Rotational waves


Lord Rayleigh ( John Strutt) made some influential notes about wave motion throughout his life. Bearing in mind he was born just before Quaternions were announced and Grassmann published his Ausdehnungslehre to a dismal response, and was in university at Cambridge about the time Maxwell published on Electromagnetism, using Quaternions and MacCullaghs curl potential, we can see he was right in the thick of the rests early attempts to model 3 d rotation mathematically.


It was really down to a few doughty souls to progress physics of the wave to its prominent position vis a vis the corpuscular dynamics of chemistry, which was making noteable headway in the industrial setting.


We have seen how Arago and Fresnel created a huge rift, with young , in the philosophical explanation of matter in the aether or plenum. While Newton provided a consisten theoretical model based on corpuscles , it was evident that it was not physical or empirical. At the same time the Wave theory was not physical with regard to light. Youngs experimental double slit interference patterns were not convincing enough , and it was the influence of Fresnel and Arago that enabled the results to make headway in the broader scientific, non chemistry based community. These tended to be more mathematically minded scientists who could understand the sine  graph, intruded by Euler as a model of a wave.


The notion of a wave is very rarely examined. One is usually immediately programmed to consider the circular functions of Euler as a wave. Thus a disconnect with physicality is immediately taught. Scientists no longer see any real wave, but rather approximations to the ideal sine graph! However in this process the ideal sine graph is misconstrued as a wave and so it’s true meaning is lost even as it is plainly laid out before the students eyes.


Firstly let us remove the blinkers. 


Euler took a circle of unit radius, that is its radius was defined as 1. Then he defined it’s semi circle or hemi arc as [tex]\pi[/tex] to about 30 decimal places. Thus he was able to draw an axis marked off in units of [tex]pi[/tex]. Thus this axis represented the rotation of a point around the circle or the motion of the centre as the circle rolled in that axial direction . In each case the circle was in dynamic motion called rotation.


Thus the sine graph represents not a wave motion , whatever that may be , but a rotation motion.


Now let us turn to wave motion. It must be observed that wave motion, vibration and periodicity are tautologically the same perceived behaviours. Any difference lies in the observers intention or purposes. Thus in the context of a sea wave the perception of a rolling body of water traversing the surface of the sea and rolling out onto the beach gives way to the undulatory motion of such waves on the personal stability of the observer. Indeed the bobbing motion of floating objects predominates over the passage of a rolling wad of water beneath ! 


Waves are observable on the surface of flats flowing rivers, but there the current predominates the observers senses and little mention is made of them. So what are the causes of these mounds of water in the surface of a dynamic fluid? It turned out not to be bobbing at all , but complex vortex behaviour. Both Lord Kelvin and Helmholtz regarded this as a groundbreaking phenomenon and they set out to describe a kinematics of vorticity. A first attempt.


This was a major influence on Stokes, Navier and Rayleigh, but Maxwell was conceptually in advance of these 2 great mathematical physicists. He wanted the vortices to act like gears nd springs and transmit strain. He opted to use Hamiltons Quaternions to express his ideas. Lord Kelvin was not amused. He like many scientists in his time felt this use of the imaginaries was Jabberwokky. A term coined by Lewis Carol, a prominent traditional Mathematicin, who derided this kind of Alice in wonderland mathematics in his book of the same title.


Consequently Maxwell was forced to recent, and in a remarkable turn around went from prise of Quaternions to a dire denouncing of them! This was at the behest of Lord Kelvin who was developing the ideas of vectors set out by a young American student of thermodynamics called Gibbs. It is a dark but not unfamiliar tale of underhand tactics. As a result, overnight research into Quaternions was shelved in America after a fateful conference on the issue of how physics should be taught. 


Maxwells statistical approach to gases suited Lord Kelvins own Kinetic theory and so statistical Mrchanics was developed by Gibbs to great effect, but the mathematics of fluid mechanics and ths Elrctromagnetism based on that floundered. This was because Maxwell expressed all the main concepts in terms of Quaternions. The fledgling vector algebras were not sufficiently graped to be able to compete with this elegant description. In addition, the Curl of a vector field was developed by McCullagh a mathematician in the same tradition as Hamilton, who used Quaternions to formulate his ideas, and the relationship with Knots and the properties of vortices in space.


The second tautological concept of a wave is periodicity. Thus when we experience the unwise everyday we apprehend periodicity, but hardly intend to call it a wave! It is clearly a rotation which involves very large scales of distance and time. Nevertheless we have to cknoledge that repeated variation which immediately makes it sn logos to regular bobbing up and down as in wave motion.


Periodicity reveals to me the essential rotation that is evident in a sea wave is lo evident at a much larger scale in astronomical terms. Astronomers since Eudoxus have modelled these circular motions to give. Apparent relative motions of planets. These motions were very wavelike and hence planets were called wanderers!


We now know that our solar  system wanders in the milky way galaxy on some spiralling rotating arm of the galactic structure. This wavelike motion is on a time scale of tens of thousands of years and on a displacement on sn astronomical scale .


My third example of the notion of wave motion is vibration. Typically we think of a piano string or a washing machine . We are told to think a piano string vibrates up and down. In fact it vibrates round and round! Despite precise plucking or striking the mechanical behaviour of taught wires in vibration is rotational. These rotations may be elliptical rather than circular but they are not up and down like a slow moving tension curl in a skipping rope.


While it is always possible to dampen the elliptical motion ofa vibrating string by placing constraints, this only emphasises the point. Vibrations are helical waves travelling bidirectionally in a tensile medium.


It really does not matter what scale you go to vibration or wave motion is due to rotational motion .


It is clear that rotation at any scale is almost similar. Thus we can expect the same mathematical formulae for wave motion to apply at ll scales.


Schroedinger’s wave equation is simply derived for rotating systems at ll scales. The idea that an atom is a planetary system look alike makedps this expectation almost inevitable. However we must not confuse rotation with planetary systems. A much more general graph of a rolling circle is called a is complexes of these that better describe arbitrary rotation in space. We shall see that means regionality is inherent in rotational motion, as is integer relationships between regional complexes.


These regional complexes define a fractal Geometry and a fractal distribution


<From topic Light page 11>


Why Rayleigh waves are the general wave wave mechanics

This is a 2d Rayleigh wave. We will see how a Rayleigh wave is the general wave notion we should use in physics and how it avoids the ultraviolet catastrophe.

This is the math that Rayleigh used with a colleague to derive the ultraviolet catastrophic equation. Notice it does not use Rayleigh waves.

The Planck derivation is in the first video

A clearer depiction of a Rayleigh wave , evident at a surface boundary where freedom of movement I greater.

So we see the the many modes of wave propagation are generalised in the spherical wave with an exponential description. At any surface boundary love and Rayleigh waves are generated.

Assuming a sine wave node at a black body boundary is thus a mistake, Planck by assuming a spring behaviour was closer to boundary reality conditions.

The propagation beyond the boundary , the black body glow will therefore be a spherical wave propagation. Such a propagation necessarily occurs in discrete frequency modes which Planck called quanta without understanding why his spring model was better than Rayleigh node model.

How do we go from springs to probability?

The answer is interesting and illuminating.

De Moivre originated probability theory. De Moivre, sir Roger Coates and Newton were a powerful research team exploring Newton’s ideas and concepts. Newton was a master of infinite series and of the math of the geometry of the unit circle. Consequently he was able to accept and use the negative quantities of Brahmagupta and the other Indian mathematicians along with the Algebraic concepts of Bombelli regarding the square root of -1. .he was a student of Isaac Barrow and learnt Pythagorean geometry through his influence. He consequently formed and solved many multinomial equations and set out a basis for polynomial theory. His understanding of difference equations and expressions in calculating the trigonometric and natural and Briggs Logarithmic tables was unexcelled by his peers, . He took on De Moivre as a personal disciple and later sir Roger Coates. Both Coates and De Moivre collaborated on the theory of roots of unity in other words the discretisation of the unit circle. . Coates went on to propose the Coates Euler theorem in its logarithmic form decades before Euler proposed it in his exponential form.. of course he died before he could explain it to Newton as a generalisation of Newton’s force laws . Later Boscovich completed this line of research , really establishing the force relations as Fourier type expressions’ which are based on rotational dynamics.

However DeMoivre took the unit circle ideas in the direction of establishing probability theory which he.did , barely establishing preeminence over another developer.

◦ These ideas of probability or expected outcomes were applied by Boltzmann to physical phenomena where populations of agents could reasonably be expected. Needless to say it was not popular with his peers who wanted exact or precise solutions. However Gauss showed how by using these ideas he could precisely predict the orbit of a comet which was only sporadically glimpsed and often immaculately measured. . By applying Boltzmann normal distribution curve to these varying data he was able to determine a bell shaped distribution which gave the probability of a range of measurements. , meanwhile Fourier was demonstrating how trigonometric functions could interpolate any polynomial and indeed any curve shape. Thus a population of sine or cosi e functions could describe a set of experimental data . Lord Kelvin promoted this view as the way forward in physics especially for the growing molecular description of material behaviour, At the same time Maxwell was using the Gauss Boltzmann normal distribution probabilities to characterise gases and the velocity of their Dalton molecular structure. All at that time accepted the aether continuum and so vortices in a continuum as proved by Helmholtz became the favoured corpuscular model for the atom. . There were no electrons until JJ Thompson demonstrated a ratio Metternich mass and so called EMF which could just as well be explained as magnetic induction force given a dynamic magnetic vortex.

So Rayleigh in relating radiation energy to frequency assumed a purely sinusoids wave. . Such a wave as a standing wave can have any frequency that fits the cavity. . In his reasoning the frequency obscured the wavelength. We can see that only half wavelengths can be counted, but not all frequencies will have half wave lengths that will fit a cavity. These were assumed to be destructively cancelled. . So right there we have discretised frequencies. However by averaging they obscured this discrete condition.

Planck on the other hand was considering populations of springs . He therefore had little choice than to start with a generalised Fourier description with its discrete frequencies. . Using the Maxwell Gauss Boltzmann De Moivre ideas an accepted energy probability curve was found, . He could not average away the discrete frequencirs. Instead he had to use the standard series sums to simplify and this gave a different form to the description. It also tied in his results with the different series used to depict the wavelength lines in rhe spectral analysis of emission spectra.

We see that classical exact formulations were not up to the job, but classical probability theory and Fourier analysis were.

Ironically Rayleigh himself pointed to flaws in the wave mechanics of his day relying too heavily on the simple sine wave. He expressed his wave mechanics in the complex Fourier form, and so predicted Rayleigh waves.Love then predicted love waves. But Planck by focusing on the modes of spring oscillations unwittingly uncovere the physical explanation of these modes of oscillation and the quanta required to isolate them.

Quanta are interesting. Because we do not understand the arithmoi we fail to grasp that quanta or units have to be carefully distinguished. So quanta in relativity are units of space times time. In rotational dynamics they are u its of h times frequency. In Newton’s principles quanta are units of density times volume and J J Thompson discovered a quantum that is units of mass times deposition time, called the EMF. Both electric and magnetic induction are used to establish this quantum which is why we call such waves electromagnetic.

How to divide a plane circle into 360 approximately equal arcs

 I accept that one can divide a circle into 12 approximately equal arcs.

I will use the chord associated with this 1/12th arc to divide the 1/12th arc into 5 approximately equal sub arcs. 


Expand the circle into one with 5 times the radius. 

Mark off the 1/12th arc on the larger circle by a radial projection. 

Using the cord in the smaller circle. Mark off 5 arcs on the larger circle and the chord on the larger diameter at the circle. 


There will be a shortfall in the arc .


Project this back radially onto the smaller circle . This will mark off a 1/5th correction arc and it’s associated chord. 


Use this correction chord to extend the initial chord marked on the diameter.

Use this chord in the smaller circle to extend the arc , and thereby form a new chord. 

Mark this new chord on the diameter of the larger circle.

Use this chord on the larger circle to extend the initial arc there, and thereby form a new chord. Mark this off on the diameter.

I now can compare 3 corrective chords on the diameter.


Use the largest chord that falls short or the least chord that extends over the 1/12th arc when applied 5 times. 


Repeat the correction method until one is satisfied they have the best chord to divide the 1/12th arc into 5 sub arcs.


Project this sub arc back onto the smaller circle  to divide it into 60 approximately equal arcs. 


Bisect these arcs to divide the circle into 120 arcs. 


I will now use the chord associated to the 1/120 th arc to trisecting it. 


First expand this circle to one 3 times the radius.

Now use the chord to trisect the arc in the larger circle.


Follow the correction process above to  find the chord that trisects the 1/120th arc. 

Simple Approximate methods for dividing the circle or a circular arc

Simple approximate methods for dividing circles and arcs
These methods are based on bisecting an angle, bisecting a line, and equilateral triangles.
We can demonstrate how to divide a circle into quarters by bisecting a line. By this method we can find the chord that quarters circular arc
We can demonstrate that we can divide the circle into 6 equal parts by using equilateral triangles. By this we can find the chord that divides the circular arc into sixths (1/ 6) 
Rotating these two chords onto a diameter, the same diameter, enables us to compare the two chords.
The chord that divides the circular arc into five equal arcs lies between these two chords on the diameter. We can now find this by trial and error, and using Euclid’s algorithm or method of division. We will know by this process that dividing the circular arc equally does not divide the diameter into equal parts.
To trisect semicircle. If we didn’t know that the radius trisects a semicircle, we can proceed in the following manner.
Construct the semicircle, and then construct a semi circle on the same centre which has three times the diameter. Bisect the largest semicircle. This gives us the cord that divides the semicircle into two. Mark off the diameter of the smaller circle on the damage of the larger circle and also Mark off this chord. We know that the chord which trisects a larger semicircle lies between these two chords. We can find it by trial and error and Euclid’s method .
In the first description we have one method of trial and error. The interpolation of the correct chord is done on the diameter.
In the second description we have two methods of getting the correct chord : one is by using the Arc and the other is by using the chord. Using both together will give us a faster convergence.
This method is worked in the arc and the diameter of the larger circle using estimates and corrections from the smaller circle. The estimates improve as the curvature difference becomes less noticeable in the arc projections.
So in the case of the semicircle being divided into three, we use a diameter of the semicircle as the chord estimate in the larger circle. We Mark this off on the diameter of the large circle and then step it off around the circle. It will be too short ( because the curvatures are different) . We then take the shortfall in the larger circle and project it back down on to tthe small circle to give us a one third estimate of the arc length in the larger circle, And the chord  that marks it off.
We can use this chord in two ways; on the diameter to extend the first estimate of the chord or on the circle to mark off an estimated third arc length on the circle, enabling us to then draw a revised chord on the circle. Rotating the chord down onto the diameter we can compare the two new chord estimates. The larger one will be used to mark off the new trisection. first. If it is too long then the shorter one will be used.
Again the shortfall will be projected down onto the smaller circle in order to obtain a estimate of the third of the shortfall arc  and the associated chord . We know by this process the bounds between which the desired chord  lies as marked out on the diameter of the larger circle. We quickly obtain an accurate approximation if not an accurate result.
In this case we check the method because the result should be the radius of the larger circle.
The method is applicable to any number of divisions and any general arc.

Circular Proportions

The trisection of an angle is a famous problem used to encourage innovative thinking. However recently , since the algebraic proof of impossibility, it has been used to brainwash vulnerable mathematicians into a hopeless conformity.

The solution was clearly found by the ancient Sumerian and Akkadian peoples , the Dravidian and Harrapan Indus Valley civilisations and the Mongol chinese steppe and Plain civilisations, all of whom had the wheel and the 60 modulo arithmetics.

The issue is a pragmatic metrical one, and relies on skillful Neusis, as well as expertise with circles

Such an expertise is now called sacred geometry, but it is a science of spherical and circular relations. Of all the forms we have explored it is the circle that encodes proportion in its simplest form: one perimeter to 1 diameter!

We all accept that a circle can be patterned by six petals formed by 6 overlapping circles. We accept the number 6 because we see symmetry. This means we cannot distinguish the 6 forms we see in the pattern by any known or used measurement; accept by calculus! In that branch of ” precision” we find pi to be not 3 but 3.1415… Because our calculation is not based on observation but by a division process!

The difference is profound. Do we trust our eyes or our formal calculation process? Both, because as it turns out our ancestors did not need precision. 6 was good enough for them even though we know that it should be 6.28…

When your compasses do not quite meet the diameter we are taught to do it again until it does, because the radius must step 6 times into the circumference! It does not, but by convention we say it does.

The pattern of 6 is so compelling , we want 6 equilateral triangles as a constructed Constant of space. Construct them in a circle and they fit, construct them in a tessellation and they fit, but they do not precisely fit a circle anymore! The construction in the circle has slightly distorted the plane forms.

Using a constant radius we can construct the sacred geometrical flower pattern. That is when we can start to set out proportions . While we ca crowd 6 around a centre of a circle with 1 radius we can crowd 9×6 around a circle with 3 times the radius! This means we can trisect the diamond made of 2 equilateral triangles . But we have to use the chord length of the 1/3 rd circle to step these off on the larger( 3 x ) circle.

This proportion exists in this set up because circles are proportions. Without a rigid measure it is fiddly to do it is much simpler with a set of measuring tools that can retain and transfer these lengths to the proper positions

There exists a circle for which this length is the precise chord which trisect the arc into 3 similar sectors. Finding it by trial and error can be made easier by using the sacred geometry to narrow the search down. The Neusis becomes simpler and more precise.

Draw an angle and make the limbs or rays long enough to step off 3 radii. The radius is the semi circle drawn at the vertex of the angle, extend your compass to 3 radii and draw the semi circle,

Using a pair of rigid divider measure the chord of the angle in the smaller semi circle.

Step this off on the larger arc until step 3( which is too small ) and step 4 (which is too large).

Leaving the dividers fixed , now use the intersection of the upper ray between steps 3 and 4 with the semi circle as the centre for a circle that has a radius given by the displacement to step 4. Retaining that radios go to step 3 and mark an intersect toward 4 as the centre of a second circle through 3

Now using the point of intersection of these circles with the ray draw a semi circle from the vertex. Using the dividers step off to point 2 along this arc. Setting your compass to the displacement from the point on the ray cut by this arc( the same as that cut by circles at 3 and 4) now draw a circle that intersects circles 3 and 4

The circles are a probability space . Where they intersect is probably the point for the radius of a semi circle which can be trisected by the dividers precisely.
There may be 2 intersections that are clear. Choose the one that fits best.
Where these circles cut the upper ray is a point which was used to draw a semi circle. That semi circle will be unable to contain more than two steps within the arc of the angle.

The demonstration relies on neusis so be as accurate as possible.

The empirical deduction is that the 3 x radius semi circle is going to present an arc( the angle) which being less curved will be too big, by a proportion . Points 3 and 4 are used to narrow the space that the sought for circular arc must pass through. By using the 4th point as a radius displacement and drawing a circle the bounds can be seen to decrease until the circle cuts the upper ray. Thus any circle drawn from the vertex passing through that circle has a high probability of approaching the required radius from above.

The second circle passing through point 3 from a marked centre has the probability of a circle approaching the correct circle from below. Thus both bound a circle which will likely contain 3 to 4 steps

The second circle will bound a circle that is likely to contain 2 to 3 steps

Where they intersect has a high probability of being the correct circle requiring precisely 3 steps to equal the angle.

Clearly the dividers must be kept rigid and the stepping off done as accurately as possible.

Should the result not be “perfect” then the two guiding semi circles, just drawn , can be used to repeat the method.

You will find that if the angle is a 60° or 90° or some multiple of those the circles at 3 and 4 will very nearly coincide. Do not neglect to differentiate the points of intersection.

At first I thought this was a method of approximation relying on proportions un related to sacred geometry, but when I saw that the circle count was 4:3 for the 120° I could see then the sacred geometrical pattern peeking through. The first radius would cut the smaller circle into 6, but the chord was cutting the 3 x circle into approximately 12. The circle I sought would be cut precisely into 9 by that same chord.

These are empirical findings, the sort every geometer should be looking for as a matter of professional expertise!



We mathematicians have become obsolete.

We were doomed to die out as soon as “we” were born in the medieval period as a class of astrologers whose main task was to calculate the positions of planets and stars and the length breadth weights and measures of the empire.

Prior to that only Astrologers were given that respect and position within society, based on the Sumerian and Egyptian empires and their need for insight from the heavenly gods as to the opportune time for any and every venture. Such shamans using various practices were called sages and seers , whose visions were highly sought after and whose interpretations were regarded with awe.

The quest was to see, to visualise how any event would carry out its due process within the heavenly cycles.

It was the Pythagorean school who developed the ancient arts in Astrology and shamanism into a single or Monas-tic tradition. They collected together the worlds knowledge of Astrology and systematised it gradually into a coherent system based on the Mosaics of their Mousaion or temples to the Musai. 

Such temples collected all the traditions and curated them , bringing together a vast worldly knowledge of Astrology and the Arts. To study at one of these Monasteries was to devote ones life to ceaseless research and curation, finding thought patterns that gave you mastery over the worlds astrological and Artistic knowledge, finding those patterns by a gift of the Muses!

The Aristotelian Lyceum replaced that whimsical tradition by hard logical and taxonomic learnings. For that reason alone the position of Mathematikos was doomed!

Prior to Plato extending the reach of the Pythagorean school and praxis into Athens Greece , Pythagoras was reputed to have established a Mousaion in Sicily under the Patronage of a local chieftain who provided the “Monks” with protection in return for their public services of education. Those that came to listen soon divided into the Akoustia and the devotees. 

The Akoustia came to listen to the public teachings but the devotees came to learn and to be inspired of a Muse so as to teach and express the gift of that Muse.

As such there was no formal curriculum as in the Taxonomically driven Aristotelian Lyceum , instead discourses on topics were given by the Mathematikos, those qualified to teach, and debate and discussion ensued. The Monk or devotee was set Koans, test designed to bring them into close vicinity of the Muses. The Musai would then whimsically impart gifts to the Devotee. What they may be would become apparent as the gift was expressed.

One became a Mathematikos, not by examination and jumping over hurdles, but rather by Merit, that is by the acknowledgement of other Mathematikoi and even non Mathematikoi that one had the gift of a Muse, and particularly the gift of astrology and it’s counterpart geometry.

A Mathematikos was thus a gifted person who could reason by the astrological and geometrical patterns of the heavenly Musai, in particular circular or periodic proportions.

The development of the Arithmoi which was the distinguished Pythagorean name for what we now call mosaics was key to all astrology and geometry, for by these fundamental patterns all proportions were distinguished .

When Aristotle failed to attain the Status of Mathrmatikos, it was not because he was not gifted, but rather because he did not merit it among the Pythagoreans. He openly disputed with them over the logic of their principles and while he remained a firm Platonist he broke away from some key Pythagorean teachings. 

Nevertheless others found him to be remarkably gifted and Learned and Philip of Macedonia employed him to tutor his 2 sons Alexander the great and Phillip. Alexander was very taken by Aristotle, regarding him as his Secret diviner, for him alone. Thus he was annoyed to hear that Aristotle had established an Academy to rival the Athenian Platonic outpost of Pythagoreanism.

The fates of empires lead to the dissolution of his academy while the Platonic academy survived until Roman times and even beyond in some meagre way enamoured by enthusiasts. Aristotles one salvation was the Arabs and Islamist rulers who defeating the Hellenists found Aristotles ideas to have been spread by Alexander everywhere. Thus they adopted them as clearly important to developing a strong Empire. 

Later when Rome sacked Athens and the Platonists fled, Pythagorean literature came into the possession of the Islamic scholars, who mistaking it for a version of Aristotelianism mishandled its provenance and bound together two formerly incompatible traditions!

The Pythagoreans were moved to display their philosophy, the Aristotelians were moved to taxonomise it! What we call geometry was and is a counterpart to Astrology, but the principles of both arose out of the Artisanship of the temple builders and mosaic layers, the Tekne or Mechanics in Greek society .

Because of their low status, but absolutely essential skills they were a distinct slave class or worker class. As such they were employed by the wealthy with reasonable respect to their skills, but no high borne would deign to learn such skills! That is until they became a Pythagorean!

In India the Temple builders were again a distinct group and they handed down trade secrets in Ganitas and Sutras that were the aphoristic stock in trade of the Indian temple Astrologers

The Arab empire firmly combined these 2 streams of similar wisdoms into Geometry and Algebra . Between the 2 the Arithmoi sat as a peculiar  skill which eventually in the Arabic universities became Arithmetic of Numbers, and Number theory. It was also called Kaballah by certain foreigners from mQuaballah to calculate, and even called Gematria( from geometria) or later Numerology.

This is the background to what Medieval scholars of the 14 th century began to call Mathematics. From arithmetical arts to arithmetical arts mainly employing the Algebraic rhetoric to mathematics seems to be the uneven course of the spread of this idea that 2 dysfunctional systems can be combined into one subject area.

Finally in the 1800 the crisis was reached, and geometry and mathematics was about to collapse. Algebra came to the rescue and from it analysis, and the problem limb Geometry was left to wither away!

However Hermann Grassmann thought this was a great injustice. He separated the geometry from Arithmetic and pointed out that that just left Formal expertises which were now inter communicant with a vibrant Geometry, Kinematics and Phorometry and Mechanics.

It is the Mechanics who in Archimedes time displayed the circular proportions on the Antikythera mechanism. It is the mechanics who displayed the polynomial calculations on mechanical computers and it is the mechanics and now Technologists who have refined the mechanisms to display the proportions on these mosaic screens,just as the Pythagoreans did.

The movement of the astrological bodies need only to be proportionately displayed, that was the goal of every Pythagorean. By so doing all wisdoms and knowledge can be organised and cohered or adhered to the great cycle and periods of the heavenly gods, and the opportune time for every venture clearly displayed, where possible.

We have no need of Mathematics so called now, because we can see on a computer monitor all we wish to envision, visualise and more!

What we require are those gifted individuals and mechanics who can build curate and maintain our new temples of the Musai.