Numbers, Constants and Phenomenology

  1. Preface
  2. Primacy of low whole numbers
  3. Why numbers manifest living planets
  4. Numbers, Constants and Phenomenology
  5. Phenomenology as an Act of Will

Please enjoy the text below which is ©2023 Richard Heath: all rights reserved.

We have seen that the early numbers define the world of musical harmony but other important patterns arise within the ordinal numbers such as,

  • the Fibonacci approximations to the Golden Mean (phi = 1.618),
  • the Exponential constant (e = 2.718, the megalithic yard), from which trigonometry of the circle arises naturally,
  • the radial geometrical constant, (pi = 3.1416) as approximated by rational fractions {π = 22/7 25/8 63/20 864/275} and
  • the triangular progressions of square roots, as another development of the early numbers (in space) as geometry, also approximated by rational* numbers (rational meaning “integer numbers that can form mundane ratios”).

The transcendent (or irrational) ratio constants * (first mentioned in the Preface) are the visible after-effects of the creation of time and space. They must be part of the framework conditions for Existence, these also creating harmonic phenomenon that are not transcendent; these relying instead (as stated) on the distance functions of ordinal numbers: their distance from one and their relative distance from each other, lying beneath the surface of the ordinal numbers. Ordinality, to modern thought a universal algorithm for such distances, explains or defines what is harmonious in the physical world and in what way. Significant distance relations, such as those found in the early ordinal numbers, must then be repeated at ever greater doubling, tripling and so on {1 2 3 4 5 6} => {45 90 135 180 225 270}, where units can be scaled up by any number to become the larger structures, within any greater micro-cosmos. This is especially seen within ancient number science and its primary context of octave doubling, where what lies within octaves vis-à-vis scales and octaves within octaves, requires the right amount of up-scaling, as in the cosmology of Will (and not of Being), presented by G.I. Gurdjieff from 1917 onwards.

The illusion of number is that one can never penetrate the ubiquitous unitary distance of 1, the unity which becomes the ordinals which are so many exact assemblies of one; and of their ratios, so that one is not a number nor a transcendent ratio but rather is Number is the primordial Thing: a transcendent wholeness, found in every unit that causes relatedness through intermediate distance, or proximity. One is like Leibnitz’s Monads applied to the cosmic enterprise of universe building, as a fully quantified Whole and its Parts.

“The Absolute, that is, the state of things when the All constitutes one Whole, is, as it were, the primordial state of things, out of which, by division and differentiation, arises the diversity of the phenomena observed by us.”

Gurdjieff. In Search of the Miraculous page 76.

In Search of the Miraculous* had an 11-fold “Diagram of Everything Living” (shown above), in which the Universal Will for the universe was represented by an equilateral triangle within a circle (top left). The triangle expresses the equal division of the circle into three parts, and it is only through seeing the parts of Gurdjieff’s cosmology, in that book of lectures and (differently) in his later writings (and other sources) that one can arrive at a simple explanation for this triangle inside a circle (see section X), which also figures in his famous Enneagram (see below). The two circles demonstrate doubling in size, using the triangle which expresses division of both circles into three parts, which in Just intonation are three large intervals {4/3 5/4 6/5} which together equal 2. The important point here is that all the notes of an octave are related by the rational distances between the numerators and denominators of ratios, thus automatically increasing the relatedness of an octave’s parts, at whatever scale. This is what the ordinal numbers achieve from the very outset, of their seemingly primitive ordinality {1 2 3 4 5 6}, which leads to an intense and well-behaved type of relatedness. By the number 24, the octave of eight notes appears and, within this octave and the next {24 48 96}, seven modal scales emerge, from each interval of the major diatonic, in order (section 2.8).

In recent centuries, alternative voices to science have been talking about this sort of phenomenology. The European School of philosophers, pursued phenomenology rather than rational introspection. From Wolfgang Goethe (1749 – 1832) onwards, they wee discovering that science was not, in principle, a participation of the human senses in understanding the world but rather had become the application of instrumentality instead of the natural senses (next section). Moments of scientific genius, discovering new mathematical laws of nature, are remarkably not focused on how they were achieved, but on what they revealed within the physical world (as a technology to create a human “good”, exploiting new understandings of nature to explain then control nature). This change of focus had collapsed the original genius of a participatory moment, in which the fundamental process of understanding for its own sake, is a part of the world understanding the world. The problem was dramatized in Goethe’s Sorcerer’s Apprentice, where an apprentice magically tinkered disastrously with the world of his Master.  Phenomenology lacks the apparent usefulness that science provides since it seeks to understands the world rather than control it. In seeking to control the world, the world soon comes to control what the human world is like and what humans, now a world, want to do.

The failure to understand the world in the right way, on an individual level, suppresses the human destiny for which nature and the whole universe was probably created. It is easily done, to revert to a life where the world no longer participates within us but we benefit from the exploitation of parts of the physical world. The meaning of the world becomes our description of it, rather than the world’s meaningful action through human beings. It is therefore true to say that today we use numbers but don’t understand them* (in their pychoactive role of expressing meaning in the world) as the manifestation of the Universal Will for this universe. Holding to this is useful when wishing to understand the ancient number sciences, as they must have originally got their insights by looking towards the being of numbers as the native framework of the Universe. Numbers are, for example, made up of prime numbers, just as we think atoms are made of subatomic particles such as protons, neutrons and electrons.

Numbers can be explained as a tool for the universe-creating Will. The will of numbers is that, within the creation, “even God cannot beat an ace with a deuce [that is, a two]”

<EN> I do not remember which of us was first to remember a well-known, though not very respectful school story, in which we at once saw an illustration of this law. The story is about an over-aged student of a seminary who, at a final examination, does not understand the idea of God’s omnipotence.

‘Well, give me an example of something that the Lord cannot do,” said the examining bishop. “It won’t take long to do that, your Eminence,” answered the seminarist. “Everyone knows that even the Lord himself cannot beat the ace of trumps with the ordinary deuce.” Nothing could be more clear.

Ouspensky, In Search of the Miraculous, page 95

That is, numbers are a foundational set of rules based upon their distances from each other, in a single dimension or rather, in any direction away from one. This is perhaps why the strings of string theory are numerical and, in manifestation, how long chains of DNA proteins determine living forms. Numbers must be obeyed for the universe to be actual. Our world appears as we think it is and, the facticity behind material causes lurks within phenomena as more than a cause-and-effect “explanation” using physical laws. One can return to the notion that nature has provided, through numbers, shortcuts between numbers which, in the physical world, can connect things of a similar sort together, in a special way that means that the connection of the whole to the parts remains intact within phenomena.

The question is, how could numbers have created the context for a Living Planet and be knowable to human beings without an instrumentality other than the sensorium made up of senses + intelligence.

Phenomenology as an Act of Will

A House that is your Home

This graphic demonstrates how the inner geometry within numbers can point to significant aspects of Celestial Time or here Space regarding the relative sizes of the Earth and the Moon, namely 11 to 3 according to pi as 22/7.

In some ways one cannot understand numbers without giving them some kind of concrete form as with seeing them as a number of identical units. Sixteen units can make a square of side 4 since the square root of 16 is 4 and 6 is factorial 3 (3! = 1 x 2 x 3 and 1 + 2 + 3) which is triangular, so together they make 22, and if the triangle to placed on top of the square, like a house and its roof, then the house is 7 tall. If you want an accurate approximation to pi of 3.14159 … (pi is transcendental), the 22/7 is good and the house defines it.

This adds another mystery to this form of pi often used in the ancient world where numbers were best handled as whole numbers and ratios of these. This pi allows a circle of diameter 11 to be set within a square of side 11, whose perimeter is then 44. This can be seen in the diagram as made up of 16 yellow squares and 6 blue ones, centered on the circle and making 22 squares in all.

If one looks to the end of the 7th square, as a radius, then 22/7 will deliver the dashed circle (red) of circumference 22 and hence equal to the house number (16 +6 = 22) just 1.5 units beyond the first circle (green). This is called the equal perimeter geometry and a small circle radius 1.5, diameter 3, will “orbit” the inner green circle and the ratio between the circles is obviously 11 to 3, and this is exactly the ratio between the mean Earth and the Moon.

It was thought, by John Michell, that the model was well known in the megalithic since simple experiments in geometry, as above, delivers the relationship between a circle’s diameter and its circumference with very small whole numbers. My own work finds it is indeed prevalent within the design of later buildings, for example in domes, circular windows, and sacred pavements. If so, such buildings became sacred spaces as models of the Earth and moon. Many examples are explored and interpreted in my Sacred Geometry: Language of the Angels.

Numbers of a Living Planet: Preface

The image above is Kurma avatara of Vishnu, below Mount Mandara, with Vasuki wrapped around it, during Samudra Manthana, the churning of the ocean of milk. ca 1870. Wikipedia.

  1. Preface
  2. Primacy of low whole numbers
  3. Why numbers manifest living planets
  4. Numbers, Constants and Phenomenology
  5. Phenomenology as an Act of Will

Please enjoy the text below which is ©2023 Richard Heath: all rights reserved.

It is impossible to talk of a creation outside of the time and space of Existence, though from it, other dimensions can be inferred such as an “Eternity” visible in the invariances of numbers and structures. It is this higher dimensionality that leads to

  1. The recurrence of celestial time periods,
  2. The mental powers to recognise manifested patterns,
  3. The use of spatial geometries of alignment,
  4. The numerate counting of time,
  5. A phenomenology which is neither factual nor imaginary.

The quantification and qualification of Existence, adequately conducted, reveals harmonious structures within time and space, especially in the spacetime of our planetary system, when this system is as seen from our planet. The harmonious nature of our planetary system helped the late stone age to develop a large numerical and geometrical model of the world through counting astronomical recurrences. This model, which shaped ancient texts, implies that solar systems may have an inherent intelligence which makes them harmonious.

Harmony in a planetary system must therefore employ invariances already present in the number field, by exploiting the recurrent orbital interactions between planets and large Moons, this in a connected set of three-body problems. Before our exact sciences and instruments, prehistoric naked-eye astronomers could understand the planetary world by counting the duration of planetary time cycles: the subject my books explore. Through counted lengths of time, the megalithic age came to understand the invariances of the number field and so evolve an early and distinct type of numeracy. This numeracy lived on as the basis for the ancient Mysteries of the early civilizations, embodied in their Temples and in the Pythagorean approach to ordinal numbers and geometries, expressing the “number field” in two or three dimensions, areas and volumes. (see Sacred Geometry: Language of the Angels for an introduction to this)

That is, this early human numeracy naturally manifests within the maths governing rotational systems, this involving key transcendental* constants such as π, these regulating what is actually possible, mathematically, within dynamic planetary systems that are gravitational attractors of each other: these constants include pi {π}, √-1 {i}, e, and phi {φ}.  The first three { π, √-1, e} are surprisingly well-organized rotational frameworks making the behaviour of vectors relatively simple using geometry. For example, the lunar year of twelve lunar months has become strongly resonant with the two outer gas giants, Jupiter and Saturn. The Golden Mean (or Phi {φ})1 can be approximated by orbital ratios between planets through exploiting the Fibonacci number series2, most visibly in the orbital recurrence of Venus and the Earth, seen in the 8/5 {1.6} relationship of its synod* to the solar year. Phi φ is also expressed in living forms of growth, since growth is often based upon the present size of a living body and what it has previously eaten.  Fibonacci ratios are ideally suited to creating the “strange attractors” which can create stable patterns out of otherwise chaotic orbital interactions.

1 My use of curly braces is borrowed from a stricter world of set notation. It offers an ability to place groups of numbers, symbols and other non-grammatical element next to their grammatical context.

2 The series reinvented by Fibonacci uses addition of two previous number to create the next number. His version of that algorithm is {0 1 1 2 3 5 8 13 21 34 55 and so on}. These numbers are found within natural form of life, where such numbers can be generated from two previous states or when two counter rotating spirals of seeds will fill the surface of an egg shape with maximum packing. More on this later.

Through universal mathematical laws and constants, rotational and recurrent systems will effectively provide numerical shortcuts* (J.G. Bennett’s null-vectors) expressing Musical or Fibonacci ratios, and without those ratios being available, relationships within existence would be more complex, less synchronous, and truly accidental. Harmonic shortcuts have therefore given the planetary world a simplified mathematics when viewed from the surface of the earth, within the geocentric pattern of time. This synchronicity provided the stone age with a path towards a direct numerical understanding of time through phenomena (that is, a direct visual and countable phenomenology).

In this way, the megalithic cultures of prehistory found that the geocentric planetary system expressed numerical invariances (these already within the number field itself) thus making the time world of the sky unusually harmonious and intelligible. This contrasts with the now-popular modern notion that, while the solar system is a large and impressive structure, its origins come only from the mathematical laws of physics, these forever operating in a mechanical way. That is, the modern way-of-seeing planetary time is heliocentric and causal and this has hidden an ancient view, gained through the megalithic study of the phenomena in the sky using megaliths as large instruments with sightlines to the horizon events of sun and moon, to simply count of time-as-length and, evolve a very basic numeracy based upon numerical lengths (a metrology) and triangular geometries to compare lengths.

Megalithic methods employed the properties of circles, ellipses, squares, rectangles, and right triangles before the analytical geometry of Euclid, Greek math, or ancient near-eastern arithmetic. This was only possible because key parts of the mathematics of complex numbers, for example, are directly visible in the form of the right triangle and unit circle; as the natural form of two vectors: a length at a given angle (or direction) and another length at different angle gives access to ratios. A right triangle can therefore express two vectors of different length and differential angle, and this applies to a pair of average angular rates in the sky, without knowing the math or physics behind it all. If the two vectors are day-counts of time, then the right triangle can study their relationship in a very exact way. Such a triangle may also have been seen as the rectangle that encloses it, making the diagonal (vector), the hypotenuse of the triangular view.

The properties of the imaginary constant i (√-1) represents, through its properties, the rotation of a vector through 90 degrees. It is this that gives the right-angled triangle its trigonometric capacity to represent the relativity of two vector lengths. My early schoolroom discoveries concerning vectors in applied math classes, that right triangles can represent vectors of speed for example, was without any knowledge of the mathematical theory of vectors. This geometry enabled prehistorical astronomy to study the average planetary periods as vectors. That is, rotational vectors enabled the sky to be directly “read”, from the surface of the third planet, through simple day-counting, comparing counts with right triangles, and forming circular geometries of alignment to astronomical events found on the horizon; all without any of our later astronomical instrumentation, maths, or knowledge of physics.

Physics has not yet explained how the time constants between the planets came into a harmonious configuration, because it is unaware that this is the case. The mathematization of Nature, since the Renaissance, has hidden the harmonious view of geocentric planets and all preceding myths, cosmologies and beliefs were swept aside by the heliocentric world view (see Tragic Loss of Geocentric Arts and Sciences, also C.S. Lewis’s The Discarded Image).

The modern approach then emerged, of blind forces, physical laws and dynamic calculations. That is, while the simplifying power of universal constants is fully recognized by modern science (these having made maths simpler) the idea that these simplifications came to be directly reflected in the sky implies some kind of design and hence an intelligence associated with planetary formation.

Furthermore, modern way of seeing things cannot imagine that the megalithic could conducted an astronomy of vectors (using geometrical methods while not understanding why they worked) and that this empowered a simple but effective type of astronomy, without our mathematical or technical knowledge. This is an anachronistic procedural heresy for the history of Science and also for the present model of history, where science for us is the only science possible, evolving out of near-eastern civilization after the stone age ended.

Foundational myths of modern civilization are threatened by the notion that the world is somewhat designed by a higher intelligence. Until these subconscious conflicts of interest are overcome, prehistory will remain the prisoner of modernity where mysteries remain mysteries because we don’t wish to understand.

2. Primacy of low Whole Numbers

Chalk Drums to Symbolise Pi and Layout Monuments

December 2016 in numbersciences.org Hits: 3872

Three Folkton Chalk Drums found in a young girl’s grave
©Trustees of the British Museum ]

Perhaps as early as 4000 BC, there was a tradition of making chalk drums. Three highly decorated examples were found in a grave dated between 2600 and 2000 BC in Folkton, northern England and one undecorated chalk drum in southern England at Lavant in an upland downs known for a henge and many other neolithic features discovered in a recent community LIDAR project. The Lavant LIDAR project and the chalk drum found there are the first two articles in PAST, the Newsletter of The Prehistoric Society. (number 83. Summer 2016.) It gives the height and radius of both the Folkton drums 15, 16 and 17 and the Lavant drum, presenting these as a graph as below.


Adapted graphic showing diameters in inches (above in red) as well as mm, and the possible PI relationships for the chalk drum diameters, key to the fact that such drums can be rolled. In line with megalithic numeracy, the simple yet accurate value of 22/7 for PI is shown.
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π and the Megalithic Yard

The surveyor of megalithic monuments in Britain, Alexander Thom (1894 – 1985), thought the builders had a single measure called the Megalithic Yard which he found in the geometry of the monuments when these were based upon whole number geometries such as Pythagorean triangles. His first estimate was around 2.72 feet and his second and final was around 2.722 feet. I have found the two megalithic yards were sometimes 2.72 feet because the formula for 272/100 = 2.72 involved the prime number 17 as 8 x 17/ 100, and this enabled the lunar nodal period of 6800 days to be modelled and and the 33 year “solar hero” periods to be modelled, since these periods both involve the prime number 17 as a factor. In contrast, Thom’s final megalithic yard almost certainly conformed to ancient metrology within the Drusian module in which 2.7 feet times 126/125 equals 2.7216 feet, this within Thom’s error bars for his 2.722 feet as larger than 2.72 feet.

This suggests Thom was sampling more than one megalithic yard in different regions or employed for different uses. Neal [2000] found for Tom’s statistical data set having peaks corresponding to the steps of different modules and variations in ancient metrology, such as the Iberian with root 32/35 feet and the Sumerian with root 12/11 feet. It is only when you countenance the presence of prime numbers within metrological units that one breaks free of the inevitably weak state of proof as to what ancient units of measure actually were and, more importantly, why they were the exact values they were and further, how they came to be varied within their modules. However, the megalithic yard of 2.72 appears to outside the system in embodying the prime number 17 for the specific purpose of counting longer term periods which themselves embody that prime number.

The discipline of using only the first five primes {2 3 5 7 11} must have been accompanied by the perception that only if primes were dealt with could certain ends be served. This is crystal clear when we come to musical ratios in which the harmonic primes alone are used of {2 3 5} with an occasional “passenger” of the prime {7} as in 5040 which is 7 x 720, the harmonic constant.

Using 2.72 feet to count the Nodal Period

The first remarkable characteristic of 2.72 feet is that 8 x 17 in the numerator means that the approximation to π of 25/8 = 3.125 can, in (conceptually) multiplying a diameter, provide an image of 25 units on the circumference of a stone circle. For example a diameter of 2 MY would suggest 17 MY on the circumference, which is quite remarkable. Further to this, we know that the 6800 days of nodal cycle is factored as 17 x 400 and that the MY was shown (acceptably) to have been made up of 40 digits (in conformance to the general tradition within metrology that there are 16 digits per foot and 40 for a step of 2.5 feet, which a MY traditionally is). The circumference of 17 MY is then 17 x 40 digits which means that a diameter of 20 MY would give a circumference of 17 x 400 digits equalling 6800 digits as a countable circumference in digits per day.

This highlights how prime number factors played a role, in the absence of arithmetical methods, in solving the astronomical problems faced by the late stone age when counting time periods in days.

The Broch that Modelled the Earth

Summary

In the picture above [1] the inner profile of the thick-walled Iron-Age broch of Dun Torceill is the only elliptical example, almost every other broch having a circular inner court. Torceill’s essential data was reported by Euan MacKie in 1977 [2]: The inner chamber of the broch is an ellipse with axes nearly 23:25 (and not 14:15). The actual ratio directly generates a metrological difference, between the major and minor axis lengths, of 63/20 feet. When multiplied by the broch’s 40-foot major axis, this π-like yard creates a length of 126 feet which, multiplied again by π as 22/7, generates 396 feet. If each of these feet represented ten miles, this number is an accurate approximation to the mean radius of the Earth, were it a sphere.

The two ratios involved, 22/7 and 63/20, each an approximation to π, become 9.9 (99/100) when they are multiplied together, as an approximation to π squared.  Figure 1 shows that these two ratios, if 22/7 differently used as its reciprocal 7/22, also generates the ratio between the mean and polar radii of the Earth, since 63/20 x 7/22 = 441/440. The ancient Meridian length could be calculated from 396 when multiplied by using the most accurate rational π noted by Fibonacci as 864/275. The 396 units, of 10 miles per foot, was a practical distance to have realized in the megalithic without arithmetic, to store the 3960 mile mean radius of the earth, since the mile of 5280 feet is 4/3 of 3960; that is, 396 x 4/3 equals 528, implying that this model was conceived of within a decimal framework but without the base-10 positional notation of arithmetic. We show that the methods of calculation used can only have seen numbers-as-lengths as being composed of factors of just the first five prime numbers {2 3 5 7 11} and that this limitation upon numbers created a metrology in which fractional units of measure could manipulate lengths to multiply and divide them through addition and subtraction of the powers of these primes.

Marc Calhoun’s picture from the Island (picture from his blog)

Contents

  1. Summary
  2. Introduction.
  3. Main Thesis.
  4. Pre-arithmetic Calculation using Powers of {2 3 5 7 11}
  5. Combining Prime Number Composites.
  6. Appendix 1 Extract from Science and Society in Prehistoric Britain.
  7. Appendix 2: Preface: The Metrology of the Brochs.
  8. Metrological Bibliography.
Continue reading “The Broch that Modelled the Earth”