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Units within the Great Pyramid of Giza

There is a great way to express pi of 22/7 using two concentric circles of diameter 11 and 14 (in any units). Normally, a diameter of 7 gives rise to a circumference of 22, when pi is being approximated as 22/7 (3.142587) rather than being the irrational number 3.141592654 … for then, the 14 diameter should have a circumference of 44, which is also the perimeter of the square which encloses a circle of diameter 11.

The square of side 11 and
the circle of diameter 14
will both have the same perimeter.

Figure 1 The Equal Perimeter model of two circles, the smaller of which has an out-square of equal perimeter to the greater circle

This geometry, first found by John Michell [1933-2009], and by others after that, within the monuments and artifacts of the megalithic, ancient and medieval worlds: notably Stonehenge where the mean diameter of the Sarsen Lintel ring is 14 units while the concentric bluestone circle appears to have had a diameter 11 of the same units. One of a handful of known models known to the ancient world, this work continues on from my forthcoming book, Sacred Geometry: Language of the Angels.

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St Pierre 1: Jupiter and the Moon

The egg-shaped stone circles of the megalithic, in Brittany by c. 4000 BC and in Britain by 2500 BC, seem to express two different astronomical time lengths, beside each other as (a) a circumference and then (b) a longer, egg-shaped extension of that circle. It was Alexander Thom who analysed stone circles in the 20th century as a hobby, surveying most of the surviving stone circles in Britain and finding geometrical patterns within irregular circles. He speculated the egg-shaped and flattened circles were manipulating pi so as to equal three (not 3.1416) between an initial radius and subsequent perimeter, so making them commensurate in integer units. For example, the irregular circle would have perimeter 12 and a radius of 4 (a flattened circle).

However, when the forming circle and perimeter are compared, these can compare the two lengths of a right-triangle while adding a recurring nature: where the end is a new beginning. Each cycle is a new beginning because the whole geocentric sky is rotational and the planetary system orbital. The counting of time periods was more than symbolic since the two astronomical time periods became, by artifice, related to one another as two integer perimeters that is, commensurate to one another, as is seen at St Pierre (fig.3).

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Durrington Walls and its massive circle of Pits

Recent analysis of animal bones within Durrington Walls indicated, to the archaeologists involved, that people had travelled there from all over the British mainland, along with animals then eaten inside the henge[1]. But what would these people be doing there? It had earlier been suggested that an elite responsible for building Stonehenge lived in a wooden roundhouse within the henge ([2] see figure 1). So, people may have come from elsewhere to help the building works now found between Stonehenge and Avebury.

Figure 1 Reconstruction of the likely roundhouses within the Durrington henge, based on post-hole evidence [2]

More recently, pits have been found [3] within a circular strip that I notice lies between 3168 feet and 4038 feet from Durrington Walls, a boundary 864 feet wide. The pits may contain the material remains of the building elite and perhaps of those workers who died, functioning like nearby barrows but vertically.

This post aims to explain why this might have been done according to a significant geometrical pattern. In the megalithic, numbers played an active role and this perhaps inspired the myth of Atlantis recorded by Plato – the classical Greek writer who transmitted the ancient notion that numbers had a causative role in forming the “world soul”, rather than our usage for number: a means to quantify things within civilized societies or laws of nature.

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Fields, Racetracks and Temples in Ancient Greece

The fields of ancient Greece were organised in a familiar way: strips of land in which a plough could prepare land for arable planting. Known in various languages as furlong https://en.wikipedia.org/wiki/Furlong, runrig, journel, machen etc, in Greece there was a nominal length for arable strips which came to be associated with the metrological unit of 600 feet called a stadia. The length of foot used was systematically varied from the foot we use today, using highly disciplined variations (called modules); each module a numeric ratio of the Greek module, whose root foot was the English foot [Neal, 2000]. These modules are found employed throughout the ancient world, lengthening or reducing lengths such as the stadia, to suit geometrical problems; such as the division of land into fields (figure 1).

Machine generated alternative text:
Ill* 
Oxgang = 15 Acres 
4 Rods
Figure 1 The land area of an acre seen as the amount of land tillable by one ox in a ploughing season
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Fibonacci in Jupiter’s 12-fold Heaven

The Fibonacci series is an ideal pattern, widely found within living systems, in which the present magnitude or location of something is the product of two previous magnitudes or locations of it. The next magnitude will again be the sum of the last two magnitudes in what is, an algorithmic pattern producing approximation to the Golden Mean (designated by the Greek letter φ,’phi’). As the series gets larger, the ratio (or proportion) between successive magnitudes will better approximate the irrational value of φ = 1.618033 … – which has an unlimited fractional part whilst the virtue of the Fibonacci numbers within the Series is that they are integers forming rational fractions.

Jupiter taken by the Wide Field Hubble Telescope by NASA, ESA, and A. Simon (Goddard Space Flight Center)
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The Avebury Square within it’s Southern Circle

Soil resistance work (“geophys”), by archaeologists from the University of Leicester, of the land inside the southern stone circle of Avebury Henge, has revealed more about the Obelisk and lines of standing stones, which appear to have formed a near-square rectangle. Information can be hard to obtain when work is yet to be published but a press release to the Guardian and others many months ago (December 27th 2019) enabled figures from a media set to be viewed with public access. This has enabled me to so some site interpretation, using the (as-yet damned) numerical technique called ancient metrology. My results are fascinating and build upon the megalithic use of counting time as length to track important time-cycles such as the Nodal Period of 6800 days, between lunar maxima and minima, and the Metonic period of both 19 years and 235 lunar months, within which all of the varied orientations of Sun Moon and Sky are recycled.

Figure 1 Avebury Henge from the North with major features labelled
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Film of John Michell at Lundy Island

This is a film by me of John Michell before his death. It was made on Lundy Island at which time he was working on some of his last published ideas about the British Isles from the perspective of sacred geometry and metrology, both fields in which John made outstanding contributions including The View Over Atlantis, Dimensions of Paradise and Ancient Metrology. It is published here to enable those who did not to experience the unique presence of John Michell, itself conducive to understanding his work.

originally published Monday, 28 May 2012 at 10:58
It was read 478 times

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The Golden Mean compared to PI

In reviewing some ancient notes of mine, I came across an interesting comparison between the Golden Mean (Phi) and PI. They are more interesting in reverse:

A phi square (area: 2.618, side: 1.618) has grown in area relative to a unit square by the amount (area: 0.618) plus the rectangle (area:1 ). This reveals the role of phi’s reciprocal square (area: 0.384) in being the reciprocal of the reciprocal so that in product they return the unity (area: 1).

On the right, the phi squared square showing how the reciprocal of phi and its square uniquely sum to unity (area: 1), a property that is scale invariant between structures who share the same units and grow according to the Golden Mean.
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Astronomical Time within Clava Cairns

In North East Scotland, near Inverness, lies Balnuaran of Clava, a group of three cairns with a unique and distinctive style, called Clava cairns; of which evidence of 80 examples have been found in that region. They are round, having an inner and outer kerb of upright stones between which are an infill of stones. They may or may not have a passageway from the outer to the inner kerb, into the round chamber within. At Balnuaran, two have passages on a shared alignment to the midwinter solstice. In contrast, the central ring cairn has no passage and it is staggered west of that shared axis.

This off-axis ring cairn could have been located to be illuminated by the midsummer sunrise from the NE Cairn, complementing the midwinter sunset to the south of the two passageways of the other cairns. Yet the primary and obvious focus for the Balnuaran complex is the midwinter sunset down the aligned passages. In fact, the ring cairn is more credibly aligned to the lunar minimum standstill of the moon to the south – an alignment which dominates the complex since, in that direction the horizon is nearly flat whilst the topography of the site otherwise suffers from raised horizons.

Cairns at Balnuaran of Clava. plan by A. Thom and pictures by Ian B. Wright
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Use of foot ratios in Megalithic Astronomy

The ratios of ancient metrology emerged from the Megalithic innovations of count&compare: counting time as length and comparing lengths as the longest sides of right triangles. To compare two lengths in this way, one can take a longer rope length and lay it out (say East-West), starting at the beginning of the shorter rope length, using a stake in the ground to fix those ends together.

The longer rope end is then moved to form an angle to the shorter, on the ground, whilst keeping the longer rope straight. The Right triangle will be formed when the longer rope’s end points exactly to the North of the shorter rope end. But to do that one needs to be able to form a right angle at the shorter rope’s end. The classic proposal (from Robin Heath) is to form the simplest Pythagorean triangle with sides {3 4 5} at the rope’s end. One tool for this could then have been the romantic knotted belt of a Druid, whose 13 equally spaced knots could define 12 equal intervals. Holding the 5th knot, 8th knot and the starting and ending knots together automatically generates that triangle sides{3 4 5}.

Forming a square with the AMY is helped by the diagonals being rational at 140/99 of the AMY.
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A Lecture introducing Sacred Number and the Lords of Time

… given as the John Michell Memorial Lecture for
at Megalithomania 2015 in Glastonbury

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John Michell’s Perpetual Choirs

15 April 2017 Views: 10450

In 1972  John Michell inferred an enormous ten-sided form nearly sixty three miles across, in which important historical and neolithic sites had been intended as ten vertices around an ancient centre, signified by a Whiteleafed Oak.

Figure 1 The Decagon of Perpetual Choirs, anchored upon Stonehenge, the Solstice sunrise in summer and set in winter

Michell had previously [1991] developed the idea of the enchantment of the land as an actual practice; land areas were enchanted by using a geometrical pattern integrated with myths and ritual calendars, enacted within that framework. This  was long before, around 930, such a pattern was being established of thing-places in Iceland. The idea of thing places is still find-able in English names such as Goring, the centre northeast of Stonehenge, where the summer solstice sun arose.

“Perpetual choirs were a Celtic institution, from pagan into early Christian times. In Iola Morganwg’s Triads of Britain, translated from Welsh, it is stated that ‘in each of these three choirs there were 24,000 saints; that is,
there were a hundred for every hour of the day and the night in rotation, perpetuating the praise and service of God without rest or intermission.’ ”  – The Measure of Albion

“Three of the choirs were located at Stonehenge, at Glastonbury, and near Llantwit Major in Wales. Others appear to have been at Goring-on- Thames and at Croft Hill in Leicestershire, a traditional site of ritual,  legal, and popular assemblies.” The Dimensions of Paradise


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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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Thornborough Henge as Moon’s Maximum Standstill

The three henges appear to align to the three notable manifestations to the north west of the northerly moon setting at maximum standstill. The distance between northern and southern henge entrances could count 3400 days, each 5/8th of a foot (7.5 inches), enabling a “there and back again” counting of the 6800 days (18.618 solar years/ 19.618 eclipse years) between lunar maximum standstills.

Figure 1 The three henges are of similar size and design, a design most clear in what remains of the central henge.
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Earth and Moon within Westminster’s Coronation Pavement

Our modern globes are based upon political boundaries and geographical topography yet they have geometrical predecessors, which described the world as an image, diagram or schemata. The original idea for the form of the world was summarised within a simple two dimensional geometry, like an eastern mandala or yantra.

Such a diagram was built into the Cosmati pavement at Westminster Abbey, built by Henry III and dedicated to the Saxon King and Saint Edward the Confessor. This exotic pavement became the focus for the Coronations of subsequent English then British monarchs.


Figure 1 Photo of the Cosmati Pavement at Westminster Abbey
[Copyright: Dean and Chapter of Westminster]
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Musical Tones of the Outer Planets

My crucial entré to planetary harmony came when I noticed musical ratios in the synodic time periods of Jupiter and Saturn relative to the lunar year. This approach differs from the norms for “harmonies of the spheres” (a.k.a. Musica Universalis) which are geometrical and spatial, rather than temporally harmonic.

The planetary harmony I found within synodic periods became the subject of my new book The Harmonic Origins of the World (pub. 2018). These synodic ratios have been parts of my work from c. 2000, then expressed as “matrix diagrams” (Matrix of Creation, figure 2 below). In my new book, I show how ancient tuning theory seems to have presented the same information, in a different type of matrix (see figure 4).

Below I connect the outer planets using two additional (and useful) kinds of diagram, the right-angled triangle (figure 1) and the Pentad (figure 5), the latter developed in the 20th century within a discipline called Systematics. 


Figure 1 The harmonic ratios between the nearest two outer planets and the lunar year. The four square rectangle with side length of four, when equal to the lunar year gives, geometrically, the solar year as its diagonal length. The outer planetary synods are longer than the solar year as the planets have moved ahead of their last opposition to the sun. Such oppositions are marked by an outer planet appearing to travel in a loop, amongst the stars
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Cretan Calendar Disks

I have interpreted two objects from Phaistos (Faistos), both in the Heraklion Museum. Both would work well as calendar objects.

One would allow the prediction of eclipses:

The other for tracking eclipse seasons using the 16/15 relationship of the synod of Saturn (Chronos) and the Lunar Year:

paper: Lunar Simulation at Le Manio

Our survey at Le Manio revealed a coherent arc of radial stones, at least five of which were equally long, equally separated and set to a radius of curvature that suggested a common centre. It appears the astronomers at Le Manio understood that, following three lunar sidereal orbits (after 82 days) the moon would appear again at the same point on the ecliptic at the same time of day

paper: The Origins of Day-Inch Counting

ABSTRACT
This paper presents the theory that in the Megalithic period, around 4500-4000 BCE, astronomical time periods were counted as one day to one inch to form primitive metrological lengths that could then be compared, to reveal the fundamental ratios between the solar year, lunar year, and lunar month and hence define a solar-lunar calendar. The means for comparison used was to place lengths as the longer sides of right angled triangles, leading to a unique slope angle. Our March 2010 survey of Le Manio supports this theory.

Le Menec: Start of Carnac’s Alignments

The Meaning Of Le Menec

“Alignments” are long rows of stones, that run in parallel for long distances through the landscape. The alignments in Carnac, Brittany, often have a starting point in what the French call a cromlech. Based upon a circular geometry, these monuments are made up of stones following arcs to form a single compound shape. The stones of a cromlech can be touching or they can be spaced out and in some cases, stones might have been removed during the historical period but in some cases also, gaps in the “walls” of a cromlech were probably intentional and are there on purpose.
Originally published July 2012

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