Universal Hyperbolic Geometry as Fluid Mechanics.
I am taking Normans course, because i am looking for the fractal foundation to Fluid mechanics. My intuition is that the arbitrary forms in Hyperbolic geometry are what model natural physical forms and behaviours. Thus replacing the straight line of projective geometry by the hyperbolic line, the sphere by the spheroid or ellipsoid i can synthesise what i can call technically hyperbolic trochoidal surfaces and forms.
Since these are generalisations of the usual Euclidean surfaces and forms many of the formulae will pass stright across, However, if i want to represent the hyperbolic in the Euclidean set up i would have to add or subtract corrective factors.
Normally we go the other way, that is to say what variation from the Euclidean to we need to get the hyperbolic. This however assumes the euclidean as the norm, which i thik is clearly not the case. Thus taking hyperbolic geometry as the norm i need to find out what transformations project it into the Euclidean forms.
This is procedurally me measuring with a hyperbolic tool sett. the natura; on, then measuring with a Euclidean tool set and noting the difference . Visually it is like projecting the observed motion on an elliptical surface onto a spherical sorface in order to understand why our equations etc are wrong!
"The forms" may not be different, but the data points may not match the predicted.There is a historical precursor for doing this and it is Kepler! He found the ellipse to be a better predictor of planetary behaviour than the circle! This is a substantial precedent and a big clue.
Many will point out that essentially this was the move Einstein made, to switch the fundamental Geometrical spaciometry! from Euclidean to nonEuclidean. But that is the reason why some do not understand it. There is no non Euclidean Geometry! Euclid did not teach geometry, he taught Philosophy. Thus All we do to measure in a hyperbollic geometry is to use a hyperbolic protrator, and a hyperbolic ruler!It is the same space, only our tools of measurement have changed!
Few take the time to realise even that our natural visual perspective is a projective geometry. Our natural thought processes include vanishing point and horizons and parallax due to perspective and focal point processes. Thus to have a spaciometry that does not allow perspective is bound to creat unnatural results and issues. Projective geometries are therefore fundamental to our apprehension of the physics in our experiences, and that naturally leads to hyperbolic geometries.
This is why Grassmann took an interest in Phorometry!
The importance of standard forms for logical structure goes in this wise. The infinite variety of form can be reduced to a finite few. Studying these few gives deep spiritual insight into the infinity. Structuring the few is structuring the infinity!
The essential spaciometric standards are the point, the sphere as a surface and a solid, the circular disc as a plane surface , the straight line as a special collection of points in the circular plane.
Combinations of these standards produce other standards, and all standards have a static and a dynamic form.. These standards do not exist without us to apprehend or create them with the tools implicit in their construction.
Many of us are mislead into confusing dimension and logical or systematic structure. That is because we do not usually think deeply about the fundamental sequential nature of our own subjective processing. Therefore we do not se how that leads naturally to categorisation and taxonomy. We use a fractal pattern to generalise or apprehend complexities. To illustrate: from the standard forms i may complexify by taking the solid object as a point! Theregore my next logical standard wuould be a soherical or solid formation of such points etx.
Once i have reached the solid of solids, i may repeat the structure again by regarding this as a point and structure or synthesise accordingly. Fractally it seems natural to call these levels or layers of fractality. Calling them dimensions just seems to mess with the head!
The use of the projective geometries as "vectors is something both Hamilton and Grassmann fundamentally agreed upon.
Further insights from Normans universal hyperbolic geometry course have lead me to understand the basic subject boundary called geometry as arising from Apollonius critique of Euclids writings in he Stoikeioon and Optics. Because Apollonius is some 50 years after Euclid , we see the effect of the Aristotelian peripatetic pedagogues on the interpretation of the Curriculua of the Platonic Academy. Essentially, each pedagogue and their schools argued their primacy, for financial and patronic gain. Consequently the shape of the curricula was determined by the highest bidder!
By the time Appolonius and Archimedes and Pappus came to study, there was a serious disconnect between those topics in the curriculum and their logical structure. Also the utopian ideal was hostage to fortune, and it was not "education to build a utopian state", but a more pragmatic education to get a state ahead in the game! . Astrology, even at this time was so specialist, and only a few positions with patrons or stipends would ever exist, that It was essential to aim the astrological education at things like land survey and weights and measures! Thus we see a pragmatic shift towards a mundane understanding of astrology, made glorious by Apollonius.
Newtons Geometrical frame through Barrow was thus Apollonian, not Euclidean.
The Apollonian geometry is implicitly dynamic. It represents 2 fundamental dynamisms: translation and rotation. In comics it does this in the most remarkable way: Apollonius projects the 3 d motion onto a plane! The projection is missed, because the notion of cutting a conic was first employed.. This notion reverses the action mentally, that is a plane is projected through a conic pair!
For possibly 2 millennia comics were viewed as cutting a cone. The alternative view of shadow casting a cone onto a plane was there, in Apollonius work, in the Indian mathematical works, but it's importance was ignored!. It was not until Newton that the dynamics of the comics were understood and utilised, and not until The Grassmanns that the Schatten or shadow projections of the comics were formally described as an Algrbra.
Hamilton grasped this aspect of Apollonian geometry and wrote a groundbreaking correction to a then extant theory in optics on "pencils" of rays. In so doing he introduced the word " vector" to describe the projection of these lines into another.
The word vector as a projection was misunderstood, it was so misunderstood that Gibbs was able to take it, and make a mistaken definition of his own, ostensibly based on Grassmanns analytical method and use. Historical documents show that Gibbs frankly admitted to not understanding what Grassmann was on about, and he seemed to have a flat out rivalry with Hamilton!
Both Hamilton and Hermann Grassmann knew that a vector was a projective product. A vector comes about as a result of a projection onto another line or a plane. But Apollonius also projected the circular arc onto a line ( straight or circular). Hamilton called these types of projections " versors". I have yet to come across these in the Ausdehnungslehre, except for the Senkrecht Strecken described in the Ebb and Flow paper, which are likely candidates, but not distinguished from the general " projection onto."
When the Apollonian dynamic geometry is viewed in this way, there is a distinction made regarding "projection onto" and "projection by" :projection onto is performed by rays of a "pencil" like light rays from a single point source( thus an optics distinction) falling onto a plane, while "projection by" is made by parallel lines falling onto a plane etc. for a long time these were distinct cases. They were united when the notion of a point at infinity became an acceptable theoretical, non Archimedian construction!
Once that was done Archimedian sense had to be made of it and that was finally achieved by Beltrami , and fully realised by Norman Wildberger..
The projection onto represents dynamically a rotation dynamic. Thus a point source emits rays in all directions by a implicit rotation of a single lineal ray. The projection by represents dynamically a translation dynamic. Thus a pair of parallel entities push another entity in their way along the psrallel , ( mutual) path .
The effect when both types of projection fall onto is determined by the surface they impact, but the translation is inherently shape preserving, the rotation inherently shape deforming.
While I do not subscribe to the fundamental forms being the point and the straight line, I recognise that the point and the sphere are the fundamental s, I have to acknowledge 2 dynamisms as also fundamental: translation and rotation! Without these 2 explicit action potential and kinetics, no sense can be attributed to a constructed reality.
Given something, and that something being as vagues as a fractal distribution of spatial regions as points, then I must also be given the actions of rotation of individual or collections of fractal regions, and also translation of individual or collections of fractal regions. Given this power I can immediately combine the 2 actions into a new action set called projection onto and projection by. With these givens I can develop from 2 arbitrary points a spherical structure that can define a circular disc plane, dual intersected points, and straight lines of such dual intersected poits ( or isos). I can define triple intersected points as where circular planes meet , and the notions of co planarity collinearity con currency . In rotation I can define "Ortho" in circular disc rotations, and from that a notion of perpendularity, which in der Raume is a physical propensity to oscillate to project onto some centre, but which in the plane is denoted as perpendularity or verticality.
There is more but the fact that Apollonius illustrated this dynamic scheme shortly after Euclid established his course, testifies to the lack of information that was transmitted to us . This is the more poignant as it became clear that the comics as a model of gravity falls out naturally and easily from Apollonian geometry. This is in fact what Newton proposed in some detail.
Today, Einstein proposed a hyperbolic geometry as a model for gravity, not realising that the 2 are the same thing, as Norman directly suggests.
My insight or hunch therefore that it is the geometry for fluid mechanics is founded on these meditations