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"I swear," said the sphere
"That my space is round!
But though i look
In every nook
This truth cannot be found!"

"I see your problem", said the cube
"And it, i think, is due
Perhaps an artefact, i say
Of a certain point of view!
Should you propose
To decompose
I think you will find its true."

Wear your cone!
You silly sphere,
Just like a dunce's hat.
To think that You could find what's round, like squareness
Just like that?

One learns from meditating on the odd couple shunaya and monad the structure of my model of the set notFS which I have called FS
And we propose it not as fact but as some concept more fundamental than that

That shunaya emits the roots of unit measure but not as some have taught. The rectilinear and the round cannot be found in shunaya except as members of a dynamic trochoidal class of rotating and rotational symmetries.
The spaciometric version of the roots of unity I will research by and by, but this we know
The yoke of roots of unity are their dynamic rotational symmetry, so that we might say the fluctuating void has rhythm, and swirling in it are rotations of a root of unity. But the roots appear and disappear through the resonance of their yoke, or rather stand as entities of a standing rotational wave.

Therefore the unity of our choice comes with it's own resonance and scalar aggregational rules, it's own combinatorial structure and it's permitted permutability. And it's ceaseless rotational frequency.

The rotation is thus quantised but as we know there is no basis to unity, thus we quantize it by our own sensors that measure, by our own tools that we use to measure.

In this way we feedback loop to ourselves and our unit limitations. The fluctuations of the void appear from our iterative convolutions in processing our signal response to shunaya. The roots of unity inform us that this fluctuation is rotational and trochoidal, combining radial and rotational, indeed defining radial and rotational.

Therefore we imagine the motions of bodies in space, and space itself to arise from these rotational symmetries, like traffic jams and synchronous swimmers, to appear from the relative synchronicity of yoked roots of unity, and to move in coordinated fashion as if a wave, switching on what comes before and switching off what falls behind: a kind of animation through pixel manipulation or rather spacel or voxel manipulation.

And so our prime elemental substance lies in the relative rotational motion of regions of space even below the Planck length, these relativities being perceptible to us through our iterative and convoluted perception processing as attributes to space.

All motion and all stasis relies on these fundamental rotational symmetries which are the roots of unity dynamically manifesting to our sensors in a vast computational flux.

"This", as the saying goes.." this is how we roll!"

Rereading Kujonai i realise that he proposed a tetrahedral tiling, not just a triangular one. I think this has been done in the tetrahedral folding of the mandelbrot thread.

In any case i reread Kujonai in the light of yoked decompositions of Shunaya.

When one has looked at the struggle of scientists and artists over the ages to measure shunaya it  is a blessed release to see all their work come to fruition in some elegant and useful form. It is a thrill to look back over the tortured mutterings of feverish minds and gay abandon of playful minds, to look a chancres and dancers, and singers and scientists, at astrologers and philosophers, at artisans and housewives at those in plenty and those in penury mendicant and monastic, egregious and generous,in short all and any human engaged in human survival, and see that they all have contributed to the formulation of the notion of the roots of unity as a measuring tool.

Thus it comes to this.

     is rate of rotation quantity x radian arc length quantity

And all are unit and scalars of units and yoked decompositions of shunaya into roots of unity based on the unit sphere, the dynamic unit sphere,moving relative to any other spatial form.

We need one element for this tool and that is a equilibrium reference from which we can construct all other equilibrium references of relative motion. This I have called a pole as in a pole or a pole star or an pole of orientation. This is a particular radial  of equilibrium in the measure or in the form being relatively measured.
 
The rate of rotation quantity is the root of unity quantity and governs the rotational symmetries of the dynamic sphere; in regards of unit: is to measure dynamic equilibrium rotational states. As with all unities  it is a set of nested yoked unities, but scalar ratios do not apply, these are roots of unity and so by definition are not scalable.

They can however carry a radial scalar that scales only radially, thus providing axial extensions and axial motion and taking orientation from the root of unity.

They also carry a special scalar of sorts called a radian, and this serves to transform one root of unity into another. The radian scalar is motion scalar dividing the rate of motion into smaller unities that relate by rotational symmetry fractal scales to the standard rotational magnitude.

De Moivre  formula gives us the roots of unity in terms of i  because all roots we may choose can ve written in terms of each other, because they are roots of unity.

So defining √-1 as +And- 1 and denoting it by i and laying out clear rules for it's usage we come to the quintessential algebra of spaciometry, to the realm of Wessel and Hamilton, the realm of quaternion algebras..

Many have gone on beyond this, but as pioneers not knowing where or how they would be going, not knowing that they were dissolving into the mists of shunaya.

The radian arc measure is a mod(2 pi) arithmetic but we can utilise other arc measure in a related way but of course not based on the unit circle. Thus a transformation exists through similar circles to any arc measure we require, based on mod arithmetics.

As simple as it is we have confused ourselves by using the notation with a multiple meaning. So y stands for a direction a value in that direction and a variable value in that direction! Thus  3 y we know by teaching is a scaled value in the direction of y  with the variable value of y being scaled.
3i however is different, and the difference is the additional rules we attach to i . These different rules relate to i being + and- in sign but in reality to i being a 4 th root of unity while other values are a 2 th toot of unity. We are not dealing with the same units, so as we know from basic algebra we do not add apples and pears.

We stick by this rule throughout all or measuring, so we do not add metres and centimetres or pounds and pence or even whole numbers and decimal fractions. We always only aggregate like units and we note next to them other units that we aggregate the result is a string of units across the page. The clever thing the Indians did was to make a pattern of these different units so that using mod arithmetic we could move to the next unit.

It is this pattern that shapes our world and our perception of it, for we see  wheels within wheels and structures within structures related by dynamic iteration and geometric convolutions. We see the structure of our fractal universe and we know that it is an illusion, a playful pattern of our choosing. We have other aggregate systems, and we can create yet others.

Choose your own and enjoy the ride!

prosthapahaeresis inspired calculators of all sorts, particularly astronomers,but more particularly John Napier, Napier was fond of playing and calculating,and very fond of trigonometry, spherical trigonometry to be precise through which he learned or rediscovered trig identities and the rules of sign. No wonder it appears to modern mathematicians that he was looking at imaginary quantities!

Among other things the notion of a ratio being viewed and dealt with as a "number" is introduced through spherical trigonometry.
"  Al-Jayyani (989-1079), an Arabic mathematician in Islamic Iberian Peninsula, wrote what some consider the first treatise on spherical trigonometry, circa 1060, entitled The book of unknown arcs of a sphere,[6] in which spherical trigonometry was brought into its modern form. Al-Jayyani's book "contains formulae for right-angle triangles, the general law of sines and the solution of a spherical triangle by means of the polar triangle". This treatise later had a "strong influence on European mathematics", and his "definition of ratios as numbers" and "method of solving a spherical triangle when all sides are unknown" are likely to have influenced Regiomontanus....."

Every aspect of complex numbers and sign rules are found in spherical trigoonometry. We also find Napiers interest  in calculation expressed by Napier's pentagon

".... To solve a geometric problem on the sphere, one dissects the relevant figure into right spherical triangles (i.e.: one of the triangle's corner angles is 90°) because one can then use Napier's pentagon:
 
Napier's Circle shows the relations of parts of a right spherical triangle

Napier's pentagon (also known as Napier's circle) is a mnemonic aid that helps to find all relations between the angles in a right spherical triangle.

Write the six angles of the triangle (three vertex angles, three arc angles) in the form of a circle, sticking to the order as they appear in the triangle (i.e.: start with a corner angle, write the arc angle of an attached side next to it, proceed with the next corner angle, etc. and close the circle). Then cross out the 90° corner angle and replace the arc angles non-adjacent to it by their complement to 90° (i.e. replace, say, B by 90° − B). The five numbers that you now have on your paper form Napier's Pentagon (or Napier's Circle). For any choice of three angles, one (the middle angle) will be either adjacent to or opposite the other two angles. Then Napier's Rules hold that the sine of the middle angle is equal to:
the product of the tangents of the adjacent angles
the product of the cosines of the opposite angles

The mnemonic for remembering the trigonometric function to use is that the first vowel of the adjective describing each angle (e.g., i for middle) is the first vowel of the name of the function. As an example, starting with the angle , we can obtain the formula:


Using the identities for complementary angles, this becomes:
cos(B) = tan(a)cot(c) = cos(b)sin(A).

See also the Haversine formula, which relates the lengths of sides and angles in spherical triangles in a numerically stable form for navigation......"

Napier found inspiration for calculation in trigonometry and using prosthaphaeresis. Because he was interested in astronomical calculations he wanted to work out the ratios for a very large hypotenuse right angled triangle giving him very accurate values for distances calculated thereby. He based his idea on rates of motion, because the idea was dynamic, not static, he was trying to show how the sine varies dynamically with change of angle

" John Napier, the Scottish mathematician, published his discovery of logarithms in 1614. His purpose was to assist in the multiplication of quantities that were then called sines. The whole sine was the value of the side of a right angled triangle with a large hypotenuse, say 10^7 units long. His definition was given in terms of relative rates. The logarithm, therefore, of any sine is a number very neerely expressing the line which increased equally in the meene time whiles the line of the whole sine decreased proportionally into that sine, both motions being equal timed and the beginning equally shift.

In modern terminology, L is the logarithm and X the sine. (In modern notation X would be r sin .)  "


We therefore have a clear and expected link between trigonometry and napierian logarithms which Cotes noted as

Changing the angle changes the sin but of course for a constant hypotenuse of 10^7 this changes the cos and the changes are proportional, but particularly identified by the sin. by inventing radianss Cotes was able to notate these relations succinctly and elegantly.

In Cotes form the i actually acta as an identifier signifying which variables are related and the + acts as an aggregate relation indicating a co-relationship between these variables and cos. In Cotes time despite Bombelli the use of √-1 was fraught with misapprehension, Euler often made mistakes using it, so for Cotes to perceive this relation before Euler is due to a different mentality, a mentality based on trigonometry not infinite series. In that mentality √-1 has the meaning of the sign value of sin as its angle rotated through 2π radians.

Thus as we know Trigonometry is at the heart of all methods of calculation especially roots of unity, logarithms and so called "imaginary" quantities.

It is these roots of unity and their absolute link to logarithms of all sorts that i think Muses envisioned as a new order of calculus.

Muses words. and his numbers

Of sines cosines and logarithms and spherical trigonometry in the measuring of shunaya by scientists of all ages. The secrets of the   Yoked roots of unity and how Napier played in the dynamic space of shunaya.

What shall we be riven with?
What nails shall we drive into our souls?
To pin us to the masts of ships
To fix us to their poles?
That commerce may continue thus
Unabated by the storm?
Nay!
That sailing through the universe
We may calculate it's form!

Only a passionate scotsman in love with his god would think to waggle a huge Bombelli vector in space just to calculate the sine for the benefit of his honourable astronomer friends! And only an intense irishman would follow his lights and make a 3d Bombelli vector that reaches not only outer space but inner space!