Signature of the Celestial Spheres - Hartmut Warm

Recto

Jedi
***Copyright Notice***
Signature of the Celestial Spheres (SotCS) : Discovering Order in the Solar System, by Hartmut Warm, © Keplerstern Verlag, Hamburg, 2004 ; This translation © Rudolf Steiner Press 2010. Parts of this material (charts and excerpts) are reproduced in the following summary for educational and research purposes only. The book is available for purchase on the publisher’s website and Amazon.


More information can be found on the book’s official website. A CD (transposing the solar system’s order into music), DVD (80min lecture by the author, Hartmut Warm) and computer program (simulations of planetary interactions), are also available over there in order to further explore the topic of the « Harmony of the Spheres ».

My (amateurish) summary include quite a few excerpts and charts as to give interested individuals the basic information they may need in asserting whether the author handle this subject seriously and in a scientific manner, or not. Indeed, no need to add further noise to the modern takes on the « Harmony of the Spheres ».

Introduction


This book aims at revisiting the theories and observations of Kepler in De Harmonices Mundi, 1619, using modern astronomical data, statistical analysis and computer software. Kepler, beyond discovering that planets traveled in ellipses, was convinced that some kind of higher/divine order (i.e. not simply stemming from the laws of Physics) shaped the cosmos. His first model, was based on the interlocking of Platonic solids representing the harmonic ratios in our planetary sysem. It was however based on crude and incomplete astronomical observation and needed further work.

Later, using the, very precise at the time, astronomical data/observations gathered by Tycho Brahe, he started calculating, then studying, the various parameters responsible for planetary movements and compared them. Kepler there found a correlation between some of those values and the intervals of musical notes. Thus finding the harmony he intuited. For more details one can read De Harmonices Mundi and/or read a pedagogical work-through like this one.

As explained in SotCS, the jump from astronomy to music wasn’t that uncommon at the time since in medieval university, a course known as the quadrivium combined the subjects of music, arithmetic, geometry and astronomy. Considering modern astronomy as impoverished, the author follows the old tradition by blending astronomy with geometry, music and arithmetic throughout the book.

Warm mentions Pythagoras only a few times in the course of the book. It is quite fortunate since Pythagoras’ life and teachings are a real can of worms, as explained by Laura in this post.

Data Sources


Since experimental research is as good as its data input (i.e. garbage in, garbage out), lets lay out the general approach taken by Warm. From the book’s official website, it reads :

Planetary positions and data from Mercury to Neptune used in “Signature of the Celestial Spheres” have been calculated according to the VSOP Planetary Theory (Variations Seculaires des Orbites Planetaires, long-term variations of the planetary orbits) developed by P. Bretagnon of the Bureau des Longitudes in Paris and published in 1982, revised for practical application in 1987 in collaboration with G. Francou (many thanks from the author). The data and calculation procedures needed for application of the VSOP Theory may be found (in a somewhat abbreviated form) in “Astronomical Algorithms” by Jean Meeus. These excellent publications make it possible in principle for anyone to calculate the planetary orbits for periods of several thousands of years to a fantastic degree of accuracy (< one arcsecond = 1/3600 degree) with the use of a simple PC. Jean Meeus’ abbreviated version is a little less accurate, but completely sufficient for the purposes of this book (the differences are discussed in detail in the appendix to “Signature of the Celestial Spheres”).

By looking at the said appendix, we find indeed plethora of details, among which :

p342-343
Calculating the planetary orbits

[…] This leads us to conclude that the deviations arising in the results of the VSOP calculations at the edges of the marked area of +- 7000 years can also not be very much greater and certainly remain considerably below 0.5°. Furthermore the calculations based on mean orbital data provide results that are sure to be qualitatively true for several millenia. This is shown by the fact that movement pictures arising from conjunction data over about 30,000 years still show figures that are principally the same.

Calculating Pluto’s orbit

There is no need to discuss further here since we can rely on NASA’s Ephemerides DE406 (JPL Planetary and Lunar Ephemerides) being among the most accurate calculations of planetary movements ever.

Calculating the Moon’s orbit

In the ELP 2000-85 version (a further improvement of ELP 2000), a maximum error of c. 20 arcseconds is given for all historical times, going back to 1500 BC. Even in ELP 2000/82 the error is already very much lower for the initial centuries. In the version he has published – although this is shorter by a few terms but contains elements of the improved version – Jean Meeus states an accuracy of 10 arcseconds without , though, specifying any validity period. However, from the other data given one can assume that this includes the stated accuracy for at least several centuries. (The longest period for the depictions regarding the Moon in Chapter 9 is c. 180 years.)

As a rule of thumb, every reasoning, calculations and data used in this book are thoroughly referenced in the appendix. It is great for independant researchers wanting to reproduce the results and/or carrying out their own study.

Definitions


Before jumping into the meaty part of the subject, I thought I would write down the relevant definition of the terms used by Warm. At least to give context to some of the book’s findings I’ll share thereafter.

At the very beginning Warm defines inner and outer planets as follow.

p13
The conformities mentioned and a fundamental degree of structure in the planetary system become very obvious when the different elements are summarized as follows :

planetary_parameters_light.jpg

This depiction offers us a meaningful division of the planets into three categories :
  • The innermost and outermost taken together as marking the boundaries ;
  • Venus, Earth and Mars (the exceedingly complex attunement of which will be discussed in detail in Chapter 7) ;
  • The four great gaseous planets (which are also finely attuned to one another, as we shall see).

Mercury and Pluto, while being part of the inner and outer groups respectively, are often (yet not always, as they are still part of the whole system) left aside when describing the intricate interplay of the planets of each group. This is mainly due to their somewhat erratic behavior and eccentricities.



Here’s a list of the most commonly used terms. They aren’t always defined as we traditionally know them (e.g. conjunction/opposition) ; while others are neologism.

Glossary
Conjunction : in this book this term always refers to the heliocentric view of the universe and therefore exclusively denotes the situation where two planets are on the same side of the Sun and in line with it.

Linkline : (in German Raumgerade) The imagined line between two planets or celestial bodies at a specific point in time. Geometric figures depicted in terms of linklines are usually heliocentric.

Octavation : The process by which any numerical ratio is brought by repeated multiplication or division by 2 into the space of a reference octave (usually the numerical region between 1 and 2) so that even widely seperated ratios can be compared directly.

Opposition : In this book this term refers exclusively to the situation of two planets on opposite sides of the Sun and in line with it.

Resonance : In physics this is an amplifying effect that can arise when the ratios of two oscillations, or two quantities perceived as oscillations, are small whole numbers. When referring to planetary orbital periods this is the same as commensurability.

Sideral period : Actual orbital period of a body in the solar system in relation to the fixed stars.

Silver section : A value that occurs repeatedly in connection with the golden section and in a number of geometric constructions. For example if the number 1 is divided into the golden section, the smaller part has the value of 0.3819… In the present work the number 1.3819… is termed the silver section.

Synodic period : The synodic period refers to the movement of a planet as seen in relation to the Sun from the viewpoint of another planet. For example the synodic period of two planets is the average time between two consecutive conjunctions or oppositions.



The following definitions are embedded in the analysis of Venus. However those concepts are applied to other celestial bodies throughout the rest of the book.

p145-146
The following sketch shows the main features of these two movements combining orbits round the Sun and the planet’s own rotation :

VSV_light.jpg

Here and in what follows the term ‘Venus axis’ refers not to the rotational axis but to an axis orthogonal to this in the equatorial plane of the planet. This corresponds to a point on the surface which is crossed by the horizontal axis. By definition the ‘Venus axis’ is here so determined that at zero hours on 1 January 2000 it is directed towards the zero point of the reference system. Any other angles would be just as good and would - in this and the following depictions – lead to the same results, merely displaced by a corresponding measure of time. Venus is at 181.79 at the starting point, i.e. the ‘Venus axis’ was directed to the Sun several hours ago or, respectively , it was midday at the relevant point on the surface. The positions 0, a-c show the forward movement against the retrograde direction ; the point 1-5 show the sequence of positions at which the ‘Venus axis’ once again points directly towards the Sun. The period of time during which this takes place is 116.7506 days. (See Appendix 3.4 for the calculation.)

This too, is a mean value ; but owing to the small eccentricity of Venus the variation is only about one day. In relation to the orbit, the duration of a Venus day amounts to c. 13/25 (12.9896 :25) which leads to a not very resonant 25-pointed star-figure. This arises by plotting the ensuing positions of Venus at exactly midday – or at any other fixed time such as midnight or the onset of twilight. In the following I shall use the term ‘Venus-Sun-View’ (VSV).

All needed terms are defined at the end of the book in a glossary. It is a nice touch which shows the instructional approach of the author (same regarding the inclusion in the appendix of the explanation of each reasoning/method used).

Methodological Framework and Statistical Analysis

In tandem with the data sourcing part, giving precise structure to the subject of inquiery as well as validating it (here using a statistical method), represents an essential step in every experimental investigation.

Warm describes the methodological framework used in his approach in the following way :

p58
We shall enumerate the criteria by means of which we can reach a reliable estimation of whether the celestial harmonies – or any other possible conspicous order of the planets – determined by a specific model can be taken seriously :
  • There must be a system on which the model is based. This could be a musical system or a mathematical law such as the Titius-Bode law or a geometrical model like that of the Platonic solids in Kepler’s Secret of the Universe.
  • The assertions must be uniform and provable statistically. This means that any statement relating to this matter must contain at least a certain number of coherent and exactly known values such as, for example, the distances of the planets from the Sun.
  • The model must be clear and as simple as possible. Kepler’s early ordering of the planets would surely soon have been forgotten if, instead of the Platonic solids, he had only used other solids constructed in any possible way even if the conformity with the actual situation had been better.
  • Conformity with the basic system of order must be as accurate as possible. Statements claiming that the actual conditions always differ slightly from the ideal, as one sometimes reads in the literature, are of little help here.
  • And now comes the be all and end all of our whole consideration : there must be an adequately clear deviation from a random distribution. And we must also be clear that when such deviations are found we may still be a long way from knowing what their cause may be.

Since statistical analysis validation is described here as the cornerstone of any serious attempt to formulate a correct scientific model, let me add some of Warm’s explanations concerning his statistical methodology :

p62
A planetary interval will always lie somewhere between two notes of whichever musical sytem is used for purposes of comparison. So there will always be a note that is closest to this interval and another, neighboring one that is the second smallest distance away. There are only two theoretical exceptions to this : either the planetary ratio coincides exactly with the ideal value, or else it lies exactly halfway between two notes and is thus equidistant from both. We can imagine that these two situations are equally unlikely to occur. So the mean probability for the members of an accidentally occuring series of values to be examined would also be the mean between the extremes. Just a clearly - or, as statisticians say, ‘significantly’ – deviating distribution could indicate that the ratios in our solar system genuinely accord with a musical order.

Armed with this testing approach and the t-test (used for relatively small number of random samples), the author set out to either prove or disprove Kepler’s theory. The planetary parameters used for comparison then where the periphelion to periphelion and the aphelion to aphelion angular velocity ratios. After reducing the intervals to the octave space (i.e. between 1 and 2 using octavation), it appears that the correlation with the major scale is only missing one note (same for the minor scale).

However, Kepler’s results (even using modern data) shows a mean of c. 0.4, which has 10.6% likelihood that these values are arising from randomly distributed data. When extending Kepler’s results to all planetary position out to Saturn, and out to Pluto, Warm’s find even poorer test results (c. 0.5, which has a 20% likelihood of being random).

True validation of the model would come under 1% likelihood (the lower the better). Being far from it, we can only accept to put the initial theory of Kepler to rest. However not all parameters and ratios were tested and Warm then set out to find, through comprehensive testing, which planetary parameters might bring a statisticaly significant result.

He manages to identify the velocity at aphelion as a good candidate. By testing for all velocities interval at every possible angle which the planet can attain in their orbits, he concludes that the optimum is at the aphelion and the most inharmonious at about 20° from periphelion.

Since the ellipse (i.e. the way the hypothetical harmony in the solar system currently manifests itself) are built from two parameters, there must be a second one representing harmony besides the velocity at aphelion, he argues. After a few more test and filtering, Warm finally settles for the ratio between velocity at aphelion and the velocity of the planet when its distance from the Sun equals its semi minor axis b. The probability of being an harmonic ratio and not noise is 0.00000096 before adjustment. When calculating all possible planets combinations with these parameters as ratio, and finding a close correlation to a musical scale, one finds that the overall probability that the values investigated reveal harmony purely by chance is c. 0.062% or 1 :1600.

There is order in the solar system at last ! :dance:

Unfortunately, most of the book’s results are charts of planetary positions or linklines, from an heliocentric or planet-centric point of view. Therefore, the parameters involved in these results are mainly the orbital period and the rotation period (among others), neither of which demonstrated any kind of order statistically significant. Warm addresses this point in the following paragraph :

p189-190
But a few preliminary remarks are needed before proceeding any further. I can tell you in advance that we will be seeing a variety of very astonishing geometrical figures. Questions will arise concerning the probability of such movement figures being formed. In view of the abundance of possibilities to be taken into account, an exact mathematical approach to this question would no doubt form the basis for a degree in applied statistics. However, it is possible to simulate the formation of such pictures with fictitious planets by using a computer. This shows that it is not at all that unusual for a well-ordered figure to arise when three random orbital periods are linked together as mentioned above. A rough estimate arising from a series of such tests shows that at a little below every tenth time a very clear figure of small whole numbers, and that a further 25% of the random constellations lead to fairly recognizable images based on numbers up to the number twelve. This would correspond more or less to the conditions in our solar system. In the same way regular figures can arise in the actual planetary system, for example when one determines the interval of time through the sideral periods instead of the conjunction intervals. However, actually quantifying the probability of the figures being formed is difficult for two reasons. Firstly, one repeatedly arrives at images which lie somewhere between copperplate perfection and a chaotic tangle of lines, so that the subjectivity of the observer plays a not entirely negligible role. And secondly, to arrive at a statistically more or less safe assertion one would have to investigate a number of artificial planetary systems and ascertain whether the different figures are subject to an overarching order in the way they interact. It is easy to see what a time-consuming exercise this would be, even for a single system of virtual planets. So in this case I have no statistical estimate to offer, since I have preferred to use the time available to me to investigate what is going on in our own, real planetary world. Whether the symphony we are about to hear is, as it were, the product of a cosmic wind playing on an Aeolian harp or whether some other creative force was involved in its creation can in any case be neither proved nor disproved by means of numbers. But we have shown sufficiently clearly in Chapter 4 that the musicality existing in our solar system is highly improbable.

Towards the end of the book, after laying out all of the intricate relations in our solar system, Warm tended to believe that the probability of such complexity arising by chance is very low.

Observations and Commentary


At last, we have come to the fun part of this summary ! If it were just me I would have included (almost) the whole book in this section, thus defeating the purpose of a summary. I’ll temper my spirit by discussing only the landmarks that impressed me the most during this little trip through our solar system.

Mercury/Pluto

I’ll start with the connection between Mercury and Pluto, which puts them firmly within the order of the planetary system. This link is based on Keplar’s steller solids, which are obtained from Platonic solids by extending their edges. The process is akin to transform a 2D pentagon into a pentagram, but this time in space.

The solids used are the dodecahedron-star (comprised of 12 points and 20 corners) and the icosahedron-star (comprised of 20 points and 12 corners). On a side note, Kepler gave the dodecahedron-star the nickname ‘Hedgehog’ ; Warm supposes that it may have been his favorite. The icosahedron-star can morph into the Hedghog by extending its edges likewise, the dodecahedron-star can morph into the icosahedron-star by extending its edges.

By doing the following tranformation : ico*Hedgehog*Ico ; We find that the ratios born out of spheres which alternately separates the stellar solids (i.e. the outer sphere of the solid in question and the sphere surrounding the Platonic solid within it), are stunningly close to ratios of Mecury/Pluto parameters. Those parameters are uninfluenced by the planets’ eccentricities.

RadiusAreaVolume
Ico*Hedgehog*Ico10.098631101.982351029.88213
Mean velocities vMean distances aOrbital periods T
Pluto/Mercury (at v : Mer/Plu)10.09960102.001851029.27080


Beyond the degree of accuracy linking some parameters of the innermost and outermost planets together, I was quite astonished at the matching of spatial parameters (e.g. radius and volume) with time (e.g. v and T). Another (perhaps) interesting observation is made by taking Kepler’s Third Law, T²=a^3, and putting it against the Square-cube law :

When an object undergoes a proportional increase in size, its new surface area (in our case a) is proportional to the square of the multiplier and its new volume (in our case T) is proportional to the cube of the multiplier.

Its funny to see the two exponents (2 and 3) switched around, however this concerns the multiplier so this idea may very well be out in left field. Since I’m not well versed in Mathematics (and even less in Geometry) I won’t be able to give insightful comments on that matter unfortunately.

One last comment I could make is regarding the ico*Hedgehog* ico*… transformation. At one point Warm says that those metamorphoses can be infinitely reproduced both inwards and outwards. Which reminded me of this answer in the 11/14/98 session :

Q: (A) To prism?! Visual spectrum? I don't know what it
tells me. I never came across any relation to prism. But,
what is this 4th dimension? Is it an extra dimension beyond
the three space dimensions, or is it a time dimension?
A: Not "time," re: Einstein. It is an added spatial reference.
The term "dimension" is used simply to access the popular
reference, relating to three dimensions. The added
"dimension" allows one to visualize outwardly and inwardly
simultaneously.

Funnily enough, 4th Density from what I’ve understood is characterized by variable physicality, allowing endless transformations/shape-shifting at will.

Inner Planets Pentragram/Pentagon

At the very beginning of the book, we are showed that some inner planets seem to be asssociated with a number that follows them in each interaction. Mercury embodies the number 3 (through its unusual rotation), Mars the number 4 (through the square) and Venus the number 5 (through the pentagram). Yet, the number 5/pentagram (and its multiples 10, 15, 20, etc) seems to be the symbolic emblem of the inner planets (reaching even Jupiter), with the interaction Venus/Earth being the main driving force behind it.

It is to note that the orbital periods ratio of Venus/Earth approaches the silver section (with a precision of 0.027%), which is connected to the golden ratio. The latter takes part in the construction of the pentagram, hence the expression of the number 5 in relation to Venus/Earth interactions.

Now, a quick illustration of this force ruling the inner planets followed by a comment of the author :

p153
pentagram_light.jpg

What we see in this picture is utterly sublime.

Beginning with the double pentagram or the inner decagon, a metamorphosis of contracting and expanding pentagons and pentagrams runs its course until in the final picture (where the views, i.e. every fifth view, of Venus to the Sun coincide with its opposition to Earth) the number ten appears once more, dominating the image with its rays. The pentagram has disappeared, but i twill reappear again, as surely as inhalation follows expiration. After c. 550 years half a cycle has been completed ; i twill now be repeated in the opposite direction until it once more comes close to conjunction. One cannot help but comparing this with living, rhythmical processes such as the beating of a heart or the opening and closing of a flower. And this play of forces is made possible because the rotation period of Venus differs in small measure from a resonance tied to Earth. If full synchronization were present, a specific point on the surface of Venus plotted in relation to Earth would always produce an identical pattern.

In his book on the golden section, Walther Bühler introduces us to a series of metamorphoses of the pentagram. The points of the pentagram are bent over inwards until they reach the opposite corner of the inner pentagon. This causes the original pentagram to disappear while two smaller ones come into being. It takes quite an effort of geometrical imagination to picture this metamorphosis to oneself, but a great deal more mental agility must have been needed to create this development out of the geometrical possibilities inherent to the pentagram. Bühler sees this process as being symbolice of the mysterious ability of living things to divide like individual cells or to proliferate like the higher organisms, to pass away while leaving descendants to follow, thus maintaining the chain of life. The 10-pointed/5-pointed star-figure metamorphosis that rules in the interplay between Venus, the symbol of love, and Earth, the human planet (whereby Venus plays the greater part), seems to me to be showing us not only a geometrical symbol of life but also a picture of the way in which celestial bodies, geometry and the characteristic marks of life are all inextricably interwoven with one another.

The comparison to a living being (and seeing it in the pictures) left a marked impression in my mind. It echoed a reflexion of mine while reading the book regarding the similarity between the oscillatory nature of eccentricities, and biorythms. The most striking example that comes to mind is the « heartbeat/pulse of the solar system », made up of the conjunctions of Jupiter and Saturn, the wheightiest planets in the solar system. Both are pulling the Sun and move it around a theoretical center. Perhaps more bridges could be/have been already built in that direction ?

Since living beings were brought into the mix, we could easily draw a parallel with the topic of intelligent design. Warm often argues for a higher (divine even, just as Kepler in his time) inspired order in the cosmos, criticising materialism and its theories in the process (e.g. Big Bang).

Outer Planets Hexagram/Hexagon

By studying the outer planets, Warm quickly shows that there is a mirror situation of the Venus/Earth relationship. This time Jupiter/Uranus lead the show and imprint the number 6/hexagram (and its multiples 12, etc) as the symbolic emblem of the outer planets.

There are also numerous links between the two planetary groups. One related to this example would be that during the same period of 10,359 years, Venus/Earth forms 6^4 pentagrams and Jupiter/Uranus forms 5^3 hexagrams (with a precision of 0.005%). This is mainly due to the almost exact 84 :1 ratio of the orbital periods of Uranus and Earth. Such precision staggers the mind, yet it is far from being an isolated occurrence in the solar system (Warm’s book contains dozens of similar trivias).

p109
hexagram_light.jpg

This figure in its entirety turns slowly around the Sun, is probably rather a surprise. Because of the number of lines that makes up the whole picture, the stepping interval was set to depict just three sequential hexagrams, whereby in this example, too, the formation as such is independent of the exact stepping interval. The period of 82.87 years needed for one hexagram corresponds to six Jupiter/uranus conjunctions. The formation of such a hexagram is indeed astonishing since rather than being drawn by a continuous line this figure is composed of two equilateral triangles. Neither the continuous planetary movements nor the depictions of the constellations at specific intervals of time, such as the conjunctions, can lead to the formation of a hexagram. So the overall geometrical picture of the interplay between Jupiter and Uranus reveals something that would be very difficult, if not impossible, to detect solely by means of an arithmetical consideration of the resonance.

It is also worth noting that after the pentagram in the inner region, which frequently used to be associated with the human being, we here have another sign, now in the outer planetary system, which was formerly given a mystical interpetation expressing , for example, the interpenetration of two polar principles. These two star-figures have probably been the most frequently used to symbolize the order in the cosmos (hexagram) and the position of the human being within it. In the planetary system these are the only resonant (or should we say clearly recognizable and also physically provable) images of a relationship between two planets up to the formation of the 12-pointed star, where a limit is reached even from the purely geometrical point of view.

There could be more study to be done, comparing those planetary relations with symbolic representations of geometrical figures, numbers and planets throughout the ages.

Geometric Signature of the Celestial Spheres

The last chapter of the book is dedicated to constructing a geometrical equivalent to the « musical scale » correspondence of the ratios of the semi-minor axes of the planets. Following the methodological framework layed out above, the rules are to use only simple geometrical figures (e.g. circle, triangle, square, pentagon, etc.) in order to build what Warm calls the « Signature of the Celestial Spheres ».

The Sun is here put at the center :

p290
All this makes it obvious that our central star is indeed included among the planetary ratios. And it is therefore perfectly possible to imagine that the foundation stone for the structure we have been discussing was already laid when the angular momentum of the young Sun was transferred to the planets or to their primordial substance, because the semi-minor axis b, among other parameters, is included in the calculation of the angular momentum of an elliptical orbit. So in accordance with this a specific arrangement of the b-axes corresponds to the distribution of angular momentum in the planetary system. In this sense the part played by the semi-minor axes must be seen as fine tuning, since the momentum of an orbit is primarily determined by the mass and the mean distance from the Sun of the planet in question. Furthermore, by far the greatest part of the angular momentum results from the orbit, whereas in the case of the Sun it arises solely from the rotation. In this sense it is therefore not arbitrary to relate to the radius of the Sun to the semi-minor axes of the planets, since in the Sun’s case it is its size which determines the angular momentum. Quite the contrary : in this sense the diameter of the Sun has proved to be geometrically and possibly even the physically fundamental basic measure of the harmonious order of the ratios – referred to elsewhere as the progression of distances. Having once arrived at this conclusion it need not take us long to exclaim : Of course ! What other measure, if not that of the Sun, can we expect to provide the foundation for the ratios of the planetary system ?
signature.jpg
Figure 4.6 The signature of the celestial spheres. Joining the individual elements from Figure 4.4 to an overall picture. Broken lines show necessary quadratures of circles, more fully explained in Signature of the Celestial Spheres.
© Keplerstern Verlag

One astonishing connection made during the process (and reported on the corresponding page of the official website), is that a 12-pointed star-figure contains area ratios which correspond to the intervals of the semi-minor axes. 12 being, according to Warm, the number of perfection in the solar system (a whole chapter is dedicated to it).

By running again a brief statistical analysis, the original probability of 1 :1600 is adjusted with this method to 1 :140,000 at the very least. The presence of order is then confirmed by the second, geometrical approach. Perhaps a statistical analysis of the linklines and conjunction charts will reveal the same tendency for other parameters ?

Conclusion

This book is definitely worth buying, firstly for the wonderful tour of our solar system it offers (astronomy, arithmetic, music and geometry all in one package). Secondly, it may be regarded as a database of many hidden interconnections of the celestial bodies composing our solar system. It would be then a great resource to advance work in various fields. Off the top of my head, here are some examples :
  • Performing a thorough statistical analysis of all the data presented in the book.
  • Now that the « How » is answered, why is there order in the solar system ? Where do the rules come from ?
  • A link to 4th Density geometry can be made factoring in the time component of the figures obtained. The dedicated thread could be a good starting point in that regard.
  • As said above further studies done on symbolism of the planets, geometrical figures and numbers may add to the topic.
  • Again, as said above a connection to biology may open new perspectives.
  • Since half of the charts are obtained from linklines (sometimes even their center of gravity), a connection to gravity (and its true nature) is immediate.
  • Warm on two occasions mentions Chladni plate experiments (more specifically Hans Jenny work in that area) as a potential clue to the force behind the accretion of planets and their rotation (through some vibration/shockwaves).
  • How come Venus got in such an harmonious place within our solar system (i.e. distance to the Sun, orbital period, rotation period) if it came as a comet and knocked Mars close to Earth in the process ? Not to mention it only happened c. 10,500 B.C. Perhaps our solar system can « regroup » into a coherent whole in a matter of centuries, like a self healing organism.
  • Same thing about Kantek. Its destruction (c. 77,000 B.C.) must have wrecked havoc among the nearby planets. Was our solar system even more harmonious before that ? Its remnant, the asteroïd belt, still fits nicely within the Titius-Bode Law, meaning Kantek was also part of this web of planetary interconnections.
  • A comparison between cyclical catastrophies/comet cluster coming into our solar system/Sun activity with the planets’ interactions cycles (i.e. how much time is needed to draw a specific shape/pattern) may reveal interesting connections. After all, the same laws may be involved to some degree.
As in every scientific investigation, this book raises more questions than it can answer. However it covers thoroughly the initially layed out problematic and makes a deeply valuable contribution to the field of astronomy, IMO.

Since the book is as entangled in its construction as our solar system, I hope my summary of it was clear enough. I’ll finish here as my draft is already 12-page long. Perhaps a subtle message from the universe :-)
 

Recto

Jedi
Ooops, it appears I tripped up when crafting my post. I'll edit offline in order to avoid posting half-written material. The next post should be the one !
 
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Recto

Jedi
I came across the work of Buckminster Fuller (famous inventor known most for his discovery of geodesic structures) recently and I think it may contain some answers (or at least leads) to a couple of questions raised by my first post. Namely :
  • Why is the icosahedron-star the bounding structure of our Solar system (linking Mercury and Pluto) ?
  • Why sixfoldness and fivefoldness are so prevalent in the patterns of the stars’ interactions ?
  • Is there a link to the electric model (as layed out in Secret History vol. 3) of our solar system ?
Lets first address the first question. In Buckminster Fuller An Auto-Biographical Monologue/Scenario by Robert Snyder, we find Fuller's approach to geometry (more like a new field, the field of Synergetics) and the place of the icosahedron in it :

Euler, the mathematician, made a very extraordinary contribution to humanity when he conceived and announced that all pattern of universe can be broken down into three clearly rememberable and differentiable aspects. He said look at any painting, any drawing, research your memory, you'll find everything breaks down into lines,areas, and crossings.

Mathematically you can describe all pattern phenomena, all conceptuality, all of thinking, by just what we call angles and frequencies.
Those lines of Euler's take some time to be generated. How long they take to be generated is measured by cycles. You look at your watch, you look at so many seconds:there are so many cycles, so many heartbeats. So you go in this direction on this line for so many heartbeats, so many seconds. Then you change your direction. Changing direction, you have to say, "What is the angle of change?"

Now in order to start talking angle at all, you have to have some line of reference. The line of reference, say, is between your head and your feet. So there's that line, that axis. So we find all phenomena in universe can be described mathematically by angle and frequency change. Now that we know that time is measured by cycles and lines are so many cycles long, we begin to think about patterns in a very mutable kind of way.

For example, here is a necklace, and it is a necklace because I can drape it over my neck and shoulders. It drapes because the angles are varying; the lines are staying the same and aren't changing; what is changing are the angles. I'm going to take out one of the beads. The necklace is still nice and flexible and still drapable. I put my head through and it bends all over the place. I'm going to take out one more. And it's still flexible with a V in front and a V in back. You and I tend to call what we now have left, a square. I can still put my head through it. When I was in school the teacher said the basis of geometry is the square. But the only reason it held its shape was because the blackboard was holding its shape. It had no integrity whatsoever. A little child doesn't like that at all. So he says I'm going to take one more out.

Suddenly, a very extraordinary thing happens-it's no longer flexible. It won't change its shape! Boy! I can put it over my head here, but it doesn't flex or drape. This is what we call a triangle—three angles. And the triangle is the only stable structure. The angle won't change, and it was all in the angles. So the triangle turns out to be structure. It consists of six completely independent parts: three of these flexible angles, and three of these push-pull compressions. One pair of these sides work like levers; the further they come out here, the more work they can do. We come to the very ends of these levers, and we put another push-pull member in here, and it stabilizes the opposite angle.

The triangle is the only inter-self-stabilizing set of events. Triangle is structure,structure is triangle. So when I want to build something, and really make it work, I've got to use all triangles. Most people think of a building as cubical, and it hasn't any structural stability whatsoever. The angles are all unstable. The only reason they stand is that we put nails in the corners. So I've got to find a way whereby everything gains in stability. I'd like to make what they call a basic structure; and I'd like to make it into a system where I have an insideness and an outsideness. I can't get something that has an insideness and an outsideness unless I have one more point. And that gives me the tetrahedron: tetra is four in Greek, so that gives us a 1-2-3-4-sided figure.

We'd like to find its relationship to the other basic structural systems. When you take an action, for instance you step forward-you push the earth backward. Or a car starts up on a gravel path, it kicks the stones backward. Every action has reaction. Now not only does every action have a reaction, it also has a resultant. Because we now know about the speed of light, we know there is a time lag between action and reaction and resultant. You jump off a boat and it takes a little time for you to hit the other boat and you've pushed the boat You jumped off, pushed the boat you land on-so there's action, reaction, resultant. This is true of every experience, every event, in the universe.

Thus, every event in the universe has three parts: action, reaction, and resultant. These are energy events and we call them vectors. The vector depends on how much energy is being expended, what its mass is, in what direction it's going, and at what velocity. All our experiences involve energy; all the physical universe involves it. These energies are operating at various directions. A vector is an energy action in a specific direction, like a thrown spear.

There's some quantity of the energy as mass, and that energy action is going in some direction, so it has a velocity relative to all our other experiences. We then take the mass and multiply it by the velocity, and that gives us the length of the line of a vector. And it's going in a specific direction in relation to our other experiences. A vector has a specific angle, and direction, and specific length.

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There are two fundamental kinds of energy events—proton and neutron. The proton has its energy side-effects; the proton has its electron and its anti-neutrino; and the neutron has its neutrino and its positron. And each one of those is called in physics one-half quantum, one-half of Planck’s constant, one-half spin—any of those three.

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Now, I'm going to put one half quantum together with another half quantum, and we find we must always put it together in an absolutely consistent way, joining the positive ends with the open angles—male goes to female.

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We put it together, and suddenly we come to our old friend, the tetrahedron, which has four triangles and six edges, or vectors, and one unit of quantum. There are two other structural systems in the universe besides the tetrahedron. The octahedron with twelve edges, or vectors, and two units of quantum and the icosahedron with thirty edges, or vectors, and five units of quantum.

So we now see a very important conceptuality beginning to characterize physics and all structural understanding. Remembering that a basic unit of quantum has six edges -or six vectors- a basic energy event has six vectors. if I use the volume of the tetrahedron as unity, this is the one that gives me the most volume with the least structure. It gives me the sharpest, the greatest strength, because these three legs,like any tripod, are much more vigorous in their support. If you begin to flatten outlike that, like your own legs spreading out, it gets weaker and weaker. The octahedron has four volumes, and here I get two units of quantum. The icosahedron has almost-pretty close to-twenty full volumes, I'm getting twenty units of volume for five units of quantum invested. So we get the most volume with the least quantum in the icosahedron. So that becomes the very basic structure in nature. I use it for geodesic domes, and nature uses it for all the protein shells of all the viruses. The icosahedron is still very strong because it is triangular or basic structure. So this is the one that gives you the most volume, and I can fortify any one of those by putting a little local tetrahedron in there, to give it the greatest strength. That’s the reason I make my geodesic structures that way.

So here we have defined the triangle as the fundamental basis of structure and tetrahedron, octahedron, and icosahedron as the three prime structural systems of Universe. With the icosahedron giving the most volume with the least quantum.

In the book A Fuller Explanation The Synergetic Geometry of R. Buckminster Fuller, Amy Edmondson offers a fantastic commentary on the work of B. Fuller regarding geometry and gives us further insights about our initial question :

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Geodesic Design in Nature

Fuller points out that an extremely high-frequency geodesic polyhedron provides the true model of physical systems which we interpret as spheres, as for example a soap bubble. The notion of a continuous surface equidistant from a central point is scientifically unacceptable, that is, inconsistent with physical reality; on some level of resolution all "spheres" consist of discrete quanta-untold numbers of energy events interconnected by an even greater number of vector-relationships, or forces. He has clarified this particular misconception countless times:

« 535.11 Because spherical sensations are produced by polyhedral arrays of interferences identified as points approximately equidistant from a point at the approximate center, and because the mass-attractive or -repulsive relationships of all points with all others are most economically shown by chords and not arcs, the spherical array of points produces ... very-high-frequency, omnitriangulated geodesic structures .... »

Our eyes cannot see individual molecules in the delicate transparent soap bubble, nor can we detect the chordal chemical attractions between molecules. Nevertheless they exist, explains Fuller, and it is our responsibility to understand and teach the truth about Universe. Once again, his goal is to provide tangible models of otherwise invisible phenomena. Of all possible solutions, a high-frequency triangulated shell with icosahedral symmetry provides the most efficient method of enclosing space with a minimum of material and effort. Accordingly, nature relies on this elegant design in many situations calling for protective enclosures, regardless of scale. Examples include the small sea creatures called Radiolaria, the fibrous web of the eye's cornea, and the protein shell of many viruses.

With that, the icosahedron becomes the « most efficient method of enclosing space with a minimum of material and effort. » It may be part of the answer to our question regarding the general structure of our solar system.

On the topic of five- and sixfoldness the following passages add some more relevant information :

p165
Fivefold symmetry dominates the icosahedron, distinguishing it once again from the cosmic hierarchy, with three-, four-, and sixfold rotational symmetries. This is another sign of the icosahedron's nonconformism. It's full of fives: to begin with, the obvious five triangles around each vertex, determining the symmetry about each of its long axes. Then, its thirty edges fall into five sets of six orthogonal edges, that is, three parallel pairs of mutually perpendicular edges.

p207-213
The vertices of regular and semiregular polyhedra lie on the surface of an imaginary sphere, which is to say that all vertices are equidistant from a polyhedron's center. Given this fact, we can picture spherical versions of each polyhedron, in which the polyhedral edges have stretched outward to become great-circle arcs and the faces have expanded into curved surfaces, as if each shape had been blown up like a balloon.
[...]
Great circles reveal a new aspect of polyhedral "intertransformability," a novel (if obscure) means of detecting symmetrical relationships among systems. We can anticipate the emergence of decidedly unfamiliar patterns in this game for their resemblance to the source polyhedron is sometimes subtle. Finally, in pursuing this study we discover new variations on upper limits and minimum cases. Great circles thus provide another tool with which to detect inherent spatial constraints.
[...]

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Outlining 12 pentagons and 20 triangles (as opposed to 24 isosceles triangles), these arcs present the most symmetrical arrangement of six great circles. When a vector equilibrium contracts into an icosahedron in the jitterbug transformation, the radius-edge-length equivalence is lost, but the distances between adjacent vertices on the surface are suddenly all equal. As an icosahedron produces the most symmetrical distribution of twelve vertices on a closed system, it follows that the same is true for their corresponding six great circles.

p223
Quoting Fuller :
« There is a basic cosmic sixness of the two sets of tetrahedra in the vector equilibrium. There is a basic cosmic sixness also in an octahedron minimally-great-circle-produced of six great circles; you can see only three because they are doubled up. And there are also six great circles occurring in the icosahedron. All these are foldable .... This sixness corresponds to our six quanta: our six vectors that make one quantum. (842.05-6) »

As hinted in the description of the icosahedron and its relation to the tetrahedron and triangle, the fives and sixes are most likely explained by the fundamental structure of the isocahedron itself which «bounds» our solar system. The "basic cosmic sixness" as Fuller puts it still remains intriguing. Another notable thing is the edges of the icosahedron projected on a sphere (see fig. 14-7; a.k.a. the six great-circles) represented in 2D resembles strikingly to the interaction patterns of the stars (e.g. Venus-Earth). Many other edge-projection figures bear resemblance to some interaction pattern brought up in Warm’s book.

Finally, discussing the third question, such a possible connection truly surprised me. Here is the relevant excerpt :

p229
Icosahedron as Local Shunting Circuit


"The vector equilibrium railroad tracks are trans-Universe, but the icosahedron is a locally operative system" (458.12). This distinction, introduced by the jitterbug transformation, has particular significance in Fuller's great-circle theories. The VE is integral to our space-filling network of equivalent vectors; however, once the VE contracts into the icosahedron with its slightly shorter radius, it is disconnected from that universal IVM network. It loses its contacts. "Energy" is free to travel endlessly throughout the railroad tracks of Universe, sliding from sphere to sphere along the economic great-circle paths, until it runs into an icosahedron. The icosahedral 31 great circles are not" trans-Universe" lines of supply; their function is to disconnect energy from the closest-packing railroad tracks and direct it into local orbits.

« The icosahedron throws the switch: The icosahedron's function in Universe may be to throw the switch of cosmic energy into a local shunting circuit. In the icosahedron energy gets itself locked up even more by the six great circles-which may explain why electrons are borrowable and independent of the proton-neutron group. 458.11 »

Fuller suggests that there might be a meaningful connection between the icosahedron and the electron, because the tiny negative charge is readily transferred from atom to atom in molecules and crystals. The icosahedron's independent role, in Fuller's view, "shunting" energy into local circuits (that is, able to disconnect energy charges from a bigger matrix) is suggestive of the electron's role:

« 458.05 The energy charge of the electron is easy to discharge from the surfaces of systems. Our 25 great circles could lock up a whole lot of energy to be discharged. The spark could jump over at this point .... If we assume that the vertexes are points of discharge, then we see how the six great circles of the icosahedron-which never get near its own vertexes-may represent the way the residual charge will always remain bold on the surface of the icosahedron. »

Lacking contact points, the icosahedron is a free-floating unit in Universe; so is the electron. The suggestion of a relationship remains just that, a tantalizing parallel, seeds perhaps of future investigation, but certainly among Fuller's more abstruse parallels.

The "railroad tracks" reminded me of the Birkeland currents and the "local orbits" of the planetary system. Again by its structure our solar system (or solar systems in general) might be able redirect this energy flow, creating some kind of isolating bubble similar to an electron, which is quite amazing.

As appealing as it may be all of this remains wild speculations without more data, testing and research. The study of remote star systems in similar depth to our own being impossible at the moment sure doesn’t help in that regard. The work of Buckminster Fuller (see Synergetics or Everything I Know) could be a good starting point with promising leads though.
 
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