# Stonehenge mathematics

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Just like any other construction the Neolithic architects and engineers who built Stonehenge knew exactly what they wanted long before the foundations were dug. The stones were positioned in respect of a vision of their vision - a superb premeditated geometric structure. It was also largely prefabricated, just look for example at the complex jointing in the lintels, they can only have been created off-site and must have been trial fitted on the ground (which also indicates just how much thought and planning went into the design). As for all the so called alignments, it has only one, an axis of symmetry; that of the midwinter sunset and midsummer sunrise. Now where does this leave Stonehenge studies? Well exactly where it belongs, as investigation into the mindset behind its construction, a mathematical and geometric problem as much as an archaeological or ‘astronomical’ one.

In a recent publication ‘Solving Stonehenge’ it is revealed how every stone conforms to a precise mirrored symmetrical plan, and that markers must have been placed in exact positions against which the centre inner faces of the uprights were set. We are talking of course about the iconic sandstone (sarsen) monument, but there is evidence that geometry played a role in every phase of construction from around 3000 BC to c. 1600 BC. Mathematicians should now become involved in exploring the potential significance of the geometric arrangement, especially that of the central ‘horseshoe array’ formed by the trilithons. The prehistoric surveyors used ropes and pegs, for which we can read ‘straightedge and compass. We start with a sight line, towards the midwinter sunset, then a circle, then a triacontagon (beginning with a hexagon) against the vertices of which the exact centre of the inner faces of the circle stones were positioned (and by default the joints of the lintels). From these same 30 vertices the locations of inner faces of the 10 Trilithon upright were established; all that is with the exception of the two uprights of the Great (midwinter) Trilithon; the better faces of which look outwards towards the winter solstice sunset. Come on guys what is the significance of how the Trilithons were arranged, and why certain vertices were chosen, is there a numeric sequence to be found?……

An image showing the method seemingly used by the prehistoric to set out the trilithons can be found here: http://www.solvingstonehenge.co.uk/trilithon_geom2.html from that you can work out the rest.

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Architects and Engineers did not build Stonehenge just like Donald Trump did not build Trump Tower . Stonehenge was built by slaves ( IMO ) .

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The current thoughts on the great prehistoric monuments in NW Europe suggested that the building of the henges and avenues was the work of the entire community (admittedly prob not the leaders) in a form of social enterprise. Whilst the early tribes did hold slaves it is not thought that the monuments were built quickly by forced labour, but more likely done over generations in a festive spirit. As someone who has saved up cash and holiday to go and build a bridge and a school - I can empathise with this concept of a community project.

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I'll hazard to state that , I could draw a few circles and lines to intersect in simple ways and before I know it , there would be a lot of Euclidian geometry theorems applicable to the situation .

The converse of this statement which I would ask about , is , if I apply a lot of Euclidean geometry theorems to this situation , will I just finish with a few circles and lines that intersect in simple ways and thus , that is the way that they were originally selected 5000 years ago , nothing complicated , just simple ? .

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Architects and Engineers did not build Stonehenge just like Donald Trump did not build Trump Tower . Stonehenge was built by slaves ( IMO ) .

Slaves could have not possibly have built Stonehenge without any technical input, and Donald Trump or even Donald Duck could never have built Trump Tower without a design team! Ancient Egypt had its own class of architects and surveyors, it’s documented, we just don't have written records for Stonehenge. I have to stress that those stones (weighing up to 50 tons) are set accurately to just a few centimeters so that their inner faces formed an accurate circle; and the tops of the lintels were near perfectly level. I honestly would expect more appreciation and thought from a math’s forum!

I'll hazard to state that , I could draw a few circles and lines to intersect in simple ways and before I know it , there would be a lot of Euclidian geometry theorems applicable to the situation .

The converse of this statement which I would ask about , is , if I apply a lot of Euclidean geometry theorems to this situation , will I just finish with a few circles and lines that intersect in simple ways and thus , that is the way that they were originally selected 5000 years ago , nothing complicated , just simple ? .

That’s exactly the point, it IS simple, no complicated sight lines or astronomical calculations, just simple geometry.

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I didn't say that slaves did the work without technical input . There was obviously somebody around to say , move that an inch over here or move that an inch over there . They weren't the muscle though .

I've seen hundreds of megalithic tombs and stone circles and tried to think how they could be aligned to sunrise and sunset and moonrise and moonset ( I'm not a professional ) . I've seen 50 ton stones on top of 400 metre high hilltops and there is no obvious place nearby where they could have come from . You have my appreciation and thought , I just wanted to get the ball rolling .

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It might be important to understand Stonehenge in relation to two other circles. One is a wood circle representing life and another is blue stone circle representing death. Those blue stones could have represented ancestors and were moved a long distance. I believe your math needs to begin with the blue stone circle, where to original people brought their ancestors. Stonehenge would have come later, and if I remember correctly, the origin stones of Stonehenge were outside the present circle, and were brought inside at a much later date. For sure it was not the first circle, and we could assume things were learned in the process of building these circles, including a more careful observation of sun rise and sun set, which connected with life and death and associated with the river, moving from the circle of life (wood) down the river to the circle of death (blue stones). Not so different from the Egyptian river crossing between life and death. That is getting outside math, but for trying to figure out they came to the math, we need to evaluate the other circles.

http://www.dailymail...tle-sister.html

I also want to say, there might be a problem trying to understand these things with today's knowledge of math. Today's math begins with the Greeks. They learned from the Egyptians, who had perfected measuring land and building, but did so without math. Hum, how do I say? The math you all are doing is a totally different brain function, and is based on theory, which appears to be a Greek invention. What the Egyptians and others did was a skill, but not math. They could not explain what they were doing in the terms you use. To understand what they were doing, you need to do it. You need to know with your gut, not the math brain function in your head. Kind of like the difference between hearing about sex in a sex ed. class and doing it. A very different knowing.

And about the slave thing. I would assume the leader was an on hands guy, leading by his actions. These early leaders had to rely on willing followers, and therefore, had to be personally liked. These circles are places of life and death, and ancestors, and of importance to everyone. These are works of love and community, connecting the people who made them with life, death and each other.

Edited by Athena

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I honestly would expect more appreciation and thought from a math’s forum!

Maybe because the OP is spammy and it's claims are overblown.

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I agree with an earlier poster that many apparent mathematical relations can be superimposed by imagination on any random set of objects, which then look as if they were designed in some sophisticated way. Something like that was done with the people trying to merchandize the mystery caves of New Hampshire as a North American Stonehenge.

I always wonder why Stonehenge would have such special mathematical sophistication when so many of the other similar stone structures built around the same time by similar cultures around Europe show little or no sophistication at all. Some are not even alligned with any astronomical orientations.

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I agree with an earlier poster that many apparent mathematical relations can be superimposed by imagination on any random set of objects, which then look as if they were designed in some sophisticated way. Something like that was done with the people trying to merchandize the mystery caves of New Hampshire as a North American Stonehenge.

I always wonder why Stonehenge would have such special mathematical sophistication when so many of the other similar stone structures built around the same time by similar cultures around Europe show little or no sophistication at all. Some are not even alligned with any astronomical orientations.

Perhaps for the same reason the Great Pyramid of Giza is better than the first pyramid built. The Stonehenge we see today is not the original one, but was reconstructed long after the first circle was made. So perhaps measurements were made from the existing circle and corrections were made. Or perhaps earlier experiments with stone circles perfected the skill. If this were a history forum I would look for more information. It is a very interesting subject, but obviously not for the people posting here.

It could be interesting to measure Stonehenge and see if there is evidence of a tool like a Golden mean calipers having been used.

Edited by Athena

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I just tend to be sceptical about the general expectation that ancient peoples were much more advanced than we have assumed. At one time people were amazed to find that many lengths of buildings made by the Ancient Eqyptians were multiples of pi, thus indicating that they had discovered this mathematically sophisticated value before the Ancient Greeks. It was later discovered that the Egyptians measured large distances over land by using a cycle on a pole and counting the number of cycles, thus making distances a mulitiple of pi, even without the Egypticans knowing the number.

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I just tend to be sceptical about the general expectation that ancient peoples were much more advanced than we have assumed. At one time people were amazed to find that many lengths of buildings made by the Ancient Eqyptians were multiples of pi, thus indicating that they had discovered this mathematically sophisticated value before the Ancient Greeks. It was later discovered that the Egyptians measured large distances over land by using a cycle on a pole and counting the number of cycles, thus making distances a mulitiple of pi, even without the Egypticans knowing the number.

Excellent point, and this is the big difference that the Greeks made. I don't think anyone, before the Greeks ventured into math, had what we would call math. I have attempted to explain this, but often my words fail me. They obviously had ways of measuring and accomplishing building feats, but this was not exactly math. I don't think it becomes math, until the Greeks asked "why is this so" and began developing explanations for things like Pi. Egyptians imagined gods had powers, and Greeks gave us a secular, mathematical explanation of the powers.

It is the difference of my X being able to figure out a practical, hands on, physics problem, and not understand theory. Like people built bridges without the math to build a bridge. They know something, but it is not formal mathematics and theory. On the other hand, if the bridge had not already been built, someone with knowledge of theory, may not be able to build the bridge. Seriously, that instinctive problem solving ability is separate form knowledge of theory, and there is concerned that reliance on theory can prevent a person from intuitively knowing how to do something. We promote only those who comprehend theory, and exclude those who intuitively know, but are not good at learning theory.

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If you look at the profoundly cumbersome way the Ancient Egyptians handled fractions; the remains of intertwined ropes which testify to the extremely ordinary way they moved the blocks which composed the pyramids; the standard templates they employed for drawing prescribed shapes to compose images of humans and animals on temple walls; the stupidly expressionless faces of people they depicted; the bizarre mix of a little empirical insight and a lot of sheer magic in their medical texts; and their way of indicating fear in depicted figures by showing them defecating, they don't seem very smart, for all their practical engineering skill.

There have been recent attempts to try to attribute all of Ancient Greek wisdom to Egyptian influence (e.g., the book 'Black Athena'), arising out of the odd notion that this would somehow empower oppressed Blacks against modern Whites by its remote implication of Black racial superiority (though the Egyptians depicted themselves as tan, and depicted Sudanese as Black in contrast), but the uniquely critical, sceptical, and enquiring Greek style of thinking just doesn't match with that of the more superstitious Egyptians.

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practical engineering skill.

I wanted to put a link about "empirism" but the word (empiricism) has a different meaning in the english language.

Practical skill, yes, knowledge from experience: a great force we are losing bit by bit.

Any profession has its little mysteries, owned only by the professionals. Old professions have dissapeared, skills disappeared.

How to scarve a rock without a jackhammer, or moving a 20 tons object without a crane, that seems inconceavable. But they did. Not by calculating, but through practical knowledge.

---------------------------

I remember some years ago at a work of restauration of a neoclassical building, a worker standing sceptical in front of a little problem: the first step of the outer staircase to the building had moved down from its place about 10 cm. It was a solid block of pentelic marble about 3 metres long, 20 cm height, 40 cm depth. estimated weight about half a ton (500kgs). He told me not to worry about. I went on my inspection and after half an hour came back to see what to do. The problem was already solved. The man all alone had lifted the marble block and put it back, just like that. With a lever I guess, he wouldn't tell me, only smiling.

-------------------------------

Some other fabulous artisan, almost illetrate. He sells and cut marbles.

One day at the office, explaining how he solved one of his problems.

He is paid following the square meters of delivered products. When the product is rectangular, all well. When it has a peculiar polygonal scheme, that is still o.k., he knows the formula for a triangle (basis by heigth divided by 2). But when you cut a circle, how to calculate the surface?

His thought (I hope it was a true story):

"When I cut a circle, I can put a rope all around and get very easily its perimeter"

"then I thought with my practicle mind, that each line coming from the center and going to the perimeter (the radius) is a very tiny tiny triangle, and that there are as many such triangles as the perimeter is long. And then, if I get the surface of all these tiny-tiny triangles I get the surface of the circle".

Which is very correctly perimeter by heigth divided by 2 or $(2{Pi}R)R/2$ = ${Pi}R^2$

(trouble with Pi in LaTeX)

Edited by michel123456

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$\pi$

\pi Pi doesn't need curly brackets - just a backslash. Like the way of measuring area.

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As an example , Trump tower was built within the last decade ( I think ) and it could not have had any influence on the thinking of those who built the Empire State building because the Empire State building was built first . Stonehenge was built a long time ago and it is very difficult to know which other structures of that time may have influenced it's construction or those it may have influenced . People move around from place to place and see with their eyes the modern ways . What is the current best reputed estimate for the period of building of Stonehenge ?

Edited by Hal.

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Stonehenge was clearly built way before the pyramids, as long as we rule out those strange theories about the base of the Sphinx showing environmental degradation indicating that it was built 15,000 years ago.

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$\pi$

\pi Pi doesn't need curly brackets - just a backslash. Like the way of measuring area.

Thanks.

that is not the way it is presented in the first posts of this thread about LaTeX tutorial.

Stonehenge was clearly built way before the pyramids, as long as we rule out those strange theories about the base of the Sphinx showing environmental degradation indicating that it was built 15,000 years ago.

From wiki Stonehenge

Archaeologists have believed that the iconic stone monument was erected around 2500 BC*, as described in the chronology below. One recent theory however, has suggested that the first stones were not erected until 2400-2200 BC,[2] whilst another suggests that bluestones may have been erected at the site as early as 3000 BC (see phase 1 below).

The earliest known Egyptian pyramids are found at Saqqara, northwest of Memphis. The earliest among these is the Pyramid of Djoser (constructed 2630 BCE–2611 BCE) which was built during the third dynasty.

And if you dig a little about megaliths, you will find that the earliest structure of this kind is in Turkey dated 9th millenium B.C.

* there were wooden structure at Stonehenge as early as 8000 B.C. but evidently these are the remains of the telegraph they used at that epoch.

Edited by michel123456

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The wooden circle and a circle of blue stones and connected with Stonehenge. The wooden circle represents life. The blue stones are death, and the river connects them. The procession would begin with the wooden circle, go down the river to the blue stones, and then to the Stonehenge.

Ten represents a recapitulation of the whole. It holds within itself the two parents (one and two) and their seven children (three through nine. Ten is a portrait of the whole family of archetypes....

Ten is the Pythagorean "perfect number", symbolizing both fulfillment and new beginnings...

Egyptian line groupings led to the arch of ten, also represented by the glyph of the penis, indicating power to create numbers beyond ten. Other glyphs (spiral, flowering plant, finger, frog (reference to myriad tadpoles), man in praise, and the eternal circle joined with a line) represented the greater fertilizing powers of ten.

More important is nine represents the horizon. So when the body is taken on the journey from life to death and through the opening facing the sunset, the soul is now ready to go into the horizon.

Nine was a central number of Celtic tradition, expressed as threefold aspects of the triple goddess, in myths of nine Celtic maidens and nine virgins attending Bridget. The sacred Beltane fire rites were attended by a cycle of nine groups of nine men. The culture is filled with references to nine.

Quotes from "A Beginner's Guide to Constructing the Universe"

For the Native American,Mayan and Aztecs there are nine cosmic levels. For the Chinese 9 was a paradigm of cosmic structure and process, mirroring the supreme order of the universe.

I think those who present Stonehenge as a great human feat are more respectful of the dignity of man,and that those who argue, not much mental effort went into the circles, do not have enough information, nor enough respect for the human mind and spirit. I think we might be making misleading divisions between ancient math and the abstract math started by the Greeks.

Edited by Athena

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You make some very good- and interesting points. Anyone who really want to know what communities contemporary with Stonehenge knew about geometry could look at the 'Bush Barrow Lozenge', a remarkable gold sheet decorated with motifs derived from circles and hexagons. There was also a smaller lozenge shaped artefact from the same burial site that was based on an hexagon, and from a burial mound at Clandon (Dorset, England) a similar geometric artefact was based on a decagon; these objects are almost 4,000 years old.

I wanted to put a link about "empirism" but the word (empiricism) has a different meaning in the english language.

Practical skill, yes, knowledge from experience: a great force we are losing bit by bit.

Any profession has its little mysteries, owned only by the professionals. Old professions have dissapeared, skills disappeared.

How to scarve a rock without a jackhammer, or moving a 20 tons object without a crane, that seems inconceavable. But they did. Not by calculating, but through practical knowledge.

---------------------------

I remember some years ago at a work of restauration of a neoclassical building, a worker standing sceptical in front of a little problem: the first step of the outer staircase to the building had moved down from its place about 10 cm. It was a solid block of pentelic marble about 3 metres long, 20 cm height, 40 cm depth. estimated weight about half a ton (500kgs). He told me not to worry about. I went on my inspection and after half an hour came back to see what to do. The problem was already solved. The man all alone had lifted the marble block and put it back, just like that. With a lever I guess, he wouldn't tell me, only smiling.

-------------------------------

Some other fabulous artisan, almost illetrate. He sells and cut marbles.

One day at the office, explaining how he solved one of his problems.

He is paid following the square meters of delivered products. When the product is rectangular, all well. When it has a peculiar polygonal scheme, that is still o.k., he knows the formula for a triangle (basis by heigth divided by 2). But when you cut a circle, how to calculate the surface?

His thought (I hope it was a true story):

"When I cut a circle, I can put a rope all around and get very easily its perimeter"

"then I thought with my practicle mind, that each line coming from the center and going to the perimeter (the radius) is a very tiny tiny triangle, and that there are as many such triangles as the perimeter is long. And then, if I get the surface of all these tiny-tiny triangles I get the surface of the circle".

Which is very correctly perimeter by heigth divided by 2 or $(2{Pi}R)R/2$ = ${Pi}R^2$

(trouble with Pi in LaTeX)

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these objects are almost 4,000 years old.

1900-1700 B.C.

Thanks, I was not aware of these findings.

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Below is a quote from an argument that Stonehenge could be constructed without math. This explanation could also be interesting to those interested in Babylon math, and why several cultures determined the year has 12 months.

http://scienceblogs....d_stoneheng.php

Research has revealed that before the Sarsen Circle of upright stones was erected, a 285 foot diameter circle of 56 chalk holes, 3 feet in diameter, was created. (These are called the Aubrey Holes, in honor of John Aubrey). A CBS TV program in the 1960's ran a computer analysis of the Aubrey circle. They declared that Stonehenge's location--latitude 51 degrees 11 minutes, was a very special location for eclipses of the moon. This location produces moon eclipses in the repeating sequence of 19 years, 19 years, and 18 years.

Adding 19+19+18=56. Thus if the white 3 foot diameter chalk holes were covered by a black stone, that was moved around the circle in synch with the passage of moon cycles, the black stone would arrive at the heal stone position, on the exact day when a moon eclipse would occur. (Eclipse computer.) (S.I.D.)

How could this stone computer have been created without the precise knowledge of the celestial mechanics of this unique geographic location? Certainly this was not the work of the early tribes that lived on this Salisbury Plain, thousands of years ago.

I am okay with the idea that math was developed later by the Greeks, but we might consider the geometry involved, pre math and pretty impressive. It is possible the builders of Stonehenge were working with the vesica piscis which is basic to sacred math.

Also, we might agree Stonehenge is not a great math feat, but it is a pretty impressive engineering feat.

You can click on these images of Stonehenge and get more information.

This comes from that link, Analysis of the results provides a best estimate for the date of the construction of the Y Holes is 1640 - 1520 cal BC. Given the archaeological arguments for the contemporaneity of the two circles and the curation demonstrated in Y Hole 30, the date from the Z Hole (2030 - 1740 cal BC) can only be regarded as a terminus post quem for these holes (see probability distributions).

Now around 1640 the Minoan eruption hit. This was so devastating, as earthquake and tidal wave, some think it is the cause of the story of Atlantis. Might it be possible these sea faring people, discovered Stonehenge and made some improvements? We know the Greeks made this trip.

Edited by Athena

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We know the Greeks made this trip.

If you mean Pytheas of Massalia's trip, it was much much later, around 325 BC.

(...) the 'Bush Barrow Lozenge', a remarkable gold sheet decorated with motifs derived from circles and hexagons. There was also a smaller lozenge shaped artefact from the same burial site that was based on an hexagon, and from a burial mound at Clandon (Dorset, England) a similar geometric artefact was based on a decagon; these objects are almost 4,000 years old.

Where do the gold come from? are there gold mines around?

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It is not really well known exactly during what periods the Ancient Greeks were making expeditions to Britain. It is generally supposed that there were long contacts between Mediterranean cultures and the tin mines near the Dorset Coast, so the general Greek-British connection was probably a long-term process rather than a matter of a few decisive expeditions, such as the great periplus.

I thought the Minoan explosion (of Santorini) was supposed to have occurred around 1200 - 1350 B.C.?

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It is not really well known exactly during what periods the Ancient Greeks were making expeditions to Britain. It is generally supposed that there were long contacts between Mediterranean cultures and the tin mines near the Dorset Coast, so the general Greek-British connection was probably a long-term process rather than a matter of a few decisive expeditions, such as the great periplus.

I thought the Minoan explosion (of Santorini) was supposed to have occurred around 1200 - 1350 B.C.?

This link could be wrong but it says:

The Minoan eruption

The most recent large eruption - the Minoan eruption - occurred in the late Bronze Age at around 1640 BC and devastated the Eastern Mediterranean.

I did not mean to imply the trip occurred only once or twice, and was thinking it occurred many times, in part because of the trade in metals. It is my understanding the people of the islands had a good relationship with the Greeks, but not the Romans. The Greeks were there to trade, the Romans were there because spreading their empire, but also on a spiritual level the Greek and Celtic beliefs were complimentary. But this is getting far from the Minoans. When people migrate they can't take the whole of their culture with them. Especially not when a volcano and tidal wave wipe out their lives. They might with only what is in their minds, and that can be limited by who is in the group. Technologies get lost in this way, and some technologies and beliefs are spread. The question is one of math and technological skills as Stonehenge was built and rebuilt over hundreds of years. Is there evidence of new skills coming at a later date?