Tag: multiple squares

Multiple Squares to form Flattened Circle Megaliths

above: a 28 square grid with double, triple (top), and four-square rectangles (red),
plus (gray again) the triple rectangles within class B

Contents

1.     Problems with Thom’s Stone Circle Geometries.

2.     Egyptian Grids of Multiple Squares.

3.     Generating Flattened Circles using a Grid of Squares.

ABSTRACT

This paper reviews the geometries proposed by Alexander Thom for a shape called a flattened circle, survivors of these being quite commonly found in the British Isles. Thom’s proposals appear to have been rejected through (a) disbelief that the Neolithic builders of megalithic monuments could have generated such sophistication using only ropes and stakes and (b) through assertions that real structures do not obey the geometry he overlaid upon his surveys.

1. Problems with Thom’s Stone Circle Geometries

Almost all of the different types of megalithic building[1] were evolved in the fifth millennium (5,000-4,000 BC), in the area around Carnac on southern Brittany’s Atlantic coast. This includes the many circles built later in the British Isles. When Alexander Thom surveyed these [2] he found them to be remarkably technical constructions, involving sophisticated geometrical ideas. It was only in the mid-seventies, when Thom came to Carnac, that the same geometries were found applied within Carnac’s stone circles which soon afterwards were found to precede those of Britain by at least a thousand years.

After an initial public and academic enthusiasm for Thom’s work[3] British archaeologists chose, with very few exceptions, to refute the entire notion that the Neolithic could have been constructing such technical geometries. As far as our History would have it, such geometries could only have been drawn after the development[4] of a functional mathematics which culminated in Euclid’s classical work on analytic geometry, Elements. Thom’s use of geometry was therefore anachronistic and Thom surely mistaken. For archaeologists to accept Thom’s geometries would have required a revolution in thinking about the megalithic; for which there was little appetite. It was easier to work instead to falsify Thom’s hypothesis with a new type work that argued against Thom’s arguments for geometry, a megalithic yard and astronomical alignments, concluding instead, for example, that “stone circles were distorted so that the audience could see all the rites; and the principals could occupy visually focal positions facing the spectators.”, clearly indicating the still current “comfort zone” within archaeology, in which unquestioned ideas about superstitious rites are used to supersede Thom’s accurate and well founded proposals, of a megalithic technical capability. The problem with inventing ancient rites as the primary purpose for stone circle building is that, whilst refuting Thom’s proposal, it cannot itself ever be proved in a scientific sense; Talk of rites as being the reason for stone circle construction is not delivering an evidence-based scientific proof and Thom’s proposals were not disproved by such ideas.

Figure 1 Thom’s site plans of two of Britain’s finest surviving Flattened Circles, above: Castle Rigg (Type A) and below: Long Meg (Type B). Castle Rigg’s axis of symmetry points (within a degree) to Long Meg, on a bearing which follows the diagonal of a two by one (east by north) rectangle, as if (despite some Lake District mountains in between) the two sites were related when built and hence contemporaneous. site plans by Alexander Thom.

Unlike many of his detractors, Thom surveyed stone circles and through this activity was to create the first (and only) extensive corpus of stone circle site plans. Through this he left a vitally important legacy by preserving their layout against further natural and man-made degradation. The geometrical overlays and typology found within Thom’s site plans have been dismissed as unlikely, on spurious technical grounds [*], usually by people with insufficient technical background in the technical issues within his work. Thom’s later work in Carnac has proven critical in providing further alternative explanations as to how the megalithic actually constructed these stone circle geometries without Euclidian geometrical methods, using instead the system of multiple squares found to be in use in the megalithic structures around Carnac[5]; this in the late 1970’s and after Thom’s surveying seasons earlier in that decade[6].

We will later show that such a system of multiple squares would have eliminated the use of a “compass” or arcing of ropes which Thom proposed to explain how different stone geometries were achieved. Instead, a grid of squares can locate the few key points on the perimeter of a flattened circle. A design method based upon a grid of multiple squares would eliminate the main objection to Thom’s proposal of: Euclid-like geometrical process was used to build stone circle geometries.

Figure 2 the geometries of Flattened Circles (left to right) called Types A, B and D

In the case of the Type A (flattened) stone circles proposed by Thom, I demonstrate below that accepting Thom’s interpretation of its shape is a necessary stepping stone to understanding how this could be achieved by a pre-arithmetic megalithic of the fifth millennium BC.

2. Egyptian Grids of Multiple Squares

The monuments of Carnac demonstrate the use of multiple squares and, because of their antiquity –one to two thousand years before the Pyramid Age – it appears likely that the later use of multiple squares in Dynastic Egypt demonstrates how such a technique could function as a pre-arithmetical geometrical framework. By the Middle Kingdom, the Egyptians had put stylus to papyrus to describe their mathematics in a document called the Rhind Manuscript. This recorded a system of geometry based around pre-Ptolemaic ideas, which included the use of a grid of multiple squares.

At Carnac, the angular extremes of sunrise and sunset, on the horizon during the year, followed the lesser angle of a 3-4-5 triangle whilst in the Rhind Manuscript one finds a “canevas” [*] or grid-based diagram, in which both of the acute angles of this 3-4-5 triangle, held primary to the Egyptians, are shown to be generated by the summed diagonal angles of either; two double squares or two triple squares. The resulting grid is then 14 squares by 14 squares, and this is exactly the grid upon which the Type A stone circles can most easily be constructed, if one excludes the use of ropes and stakes to achieve this design.

Figure 3 of a Rhind diagram showing evolution of a 3-4-5 triangle within a 14 by 14 grid of squares

Such a use of multiple squares, as a template on which to construct a stone circle geometry, raises the question of the side length used, since they all need to be identical and so the ability to create identical lengths would certainly suggest an accurate system of measures, or metrology, was in use. This leads into another bitter dispute, concerning the existence of Alexander Thom’s found measure, the megalithic yard, as being a primary unit of measure maintained accurately by the megalithic builders throughout the British Isles and Brittany. Thom did not know enough about historical metrology to see that the megalithic yard might well have been accompanied by systematic variations applied to its length or indeed, that other measures might also have been evolved. His proposal of an accurate megalithic yard, like that of exact stone circle geometries, also came to be rejected by archaeologists, who themselves knew very little about historical metrology[7] [*], pointed to cases where Alexander Thom’s hypothesis of a singular measure in use within megalithic Britain was muddied by the presence also of other standard units of measure.

3. Generating Flattened Circles using a Grid of Squares

One of the key objections for the megalithic concerning ropes to construct flattened circles is the necessity for measured radii and their centres. If Thom’s Type A or Type B flattened circles were instead constructed using a grid of squares, then some of the key points where a flattened circle’s radius of curvature changes (of which there are only four) must be points of intersection within the grid . This became clear through considering the Type A geometry and specifically its implicit double triangles, as possible right triangles.

Robin Heath has already noted[8] that these triangles are close to the invariant ratio, in their longest sides, of the (lengths of) time found between the eclipse year and the solar year, and this ratio is also to be found between the solar year and the thirteen lunar month year.

The baseline of such a right triangle is found to be 6/7 of the diameter MN of the Type A flattened circle and this implies, given the left-right symmetry of this form, that this key point at the end of the hypotenuse (where the radius of curvature changes) would sit on the corner of a grid point within a 14 by 14 square grid as a length equal to twelve grid units. The forming circle used by Thom, of diameter MN, would then inscribe the grid square.

Figure 4 Type A drawn on a 14 square grid

We also know, from Carnac, that the astronomers used a triple square to frame this right triangle which then relates the periods of eclipse and solar year. Since the vertical position of the key point is 12 units, then to left and right the key points either end of the central flattened arc are 4 units, either side of the central axis. To right and left of these triple squares can be found two four squares, that express with perfect accuracy the relationship of the lunar year to the solar year, as diagonal. These four squares have a baseline of twelve grid squares which exactly matches the number of lunar months within the lunar year.

One can then see within the 14 square grid that many multiple squares can be found, for example the triple squares either side of the vertical centreline have two four-square rectangles to the right and left (shown in red below, the ripple-squares being blue). These leave a row of 14 by 2 squares at the top which can be seen as a seven-square, the rectangle whose diagonal to side alignment is found between a double and a triple square.

Conclusions

Clearly there are alternative ways of generating a flattened circle geometry that using stakes and ropes (geometry as we know it). We know that the Egyptians used grids within square grids of constant unit size and that multiple square rectangles were clearly used at Carnac in the megalithic (c. 4000 BC) before dynastic Egypt began, and by the time of the Rhind papyrus (Middle Kingdom) a system for containing irrationality of numbers had developed a school using grids, and what could be done with them. Ever since the Ancient world, this practice of “modularizing” buildings along rectangular or triangular “lines” became a key practical method outside of algebraic maths. It is therefore highly likely that grids gave the megalithic and later builders a canvas upon which to design and achieve accurate geometries not then rectalinear.

Some other resources.

More on the practical models of such early practices see my book Sacred Geometry: Language of the Angels. For more on flattened and multiple squares, please see chapter two of Sacred Number and the Lords of Time.

see also my youTube video of a talk at Megalithomania in 2015.

[1] ] Megalithic building types include standing stones, stone circles, stone rows, dolmen, chambered and other cairns.

[2] between 1934 and 1978

[3] during the late 1960s and early 1970s

[4] over two thousand miles away in the ancient near east

[5] [AAK and Howard Crowhurst]

[6] His survey can be found

[7] Historical metrology is a scattered remnant of the metrological system employed within the British stone circles and also within the Egyptian pyramids. It is this latter application of metrology in the ancient near east which spread metrology, though such an idea has also been opposed by archaeologists working in the near east.

[8] Sun, Moon and Stonehenge by Robin Heath 1998

Music of the Olmec Heads

Seventeen colossal carved heads are known, each made out of large basalt boulders. The heads shown here, from the city of San Lorenzo [1200-900 BCE], are a distinctive feature of the Olmec civilization of ancient Mesoamerica. In the absence of any evidence, they are thought to be portraits of individual Olmec rulers but here I propose the heads represented musical ratios connected to the ancient Dorian heptachord, natural to tuning by perfect fifths and fourths. In the small Olmec city of Chalcatzingo [900-500BCE] , Olmec knowledge of tuning theory is made clear in Monument 1, of La Reina the Queen (though called El Rey, the King, despite female attire), whose symbolism portrays musical harmony and its relationship to the geocentric planetary world *(see picture at end).

* These mysteries were visible using the ancient tuning theories of Ernest G. McClain, who believed the Maya had received many things from the ancient near east. Chapter Eight of Harmonic Origins of the World was devoted to harmonic culture of the Olmec, the parent culture of later Toltec, Maya, and Aztec civilizations of Mexico.

Monument 5 at Chatcatzinga has the negative shape of two rectangles at right angles to each other, with radiating carved strips framing the shape like waves emanating from the space through which the sky is seen. The rectangles are approximately 3 by 5 square or of a 5 by 5 square with its corner squares removed.

Monument 5 at Chalcatzingo is a framed hollow shape. The multiple squares have been added to show that, if the inner points are a square then the four cardinal cutouts are described by triple squares.

The important to see that the Olmec colossal heads were all formed as a carved down oval shape, that would fit the height to width ratio of a rectangular block. For example, three heads from San Lorenzo appear to have a ratio 4 in height to 3 in width, which in music is the ascending fourth (note) of our modern diatonic (major or Ionian) scale.

Even narrower is the fourth head at San Lorenzo, whose height is three to a width of two. This is the ratio of the perfect fifth, so called as the fifth note of the major scale.

And finally (for this short study), the ratio 6/5 can be seen in Head 9 of San Lorenzo and also at La Venta’s Monument 1 (below).

MUSICAL RATIOS

If the heads were conceived in this way, the different ratios apply when seen face on. The corners of the heads were probably rounded out from a supplied slab with the correct ratio between height and width. The corners would then round-out to form helmets and chins and the face added.

And as a group, the six heads sit within in a hierarchy of whole number ratios, each between two small numbers, different by one. At San Lorenzo, Head 4 looks higher status than Head 9 and this is because of its ratio 3/2 (a musical fifth or cubit), relative to the 6/5 of Head 9. We now call the fifth note dominant while the fourths (Heads 1, 5 and 8) are called subdominant. These two are the foundation stones of Plato’s World Soul {6 8 9 12}, within a low number octave {6 12} then having three main intervals {4/3 9/8 4/3}* where 4/3 times 9/8 equals 3/2, the dominant fifth.

*Harmonic numbers, more or less responsible for musical harmony, divide only by the first three primes {2 3 5} so that the numbers between six and twelve can only support four harmonic numbers {8 9 10}

San Lorenzo existed between 1200 to 900 BCE, and in the ancient Near East there are no clear statements for primacy of the octave {2/1}, nor was it apparent in practical musical instruments before the 1st Millennium BCE, according to Richard Dumbrill: Music was largely five noted (pentatonic) and sometimes nine-noted (enneadic) with two players. However, the eight notes of the octave could instead be arrived at, in practice, by the ear, using only fifths and fourths to fill out the six inner tones of a single octave; starting from the highest and lowest tones (identical sounding notes differing by 2/1). A single musical scale results from a harp tuned in this way: the ancient heptachord: it had two somewhat dissonant semitone (called “leftovers” in Greek), intervals seen between E-F and B-C on our keyboards (with no black note between). Our D would then be “do“, and the symmetrical scale we today call Dorian.

The order of the Dorian scale is tone, semitone, tone, tone, tone, semitone, tone {T S T T T S T} and the early intervals of the Dorian {9/8 S 6/5 4/3 3/2} are the ratios also found in these Olmec Heads*. The ancient heptachord** could therefore have inspired the Olmec Heads to follow the natural order tuned by fourths and fifths.

*I did not consciously select these images of Heads but rather, around 2017, they were easily found on the web. Only this week did I root out my work on the heads and put them in order of relative width.

**here updated to the use of all three early prime numbers {2 3 5} and hence part of Just Intonation in which the two semitones are stretched at the expense of two tones of 9/8 to become 10/9, a change of 81/80.
(The Babylonians used all three of these tones in their harmonic numbers.)

To understand these intervals as numbers required the difference between two string lengths be divided into the lengths of the two strings, this giving the ratio of the Head in question. The intervals of the heptachord would become known and the same ratios achieved within the Heads, carved out as blocks cut out into the very simple rectangular ratios, made of multiple squares.

The rectangular ratio of Head 4, expressed within multiple squares as 3 by 2.

The early numbers have this power, to define these early musical ratios {2/1 3/2 4/3 5/4 6/5}, which are the large musical tones {octave fifth fourth major-third minor-third}. These ratios are also very simple rectangular geometries which, combined with cosmological ideas based around planetary resonance, would have quite simply allowed Heads to be carved as the intervals they represented. The intervals would then have both a planetary and musical significance in the Olmec religion and state structure.

Frontispiece to Part Three of Harmonic Origins of the World: War in Heaven
The seven caves of Chicomoztoc, from which arose the Aztec, Olmec and
other Nahuatl-speaking peoples of Mexico. The seven tribes or rivers of the old world are here seven wombs, resembling the octaves of different modal scales, and perhaps including two who make war and sacrifice to overturn/redeem/re-create the world.

A Musical Cosmogenesis

Everything in music comes out of the number one, the vibrating string, which is then modified in length to create an interval. Two strings at right angles, held within a framework such as Monument 5 (if other things like tension, material, etc.were the same) would generate intervals between “pure” tones. However Monument 5 is not probably symbolic but rather, it was probably laid flat like a grand piano (see top illustration). Wooden posts could hold fixings, to make a framework for one (or more) musical strings of different length, at right angles to a reference string. This would be a duo-chord or potentially a cross-strung harp. Within the four inner points of Monument 5 is a square notionally side length. In the image of Monument 1, and variations in height and width from the number ONE were visualized in stone as emanating waves of sound.

The highest numbers lead to the smallest ratio of 6/5 then the 6/5 ratio of Head 9 can be placed with five squares between the inner points and the 3/2 ratio of Head 2 then fills the vertical space left open within Chalcatzingo’s Monument 5.

Monument 5’s horizontal gap can embrace the denominator of a Head’s ratio (as notionally equal to ONE) so that the inner points define a square side ONE, and the full vertical dimension then embraces the 3/2 ratio of the tallest, that of Head 2.

It may well be that this monument was carved for use in tuning experiments and was then erected at Chalcatzingo to celebrate later centuries of progress in tuning theory since the San Lorenzo Heads were made. By the time of Chalcatzingo, musical theory appears to have advanced, to generate the seven different scales of Just intonation (hence the seven caves of origin above), whose smallest limiting number must then be 2880 (or 4 x 720), the number presented (as if in a thought bubble) upon the head of a royal female harmonist (La Reina), see below. She is shown seeing the tones created by that number, now supporting two symmetrical tritones. The lunar eclipse year was also shown above her head (that is, in her mind) as the newly appeared number 1875, at that limit. This latter story probably dates around 600 BCE. This, and much more besides, can be found in my Harmonic Origins of the World, Chapter Eight: Quetzcoatl’s Brave New World.

Figure 5.8 Picture of an ancient female harmonist realizing the matrix for 144 x 20 = 2880. If we tilt our tone circle so that the harmonist is D and her cave is the octave, then the octave is an arc from bottom to top, of the limit. Above and below form two tetrachords to A and D, separated by a middle tritone pain, a-flat and g-sharp. Art by by Michael D Coe, 1965: permission given.

The Megalithic Pythagoras

Pythagoras of Samos (c.600BC) very likely gleaned megalithic number science on his travels around the “Mysteries” of the ancient world. His father, operating from the island of Samos, became a rich merchant, trading by sea and naming his child Pythagoras; after the god of Delphi who had “killed” the Python snake beneath Delphi’s oracular chasm, now a place of Apollo. The eventual disciples of Pythagoras were reclusive and secretive, threatening death on anybody who would openly speak of mysteries, such as the square root of two, to the uninitiated. It can be seen from the previous post that many such “mysteries” were natural discoveries made by the megalithic astronomers, when learning how to manipulate number without arithmetic, through a metrological geometry unfamiliar to the romantic sacred geometry of “straight edge and compass”.

As previously stated, the vertex angles of right triangles whose longer sides are integer in length, are angular invariants belonging to the invariant ratio of their sides. To create a {11 14} angle one can use any multiple of 11 and the same multiple of 14 to obtain the invariant angle whereupon, the hypotenuse and base will shrink or grow together in that ratio: any length on the “14” line is 14/11 of any length below it on the “11” base line and visa versa.

If one enlarges the base line to being 99 then the diagonal of the square side length 99 will be 140, which is 99 times the square root of two. In choosing, as I did, to enlarge 91 (the quarter year) to 9 x 11 = 99, I encountered the cubit of the Samian (“of Samos”) foot of 33/35 feet, as follows. When Heraclitus, also of Samos, visited the Great Pyramid he gave its southerly side length as 800 “of our feet” and 756 English feet (the measured length) needs to be divided by 189 and multiplied by 200 to obtain such a measurement, giving a Samian foot of 189/200 (=0.945 feet) which is 441/440 of the Samian root foot of 33/35 feet. 33/35 x 3/2 = 99/70 (1.4143) feet but its inverse of 35/33 x 4/3 = 140/99 feet.

There is then no doubt about Samos as being a center in the Greek Mysteries since, the form of the Greek temple seems first to evolve there. For example, 10,000 feet of 0.945 feet equal 945 feet, the number of days in 32 lunar months. The Heraion of Samos (pictured above) has been shown to have had pillars around a platform (a peristyle), and an elongated rectangular room (a cella), involving megalithic yards and a 4-square geometry cunningly linking lunar and solar years, to alignments to the Moon’s minimum using the {5 12 13} second Pythagorean Triangle. (diagram at top is from figure 5.9 of Sacred Geometry: Language of the Angels).

The reason for the Samian (lit. “of Samos”) foot being 33/35 feet appears to be that as a cubit of 99/70 feet, or √2 =1.4142, it is the twin of 140/99 as 1.41. In the geometrical world such foot ratios were exact, relative to the English foot; which is the root of the Greek module and of all other rational modules, such as the Royal of 8/7 feet. Such cubits could measure across the diagonal the same number as the side length in English feet. Such measures became essential for building of rectangular temple structures in Greece and further east, but when the metrological geometry, of square and circle in equal perimeter, was the focus, 140 in the diagonal can use 99 in the base (or side-length of the square).

If we remember that the 99 length must be rooted from the shared center of the square and equal circle then, the side length of the square must be twice that, or 198. This means that the perimeter of the square must be 4 times that, equal to 792, at which point readers of John Michell’s books on models of the world will recall that the diameter of the mean earth can be presented, within an equal perimeter design, if each unit is multiplied by 720 units of 10 miles, my own summary being in my recent Sacred Geometry book , chapter 3 on measuring the Earth. This model Michell called The Cosmological Prototype, where the mean earth diameter is (quite accurately) 7920 miles.

If the square of 198 feet is rolled out into a single line, it “becomes” the mean diameter of the Earth in units of 10 miles. For this sort of reason, my 2020 book was called Language of the Angels, since this model looks like a first approximation of the mean earth size which a later Ancient Metrology would improve upon as to accuracy, by a couple of miles! That is, that the earth’s dimensions follow a design based upon metrological geometry and the properties of numbers.

John Michell finalized his Cosmological Model in an Appendix to The Sacred Center, and in his text on “sacred Geometry, Ancient Science, and the Heavenly Order on Earth” called The Dimensions of Paradise, both published by Inner Traditions.

The Best Eclipse Cycle

The anniversary of the Octon (4 eclipse years in 47 lunar months) did not provide similar eclipses and so, by counting more than four, the other motions of the Moon could also form part of that anniversary. This is especially true of the anomalistic month, which changes the changes the apparent size of the Moon within its phase cycle, recreate the same type of lunar eclipse after nineteen eclipse years. This 18 year and 11 day period is now taken as the prime periodicity for understanding eclipse cycles, called the Saros period – known to the Babylonian . The earliest discovered historical record of what is known as the saros is by Chaldean (neo-Babylonian) astronomers in the last several centuries BC.

The number of full moons between lunar eclipses must be an integer number, and in 19 eclipse years there are a more accurate 223 lunar months than with the 47 of the Octon. This adds up to 6585.3 days but the counting of full moon’s is obviously ideal as yielding near-integer numbers of months.

We noted in a past post that the anomalistic month (or AM), regulating the moon’s size at full moon, has a geometrical relationship with eclipse year (or EY) in that: 4 AM x pi (of 3.1448) equals the 346.62 days of the eclipse year as the circumference. Therefore, in 19 EY the diameter of a circle of circumference 19 x 346.62 days must be 4 x 19 AM so that , 76 AM x pi equals 223 lunar months, while the number of AM in 223 lunar months must be 239; both 223 and 239 being prime numbers.

Continue reading “The Best Eclipse Cycle”

Vectors in Prehistory 2

In early education of applied mathematics, there was a simple introduction to vector addition: It was observed that a distance and direction travelled followed by another (different) distance and direction, shown as a diagram as if on a map, as directly connected, revealed a different distance “as the crow would fly” and the direction from the start.

The question could then be posed as “How far would the plane (or ship) be, from the start, at the end”. This practical addition applies to any continuous medium, yet the reason why took centuries to fully understand using algebraic math, but the presence of vectors within megalithic counted structures did not require knowledge of why vectors within geometries like the right triangle, were able to apply vectors to their astronomical counts.

Continue reading “Vectors in Prehistory 2”

Gavrinis 1: Its dimensions and geometrical framework

This article first appeared in my Matrix of Creation website in 2012 which was attacked, though an image had been made. Some of this material appeared in my Lords of Time book.

photo For Wikipedia by Mirabella.

Gavrinis and Tables des Marchands are very similar monuments, both in the orientation of their passageways and their identical latitude. Gavrinis is about 3900 metres east of Tables des Marchands but, unlike the latter, has a Breton name based upon the root GVR (gower). Both passageways directly express the difference between the winter solstice sunrise and the lunar maximum moonrise to the South, by designing the passages to allow these luminaries to enter at the exact day of the winter solstice or the most southerly moonrise over many lunar orbits, during the moon’s maximum standstill. Thus both the monuments allow the maximum moon along their passageway whilst the winter solstice sunrise can only glance into their end chambers.

From Howard Crowhurst’s work on multiple squares, we know that this difference in angle is that between a 3-4-5 triangle and the diagonal of a square which is achieved directly by the diagonal of a seven square rectangle.

Figure 1 The essence of difference between the winter solstice sunrise (as diagonal of 4 by 3 rectangle) and southerly maximum moonrise (as diagonal of a single square), on the horizon, is captured in the diagonal of a seven squares rectangle.
Continue reading “Gavrinis 1: Its dimensions and geometrical framework”