Cologne Cathedral Facade as Double Square

image: The Gothic cathedral of Cologne by night, by Robert Breuer CC-SA 3.0

On the matter of facades of Gothic cathedrals, I hark back to previous work (February 2018) on Cologne cathedral. This was published in a past website that was destroyed by its RAID backup system!

As we have seen with Chartres, some excellent lithographs with scales can often exist online from which one can interpret their sacred geometrical form and even the possible measures used to build that form. The Gothic norm for a facade seem more closely followed at Cologne facade which has two towers of (nearly) equal height.

We saw at Chartres that an underlying geometry using multiple squares may have been used to define a facade and bend it towards a suitable presentation of astronomical time, in a hidden world view that God’s heaven for the Earth is actually to be found in the sky as a pattern of time. This knowledge emerged with the megaliths and, in the medieval, it appeared again in monumental religious buildings built by masons who had inherited a passed-down but secret tradition.

A Prologue to Cathedral Music

Continue reading “Cologne Cathedral Facade as Double Square”

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.


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


1.     Problems with Thom’s Stone Circle Geometries.

2.     Egyptian Grids of Multiple Squares.

3.     Generating Flattened Circles using a Grid of Squares.


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.

Continue reading “Multiple Squares to form Flattened Circle Megaliths”

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).


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.

A Mexican Triple Square at Teotihuacan

image: Ricardo David Sánchez for Wikipedia 

This article is from June 2012 on my past Matrix of Creation site where it was read 548 times at the time of last backup. It led to another article and so I repeat it here.

The late Hugh Harleston Jr revealed the famous Mexican pyramids at Teotihuacan as being the manifestation of a very advanced megalithic culture, the Olmec as a root culture for New World Megalithism of Mexico and South America (that led to the Maya nearly a millennium later, the Aztec and the Inca) . The Teoti city-building culture started around 200 BCE but it is not exactly clear when the great city started to be built or what it represented. However, Carnac’s megalithic geometries, its day-inch counting within monuments and evident use of circumpolar astronomy suggests important new clues in the interpretation of this sacred city’s design.

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