely. The general gnomon is not acknowledged, but rather the right gnomon is usually spoken of,the rejection of neusis, the abstraction from meros and pollapleisios as forms to mere scribbles on a page or puffs of the sounds of words, these things have , in the absence of the theurgical philosophy,steered us wrongly in our thinking.
Fortunately our science has been shaped by empirical interaction with space and this has thus borne testimony to the original conceptions of PYTHAGORAS. However, the blind, being lead by the blind has often lead them both into the ditch
Bombelli,by going back to the "fathers" reintroduces neusis and thus partially "solved" a problem in renaissance understanding of spatial measurement, √ of a quantity tha was "meno" or negative. It was not a real problem, just an ignorant use of the gematria of the greeks and the indians. Today we rehearse this misunderstanding in the ears of our children and confuse them when there is no need. We have a vector algebra we do not need a "sign" mark anymore. We need to embrace the neusis and the rotation that are obscured by this convention and straighten ourselves out mentally!.
The method of the arithmoi is to look at ways of combining appropriate arithmoi to make forms, and then to apply the laws of multiples which include rotational addition( vector combinations) and relativistic identites and ratios. For each form we will find a combination that will aggregate to the structure, and tha combination is usually called a product, but lately an algorithm. These algorithms form the products of the arithmoi.
Today the advancement of computing has placed the power of the iterative structure of nature, modeled in the arithmoi method of Euclid, at our finger ips. Now we can not only draw but really model an object, or any thing thought of as a structure,by truely iterative means that show natural growth behaviours, natural relationship behaviours and with physical attributes like objects in our objective world. CGI and fractal generators are key to this development and understanding.
When i watch a film like Avatar, i reflect on how far the Pythagorean dream has come. When i stare in amazement through the Nintendo 3ds at augmented reality in 3d or at an animation showing me ow a chain reaction is understood to work in the blink of an eye, or a run and rerun a video of a slow motion crash i have only an inkling of the yet untapped knowledge to be revealed by the products of the arithmoi.
Vector products, differential calculus algorithms,matrices are all special algorithmic methods developed to deal with the structure of the iterative calculations necessary to compute change. but the form itself is a simple linear combination of arithmoi. The two parts the computation of change and the assembling of the srfaces have particularly revealed a fascinating reality in the Mandelbulb 3d algorithmic strucures. The animation of these surfaces developing also have revealed suprises.
It is indeed a brave new world full of the harmony of the Spheres!