Author: Richard Heath

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 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 as proposed by Mackie).

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, the simplest accurate approximation to the π ratio, between a diameter and circumference of a circle, as used in the ancient and prehistoric periods., 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.

If we take the size of the moon in that model, as being 3/11 of 396 feet this would give a circle radius 108 feet and one can see that, using the moon, the outer perimeter of the brock was probably elliptical too.

Thank you for watching.

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.


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

Such a diagram came to be built into the Cosmati pavement of Westminster Abbey, this installed during the reign of Henry III as a gift from the Pope and one or more Cosmati master craftsmen. It was 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. Its presence at the heart of English then British king-making is part of what is called the Matter of Britain, one of many Mysteries as to the meaning of its design.

Continue reading “Earth and Moon within Westminster’s Coronation Pavement”

OpenAI: Problems for Research using ChatGPT

I asked the chatbot a series of questions which homed in on a key problem with the chatbot. The problem was that the chatbot thinks that an 819-day period can equal the 260-day period of the Maya calendar period of 260 days.

how does the 819-day period interact with the 260-day Tzolkin?

The 819-day period is the length of a full cycle of the Tzolkin, which is a 260-day sacred and divinatory cycle in the Maya calendar system.

The Tzolkin is formed by the combination of two smaller cycles: a 13-day cycle and a 20-day cycle. Each day in the Tzolkin is represented by a unique pair of glyphs, resulting in a total of 260 possible combinations. The Tzolkin was used for divination, determining the best times for planting crops, making decisions, and performing rituals.

The interaction between the 819-day period and the 260-day Tzolkin is that every 819 days, the Tzolkin cycle completes a full cycle of 260 days, which means that the same combination of the 13-day and 20-day cycles repeats. In other words, every 819 days, the Maya believed that the energies and personalities associated with each of the 260 possible combinations of glyphs would repeat.

This repetition was seen as a symbol of the cyclical nature of time and the universe by the ancient Maya. The Tzolkin was also used to calculate longer periods of time, such as the Haab’ (365-day) calendar and the Calendar Round (a 52-year cycle), which were used for agricultural and religious purposes.

ChatCPT is an artificial intelligence (AI) system which is language based, so that it apparently does not apparently check any mathematical consistency, and this makes it possible for such an error impossible to humans who know that one number never equals another, by definition.

In some areas this AI appears superb, for example in summarizing an issue such as

What were the deficiencies of the Short Stirling Bomber?

The Short Stirling was a British four-engined heavy bomber aircraft used by the Royal Air Force (RAF) during World War II. While it was an important aircraft in the early years of the war, it was ultimately replaced by more advanced designs. The Stirling had several deficiencies that limited its effectiveness, including:

  1. Range: The Stirling had a limited range, which made it difficult to reach targets deep in Germany. This was due in part to its large size and weight, which made it less efficient than other bombers in terms of fuel consumption.
  2. Payload: Despite its large size, the Stirling had a relatively small bomb load compared to other heavy bombers of the time. This limited its effectiveness as a strategic bomber.
  3. Maneuverability: The Stirling was not very maneuverable, which made it vulnerable to enemy fighters. This was partly due to its size and weight, which made it slower and less agile than other aircraft.
  4. Maintenance: The Stirling was difficult to maintain, which meant that it spent more time on the ground than in the air. This limited its operational effectiveness and made it more vulnerable to attack while on the ground.
  5. Visibility: The Stirling had poor visibility from the cockpit, which made it difficult for the crew to spot enemy fighters and defend the aircraft.

Overall, the Stirling was an important aircraft for the RAF during the early years of World War II, but its deficiencies limited its effectiveness as the war progressed. It was ultimately replaced by more advanced designs, such as the Avro Lancaster and the Boeing B-17 Flying Fortress.

Of course it may have found one or more perfect answers to this question.

Ancestry

But in another area, more serious errors came for ancestry research where it can confuse a name and date with another well-formed biography for inexplicable reasons. My wife has a relative Charles Samuel Weston, a family description of whom exists. He was significant as a soldier in colonial India but asking of him first gave:

Charles Samuel Weston (1821-1880) was a British engineer who made important contributions to the development of the telegraph and submarine cables.

Weston was born in Bristol, England and worked as an apprentice at a local brass works before starting his own engineering business. He became interested in the telegraph and developed a number of innovations that improved the reliability and efficiency of telegraph systems.

One of Weston’s most important contributions was the development of a new type of battery, known as the “gravity cell,” which was used to power telegraph systems. The gravity cell was more efficient and reliable than other batteries of the time, and it became widely used in telegraphy and other applications.

Weston also played a key role in the development of submarine telegraph cables, which were used to transmit messages across the Atlantic Ocean and other bodies of water. He developed a number of innovations that helped to make submarine cables more reliable and efficient, including a new type of cable insulation and a system for detecting breaks in the cable.

In addition to his work in engineering, Weston was also a noted philanthropist who supported a number of charitable causes. He died in 1880 at the age of 59.

In changing the date of death to 1909, which is correct,

I apologize for my previous response – there appears to be some confusion regarding the dates of Charles Samuel Weston.

Charles Samuel Weston (1821-1909) was a British engineer and inventor who made significant contributions to the development of the electric telegraph and the submarine cable.

Born in Bristol, England, Weston began his career working as an apprentice at a brass foundry before starting his own business as a mechanical engineer. He became interested in the telegraph and made several important improvements to telegraph systems, including developing a more reliable and efficient battery known as the Weston cell.

etc, then Weston died in London in 1909 at the age of 88.

Before this he had been an administrator who introduced taxation to India, established university, and so on. Compelling but false, and so showing an excellent ability to invent plausible biographies based upon fragments of existing biographical texts.

Charles Samuel Weston was a British civil servant who served in India during the late 19th and early 20th centuries. He played an important role in the administration of the British Raj, particularly in the areas of finance and taxation.

Weston began his career in India in the mid-1880s, working in the Madras Presidency as a deputy collector of customs and salt. He quickly rose through the ranks, serving in various administrative and financial positions in Madras and other parts of the country.

One of Weston’s most significant contributions to British rule in India was his work on the Indian Income Tax Act of 1886. This legislation established a system of income taxation in India, which helped to provide a stable source of revenue for the British government and modernize the Indian economy.

In addition to his work on taxation, Weston was also involved in various other aspects of British administration in India. He served as a member of the Viceroy’s Legislative Council and as a member of the Indian Finance Committee. He was also involved in the establishment of several government institutions in India, including the Imperial Bank of India and the Indian Institute of Science.

Overall, Charles Samuel Weston played an important role in the British Raj in India, helping to modernize the Indian economy and establish systems of governance and taxation that would shape the country for years to come.

Who is Richard Heath?

This question managed to find data despite having no Wikipedia page.

Richard Heath is a writer, researcher, and speaker who is known for his work on ancient cultures, mythology, and symbolism. He has written several books, including “Sacred Number and the Origins of Civilization”, “The Harmonic Origins of the World”, and “Matrix of Creation: Sacred Geometry in the Realm of the Planets”.

Heath’s work explores the connections between ancient sacred sites, mythology, and the principles of sacred geometry and numerology. He has studied the geometry and symbolism of numerous ancient sites around the world, including Stonehenge, the Great Pyramid of Giza, and the temples of ancient Egypt.

Heath’s work has been praised for its interdisciplinary approach, combining insights from archaeology, anthropology, mythology, and geometry. His research has also been influential in the fields of alternative history and alternative archaeology.

In addition to his writing and research, Heath has also given lectures and workshops on topics related to sacred geometry, ancient wisdom, and symbolism.

I need to say that this was remarkable unless it knows I am asking, since there are many Richard Heaths both living and dead including a congressman, Victorian author, furniture shop and so on. One can say this is a novel type of searching which is more interesting and humanistic despite being detached from some kinds of definitional rigour found in non-language artificial intelligence, for instance that 260-days cannot equal 819-days. Many Maya textbooks are not available online so that exotic periodicities, in books like

Star Gods of the Maya: Astronomy in Art, Folklore and Calendars (The Linda Schele Series in Maya and Pre-Columbian Studies) by Susan Milbrath are somewhat hidden to AI while articles such as Scientists Think They’ve Finally Figured Out How a Maya Calendar Works will feature the 819-day period, and other articles will lie behind paywalls. This latter article is not as pedagogical as I have found it necessary to be regarding numbers in ancient calendrical contexts. That is, on a Monday morning, I find it likely that the logic is itself obscure to a general audience. I will endeavour to post on this article this week.

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

Astronomical Rock Art at Stoupe Brow, Fylingdales

first published 28 October 2016

I recently came across Rock Art and Ritual by Brian Smith and Alan Walker, (subtitled Interpreting the Prehistoric landscapes of the North York Moors. Stroud: History Press 2008. 38.). It tells the story: Following a wildfire of many square miles of the North Yorkshire Moors, thought ecologically devastating, those interested in its few decorated stones headed out to see how these antiquities had fared.

Background

Fire had revealed many more stones carrying rock art or in organised groups. An urgent archaeological effort would be required before the inevitable regrowth of vegetation.


Figure 1 Neolithic stone from Fylingdales Moor | Credit: Graham Lee, North York Moors National Park Authority.

A photo of one stone in particular attracted my attention, at a site called Stoupe Brow (a.k.a. Brow Moor) near Fylingdales, North Yorkshire.

Continue reading “Astronomical Rock Art at Stoupe Brow, Fylingdales”

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.
Continue reading “Chalk Drums to Symbolise Pi and Layout Monuments”