Sacred numbers within quantified geometries.
- The Cellular World of Twelve
The foot has twelve inches just as the British shilling had twelve pence. A good case can be made for twelve as a base like 10, since there are 12 months within the year and many ancient monuments can be seen to have employed duodecimal alongside decimal number, to good effect. - Design of the Taj Mahal: its Façade
The Taj Mahal is one of the most recognizable buildings on earth. It was built by a Moghul king as a memorial for his dead queen and for love itself. The Mughals became famous for their architecture and the Persian notion of the sacred garden though their roots were in Central Asia just north of Persia. - The Moon is Key to our Survival
With the advent of many orbital missions, the Moon is threatened with orbital and other changes due to space travel. - The Stonehenge Crop Circle of 2002
One sees most clearly how a single concrete measure such as 58 feet can take the meaning of the design into the numbers required to create it. However, metrology of feet and types of feet can hide the elegance of a design. - Chartres 3: Design of West Façade
The design of the twin towers of Chartres point to an extraordinary understanding of its designers, quite unlike pre or modern understandings of the outer planets and their harmonic ratios. We have already seen a propensity for using the ordinary English footThe standard prehistoric foot (of 12 inches) representing a unity from which all other … Continue reading “Geometry and Metrology” - Chartres 1: the cosmic coding of its towers in height
The lunar crescent atop the “moon” tower’s cross. Chartres, in north-west France, is a very special version of the Gothic transcept cathedral design. Having burnt down more than once, due to wooden ceilings, its reconstruction over many building seasons and different masonic teams, as funds permitted, would have needed strong organizing ideas to inform the … Continue reading “Chartres 1: the cosmic coding of its towers in height” - Dun Torcuill: The Broch that Modelled the World
image above courtesy Marc Calhoun Script This video introduces an article on a Scottish iron-age stone tower or brock which encoded the size of the Earth. You can view the full article on sacred dot number sciences dot org, searching for BROCK, spelt B R O C H. In the picture above [1] the inner profile of … Continue reading “Dun Torcuill: The Broch that Modelled the World” - Earth and Moon within Westminster’s Coronation Pavement
Our modern globes are based upon political boundaries and geographical topography yet they had geometrical predecessors which described the world as an image, a diagram or schemata. By some act of intuition, an original Idea for the form of the Earth had become established as a simple two-dimensional geometry, very like eastern mandalas. Such a … Continue reading “Earth and Moon within Westminster’s Coronation Pavement” - Developmental Roots below 6
Square roots turn out to have a strange relationship to the fundaments of the world. The square root of 2, found as the diagonal of a unit square, and the square root of 3 of the diametric across a cube; these are the simplest expressions of two and three dimensions, in area and volume. This … Continue reading “Developmental Roots below 6” - Umayyad Mosque: Golden Rectangles from Squares
photo above of Umayyad Mosque, Damascus by Bernard Gagnon for Wikipedia CC BY-SA 3.0. In previous articles on double squares and then St Peter’s Basilica, it became clear that squares and double squares have been embodied, within sacred buildings and art, because circles can then spawn golden rectangles from them. A golden rectangleA rectangle whose sides … Continue reading “Umayyad Mosque: Golden Rectangles from Squares” - St Peter’s Basilica: A Golden Rectangle Extension to a Square
HAPPY NEW YEAR above: The Basilica plan at some stage gained a front extension using a golden rectangleA rectangle whose sides are in the ratio of the Golden MeanThe Golden Mean is that unique ratio {1.618034}, relative to ONE {1}, in which its square and reciprocal share the same fractional part {.618034}. It is associated with the synodic period of the planet Venus, which is 8/5 {1.6} of the practical year {365 days}, by approximation. It is a key proportion found in Greco-Roman and later "classical" architecture, and commonly encountered in the forms living bodies take. (1.618034) or a Fibonacci approximation to the Golden Mean.. below: Later Plan for St. Peter’s 16th–17th century. Anonymous. Metropolitan Museum. The question is whether the extension … Continue reading “St Peter’s Basilica: A Golden RectangleA rectangle whose sides are in the ratio of the Golden Mean (1.618034) or a Fibonacci approximation to the Golden Mean. Extension to a Square” - Starcut Diagram: geometry to define tuning
This is a re-posting of an article thought lost, deriving in part from Malcolm Stewart’s Starcut Diagram. The long awaited 2nd edition Sacred Geometry of the Starcut Diagram has now been published by Inner Traditions. Before this, Ernest McClainAmerican Cryptologist and Pythagorean Musicologist who decoded Plato’s cryptic numerical ciphers in The Pythagorean Plato. The Myth … Continue reading “Starcut Diagram: geometry to define tuning” - Double Square and the Golden Rectangle
above: Dan Palmateer wrote of this, “it just hit me that the conjunction of the circle to the golden rectangleA rectangle whose sides are in the ratio of the Golden Mean (1.618034) or a Fibonacci approximation to the Golden Mean. existed.” Here we will continue in the mode of a lesson in Geometry where what … Continue reading “Double SquareA unit rectangle of 1 by 2, with important use for alignment (Carnac), cosmology (Egypt) and tuning theory (Honnecourt Man). and the Golden Rectangle” - Story of Three Similar Triangles
first published on 24 May 2012, Figure 1 Robin Heath’s original set of three right angled triangles that exploited the 3:2 points to make intermediate hypotenuses so as to achieve numerically accurate time lengths in units of lunar or solar months and lunar orbits. Interpreting Lochmariaquer in 2012, an early discovery was of a near-Pythagorean triangle … Continue reading “Story of Three Similar Triangles” - The Approximation of π on Earth
π is a transcendental ratio existing between a diameter/ radius and circumference of a circle. A circle is an expression of eternity in that the circumference, if travelled upon, repeats eternally. The earths shape would be circular if the planet did not spin. Only the equator is now circular and enlarged, whilst the north and … Continue reading “The Approximation of π on Earth” - 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 … Continue reading “The Broch that Modelled the Earth” - Walking on the Moon
There are plans to walk again on the moon (above is a NASA visualization), but there is a sense in which the surface of the moon belongs to the surface of the earth, since the earth’s circumference is 4 times the mean diameter of the earth, minus the moon’s circumference. The Earth and Moon were … Continue reading “Walking on the Moon” - Seven, Eleven and Equal Perimeters
above: image of applications involving sacred geometry based upon pior π: The constant ratio of a circle’s circumference to its diameter, approximately equal to 3.14159, in ancient times approximated by rational approximations such as 22/7The simplest accurate approximation to the π ratio, between a diameter and circumference of a circle, as used in the ancient and prehistoric periods.. as 22/7The simplest accurate approximation to the π ratio, between a diameter and circumference of a circle, as used in … Continue reading “Seven, Eleven and Equal Perimeters” - The Megalithic Numberspace
above: counting 37 lunar months six times to reach 222, one month short of 223: the strong SarosThe dominant eclipse period of 223 lunar months after which a near identical lunar or solar eclipse will occur. eclipse period. There is an interesting relationship between the multiple interpretations of a number as to its meaning, and … Continue reading “The Megalithic Numberspace” - Counting Perimeters
above: a slide from my lecture at Megalithomania in 2015 We know that some paleolithic marks counted in days the moon’s illuminations, which over two cycles equal 59 day-marks. This paved the way for the megalithic monuments that studied the stars by pointing to the sky on the horizon; at the sun and moon rising … Continue reading “Counting Perimeters” - 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 … Continue reading “Vectors in Prehistory 2” - Vectors in Prehistory 1
In previous posts, it has been shown how a linear count of time can form a square and circle of equal perimeterA type of geometry where an rectilinear geometry has same perimeter as a circle, usually a square but also a 6 by 5 rectangle whose perimeter is 22, assuming pi is 22/7 or 3 + 1/7. to a count. In this way three … Continue reading “Vectors in Prehistory 1” - How Geometries transformed Time Counts into Circles
Above: example of the geometry that can generate one or more circles, equal to a linear time count, in the counting units explained below. It is clear, one so-called “sacred” geometry was in fact a completely pragmatic method in which the fourfold nature of astronomical day and month counts allowed the circularization of counts, once … Continue reading “How Geometries transformed Time Counts into Circles” - The Octon of 4 Eclipse Years
Having seen, in the last post, that three eclipse years fitted into the three-year count at Le Manio, another eclipse fact has come to light, recorded within the nearby site of Crucuno, between its dolmenA chamber made of vertical megalithsStructures built out of large little-altered stones in the new stone age or neolithic between 5,000-2,500 (bronze age), in the pursuit of astronomical knowledge. upon which a roof or ceiling slab was balanced. and rectangle. The coding of time … Continue reading “The Octon of 4 Eclipse Years” - The Strange Design of Eclipses
We all know about solar eclipses but they are rarely seen, since the shadow of the moon (at one of its two orbital nodes) creates a cone of darkness which only covers a small part of the earth’s surface which travels from west to east, taking hours. For the megalithic to have pinned their knowledge … Continue reading “The Strange Design of Eclipses” - The Integration of the Megalithic Yard
Above is a proposed geometric relation between Thom’s megalithic yardAny unit of length 2.7-2.73 feet long, after Alexander ThomScottish engineer 1894-1985. Discovered, through surveying, that Britain's megalithic circles expressed astronomy using exact measures, geometrical forms and, where possible, whole numbers. discovered 2.72 ft and 2.722 ft as units within the geometry within the megalithic monuments of Britain and Brittany. (2.72 feet), the royal cubit3/2 feet of any sort, such as 12/7 {1.714285}, 1.5 Royal feet of … Continue reading “The Integration of the Megalithic Yard” - Counting Days and Lunar Months
Megalithic astronomy achieved far more than modern studies of their astronomy have thought possible. The role of the megalithic in seeding the later religious ideas, of subsequent civilizations, has therefore caused ancient religions to be seen as having no objective basis, and to be considered works of human imagination alone. To correct for this wrong … Continue reading “Counting Days and Lunar Months” - An Angelic Geometrical Design
The above diagram contains information with can generally only be grasped by using a geometrical diagram. Its focus is the properties of a right triangle that is 4 times larger than its third and shortest side. The left hand view illustrates what we call Pythagoras’ theorum, namely that “The squares of the shorter sides add … Continue reading “An Angelic Geometrical Design” - Powers of the Golden Mean
Sheikh Lotfollah Mosque is one of the masterpieces of Iranian architecture that was built during the Safavid Empire, standing on the eastern side of Naqsh-i Jahan Square, Esfahan, Iran. Construction of the mosque started in 1603 and was finished in 1619. for Wikipedia by Phillip Maiwald The Golden MeanThe Golden Mean is that unique ratio {1.618034}, relative to ONE {1}, in which … Continue reading “Powers of the Golden Mean” - Leak Project Interview
Rex Bear talked to me by Zoom on the 11th March; about the extensive background of my new book Sacred Geometry: Language of the Angels. Below is embedded from the Leak Project YouTube channel. - The Stonehenge Trilithons (Part 2): Day-Inch Counting
In the previous article, it was shown that the form of the trilithons, of five taller double sarsens approximating to a five-pointed star, matches the astronomical phenomena of the successive morning and evening stars, as Venus approaches Earth from the east and then recedes to the west as the morning pass. On approach, the planet … Continue reading “The Stonehenge Trilithons (Part 2): Day-Inch Counting” - The Richard Syrett Interviews on Sacred Geometry: Language of the Angels
I recently recorded a podcast with Richard Syrett and will be talking with him again today (January 2nd) on Coast to Coast, starting 10pm Pacific time. In the UK, this is tomorrow (Sunday the 3rd) at 6am GMT. Both these interviews are in response to my new book Sacred Geometry: Language of the Angels, which … Continue reading “The Richard Syrett Interviews on Sacred Geometry: Language of the Angels” - Geometry 7: Geometrical Expansion
above: the dolmenA chamber made of vertical megaliths upon which a roof or ceiling slab was balanced. of Pentre Ifan (wiki tab) In previous lessons, fixed lengths have been divided into any number of equal parts, to serve the notion of integer fractions in which the same length can then be reinterpreted as to its … Continue reading “Geometry 7: Geometrical Expansion” - Geometry 6: the Geometrical AMY
By 2016 it was already obvious that the lunar month (in days) and the PMYproto-megalithic yard of 32.625 (261/8) day-inches, generated at Le Manio Quadrilateral as the difference between three solar and three lunar year counts., AMYA megalithic yardAny unit of length 2.7-2.73 feet long, after Alexander Thom discovered 2.72 ft and 2.722 ft as units within the geometry within the megalithic monuments of Britain and Brittany. which, in inches, expresses the true astronomical ratio of mean solar months to lunar months. and … Continue reading “Geometry 6: the Geometrical AMY” - Le Site Mégalithique du Manio à Carnac
by Howard Crowhurst Perched on a hill in the forest north of the CarnacAn extensive megalithic complex in southern Brittany, western France, predating the British megalithic. alignmentsIn general, to the sun and moon on the horizon, rising in the east or setting in the west. Also, a name special to Carnac’s groups of parallel rows … Continue reading “Le Site Mégalithique du Manio à Carnac” - Geometry 4: Right Triangles within Circles
This lesson is a necessary prequel to the next lesson. It is an initially strange fact that all the possible right triangles will fit within a half circle when the hypotenuse equals the half-circles diameter. The right angle will then exactly touch the circumference. From this we can see visually that the trigonometrical relationships, normally … Continue reading “Geometry 4: Right Triangles within Circles” - Geometry 3: Making a circle from a counted length
The number of days in four years is a whole number of 1461 days if one approximates the solar yearFrom Earth: the time in which the sun moves once around the ZodiacThe 12 constellations through which the sun passes in the solar year of 365.2422 days, now known to be caused by the orbital period of the Earth around the Sun. to 365¼ days. This number is found across … Continue reading “Geometry 3: Making a circle from a counted length” - Preface: The Metrology of the Brochs
feature picture: Broch of Mousa. The broch on the island of Mousa is the best-preserved of the many brochs in northern Scotland. It is thought to be some 2000 years old credit: Anne Burgess / Broch of Mousa / CC BY-SA 2.0 I wrote this preface for Euan MacKie who had resurrected his work on measures found within the brochs … Continue reading “Preface: The Metrology of the Brochs” - Sacred Number and the Lords of Time
Back Cover ANCIENT MYSTERIES “Heath has done a superb job of collating his own work on the subject of megalithsStructures built out of large little-altered stones in the new stone age or neolithic between 5,000-2,500 (bronze age), in the pursuit of astronomical knowledge. with the objective views of many other researchers in the field. I … Continue reading “Sacred Number and the Lords of Time” - A Pyramidion for the Great Pyramid
image: By 1200 BC, the end of the Bronze Age, the Egyptian map of the world (above) showed nine bows or latitudes, numbers 4 to 9 including the Nile Delta, Delphi, Southern Britain and Iceland, a map based on an ancient geodeticUnits of measures and monumental measurements relating to the numerical definition of the shape … Continue reading “A Pyramidion for the Great Pyramid” - Recalibrating the Pyramid of Giza
Once the actual height (480 feet) and actual southern base length (756 feet) are multiplied, the length of the 11th degree of latitude (Ethiopia) emerges, in English feet, as 362880 feet. However, in the numeracy of the 3rd millennium BC, a regular number would be used. In the last post, it was noted that John … Continue reading “Recalibrating the Pyramid of Giza” - Geometry 2: Maintaining integers using fractions
understanding the megalithic: circular structures: part 2 The megalithic sought integer lengths because they lacked the arithmetic of later millennia. So how did they deal with numbers? There is plenty of evidence in their early monuments that today’s inch and foot already existed and that these, and other units of measure, were used to count … Continue reading “Geometry 2: Maintaining integers using fractions” - Geometry 1: πunderstanding the megalithic: circular structures: part 1 It would require 3 and a bit diameters to wrap around the circle – the ratio of 3 and a bit diameters to the perimeter is known as “Pior π: The constant ratio of a circle’s circumference to its diameter, approximately equal to 3.14159, in ancient times approximated … Continue reading “Geometry 1: &pior π: The constant ratio of a circle's circumference to its diameter, approximately equal to 3.14159, in ancient times approximated by rational approximations such as 22/7.;”
- THE MEANING OF LE MENEC (PDF)
This paper proposes that an unfamiliar type of circumpolar astronomy was practiced by the time Le Menec was built, around 4000 BCE. This observatory enabled the rotation of the earth and eclipticThe path of the Sun through the sky along which eclipses of sun and moon can occur, traditionally divided into the 365¼ parts of the solar … Continue reading “THE MEANING OF LE MENEC (PDF)” - A Brief Introduction to Ancient Metrology (2006)
appended toSacred Number and the Origin of Civilisation There used to be an interest in metrologyThe application of units of length to problems of measurement, design, comparison or calculation. – the Ancient Science of Measures – especially when studying ancient monuments. However the information revealed from sites often became mixed with the religious ideas of the researcher … Continue reading “A Brief Introduction to Ancient Metrology (2006)” - Use of Ad-Quadratum at Angkor Wat
Ad Quadratum is a convenient and profound technique in which continuous scaling of size can be given to square shapes, either from a centre or periphery. The differences in scale are multiples of the square root of two [sqrt(2)] between two types of square: cardinal (flat) and diamond (pointed). - The Golden Mean compared to PIIn reviewing some ancient notes of mine, I came across an interesting comparison between the Golden MeanThe Golden Mean is that unique ratio {1.618034}, relative to ONE {1}, in which its square and reciprocal share the same fractional part {.618034}. It is associated with the synodicThe recurring time cycle of a given celestial phenomenon seen from the Earth. period of the planet Venus, which is 8/5 {1.6} … Continue reading “The Golden Mean compared to PI”
- Use of foot ratios in Megalithic Astronomy
The ratios of ancient metrologyThe application of units of length to problems of measurement, design, comparison or calculation. 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 … Continue reading “Use of foot ratios in Megalithic Astronomy” - Old Yard’s Mastery of the Square Root of 2
The old yard was almost identical to the yard of three feet, but just one hundredth part smaller at 2.87 feet. This gives its foot value as 99/100 feet, a value belonging to a module very close to the English/Greek which defines one relative to the rational ratios of the Historical modules. So why was … Continue reading “Old Yard’s Mastery of the Square Root of 2” - Models of Time within Henges and Circles
image: composite, see figure 1 below Presenting important information clearly often requires the context be shown, within a greater whole. Map makers often provide an inset, showing a larger map at a smaller scaling (as below, of South America) within a detailed map (of Southern Mexico). Megalithic astronomy generated maps of time periods, using lines, … Continue reading “Models of Time within Henges and Circles” - Palsson’s Sacred Image in Iceland
Extracted from The Structure of MetrologyThe application of units of length to problems of measurement, design, comparison or calculation., its Classification and Application (2006) by John Neal and notes by Richard Heath for Bibal Group, a member of which, Petur Halldorsson, has taken this idea further with more similar patterns on the landscape, in Europe … Continue reading “Palsson’s Sacred Image in Iceland” - Megalithic Measurement of Jupiter’s Synodic Period
image: Jupiter with now-shrunken red spot – Hubble Space Telescope Though megalithic astronomers could look at the sky, their measurement methods were only accurate using horizon events. Horizon observations of solsticeThe extreme points of sunrise and sunset in the year. In midwinter the sun is to the south of the celestial equator (the reverse in … Continue reading “Megalithic Measurement of Jupiter’s Synodic Period” - Megalithic application of numeric time differences
Natural time periods between celestial phenomena hold powerful insights into the numerical structure of time, insights which enabled the megalith builders to access an explanation of the world unlike our own. When looking at two similarly-long time-periods, the megalithic focussed on the difference between them, these causing the two periods to slide in and out … Continue reading “Megalithic application of numeric time differences” - Lunar Counting from Crucuno Dolmen to its Rectangle
A fuller treatment of this article can now be found in Sacred Geometry: Language of the Angels (2021). It is not immediately obvious the Crucuno dolmenA chamber made of vertical megaliths upon which a roof or ceiling slab was balanced. (figure 1) faces the Crucuno rectangle about 1100 feet to the east. The role of … Continue reading “Lunar Counting from Crucuno DolmenA chamber made of vertical megaliths upon which a roof or ceiling slab was balanced. to its Rectangle”