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There is a fundamental realisation now apparent in spaciometry which is that boundarisation is a fundamental element of the topic. I have mentioned that rotation does not exclude translation, but it is now obvious that every translation is bounded by a rotation. Thus there are these fundamental motions in the topic of spaciometry : rotation and extension but every extension is a part of a spaciometric rotation. This is not obvious if one uses an abstract entity like a Euclidean line but in spaciometry such things are abstract tools. Thus any real spaciometric form is bounded and therefore within a spaciometric rotation. The idea is useful in that motion  especially relativistic motion does not then have to be perceived as motionless just bounded. Thus relativistic motion forms that is relativistic forms are bounded by a rotational motion which I have called a boundary. When I say rotation here I am saying spaciometric rotation, but the analogous idea needs to be explored by myself to find out how much this is already in use in mathematical activity. For example I can already see a connection to the Heine Boreal covering theorem,and certain discs and spheres used in Hausdorfs dimension theorems.
which leads me to suggest that  a serviceable theory of everything formula will naturally arise from a reference framework that emphasises rotation and extension. Such a non cartesian framework already exists and i think to pursue its development in spaciometric terms and in relativistic motion terms may be useful to my quest for the foundations of mathematics.

I want to define a spaciometric locus or path that is a "curve" in a general sense not specified here.

The locus is a path that apprehends every boundary of a spaciometric form in sequential manner. this ;ocus will therefore by definition be a complex apaciometric rotation in general. I will call this a spaciometric vortex locus.

Some examples of a spaciometric vortex locus are a spiral and in particular a vorticular spiral.

While a vorticular spiral can encompass a simple form it may not deal with more complex shapes like trees etc. A combination of spiral vortices  may be needed to proscribe a complex form.

However a peano curve may be a special example of a vorticular spiral that can deal with complex spaciometric forms. Now i can define a length along a spaciometric spiral that proscribes a form as a spaciometric angle measure.of that form.

Thus a spaciometric spiral will be my tool for defining an angle measure in spaciometry where it makes sense, and i will take whatever element of a spaciometric  spiral necessary to make a useful set of distinctions and ratios.

So it becomes apparent that using a vorticular spiral locus as a tool to reference frame a spaciometric form is only the beginning. A vorticular spiral is an abstract tool, which in all instances would lead to an infinite length in being applied to a spaciometric form. In practice i wold only use a finite lengthed vorticula spiral to reference frame a spaciometric form. But now it seemed to me that i was attempting to "wrap" the vorticular spiral around the form and i already know how to do this because wrapping spaciometric forms is a common activity, especially at christmas and birthdays!

So i realised that i could use a finite spaciometric surface to wrap a spaciometric form following a vorticular spiral locus. This wrapping would cover more of the spaciometic forms surfaces in a finite  lengthed  vorticular spiral  and would be called a vorticular spiral wrap.

Thus i defined the process of applying a finite vorticular spiral to a spaciometric form as wrapping the form;  the surface that is used to wrap the form as a vorticula spiral wrap; and the adjustment of the surface dimensions to achieve the best wrap as a process of optimising the wrap, that is finding the minimum finite dimensions for the maximum finite surface area of the form.

This is a bit like Dirac's instance integral in that the vorticular spiral locus wrapping the form will be the limit case of an optimisation process which may always have the same known value.

Now the theorem would be that every spaciometric form has a finite vorticular wrapping for each relativistic point of view.

It would then be nice to show a uniqeness of an optimised vorticular wrap for a relativistic view and a fixed vorticular spiral locus.

Now the sweet thing: a vorticular spiral locus and thus a vorticular spiral wrap is a relativistic reference framework for the form. This relativistic reference framework could totally represent the form. and in general all spaciometric forms could then be represented by vorticular spiral wraps.

This then means that any general equation is only dealing with one spaciometric structure for all forms in spaciometry: a vorticular spiral wrap.

Once the properties of vorticular spiral wraps are explored i imagine that they could be easily applied to all scales from the quantum to the classical, and would represent quantum characteristics at small scales because P(r,r/n) would characterise small wraps and P(r,n*r) would characterise large vorticular wraps where n is an increasing rational number.

Now i had some difficulty in visualising the construction of a vorticular spiral because i wanted to refer to a spaciometric form that was nascent not preloaded with presuppositions like the circle form does for angle measure. I know that i cannot stray to far from the trigonometric ratios in their fundamental descriptive power, but i did not want them to be apriori to the vorticular spiral. The solution to this for me is the use of the cone shape as an apriori measure just as the orientation sphere is the apriori measure for all orientation and angle. I will call it the spaciometric orientation cone from which vorticular spiral wraps may be determined. Of course cones are tangential to interior spheres the tangent forming a circle which lmplies trigonometric ratios are inherent in the cone measure, and also extends to ellipsoidal measures of ratio for vorticular spiral maps. The full conic sectional resources are accessible through this measure.

Field equations would then only have to be developed for vorticular spiral wraps to describe relativistic motion and relativistic motion transfer at any scale within any vorticular spiral reference frame, and the self similarity within the reference frame will be built in along with the iterative spiral process which to me means that the equations will produce and capture the  truly fractal nature of all things.

i posited once that all motion could be understood as traveling wave motion, and that inertia would relate to the establishing of a standing wave in an object to which momentum was being transferred, kind of like a latent heat function found in phase change scenarios. I did not know what i was talking about then and i do not know what i am talking about now   but i hope someone else does and finds it useful.  

OMG cones are the business! If i draw a radius from the apex to the base of the cone and use this as a meridian i can spaciometrically rotate around the cone by : a circle, an ellipse, a parabola followed by a negative/ opposite parabola and 4 hyperbolas  each on an orthogonal plane to its intersecting neighbour. with parallel planes having the same hyperbola, and of course the spiral. Each of these rotating motions represents  periodic locii.

I can imagine these motions in the clock, the planets, the pendulum and the jointed pendulum, and in the watch spring( an old fashioned motive force) .

As to my exploration of finding the length of the spiral. A spaciometric investigation of the cylindrical helix led to a sketch of the conic curve shape, slightly different to a right angled triangle base 2π with a height that is the pitch of the triangle. Once  had identified a rough formula and a rough shape i found myself in the kitchen stirring rice pudding.

What's the best way to stir a pot? Well i have done this exercise many times over the years and am of the opinion that to stir in a circle is the least effective . My personal choice is to stir in a grid pattern. However today i had spirals and spiral wraps and vorticular spiral locii on my mind. The vorticular spiral locus into the centre of the pot and then out created a satisfying wave effect on the rice pudding surface as it heated up, but that is too much effort to do for long. i went for my simplest counter stir: semicircle, cross the pot then opposite semi circle, a figure of 8 type stir. Suddenly it reminded me of the yin yang symbol, a symbol i have already associated with vorticular motion. That created an interesting pattern on the rice pudding surface.

 Suddenly it dawned on me that this was like the triangular pattern i was looking for to measure the length of the spiral . After i dished up the rice pudding i considered the spaciometry of the yin yang symbol. it was then easy to see the spiral "comma" forms as two rotationally symmetrical nets for a cone!

The ratios were 1:2 for each net to the whole mandala and 1:1 for each net. In addition each net curled exactly up into a cone with each part of the inner S rotating into its rotational counterpart and the long tail curling in like a cats tail to finish off the  base of the cone.

I looked and i did not find this any where on the internet. So any chinese/ japanese monks who can tell me if this was already known  in their traditions would be welcome.

You see quite simply what i have is a net that gives me the spiral length simply and the two cones of the conic section models.

Check this out !

And in addition i have pointed to general reference frameworks being, in general, tensors. I suspect that not all tensors thus defined will be invariant under transformation, so i introduced the names only for two more types of tensor Flexors and Growors. The intention in the names are clear i hope.

According to the way i am pursuing this, i have now my standard vorticular wraps in these Spiral Tensors and the length along a spiral is the Tensor reference system which will then uniquely determine a location in space relative to the spiral form the, observer, and the conical orientation standards.

I cannot avoid the connection to DNA as a reference framework for biological structures and so i will refer to these vorticular wrap tensors as DNA tensors.

A note about my language and style.

Some of you guys and gals  may be put off, not only by the word maths, but also by the way i put words togeher, and "needlessly" complicate words instead of simplifying them. And also apparently i might be a bit of a Mr. Micawber in these tendencies. To you  i apologise.

However that is not to say i can change my style, because as i frequently point out much of what i write i do not myself fully understand. So while this is not quite mediumistic automatic writing, or dictation from spirit guides, etc it very nearly feels like it sometimes!

I explain it as a kind of channeling from the universal computation that is going on all around us to which i contribute my part by thinking. Garbage in garbage out applies here. So if i ask the right questions my brain is able to access and download the right kind of answers and process it into a form i can understand and utilise. It just so happens that efficiency demands the most information in the shortest form which unfortunately means very concise and terse language forms with as much redundancy squeezed out as possible.

As a case in point: from my understanding or rather apprehension of english grammatical forms there should exist a word for vortex whirling motion. I decided it would be "vorticular". i had no idea if the word existed or not it suited my needs, and after all these pages, although they have an open invitation to all to comment, are in the form of a journal, a diary of notes to myself.

So i will leave it up to you to find out if the word exists or not, but i could equally as well use helix, helical, conical helix, vorticose, vorticial etc. Words are tools, however, and the particular forms i choose represent a focus of investigation. And i am investigating and exploring vortices.

Another use of the yin yang mandala is to describe earthquakes. If i extend the yin yang using the semicircle spiral pattern and then join the two centres of the yin yang by a line, i can translate the two parts of the diagram along the line. So simpy i have split the yin yang by translation and made the nested semi circles obvious on both sides of the line.

Sliding these two sets of semi circles past each other enables me to create different spiral patterns.

the relation to earthquakes is the circular radiating compression wave sliding past each other either side of the slipping rift.

The ying yang 3d television system: first have a binocular camera system with each lense scanning 180 degrees out of phase onto a yin yang camera sensor. The arrangement of the pixels on the sensor and on the tv screen will  be yin yang

                r         g  b
               b g         r

The binocular signal would raster across the screen in opposite directions so that the shortest light path favoured the requisite eye. With tuning of the raster speed, the pixel distance for the yin yang screen and or camera sensor , and testing for epilepsy should produce interesting diffraction patterns that the visual system can interpret in 3d.


Now the paths around a cone can be recorded in a spiral wrap of a spherical object. The paths are s  p and d, and i would venture a fourth one called e.

The s is for a circular path and e is for an elliptical path. These would represent the s shells from the Schroedinger wave equation solutions.

The p is for parabolic and this would represent the p shells from the schroedinger equation solutions.

The d is for hyperbolic and represents the d shells

Finally there is the spiral which i have not found an analogue for in the solutions for the schroedinger yet but would guess at a spin solution. The spiral on the cone would project onto the sphere as a loxodrome.

 A yin yang spiral group would wrap a sphere, but orientation and spiral chirality are important on a cone, and the spaciometric vorticular spiral wrap produces a kind of inside out cone as well as a more familiar cone.