432 shows harmony of Sun, Moon, Earth Design

[Jiri]

No, if you divide a rectangle by drawing a diagonal between two corners, and then draw a perpendicular to that diagonal from one of the other corners, and so on, you will inevitably generate a number of proportional spaces, including series of proportional triangles which correspond to those that would be drawn inside a spiral. When you create nested series of right triangles you inevitably are working with powers of two, and the progression is inevitably logarithmic. It's interesting but not terribly arcane. The illustrator Jay Hambidge coined the term "dynamic symmetry," I think, and provided a number of details on how various spaces can be apportioned using the simple geometry of rectangles and diagonals.

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It seems that he came up with simple repetitive algorithms. Not much in that.
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He theorized that the Greeks used these proportions systematically not only in architecture but in art and craft, and measured a number of Greek vases by way of illustration. For more on the subject, I recommend you find a copy of his 1920 work, Dynamic Symmetry: The Greek Vase. My point in all this is not to sell Hambidge's theory so much as to point out that if you play around with rectangles, squares, and diagonals, it is not difficult to come up with interesting proportions and numbers.

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Without necessarily understanding what they represent in mathematics.
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I haven't spent a great deal of time on Hambidge, but suspect that if one measures a vase with a large enough arsenal of different root squares and diagonals, one will almost certainly find one that is a pretty good fit, especially if, as Hambidge occasionally did, one "corrects" errors in the vase itself. I suspect that your work is similar, in that if you try hard enough with an a priori principle, you will find a way to fit it to just about anything you find. I am not saying that this is dishonest, but I think it is illusory. Hambidge does at least have the advantage that he measured physical objects with unquestionable boundaries; that he did not need to rely on any specific units of measure (only proportion); and that he did not attribute to the designers any technology or geometrical knowledge that would insult our understanding of their history and their technology.

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Perhaps, the present day consensus insults the Egyptian learning. What I have to add to the mosaic of our knowledge of them happens to run a strong counter to the consensus, but somehow I don't feel much guilt.

Aside from the fact that there's no reason to believe the heiroglyphics mean what you seem to think they do, I'm not convinced that your choice of where to draw the outlines is meaningful. Even if it is, it's not novel. Again, I refer you to Hambidge, who provided a number of illustrations to support his theory that the Egyptians at least occasionally used systematic geometric principles to lay out their work, creating complex figures derived from simple geometric rules.

Right, simple algorithms. In contrast, one area of the Abydos Helicopter scene is a self-portrayal by the Golden Section construction. The layout of the area is easily reconstructed if we start with the rigorous construction of the Phi ratio. That's the algorithm for the area.
About this fitting a principle to anything, and having it fit, does it mean that the best algorithm for recreating that something has been found? Sounds unlikely, considering that the algorithm was imposed by force. The natural conclusion ought to be that since we can recreate the layout of the given area by the Golden Section construction, then it was originally created by the same method. Of course, this brings hostile reaction from archaeologists to whom the iconoclastic image is a palimpsest.

 

[bruto]

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It seems that he came up with simple repetitive algorithms. Not much in that.
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Without necessarily understanding what they represent in mathematics.
...
Perhaps, the present day consensus insults the Egyptian learning. What I have to add to the mosaic of our knowledge of them happens to run a strong counter to the consensus, but somehow I don't feel much guilt.

Right, simple algorithms. In contrast, one area of the Abydos Helicopter scene is a self-portrayal by the Golden Section construction. The layout of the area is easily reconstructed if we start with the rigorous construction of the Phi ratio. That's the algorithm for the area.
About this fitting a principle to anything, and having it fit, does it mean that the best algorithm for recreating that something has been found? Sounds unlikely, considering that the algorithm was imposed by force. The natural conclusion ought to be that since we can recreate the layout of the given area by the Golden Section construction, then it was originally created by the same method. Of course, this brings hostile reaction from archaeologists to whom the iconoclastic image is a palimpsest.

I don't see where the use of golden sections in layout, or any other geometric formulas relates to the content of the so-called helicopter. In any case, I don't think there's all that much controversy or all that much interest in the finding of golden rectangles and golden ratios in Egyptian or any other art. It's an old idea, still clouded by cherry picked samples, and the obvious necessity of reverse-engineering rules for which there is little or no direct evidence.

It's odd that you should apparently believe that the Egyptians understood some arcane significance to the rules they used, where the Greeks, using similar rules, did not. There's still doubt on the extent to which the Greeks or anyone else consciously or consistently used the golden ratio in their art, but its existence and the methods for generating it were well known and documented in contemporary geometry, and they certainly could have if they'd cared to.

Your conclusion that finding a golden section in the Egyptian example must mean that golden sections were used to do it is not convincing. The placement of your borders and divisions looks to me to be arbitrary, and if you start with a template, you'll find a way to fit it. But the fact remains that even if it turns out that they did use golden sections, it would not be particularly surprising or revelatory, nor would it indicate anything more arcane than the fact that they, like the Greeks, knew their geometry and (according to some) found it useful in composition.

 

[JonnyFive]

Why won't Jiri explain to me what use Osirus numbers have?

Do they model something? Are we naturally attuned to them? What's the deal and where's the proof?

I would suggest you read this, Jiri. It deals with various misconceptions about the golden ratio. I would also note that it addresses the issue of imprecise measurement and "approximation" of the golden ratio.

 

[Belz...]

Nautical miles seem to be off by less than three miles, or less than 5.556 kilometer - Superior by three kilometers to the effort of learned Napoleonic savants. Not bad for the source prehistoric civilisation.

Prehistoric ? How do you know how they measured things, then ?

I've always known about the issue of advanced ancient civilisations, or even alien astronauts.

Uh-huh. Found any evidence for that, aside from your own interpretation of vague drawings ?

There are great many facts to reconcile. We have the archaeological record, which gives us evidence of primitive societies versus evidence of advanced science, which can only be formed in advanced civilisation.

That doesn't follow.

 

[Paulhoff]

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40,008.6 / 1852 = 21,6029...

Nautical miles seem to be off by less than three miles, or less than 5.556 kilometer - Superior by three kilometers to the effort of learned Napoleonic savants. Not bad for the source prehistoric civilisation.

Prehistoric civilization means before writing was invented.

.There are great many facts to reconcile. We have the archaeological record, which gives us evidence of primitive societies versus evidence of advanced science, which can only be formed in advanced civilisation.
My take on this is that the area where AC originated and thrived must have been very small and isolated, like a large island, for instance.

Really, where was the arch used in their buildings....

Paul

 

[Jiri]

Why won't Jiri explain to me what use Osirus numbers have?

Do they model something? Are we naturally attuned to them? What's the deal and where's the proof?

I would suggest you read this, Jiri. It deals with various misconceptions about the golden ratio. I would also note that it addresses the issue of imprecise measurement and "approximation" of the golden ratio.

Johny, besides you I also owe some answers to JSFischer, whom I definitely do not wish to neglect, because he's the Mathematician here, as far as I can see. But before I get around to one thing another comes. Thank you for pointing me to Markowsky's article on misconceptions about the Golden Ratio.
http://www.umcs.maine.edu/~markov/GoldenRatio.pdf
To me it is a classic example of a hatchet job, perpetrated in the style of Marxist scholars, which I had to suffer through in my youth. Of course, such a serious accusation has to be born out by evidence, and so I am going to comment on each of the points he makes. Unfortunately, I only had about four hours of sleep, and so I'd like to catch up, have some winks, and then do a few chores.
One thing before going: By his standards of judgement, such as acceptance range criteria, my work passes with flying colors , comfortably exceeding his standards of accuracy. I am happy only with visual precision in my results.

 

[JonnyFive]

Johny, besides you I also owe some answers to JSFischer, whom I definitely do not wish to neglect, because he's the Mathematician here, as far as I can see. But before I get around to one thing another comes. Thank you for pointing me to Markowsky's article on misconceptions about the Golden Ratio.
http://www.umcs.maine.edu/~markov/GoldenRatio.pdf
To me it is a classic example of a hatchet job, perpetrated in the style of Marxist scholars, which I had to suffer through in my youth. Of course, such a serious accusation has to be born out by evidence, and so I am going to comment on each of the points he makes. Unfortunately, I only had about four hours of sleep, and so I'd like to catch up, have some winks, and then do a few chores.
One thing before going: By his standards of judgement, such as acceptance range criteria, my work passes with flying colors , comfortably exceeding his standards of accuracy, because as you may have noticed there is a lot of visual precision in my results.

Sure, prove it. You've yet to show me anything that convinces me you're doing anything other than arbitraily assigning significance to numbers that have none.

What in the world do you mean by "visual precision?" You mean, something that isn't actually precise but kind of looks precise.

Hey now, I'm a mathematician-in-training! Got a couple years left on my second bachelors (first was non-mathematics, focused on research into law enforcement systems and methodology) and then I'm so there.

Anyway, I know enough to understand the human design of numbers.

Why do your Osirus numbers adhere to the base 10 counting system, anyway? Is Base 10 special? If so, why? Also, provide evidence.

I think you just made this all up in your head and have assigned artificial significance to something that is completely and utterly meaningless.

But I still look forward to any kind of coherent arguments to the contrary. Cheers.

 

[jsfisher]

...I would suggest you read this...

Very nice reference, JonnyFive. I will admit guilt to having held at least one of the fallacies.

 

[Jiri]

Sure, prove it. You've yet to show me anything that convinces me you're doing anything other than arbitraily assigning significance to numbers that have none.

What in the world do you mean by "visual precision?" You mean, something that isn't actually precise but kind of looks precise.

Anything? I don't know what to say to you since you fail to be impressed by things that impress me. However I do detect some hostility in you towards me, so perhaps it is affecting your observations. A good example is right above as you are unhappy with my term "visually precise".

You mean, something that isn't actually precise but kind of looks precise.

..
Something that looks precise is visually precise.
Something that looks kind of precise is visually imprecise. What would be the point of being untruthful? The very nature of this thing relies on visual accuracy. You either see it, or you don't.
Why do you think I got double-lifesize blow-ups of the engraving before even measuring the Frame? Because the engraving seemed too small to allow me any reasonable exploration with classic geometrical tools. I didn't even touch some parts of the picture before having even higher magnifications, like the lines inside the torso of the girl.

..

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Here is two images of the same, what do you think their magnification could be? But, perhaps you are right, and I should replace the term 'visually precise' with 'visually exact'. Maybe visually precise is just another one of my Czech into English idiosyncrasies. Yeah, I guess it must be, so henceforth it is 'visually exact'. Thanks for helping
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Hey now, I'm a mathematician-in-training!

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Good for you lad! Me too, I have acquired a liking for mathematics since getting into this, but all my knowledge is pretty well limited to what has been imposed upon me by the above mentioned engraving. If I had a degree in math, people would definitely have a much easier time of accusing me of imposing ideas onto the engraving.
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Got a couple years left on my second bachelors (first was non-mathematics, focused on research into law enforcement systems and methodology) and then I'm so there.


Anyway, I know enough to understand the human design of numbers.

Why do your Osirus numbers adhere to the base 10 counting system, anyway? Is Base 10 special? If so, why? Also, provide evidence.

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Can you please stop evoking the name of some drug addict from California? Use "Osiris", please!

As to your question - I don't know. Is Base 10 special? All I know is that it works in this case. I am not a mathematician, remember? It was the designers of the engraving, who were mathematicians. However, let me ask you one thing: Let's say you draw two line segments next to each other, one is 355 long and one is 113. The ratio between them is 3.141592.. Are these segments decimal?
Now draw the same segments, but divide the 355 degment into segments of 339 and 16. The ratio is still the same, but is it decimal only? Just asking.

I think you just made this all up in your head and have assigned artificial significance to something that is completely and utterly meaningless.

I am neither that smart, nor that stupid.

 

[bruto]

Jiri, my problem with the above diagram is that I cannot see a compelling reason why the lines you drew should be where you drew them, unless you had already decided what you wanted to find. This is the problem with most such impositions of assumed design principles on designs whose creators are unknown. The pentagram you've drawn doesn't follow many of the actual lines of the engraving, and is extrapolated to form lines of which there is no hint in the engraving. In order to see a pentagram in that engraving, you really have to put it there yourself. Similarly, in a drawing that has no straight horizontal or perpendicular lines, the choice of where to draw the framing rectangle is up to the interpreter. I see no reason to accept your placement, or your rectangle, as being related to the design of the engraving.

You write:

However, let me ask you one thing: Let's say you draw two line segments next to each other, one is 355 long and one is 113. The ratio between them is 3.141592.. Are these segments decimal?
Now draw the same segments, but divide the 355 segment into segments of 339 and 16. The ratio is still the same, but is it decimal only? Just asking.

That doesn't make any sense to me. Can you explain what you're trying to say here?

 

[JonnyFive]

As to your question - I don't know. Is Base 10 special? All I know is that it works in this case. I am not a mathematician, remember? It was the designers of the engraving, who were mathematicians. However, let me ask you one thing: Let's say you draw two line segments next to each other, one is 355 long and one is 113. The ratio between them is 3.141592.. Are these segments decimal?
Now draw the same segments, but divide the 355 degment into segments of 339 and 16. The ratio is still the same, but is it decimal only? Just asking.

What? The ratio isn't the same, if that's what you're saying. If not, you'll need to explain it more.

How do you know they were mathematicians?

Since you're going to be snippy, could you please explain what an Osiris number is (unless it is simply a factor of 25,920) and what it does/models/predicts?

Apparently Osirus is a rapper and a very, very common misspelling of Osiris. I actually didn't know that, so you're lying about not knowing anything about American rap culture.

I'll stop being silly and play your game. I've looked at the frame web page in more detail, and I'd like to know a few things:

(NOTE: I don't know if you actually created this page, but I'll work from the assumption that you did. If not, replace "you" with "the author" or whatever gets your going)

1-Did you use the original frame as a source?
1a-If not, why not?
1b-If not, why are you convinced you are working with precise measurements?

2-What was the condition of the frame?
2a-Do you believe this affected the calculations you made.

3-What was the degree of precision of your measurements?
3a-If you did not use the original frame, how did you determine this without reference to the actual object you're claiming to measure, and factoring in the errors due to computer representation? (Note: On the web page, you claim to use mm level precision because it is the smallest unit on your ruler. If you didn't have access to the frame, what was the level of precision you claim to have on the object itself, through proper high resolution magnification or whatever... please state scales used and how you calculated them)

4-What was the margin of error for your measurements?
4a-Same as 3a, but for the margin of error.

5-What degree of error tolerance (allowable margin of error before you decide the ratios probably aren't significant anymore) are you allowing for, and why?

6-If you're using a degree of precision less than x decimals to prove that the measurements of the frame show pi or phi to x decimals, why do you believe this would have no effect on your calculations?

7-Why did you divide the frame into a pie chart, and why did you arrange it like that?

8-Some of your calculations round mm to cm to obtain a ratio, why? Note that this leads me to believe you're working with a precision of over 1 cm, which will consequently lead me to laugh at any claims you have to have measured anything in half mm!

9-Perhaps most importantly: Why did you draw the lines where you drew them? As bruto pointed out, there does not appear to be a compelling reason for this except for drawing those lines there because you wanted to create certain ratios.

Go ahead, take your time. You insist that this was all designed (ETA: By the frame makers, not by you), and that it has some significance. Now it's time to prove it, if you're so inclined. That web page is not convincing, maybe your arguments will be.

 

[Belz...]

Anything? I don't know what to say to you since you fail to be impressed by things that impress me.

Well, beign impressed by numbers, a mathematical construct of man's making, is not itself very impressive...

Something that looks precise is visually precise.

Therefore that term means nothing.

Here is two images of the same

What the hell are those ?

If I had a degree in math, people would definitely have a much easier time of accusing me of imposing ideas onto the engraving.

No. If you had one, you wouldn't be saying what you're saying. Hopefully.

Can you please stop evoking the name of some drug addict from California? Use "Osiris", please!

Yawheh - ( ( ( Osiris + Horus ) / Zeus ) ^ Heimdall ) + Quetzalcoatl = Bonkers.

As to your question - I don't know. Is Base 10 special?

Nope.

 

[JonnyFive]

No. If you had one, you wouldn't be saying what you're saying. Hopefully.

Allso, som of us numarikul tipes donet spel wurds like "Osirus" none to gud.

 

[jimbob]

Cross posted from the "can anyone recommend a good fiction book" thread:

Foucault's Pendulum by Umberto Eco.

It should appeal to those of a sceptical bent, as (IIRC) it has a fine debunking of numerology in the middle. (IRRC) relating astronomical units to the dimensions of a telephone box...

Ah yes:


Wiki link

(thinking man's Da Vinici code???)

 

[Hokulele]

Since you apparently care for precision and accuracy in your measurements, I would recommend taking measurements directly from originals, not from published photographs (particularly of the Nazca area). Any photo is subject to a host of errors including lens distortion (especially in aerial photography), perspective errors (if the camera is not perfectly centered and perfectly perpendicular to the object of interest), and scanning and reproduction errors (skew and bias in particular). If you would like serious scientific interest in your work, you will need to document your measurement process and describe how you avoid or adjust for errors in measurement.

NB - True, I am not a mathematician, but I have a background in surveying and mapping, and can offer information on measuring things to a high degree of accuracy (precision and accuracy are not the same!), especially when working with aerial photography.

 

[Jiri]

Pyrostar

Jiri, my problem with the above diagram is that I cannot see a compelling reason why the lines you drew should be where you drew them,

That's no problem, Bruto, as I can account for my reasons to draw these lines, the problem is that you drew the conclusion that I had swindled the design in (see below).
The lines have names, we can inspect them, and I can explain line by line. First, I'd like to give my account of how all these lines happened. I didn't touch the torso (of the girl) before having copies of at least 4x lifesize. At that size the engraved lines were much thicker than the exploratory thin pen lines, and it was easy to see, how the two meshed.
To draw all the exploratory lines in the torso I had used only a ruler and a pen. I had no idea what the result would be, because I was not smuggling in any angles. I had taken great care to use each engraved line's characteristics to guide me in the process of extrapolating the straight lines. That's why I had waited until the larger magnification copies, I wanted to see these characterictistics clearly, because I already believed at that time that the designers themselves were working not with the lines, but with their edges that they worked with lines not as lines but two-dimensional figures. I had the following separate methods for extrapolating lines.
a) If a line curves like a bow then imagine that you attach a string at its ends. That's one way of deriving a line.
b) Some lines are almost straight, but we find that there is only one way to draw a straight line through a given line, because the wobble in the engraved line will let throug only one straight line (without popping to the outside at some point).
c) If the engraved line is ironically too straight for deriving a straight line from it, we look for guiding elements as we extend the test lines further into the picture.
You don't have to believe me, but I just told you, how the derived lines inside the torso had been made.
.

unless you had already decided what you wanted to find. This is the problem with most such impositions of assumed design principles on designs whose creators are unknown. The pentagram you've drawn doesn't follow many of the actual lines of the engraving, and is extrapolated to form lines of which there is no hint in the engraving. In order to see a pentagram in that engraving, you really have to put it there yourself.

Again, you misrepresent what I had done. I did not draw a pentagram - the lines obtained by the method described above had formed a triangle with the y-axis. Only then was I able to measure its angles and note that the triangle could be a part of a 5-pointed star. That's when I got to the stage, where I could complete the triangle into a star. Only then did it become obvious that the star was anchored in the 0,0 point of the x,y axes.
.

Similarly, in a drawing that has no straight horizontal or perpendicular lines,

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It just dawned on me that you don't realize the x,y-axes were already there. They belong to the core Cone & Square formation, which is already there from the start. The triangle fits the formation. It grows into a beautiful Divine Proportion formation.as it integrates into the system.
Did I learn this from the engraving, or did the engraving teach me like an interactive CD?

the choice of where to draw the framing rectangle is up to the interpreter. I see no reason to accept your placement, or your rectangle, as being related to the design of the engraving.

The (Golden) rectangle is just part of the system. You really are looking at a system, Bruto. If only you could reconstruct it from memory, which is a really simple thing to do, although a number of steps is necessary - you'd also develop some appreciation for this system. I bet, not even J.S.F. could do it at this very moment. It is also possible to recreate the entire system from the engraving, which is when you'd really have an understanding of what the heck is going on.

 

[jsfisher]

A lot of what is being said boils down to this: Finding some curious relation or artifact in something does not mean the creator of that something did it intentionally.

A small example: The square is a simple, basic shape. I suspect not even DJJ would be surprised to find squares in ancient artwork and architecture. Put two squares next to each other to form a 1x2 rectangle--another simple, basic shape. No surprises here. But wait...






wait for it...







wait for it...






wait for it...






The diagonal of that 2x1 rectangle is √5, the major part of that mysterious golden ratio thing. Throw in a circle or two, and, my goodness, there's phi itself in all its miraculous glory.

Phi appearance amongst a few lines and circles is the same level of proof of intent as is an infant waving her arms in circles proof the child understands pi.

 

[Belz...]

That's no problem, Bruto, as I can account for my reasons to draw these lines, the problem is that you drew the conclusion that I had swindled the design in (see below).
The lines have names, we can inspect them, and I can explain line by line. First, I'd like to give my account of how all these lines happened. I didn't touch the torso (of the girl) before having copies of at least 4x lifesize. At that size the engraved lines were much thicker than the exploratory thin pen lines, and it was easy to see, how the two meshed.
To draw all the exploratory lines in the torso I had used only a ruler and a pen. I had no idea what the result would be, because I was not smuggling in any angles. I had taken great care to use each engraved line's characteristics to guide me in the process of extrapolating the straight lines. That's why I had waited until the larger magnification copies, I wanted to see these characterictistics clearly, because I already believed at that time that the designers themselves were working not with the lines, but with their edges that they worked with lines not as lines but two-dimensional figures. I had the following separate methods for extrapolating lines.
a) If a line curves like a bow then imagine that you attach a string at its ends. That's one way of deriving a line.
b) Some lines are almost straight, but we find that there is only one way to draw a straight line through a given line, because the wobble in the engraved line will let throug only one straight line (without popping to the outside at some point).
c) If the engraved line is ironically too straight for deriving a straight line from it, we look for guiding elements as we extend the test lines further into the picture.
You don't have to believe me, but I just told you, how the derived lines inside the torso had been made.

In other words, you just added lines where there weren't any.

 

[bruto]

That's no problem, Bruto, as I can account for my reasons to draw these lines, the problem is that you drew the conclusion that I had swindled the design in (see below).
The lines have names, we can inspect them, and I can explain line by line. First, I'd like to give my account of how all these lines happened. I didn't touch the torso (of the girl) before having copies of at least 4x lifesize. At that size the engraved lines were much thicker than the exploratory thin pen lines, and it was easy to see, how the two meshed.
To draw all the exploratory lines in the torso I had used only a ruler and a pen. I had no idea what the result would be, because I was not smuggling in any angles. I had taken great care to use each engraved line's characteristics to guide me in the process of extrapolating the straight lines. That's why I had waited until the larger magnification copies, I wanted to see these characterictistics clearly, because I already believed at that time that the designers themselves were working not with the lines, but with their edges that they worked with lines not as lines but two-dimensional figures. I had the following separate methods for extrapolating lines.
a) If a line curves like a bow then imagine that you attach a string at its ends. That's one way of deriving a line.
b) Some lines are almost straight, but we find that there is only one way to draw a straight line through a given line, because the wobble in the engraved line will let throug only one straight line (without popping to the outside at some point).
c) If the engraved line is ironically too straight for deriving a straight line from it, we look for guiding elements as we extend the test lines further into the picture.
You don't have to believe me, but I just told you, how the derived lines inside the torso had been made.
.


Again, you misrepresent what I had done. I did not draw a pentagram - the lines obtained by the method described above had formed a triangle with the y-axis. Only then was I able to measure its angles and note that the triangle could be a part of a 5-pointed star. That's when I got to the stage, where I could complete the triangle into a star. Only then did it become obvious that the star was anchored in the 0,0 point of the x,y axes.
.

.
It just dawned on me that you don't realize the x,y-axes were already there. They belong to the core Cone & Square formation, which is already there from the start. The triangle fits the formation. It grows into a beautiful Divine Proportion formation.as it integrates into the system.
Did I learn this from the engraving, or did the engraving teach me like an interactive CD?



The (Golden) rectangle is just part of the system. You really are looking at a system, Bruto. If only you could reconstruct it from memory, which is a really simple thing to do, although a number of steps is necessary - you'd also develop some appreciation for this system. I bet, not even J.S.F. could do it at this very moment. It is also possible to recreate the entire system from the engraving, which is when you'd really have an understanding of what the heck is going on.

Sorry Jiri, it just fails to convince. No matter how you slice it you come up with an explanation that clearly indicates that you inferred the system first, and then drew in your lines on the assumption that the system applied to the drawing. We could go round and round with this forever, but it is simply not convincing. If the lines have names, they are names of yours, not of some prehistoric artist, no matter how fine his work may have been.

One of the most obvious problems with this whole thing is the fact that we can see no reason why, if the creators of that drawing did base it on mathematical principles, they did not evince them more directly. They could, after all, have actually drawn pentagrams, or other geometrical figures, if this was their intention. There is no indication that this drawing was intended to communicate anything but its obvious, pictorial content. You cannot justify extrapolation without some reason other than your own conviction that there's a system. Every time you try to explain and counter what I say, you come back with another form of saying the same thing: you're working backwards, fitting the phenomenon to the theory.

Of course, if some theorists are right, and some of the mathematical proportions involved are inherently, aesthetically pleasing and apt, then it would not be entirely surprising or significant to find that at least occasionally, an artist, even a prehistoric one, might have produced a particularly well composed picture that embodies those principles. It's clear from many primitive paintings that one of the first areas in which human beings really began to separate themselves from the other animals of the world was art, and that just about as far back as we can trace mankind, there have been artists of great ability, even genius. It is a big leap, however, to infer from that that they knew the math, when we have no other reason to believe that people of that era knew any math at all, since it's obvious from the artifacts that they had no written language, and no written numbers.

I'm not aware of anything in prehistoric drawing or artifacts that suggests such geometric or mathematical sensibility. Even if the extent of the use of geometry by the Egyptians and the Greeks is questionable, we can at least know with certainty that they were familiar with some of the principles, that they had a number system, a written language, and a level of mathematical discipline on which to draw.

 

[Paulhoff]

I am still waiting to see an arch used in their buildings....

Paul