Skip to comments.Unprecedented mathematical knowledge found in (Minoan) Bronze Age wall paintings.
Posted on 03/02/2006 5:01:38 AM PST by S0122017
Published online: 28 February 2006; | doi:10.1038/news060227-3 Were ancient Minoans centuries ahead of their time? Unprecedented mathematical knowledge found in Bronze Age wall paintings. Philip Ball
Did the Minoans understand the Archimedes' spiral more than 1,000 years before him?
A geometrical figure commonly attributed to Archimedes in 300 BC has been identified in Minoan wall paintings dated to over 1,000 years earlier.
The mathematical features of the paintings suggest that the Minoans of the Late Bronze Age, around 1650 BC, had a much more advanced working knowledge of geometry than has previously been recognized, says computer scientist Constantin Papaodysseus of the National Technical University of Athens, Greece, and his colleagues.
The paintings appear in a building that is still being excavated and restored in the ancient Minoan town of Akrotiri on the island of Thera. A catastrophic eruption of the volcano on Thera, now known as Santorini, around 1650 BC, is thought to have dealt a fatal blow to the Minoan culture. The blast covered Akrotiri, on the island's southern coast, in a thick layer of ash that preserved many buildings and artefacts.
Ten or so buildings have been excavated in Akrotiri so far, including a large one known as Xeste 3, which stands close to the ancient quay. Judging from its large size and extensive wall decorations, Xeste 3 appears to have been some kind of public building, such as a temple or a place for ritual ceremonies.
The most impressive feature of the paintings found in Xeste 3 is a series of spirals, each about 32 centimetres in diameter and embellished with dots. Papaodysseus and his team have shown that these are near-perfect Archimedes' spirals: shapes tightly defined by a simple mathematical formula, in which the distance between the windings is constant.
Some spirals, such as the ones found on snail shells, are common in nature. And others can be easily made by unwinding a thread around a central peg. But the Archimedes' spiral is not like either of these. "Seemingly it does not exist in nature," the researchers say.
"This is the earliest time that such advanced geometric figures have been spotted," says Papaodysseus. "The next such figures appear only 1,300 years later." The team report their work in the journal Archaeometry1.
A feeling for maths
Papaodysseus and his co-workers admit that they cannot know how much the Theran artists actually understood about the geometric principles they used for the paintings, because no written documents from this period are known to exist.
Experiments with geometry must lie behind the construction of these paintings.
Constantin Papaodysseus, National Technical University of Athens, Greece.
But he says that, at the very least, "experimentation with geometric tools must lie behind the construction of these wall paintings, as well as an impressive feeling for geometry."
Spiral designs in Xeste 3 were first noticed years ago by archaeologists working at the site. But Papaodysseus says that most people previously assumed that the shapes were painted freehand.
His studies suggest that the curves are just too accurate for that: the edges deviate from their strict mathematical form by typically less than a third of a millimetre. Papaodysseus thinks that this precision was probably achieved by the use of stencils, which appear to have been broken up into six parts to make them easier to transport and the paintings easier to fit to a given space.
The key question is how the stencil itself was made.
Splitting a circle
The researchers point out one relatively simple way of constructing such a spiral, without knowing the precise mathematical formula for it. One could divide up a circle using a large number of radial lines with equal angles between them, and a large number of concentric circles. A series of dots moving out one radial line and one concentric circle at a time could be joined together into an Archimedes' spiral. But dividing a circle into more than a dozen equal sections is not a trivial task; try it yourself.
Papaodysseus and his colleagues find that the dots decorating the spirals seem to be positioned almost exactly on the radial lines of circles that are divided into 48 sections.
The wall paintings don't in themselves prove that the Therans knew enough geometry to bisect angles. But it certainly looks that way, says Papaodysseus.
Great Minds think alike.
Sometimes mathematical concepts are dimly anticipated before being formally defined. The spiral might have been determined through fooling with compass/protractor techniques, but its full meaning would be hard to understand without the benefit of the Pythagorean theorem. There would have to be a comparison between how deeply Archemedies understood the properties of the object and how well the Minoans understood it. You aren't going to get that from one picture.
That is true, but the interesting thing here is the minoan culture influenced the later greek culture in many ways.
Including art and mythology. Then why not mathematics?
It is ofcourse possible that Archimedes just saw the drawings and got inspired. But i leaves other options aswell.
"the powerful influence that the Minoan culture had on the later classical Greek culture"
I think this date is actually 1628BC and the same time as the Jewish Exodus from Egypt. The volcano plume would have had to be 30 miles high to be seen from Egypt..."staff by day, torch by night."
The most recent eruption of Pinatubo in the Phillipines was 26 miles high and the one in/around Alaska was greater than 30 miles high.
The Minoans had far more advanced navies than surrounding countries, it wouldn't surprise me if their mathematics were advanced as well.
Everytime I hear about some culture supposedly inventing the mathmatical concept of 0, I never believe it
"Moses called down a host of calamities upon Egypt until the pharaoh finally freed the Israelites. Perhaps he had the help of a comet impact coupled with a volcano.
A volcano destroyed the island of Santorini in the Aegean Sea (between today's Greece and Turkey) around the middle of the second millennium B.C. Researchers Val LaMarche and Kathy Hirschboeck suggest the volcano might be associated with tree-ring evidence for several years of intense cold beginning in 1627 B.C
. "Could that form the basis for strange meteorological phenomena recorded in the biblical book of Exodus? In the book of Exodus, which describes events a few hundred kilometers from Santorini, we read of a pillar of cloud and fire, a lingering darkness, and the parting of the Red Sea. An enormous column of ash must have hung in the sky over the eruption (the Israelites pillar of cloud by day and fire by night?), and the volcano doubtless caused a tsunami, or tidal wave (which could have drowned a pharaoh's army)."
"The Exodus story is traditionally dated to either the thirteenth or fifteenth century B.C. Those dates, however, depend ultimately on identifying the Pharaoh of the Oppression, and historians have never proven to which ruler that infamous title referred."
Sure you did, ever hear of a chalkboard? Erasable walls just hadn't been invented yet.
Ever hear of the math major that had a constipation problem? He got a pencil and worked the problem out.
.....But dividing a circle into more than a dozen equal sections is not a trivial task; try it yourself. ......
Absurdly simple. Divide it into six by taking radial arcs, split one and do it again from tne mid point.
Not true. Every mathematician and math instructor I have ever known "did math on a wall". Back in those days, they were called "blackboards".
I guess the dude wouldn't have got as much newsprint if he speculated "Maybe it was just an accident...?"
Another one of the "I think I found Atlantis near Cuba" crowd.
Nothing to do with atlantis and he is not suggesting it.
It isnt like finding archimedes spiral on a rock in africa or easer island, the minoans have influenced the greek in a great manner.
To find two identical spirals among two cultures of which one was the older and was known to have influenced the other (and perhaps vice versa) is enough to raise questions concerning the origin of the spiral.
And the math that it required.
Hexagons are found in both math and nature. But if I draw a hexagon, that doesn't mean I invented life in a test tube.
The parallels between math and art are well known. Sometimes art is just that - art. A testing and discovery of forms and figures.
the Pythagorean says, "all is number", I say...To all -- please ping me to other topics which are appropriate for the GGG list. Thanks.
- How many finger am I holding up?
- What about letters?
- Must have been a big number that you smoked.
- What's the number for 9-1-1?
New Ice-Core Evidence
Challenges the 1620s age for
the Santorini (Minoan) Eruption
Gregory A. Zielinski, Mark S. Germani
Journal of Archaeological Science
Volume 25, Issue 3
March 1998, Pages 279-289
13 July 1997
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"To find two identical spirals among two cultures of which one was the older and was known to have influenced the other (and perhaps vice versa) is enough to raise questions concerning the origin of the spiral.
And the math that it required."
Maybe they just saw a snail and admired the spiral shape of the shell. No math needed, but it is a perfect expression of the golden mean.
I don't have time this morning to run it down, but suspect this is in reality a very early instance of the Golden Mean. The spiral is generated by plotting values developed graphically by various multiples of Golden mean rectangles.
The Golden Mean is used in the construction of the Great Pyramid so Minoan usage might not be all that big a deal.
By the way, the spiral is incorporated into the geometry of a pine cone so it won't roll down hill.
I think we routinely underestimate ancient societies.
I don't think it is unreasonable to think that at various times in history (and prehistory) some discoveries were made that perhaps only barely missed connection with other ideas that would have launched technologies centuries earlier. I cannot imagine what was lost in the burning(s) of the library at Alexandria... things that took centuries to rediscover and build upon.
Seems like every time I connect to the internet, somebody's got a new date for the Thera catastrophe.
Just once I'd like to see "Minoan-Egyptian Dictionary found." Or "Board Game Pieces for Phaistos Disk Found."
I always thought Crete was the leader in civilization until for some unknown reason, possibly attacks from mainland Greece, they ceased to be.
What ever the reason, Crete has always fascinated me.
The Linear B Tablets and Mycenaean Social, Political, and Economic Organization
Lesson 25, The Prehistoric Archaeology of the Aegean
Revised: Friday, March 18, 2000 | Trustees of Dartmouth College
Posted on 08/29/2004 11:19:46 PM EDT by SunkenCiv
6 posted on 08/30/2004 3:43:06 PM EDT by SunkenCiv
The Phaistos Disk
various | various | various
Posted on 09/22/2005 11:12:35 AM EDT by SunkenCiv
We're currently studying Crete, the Minoans and Greece with my first grader. It really is fascinating. (We made a Mycenaean helmet out of paper mache.)
Anyway, we went to see "Greece" at the Imax a few days ago, and I wish that I had realized that Santorini was actually Thera, as a lot of the first part of the film was based there. I would recommend seeing this production when it gets to your area.
I only wish we could travel to the Mediterannean (sp) to visit it first hand. The scenery was breathtaking!
Without taking a stand on the accuracy of the claim, the article states that the Archimedes spiral is not found in nature, and it specifically mentions that snail shells do not follow that pattern.
Perfect for defending yourself against a man armed with a banana.
I remember reading about the Ionian civilization on the east coast of the Adriatic duringthe Greek era. If they had garnered a little more power and support from the Greeks, the industrial revolution could have started 2000 years earlier than the Rennaissance.
Information at this site would refute those claims.
A way-cool site; however, I don't see where it addresses the article's claim that the Archimedes spiral does not match natural patterns such as snail shells.
Archimedes certainly did not invent geometry. Ancient civilisations in Iraq, India and China demonstrate that they knew the subject well. Writings have been found, such as student excercise tablets from temples in Babylon.
One link to a discussion of mathematics in Babylon is: http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_mathematics.html
A link to a short discussion of Early Geometry: The Babylonians, Egyptians & Chinese: http://library.thinkquest.org/C0110248/geometry/history1.htm
The above link even says, "The Pythagorean Theorem, although named after Pythagoras, was actually already known in ancient times.". It goes on to say why the writers think so.
In answer to your question, I think Archimedes published material that he was taught, added some original work and called the whole body of work his own. More a plaigarist than a Korean fake.
Still the idea that math may be a lot older makes sense to me. And it does certainly makes it likely that Archimedes was at least inspired by Minoan designs, perhaps more.
It's also heavily used in finance to analyze economic cycles.
You have been watching the movie "Phi"?
Most believed Ramses II to be the pharaoh of the Exodus. That doesn't seem to fit. The latest evidence of the timeline from the Egyptian side places Ramses as late as 925bc as the Shishak that plundered Jerusalem. Others tend to favor Seti I .
Ramses was dated at one point to the 13th century BC. That is substantially later than 1627 and makes the linkage impossible with regard to the volcano theory. Other problems arise here as well. Solomon's reign would be ending in the time of Ramses II, making Solomon a Contemporary of Seti I. Solomon married Egyptian Royalty.
His wife would necessarily come from the court of Seti or Ramses. In any case, neither of them could be the pharaoh of the Exodus. Going backward, you have Solomon's father, David, sitting as Second King of Israel, then Saul who was the first. Saul has linkage with a prior pharaoh under his birth name Lebayou (sp?). This doesn't point to the pharaoh of the Exodus; but, it does rule out who could not have been.
Shoshenq I's Chief Archetect is recorded in a generational family tree that has been dated to year 26 of the reign of Darius I. Also listed in that family tree is the cheif architect of Ramses. So these aren't fixed dates for the beginnings and endings of the reigns, it just tells us who was reigning at a given snapshot in time. Shoshenq was reigning by the family tree in 776. Ramses II was in 925.
We know Seti I reigned into his 90s. And the Amarna letters
tie Saul to Ankhenaton. Saul was the first King of Israel after the Exodus. The letters to Pharaoh from Saul tie in nicely to biblical events recorded. Ankhenaton reigned under the name Amenhotep IV. This would rule out the conjecture about Tutankhamun being the firstborn son killed at passover.
Linkages have been drawn between Joseph and Amenemhet III which seem to support the later timeline - framing things in as it were. The Bar Joseph waterway from that period still bears Joseph's name. What seems problematic to me is the explosion of kings in the 13th Dynasty combined with the imprecision of dating in general. There are also a lot of assumptions made as to when many of the Kings ruled. Some dynasties seem to have been co-ruling instead of ruling in dynastic order - a kingdom divided between upper and lower Egypt as it were. Not altogether unexpected; but, a blind eye seems to prevail in attempt to stretch the timeline for reasons obvious to some and embarrassing to others when the corrections are made. I find it interesting the linkages made at Megiddo to Solomon. A building not matching his recorded building style and requirement is identified as his in preference over one that matches it precisely. Why? Because the one matching doesn't match the presumed "layer" in which Solomon is expected to be found. Faced with the contradiction, the hard evidence of the matching structure is abandoned in preference to the conjectural dating method of the layers. IMHO, this is a lot of what is wrong with science today. Ideologies get in the way of methodology. One should be following the evidence.
Anyway, I'm off to bed. Have fun with it.
"A way-cool site; however, I don't see where it addresses the article's claim that the Archimedes spiral does not match natural patterns such as snail shells."
Click on plants or animals, or most any of the items in the center column.
I see the golden ratio part .... what I don't see is any reference to the Archimedes Spiral. You can ignore me, or lead me by the nose.
Wasn't the whole idea of the pillar of smoke that it would be a directional marker that would lead them away from Egypt and toward the promised land?
I hate to rain on your theory connecting the Thera volcano and Exodus, but if the Israelites had followed the smoke they would have wound up going NORTH our of Egypt into the Mediterranean Sea.
The spiral growth of sea shells provide a simple,
but beautiful, example . . . Bottom of page
The valcano idea is tossed in because these guys want "naturalistic" explanations for things they can otherwise not explain. It's never enough for them to say "I don't know" or to take the scripture literally. People want to force supernatural into a natural explanation. When they can't they tend to avoid the subject rather than admit their ignorance. That is pride.
I like the jacketed sulfur pellets from the area of Sodom and Gamorah. I've never seen a natural explanation for them, yet there they are. Volcanos don't produce them nor is there one near by to explain them if volcanoes did produce them. And further, there is no explanation for the specific places they fell to the exclusion of others - that is unless one takes scripture at it's word and God at His.
I've never heard of 'jacketed sulphur pellets' around the alleged site of Sodom & Gomorrah. Very interesting. I'd like to know more.
Are these anything like the jacketed depleted uranium shells the US Airforce uses to pierce the armor on tanks?
I suppose God could have used depleted uranium if He wanted to--but God knows jacketed sulphur pellets were probably enough to destroy the rock huts of that rabble in Sodom and Gomorrah.
The sulphur pellets are fairly small - about the size of a golf ball as I remember. The core is pure sulphur with an outher jacket or casing that I'm not sure about the makeup of.. The interesting thing about them is that they burn so hot, they are found imbedded into solid rock that they melted their way into as it were.
There used to be some pretty good shots of these posted at Wyatt's website. I found a few.
I don't know if the casing is a result of impacting something or not. It's just something I remembered seeing in other photos.
I think the numbers I remember seeing are that most sulphur deposits are in the range of 30-40 percent pure. These pellets are 95 plus percent pure and burn at 5000 to 6000 degrees. Very nasty little buggers as it were. My understanding is that they produce ash that is denser than normal - heavier. And looking at the remains of Gamorah, that isn't tough to believe.
Here's another link with more info. This one includes some discussion of the jackets I mentioned. Also notes that the purity is higher than I remembered and that Magnesium is present in the makeup of these things.