A study of Golden Ratio 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. The observation is philosophically spoken by saying that sacred geometry is a perspectival emanation of our reality. There is a point at which, when one observes two equal line segments, they appear to be in golden ratio with each other.

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:

θ1=s/R=y/R2=y/√((x+l)^2+h^2)

θ2=s/R=ϕy/R1=ϕy/√(x^2+h^2)

Our first Postulate:

There is a point at which, when one views two equal line segments at perspective, they appear to be in Golden Ratio with each other, and L is in golden ratio with the larger of the two lengths that are viewed at perspective.

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 golden ratio 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(θb)= 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))

A quick Case Study of the Theories:

38.17=h/l+x

if x=L, what is h?

and L=2

H=2.47

What is y?

Y = .763

What is Theta t?

Theta T=22.45

What is Theta B?

Theta B=51.002

That Concludes the Extent of the study for the mean time. It looks as though we have shown the correct way of calculating our distance and height from the two equal line segments if we want to perceive them in Golden Ratio.

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