Geometric Patterns of Perception by Parker Emmerson

[DiscovererOfNewKnowledge]

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Please see my website http://www.myblogband.com and find me under piano listing under musicians for contact, also my website parkeremmerson.com is interesting.

 

 <a rel="license" href="http://creativecommons.org/licenses/by-nc-nd/3.0/us/"><img alt="Creative Commons License" style="border-width:0" src="http://i.creativecommons.org/l/by-nc-nd/3.0/us/88x31.png" /></a><br /><span xmlns:dc="http://purl.org/dc/elements/1.1/" property="dc:title">Geometric Patterns of Perception</span> by <a xmlns:cc="http://creativecommons.org/ns#" href="http://hypography.com/forums/physics-and-mathematics/16193-geometric-patterns-of-perception-parker-emmerson.html" property="cc:attributionName" rel="cc:attributionURL">Parker Matthew Davis Emmerson</a> is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-nc-nd/3.0/us/">Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 United States License</a>.<br />Based on a work at <a xmlns:dc="http://purl.org/dc/elements/1.1/" href="http://www.flickr.com/photos/30549435@N07/2860835743/in/photostream/" rel="dc:source">www.flickr.com</a>. 

 

----- If it doesn't show up---- this is to say that this work is mine and not to be tampered with or distributed, and it is attached to the Creative Commons agreement of many different countries across the world and subject to the rights therein.

 

Some have said this work cannot be copyrighted. I disagree, but am inclined to at least ask that credit be given to its discoverer: Parker Matthew Davis Emmerson

 

Although this is a geometric construction and kind of mathematical proof, I constructed it, and thus I argue that it is applicable to copyright law. No distribution, sale, or derivative work may be done without sole permission of Parker Matthew Davis Emmerson. [email protected] I am happy to discuss and allow work to be done on it under certain conditions, I just want to make sure it is protected and does not fall into malicious hands. I can copyright the diagram because I am also copyrighting the mechanism in which one folds the diagram. It transcends geometry to become art, which is copyrightable.

 

Please view the above link for access to the diagrams, etc.

 

A study of Golden Ratio Perception and Introduction to the coming paper Geometric Patterns of Perception

 

By Parker Emmerson

 

 

Through History, there have been many ways of portraying the Golden Ratio.

 

In this paper, I am going to outline how the Golden Ratio plays into our everyday perception of reality.

 

We are also going to go on a journey through Mathematics and Geometry to discover if there is any relation of our perception of reality on a daily basis. Please consult the Attached diagrams for a more in depth understanding of the symbology.

 

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.

 

Some Background Information:

 

R1=√(x^2+h^2)

R2=√((x+l)^2+h^2)

R3=√((x+2L)^2+h^2)

 

Let us also note that:

 

θ2=s/R=y/R2=y/√((x+l)^2+h^2)

θ1=s/R=ϕy/R1=ϕy/√(x^2+h^2)

 

It might be stated then that Theta 2= 1/2 Theta 1

 

Our first Postulate:

 

 

λ=ϕ(ϕy)=L

 

This creates an analogous relationship of y and ϕ That is like this:

 

y: ϕy:: ϕy: (ϕ (ϕy))

 

The trigonometric relations which follow our diagram of 2:1 arc perception of two equal line segments for the angle θk are:

 

tan(θk)= h/(L+x) = ϕy/(r2-r1)= ϕy/√(L^2-(ϕy)^2)

sin(θk)= h/r2 = ϕy/L

cos(θk)= L+x)/r2 = (r2-r1)/ ϕy = (√(L^2-(ϕy)^2))/ϕy)

 

These trig relations show us that:

 

R2-R1= √(L^2-(ϕy)^2)

 

From this, we can definitively show that Theta K is constant and that it is equal to 38.17… degrees so long as our relation of L= ϕ (ϕy) stays the same.

 

θk=38.17314 that is to say that:

 

(sin(θk)*L)/y=ϕ; (sin(θk)*ϕ(ϕ y))/y=ϕ

ϕy= sin(θk)*ø(øy)

1/ϕ= sin(θk)

ϕ=1/ sin(θk)

 

Now let us examine the other thetas, B, and T.

 

tan(θ:)= h/x

sin(θB)= h/√(((2L+x)^2)+h^2)

cos(θB)= x/√(h^2+x^2)

 

tan(θT)= h/(2L+x)=y/√(L^2-(y^2)=y/(R3-R2)

sin(θT)=h/√(((2L+x)^2)+h^2)=y/L

cos(θT)=(2L+x)/√(L^2-(y^2))

 

Theta T= 22.45 degrees

 

We may then take radius R1 and fold it onto R2, then fold R2 onto R3, and the phenomenon which is therein seen is that not only do the outer hexagons described by the visica piscis which are constructed around each radius, respectively, transform into regular pentagons inscribed in a conic at the angle where 5 equilateral triangles combine to form a cone or 5-sided pyramid, but the inner hexagons do as well.

 

Thank you, and sorry if I deprived anybody out there of discovering this on your own.

 

Sincerely,

 

Parker Emmerson [email protected]

 

I maintain copyright on these ideas so that proper use of them is made and advances are taken with precaution. I do not claim that I am the only creator of this. I believe that there is a divine power out there and that He is responsible for the gift of knowledge, the protection of truth, and the creation of the universe.

[Turtle]

Although this is a geometric construction and kind of mathematical proof, I constructed it, and thus it is applicable to copyright law. No distribution, sale, or derivative work may be done without sole permission of Parker Matthew Davis Emmerson.

 

Darned if ya do, & darned if ya don't, eh? B) Honestly though, I don't think you can copyright a geometric construction. B)

 

We may then take radius R1 and fold it onto R2, then fold R2 onto R3, and the phenomenon which is therein seen is that not only do the outer hexagons described by the visica piscis which are constructed around each radius, respectively, transform into regular pentagons inscribed in a conic at the angle where 5 equilateral triangles combine to form a cone or 5-sided pyramid, but the inner hexagons do as well. This happens only at this described combination of arc lengths and line segments in golden ratio with each other at the angle at which one would view the two equal line segments in a 2:1 ratio.

 

Thank you, and sorry if I deprived anybody out there of discovering this on your own.

 

Sincerely,

 

Parker Emmerson

 

I imagine there are quite a few ways to derive the Golden Section geometrically. Here is a thread on the vesica piscis and some of the figures one can derive from it: >> http://hypography.com/forums/physics-and-mathematics/1902-vesica-piscis-real-sacred-geometry.html

 

If you use any of our work, give credit so we don't have to come lookin' for ya. :D

Sincerely, Roger :)

[Tormod]

----- If it doesn't show up---- this is to say that this work is mine and not to be tampered with or distributed, and it is attached to the Creative Commons agreement of many different countries across the world and subject to the rights therein.

 

If you want to promote your ideas and make sure nobody "tampers" with it, posting them on a public forum is not a good idea.

 

And it's hardly in the spirit of science either. :)

[DiscovererOfNewKnowledge]

Nobody is to tamper with these ideas without my consent. I will easily give consent to other people as long as the e-mail me about it and give credit where credit is due. That credit being to me.

 

What do you mean, you can't copyright a geometric construction? Of course you can. It's like copyrighting any proof, paper, invention or idea.

 

In any event, I'm the first to discover it, so give credit where credit is due.

 

I only posted this knowledge after considerable goading into doing so.

 

I have already told several other people in person about this idea and that I discovered it.

[freeztar]

I think Tormod's advice is good. If you are truly concerned about people stealing it, a public forum is not a safe bet. The best route would be to coordinate with a professor and have it published in a perr-reviewed journal. That way it is copyrighted and documented as yours (you will have to share the credit with the professor of course).

 

I'm not sure about copyrighting geometric designs. :)

This would be a question for the patent office.

 

So...having browsed through your post, I'm not sure how the geometry correlates to perception. Can you expound upon this please?

[DiscovererOfNewKnowledge]

Yes, I'm sorry about that, however, this is an old version of the paper I'm writing and still requires a lot of redrafting.

 

It was the quickest way to get the ideas across.

 

One perceives the two equal line segments at the given angles I mention to be in a 2:1 ratio with each other. This is because the angle measures are acually in this ratio. The arc lengths that can be drawn are thus related to the golden ratio. When you fold up the radii, you get a conic of 5 intersecting equilateral triangles.

 

I'll expound on this more when I get back from class. gotta run! peace ya'll

[DiscovererOfNewKnowledge]

So, I am back and able to discuss my discovery with you if you like.

 

In regard to the perception aspect, as said in the paper, one perceives any two angle measures entering the eye as amounts of space that take up the perceptual field.

 

So, the arc lengths are just a happy occurence of Golden Ratio.

 

and it shows and proves that the Golden Ratio is present in the patterns of geometry which arise from viewing two equal line segments in a 2:1 ratio.

 

This is interesting, because never before has it been shown that the golden ratio can be described as being the ratio of two whole numbers. I have proven in someways that it can be.

 

I also would like to say that it is possible that with the application of group theory, the diagram could be turned into a quantum computer, but I would like to say that nobody is authorized to derive any further work on my discovery without first consulting me and getting my permission at [email protected]

[DiscovererOfNewKnowledge]

So, when we fold the radii onto each other, we see another expression of the Golden Ratio--- that being a regular pentagon. It only looks like a regular pentagon, however, and is actually a pentagon that comes from turning the vesica picis of the radii extending to the eye into metatron cubes, and then folding them onto each other, thus creating a cone.

[DiscovererOfNewKnowledge]

I post them in this forum to publicize the discovery and protect my right to be acknowledged as the first Earthly discoverer.

[modest]

I noticed Mr. Emmerson's pic has circles and hexagons and is supposed to be related in some way to the golden ratio. I have no idea if the following has anything to do with the proof or derivation or whatnot given above as I can't make any sense of it, but I noticed something while drafting some years ago.

 

The way to draft a golden rectangle with a compass and ruler is to make a square:

 

 

Bisect the square:

 

 

Extend the base line:

 

 

and draw a circle with radius ED, such that ED is equal in length to EF:

 

 

And that's a golden rectangle. CA and AF are two sides - I've left off the fourth corner, but you get the idea. It's a quick way to make a golden rectangle with a compass and straight edge. If the line AC is 1 then the line AF is 1.618... the golden ratio. This is commonly done and commonly (or at least used to be) commonly taught in drafting.

 

In any case, I noticed one day that centering a hexagon at E fits up quite nicely:

 

 

The radius of the circle ED is the same as the inside radius of the hexagon. :)

 

Turtle, considering a golden rectangle is an architect's favorite shape and Fuller's penchant for hexagons, I wonder if his book mentions this :)

 

~modest

 

EDIT: I forgot to mention. This observation is copyrighted. It's not to be reproduced or altered or read out loud or discussed without my permission... and giving me credit.

© All rights reserved. (™) Registered Trademark ® (Just kidding Parker ;))

[DiscovererOfNewKnowledge]

Yes,

 

That is quite interesting, and I'm sure it relates to the diagram. However, however, the most interesting part of my diagram is the interrelatedness of the two hexagons, as well as their duel transformation into a pentagon. HA!

 

also, the interrelatedness of perception, thus the title Geometric Patterns of Perceptions.

 

I hope to be able to upload a new image with appropriate labels soon so that it'll make it more easier to understand what I'm doing.

 

Hope you get a little bit of understanding out of it at least. It's quite common sensical, and you should really try to work it out for yourself, seeing as though it was discovered by a flash of intuition.

[Turtle]

...

Turtle, considering a golden rectangle is an architect's favorite shape and Fuller's penchant for hexagons, I wonder if his book mentions this

 

~modest

 

EDIT: I forgot to mention. This observation is copyrighted. It's not to be reproduced or altered or read out loud or discussed without my permission... and giving me credit.

© All rights reserved. (™) Registered Trademark ® (Just kidding Parker ;))

 

There's a door opened wide enough to sling Schrödinger's cat through. :cat: Meow! :dog: Woof! ;) We do love our drawings though New Discovererere. :) :)

 

A quick search of the index of Synergetics for mention of hexagons yields this: >>

825.20 Hexagonal Construction

825.21 Diameter: The Greeks then started another independent investigation with their three tools on the seemingly flat planar surface of the Earth. Using their dividers to strike a circle and using their straightedge congruent to the center of the circle, they were able with their scriber to strike a seemingly straight line through the center of construction of the circle. As the line passed out of the circle in either direction from the center, it seemingly could go on to infinity, and therefore was of no further interest to them. But inside the circle, as the line crossed the circumference at two points on either side of its center, they had the construction information that the line equated the opening of the dividers in two opposite directions. They called this line the diameter: DIA + METER. ...

 

The work continues & contains a diagram of a construction. >>

800.00 OPERATIONAL MATHEMATICS

From the grave, :), Bucky grants us permission to use it as long as it's not for profit. ;)

[modest]

A quick search of the index of Synergetics for mention of hexagons yields this: >>

 

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! :)

 

DiscovererOfNewKnowledge,

 

I look forward to the new image. You can also upload images to your user profile page or attach them to your post here at Hypography.

 

~modest

[freeztar]

Thanks Modest! I took quite a bit of drafting in college and I'm completely shocked that the method you demonstrated was not taught to us (especially considering that the class of 60 students was taught by two professors: one a practicing architect and the other a physicist). :)

 

The geometry makes a lot more sense to me now, but I'm still completely lost on how this relates to perception. Discoverer, you mentioned how this is a visual representation (the graphics). That makes me wonder, is this only meant to correspond with visual perception? Another Q, since our visual perception is not in 2D, how do you propose that our 3D visual perception can be constructed from the geometries of the golden mean you have presented in 2D?

[DiscovererOfNewKnowledge]

Thanks Modest! I took quite a bit of drafting in college and I'm completely shocked that the method you demonstrated was not taught to us (especially considering that the class of 60 students was taught by two professors: one a practicing architect and the other a physicist). :)

 

The geometry makes a lot more sense to me now, but I'm still completely lost on how this relates to perception. Discoverer, you mentioned how this is a visual representation (the graphics). That makes me wonder, is this only meant to correspond with visual perception? Another Q, since our visual perception is not in 2D, how do you propose that our 3D visual perception can be constructed from the geometries of the golden mean you have presented in 2D?

 

Our visual perception is 2D when we only look with one eye. The amount of space taken up in the perceptual field is due to ratio of angles at which light from two objects would enter the eye or be blocked from entering the eye. I would not say how it works in 3D. I'm only referring to how it works with one eye, but I could postulate some further ideas in 3D, but would prefer not to do so.

[modest]

Thanks Modest! I took quite a bit of drafting in college and I'm completely shocked that the method you demonstrated was not taught to us

 

No prob, after reading through:

 

Two-dimensional Geometry and the Golden section

 

I'm very surprised myself that the golden ratio is connected to so many geometric shapes. Parker mentioned a pentagon - this site shows how and why that is and also the hexagon and pentagram and equilateral triangles and explains why mathematically the ratio shows up in those shapes. I'm embarrassed to admit that I had no idea the golden ratio was so prevalent in polygons and geometry in general.

 

I'm completely shocked that the method you demonstrated was not taught to us (especially considering that the class of 60 students was taught by two professors: one a practicing architect and the other a physicist).

 

My first year studio teacher, a Grad Student... Chris Spaw as I recall, was somewhat obsessed with golden rectangles. I started to see why after reading the web page I linked. Cool stuff.

 

But, I also don't see what it's to do with perception, nor any scientific model. :)

 

~modest

[DiscovererOfNewKnowledge]

OK, if you can't see that, then you aren't looking. It's the basic model of how much of the visual field of one eye gets taken up by two objects- those two objects being a bisected line segment. Then, specifically, we set a ratio of the angles to describe the ratio of the perceptional aspects of each line segment. Then, specifically, I discovered that the arc lengths happen to be in golden ratio. I'm sorry if it's hard to figure it out, but it would be good to try and grasp it on your own. When we then fold the radii onto each other, the phenomenon happens that we transform two hexagons into two pentagons exactly.