Geometric Patterns of Perception by Parker Emmerson

[freeztar]

Our visual perception is 2D when we only look with one eye.

 

This is incorrect. Our visual 3D perception is present in both eyes individually. Covering one eye certainly reduces visual perspective/coverage, but does not equate to a net 2D visual perception.

[DiscovererOfNewKnowledge]

Yes, yes, this is because of muscle contractions in the eye. These are not nearly as important as the idea of the diagram and the idea that one perceives things with one eye as the amount taken up in the visual field---- that is the key idea and that remains true. Judging depth happens in both eyes for sure, but it is not nearly as important with one eye, nor does the fact that you can judge depth a bit change the idea of the "amount of perceptual field" key concept.

[Turtle]

Yes, the diagram, I like:

 

 

But he doesn't mention the golden ratio it makes in the hexagon between green and blue:

 

 

I'm sure this wasn't lost on him, it must just not be pertinent to what he's talking about. But, that chapter does explain how bisecting the line makes the hexagon, and it's understandable!

~modest

 

Fuller draws attention to Phi by omitting any mention. I have searched Synergetics the last hour, and while this is no evidence he doesn't ever mention Phi, I can not find it. I'm well assured nothing is lost on Fuller. :cat: I suspect that because it is a 'plane' figure ratio, it plays no role in Synergetics because for Fuller the flat plane is an illusion. Here's his jab at it following the construction of the hexagon in Synergetics. Bolding mine. :)

 

The Greeks found it a tantalizing matter that the two lines CE and C'E', which lie between the vertexes of the two pairs of equilateral triangles, seemed to be equal, but there was no way for them to prove it by their construction.

825.24 At first it seemed they might be able to prove that the increments CE and C'E' are not only equal to one another, but are equal to the basic radius of the circle AD; therefore, the hexagon ACEBE'C' would be an equilateral hexagon; and hexagons would be inherently subdivisible into six 60-degree equilateral triangles around the central point, and all the angles would be of 60 degrees.

825.25 There seemed to be one more chance for them to prove this to be true, which would have provided an equiangular, equiedged, triangularly stable structuring of areal mensuration. This last chance to prove it was by first showing by construction that the line ADB, which runs through the point of tangency of the circles A and B, is a straight line. This was constructed by the straightedge as the diameter of circle D. This diameter is divided by four equal half-radii, which are proven to be half-radii by their perpendicular intersection with lines both of whose two ends are equidistant from two points on either side of the intersecting lines. If it could be assumed that: (1) the lines CE and C'E' were parallel to the straight line ADB running through the point of tangency as well as perpendicular to both the lines CC' and EE'; and (2) if it could be proven that when one end between two parallels is perpendicular to one of the parallels, the other end is perpendicular to the other parallel; and (3) if it could be proven also that the perpendicular distances between any two parallels were always the same, they could then have proven CE = CD = DE = D'E', and their hexagon would be equilateral and equiradial with radii and chords equal.

825.26 Pythagorean Proof

825.261 All of these steps were eventually taken and proven in a complex of other proofs. In the meantime, they were diverted by the Pythagoreans' construction proof of "the square of the hypotenuse of a right triangle's equatability with the sum of the squares of the other two sides," and the construction proof that any non-right triangle's dimensional values could be obtained by dropping a perpendicular upon one of its sides from one of its vertexes and thus converting it into two right triangles each of which could be solved arithmetically by the Pythagoreans' "squares" without having to labor further with empirical constructs. This arithmetical facility induced a detouring of strictly constructional explorations, hypotheses, and proofs thereof. ...

 

I also spent a quarter hour looking at Dicoverer's drawing, and without some supporting text to describe it, I get nothing from it. ;) :)

[DiscovererOfNewKnowledge]

OK you guys, I'm downloading the GIMP image working program and should be able to come out with some better explanations of exactly what's going on for you.

 

I still maintain that you should be able to think about it on your own and get it, but I do understand that it can be a bit difficult to convey the full expression without my displaying the phenomenon to you exactly, which is what I did practically in this photo!

 

http://farm4.static.flickr.com/3284/2860971015_5eb00fe77f.jpg?v=0

 

BY-NC-ND

[CraigD]

But he doesn't mention the golden ratio it makes in the hexagon between green and blue:

 

He wouldn’t mention it, because the length of the blue line divided by the length of the green line isn’t the golden ratio, [math]\frac{1+\sqrt{5}}{2} \dot= 1.618[/math], but slightly greater, [math]\sqrt{3}=1.732[/math].

[modest]

He wouldn’t mention it, because the length of the blue line divided by the length of the green line isn’t the golden ratio, [math]\frac{1+\sqrt{5}}{2} \dot= 1.618[/math], but slightly greater, [math]\sqrt{3}=1.732[/math].

 

Yeah, I was just reading,

 

Two-dimensional Geometry and the Golden section - Phi and the Equilateral Triangle

 

and slowly coming to the same conclusion. So close :)

 

~modest

[Turtle]

Aha! Not the blue/green line aha, but the I-found-the-Synergetics-reference-to-golden-section aha. ;) Seems it is in a section of the books at the backs by Arthur L. Loeb, and moreover the online version omits the section. :cat: :) It is a series of proofs by Loeb, and according to him, Fuller didn't care much for proofs. Seems Mr.Loeb collaborated with both Fuller and Escher, as well as writing his own books on spatial patterning. Here's his obit: >> loeb

 

I'll see if he makes any connection to a hexagon in his bit on phi. :)

[Turtle]

I'll see if Mr. Loeb makes any connection to a hexagon in his bit on phi. ;) :)

 

OK. Loeb uses the icosahedron to develop/illustrate phi in relation to Fuller's geometry. Inasmuch as Discoverer does not plan on going 3-d here, I'll leave it for some other place & time. :)

[Turtle]

The Science of this paper is in the power of Observation and the art of sacred geometry. It is close to pure science in that is is a direct response to inspiration and a science of the mind. The observation is philosophically spoken by saying that sacred geometry is a perspectival emanation of our reality.

 

You have inspired me to a compass & straightedge construction. :eek_big:

 

I build Fuller's construction of the hexagon, but then follow his further instruction to extend the lines to form a triangular matrix. From there I apply C1ay's construction of a star-pentagon on the vesica piscis and then draw the diameter of the pentagon in green and the edge in blue and the ratio of green-to-blue is phi. :eek_big: :naughty:

 

YouTube - finding phi http://www.youtube.com/watch?v=o7zFfZp6WAU