the geometric pattern of perception of the moon

8/14/2019 The Geometric Pattern of Perception of the Moon

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The Geometric Pattern of Perception of the Moon

from beneath the Legs.

by Parker Emmerson

From Perception of the Environment with Tom Toleno

I. Introduction

Perception would and could only be perceived as the inception and or acceptance of the information provided by the

individual or individuals through any giving persona section of reality. In this paper, I will discuss how Gibson’s theory of the

ambient optic array, in combination with mathematical inquiry of the experiential phenomenon, will act like a thought experi-

ment for understanding the reason behind the illusion that, when you look at the moon through your legs when positioned upside

down, you will see that it looks smaller than when you see it from a standing position. Specifically, we will attempt to answer the

question of, "how can perceived change in size, theoretically relating to a change in distance, be accounted for even though thechange in distance does not actually occur"?

When people look up in the sky from a standing position, a clear and accessible visual structure presents the opportunity

for relatively accurate depth perception and size discernment. The object being perceived takes the shape of a circle or orb in the

sky, which is the moon. However, when the body is turned upside down and the eye strains to move about searching for or

focusing on external stimuli, the visual structure is depleted. It was shown that by G. M. Stratton that, for voluntary eye motions,

following a moving object was anything but predicatble. The conclusion from Stratton's experiments pertains to the phenomenon

of the moon's looking smaller from upside down beneath the legs, because it shows us that, "reception of visual structures is

possible only for the eye at rest" (The Perceptual World by Wolfgang Metzger 63). We will now use mathematics to describe the

visual structure of the variables relating to depth perception and measurement of the size of a circle or orb.

We might then ask, in what way is the perception of the moon in the sky similar to the perception of an image like a

painting? 170

First, we postulate that :

(1)

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The diagram is not drawn to scale with regard to theta or an actual transformation of this sort. This diagram is a representa

tion of a geometric structure and a system that has complex implications. If an object or wavelength progresses through the

height of the cone, it is said to be traveling through theta in terms of time. However, it should be noted that the circle does not

have to fold all the way up in order for the max height of r to be achieved for variables to maintain their correlation. This is said

to be an invariance of the system. The position is also said to have an initial radius of a max of the height of the cone during a

given instance.

Gibson uses the word invariance slightly differently. He is interested in the invariants in the ambient optic array, and says

that,

"A list of the invariants in an array as the amount of illumination changes, as the type of illumination changes, as the

direction of the prevailing illumination changes, and above all as the point of observation changes cannot yet be drawn up with

any assurance. the contours in an array are invariant with most of the changes in illumination. The textures of an array are

reliably invariant with change of observation-point,. The property of a contour being closed or unclosed is always invariant. The

form of a closed contour in the array is independent of lighting but highly variant with change of obseration point." Brightness

depends on the collection, density and flux of energy of light moving in these perceptual structures of the ambient array.

The cone is a structure that I propose may be present in every point of the ambient optic array, structuring information in

perceptual space, and one that also has specific geometric relations through the folding of spacetime. Perceptually, spacetime

folds are a part of what space-time is, and are drawn out by mass. and not a It also acts like an observable metaphor for how

discernment of the size of a massive orb in space occurs and can change with context.

Churchland noted that the first computational problem for the visual system is, "how to get a geometric description - a Primal

Sketch - out of the intensity array on the retina" (BRP 221). Intensity is measured by wavelength and frequency of light, which is

related to saturation and luminence of a pigment or radiating body. The cone model works somewhat before the retina, but

potentially within it. It can help us to see the structuring of light and how it interacts with the structuring of a perceptual reality,

because in the normal course of primal sketching, a transformation through time is occuring. This can be accounted for through

theta if theta is a function of time.

The parameters of the system for the purposes of this discussion are:

(2)

C = 2 pr

This is the circumference of our initial circle of radius r

C2 = 2 pr1

This is the circumference of our second circle,

the base of the cone, of radius r1

r ^ 2 = r1^ 2 + h ^ 2

This is the initial radius squared expressed as the slant

of the cone in terms of the height of the cone, h, and the radius

of the base of the cone, r1

r =,Hr1^ 2 + h ^ 2L

s = qr

s ê q = r

The arc length taken out of a circle at a given time is =

t = C - C2 = 2 pr - 2 pr1 = qr Ø Equation 7

r1^ 2 ã r ^ 2 - h ^ 2

r1 =,H

r ^ 2 - h ^ 2

Lh § r

t = time

1 second = 6 degrees

t = 6 q

I will now do some algebra to conclude what the height of the cone is in terms of the initial parameters. It can eventually

be reduced to a single variable.

II. Proof

2 Perception of the Moon from Beneath the Legs.nb




Solve@r1^ 2 + h ^ 2 ã r^2, hD

(3)::h Ø - r2- r

1

2 >, :h Ø r2- r

1

2 >>

We say that the amount of q r = s, taken out of the circle is the change in the circle's circumference that is the base of the cone.

The change is equal to s = 2pr-2pr 1.

Notice that q = HH2 p rL ê rL - HH2 p r 1L ê r L, because we divide by r on both sides.

We will focus on the positive solutions for the height of the cone.

SolveBh == r2 - r1

2 , r1F

::r1 Ø - -h2

+ r2 >, :r1 Ø -h2

+ r2 >>

This is the change in circumference with the substituted expression for r1 in terms of h and r.

(4)rq == s ã 2 p HrL - 2 p HHrL^ 2 - h ^ 2L = 2 p HrL - 2 pr1

q == H2 p rL ê r - H2 p r 1L ê r L = HH2 p rL ê rL - 2 p -h2

+ r 2 ì r

SolveBq r == 2 p HrL - 2 p HHrL^ 2 - h ^ 2L , hF

::h Ø -

4 p r2 q - r2 q2

2 p

>, :h Ø

4 p r2 q - r2 q2

2 p

>>

We can use this equation in several ways, and its meaning is not constrained to a single contextual interpretation during

the discernment of depth and size. Although, the mind/brain may access its truth in specific ways when context influences the

perception during observations within the ecological surrounding.

III. Environmental Context; Ecological Optics

The context of the station point and the focus of perception of the moon in the sky play a crucial role in theorizing about

the validity of a computational approach to aiding an understanding of life-world experience.

Ecological optics has its roots in differentiating between, "stimulation by light and the information in light" (BRP 160).

The study of the energy of light is in the realm of physical optics, while the light that is informative to the subject about the world

is, "the unfamiliar discipline called ecological optics" (BRP 160).

Perception occurs at h = r, because we see the moon with radius r first, then when looking beneath our legs upside down,

the eye makes syccatic movements, while maintaining a station point of reference at the center of the horizon with respect to the

subject. H can be both the radius of the moon, and the distance we are away from the moon at the same time. Thus, when light

enters our eye from the moon, it has traveled a distance h, equal to r during the perceived size measurment of the moon' s size.

Perception of the Moon from Beneath the Legs.nb 3




This is a kind of structuring of light that is present within the array of ambient light.

From the sheer visual perception, are we sure that the moon in the sky is a physical object, and not just a luminescent

orb? If we consider just to be a luminescent orb, then...

The element of persona, which is the embodiment of the human being integrates the context of their perceptions with

objects in the world. Thus, the distance up in the sky is discerned also by the workings of a cone that stretches out from thehorizon, extending into the air through perception. Giving it a physical location with relation to the ground is contour tells us its

height in the sky only insofar as we can perceive it due to an imaginary circle extending out on the ground for the base of a cone

with height extending to the sky. Though the moon's position is understood being directionally outward, any discernment of size

is related to the ecological context of a human being, because "the information does not consist of signals to be interpreted but of

structural invariants which need only be attended to" (BRP 161).

When we look at the moon from beneath the legs, we see an object in relation to contour. The object, however, is a

heavenly body, and discernment of its actual size is more difficult than objects on the ground. In Gibson’ s terminology, what

happens during the perception of the moon when standing up is that, although the object, “neither approaches nor recedes from

the point of observation, and no change occurs within the contour corresponding object” (Gibson, 103), there is a perceived

magnification. In the optical array, when perceiving the moon, we can suggest that there must be a decretion in the optical

structure, i.e. framing by the legs (deletion of optical structure relating to the contour of the moon), and therefore, a decrease of

the moon’ s edge contour from the bent over position. We can even use a continuous function in order to describe this transition

of perceptions when we compare it to the height of the moon in the sky, and realizing that its decrease in perceived size is a

function of theta.

IV. Illusory Contour and a Computational Approach





When we look from beneath our legs, not only do our eyes have to strain to look upward (creating hypocycloid-like,

syccatic eye movements, but also they switch positions and invert. The phenomenon occurs within the sphere of an individual’ s

perceptual space. From the notion of mirroring (due to inverted eye positions), we notice that the derivative of the actual

geometric element of depth inceives a perceptual illusion of depth change independently from within the space of an individual

and their translation of the existence of objects in real space into subjective experience upon reception of the external sensation

of light.

In order to model this phenomenon, we will use the hypocycloid to model the syccatic eye movements, and the derivative

of the height (due to mirroring of the individual' s perceptual space) to find the exact amount of decretion of contour around the

perceived moon when looking upside down from beneath your legs.

When the radius has an initial value of one, we can take the derivative of the height, that is the rate at which the height

extends into perceptual space. The setting of the radius to one simply allows us to gauge the distance/wavelength of ambient light

with a conic structure.

The primal sketching, which is involved in the measurement of length, specifically the height of the cone (radius of the

moon), when the moon itself is seen to be analogous to the entire array of optical flow, to a subjective perceiver, occurs through

time.

When assigning variables, we have several options, and they will give us different interpretations of the meaning of

correlations in perceptual space, because "the emergence of a percept with illusory contours represents the solution to the

probelm posed by the stimulus as to what it represents in the world" (BPR 222, (Rock and Anson, 1979)). What I am aiming to

find out, by applying the cone model of phenomenologically reducing the perception of the moon to a transformation in per-

ceived size (continuous, but limited with relation to the horizon, i.e. it gets smaller at a higher position in the sky), to a mathemati-

cal analogy of eye motion (i.e. syccatic eye motions being similar to the hypocycloid), is if there can be the removal of perceived

real contour due to an illusory lack of contour through percepts involving virtual lines, because "construction of virtual lines is a

routine part of low-level processing, which yields the basic representation of the geometry of the image" (BRP 223). If so, we

should be able to see how much contour is taken out when there is a change in the ecologically optical environment.

Since we can prove similarity of the cone having a height equal to the perceived radius of the moon and the cone helps

designate the position of the moon in the sky in perceptual space, endowed with depth through pick - up of contour from informa-

tion in the ambient optic array of perceptual space, we can understand their being related to each other in terms of the change

measured perception.

is whether or not contours in the world are actually illusory, and if

a is the initial height of the cone, b is the change of the cone through time. In essence,

as the moon gets higher in the sky, the information of its structure goes down, because it is less

related to contour on the ground. It gets to be hard to say where it is in the fourth dimension.

In[26]:= DB 4 p HrL - q HrL q H1L2 p

, qF

Out[26]= -

r q

4 p 4 p r - r q

+

4 p r - r q

4 p q

DB 4 p H1L - q H1L q H1L2 p

, qF

4 p - q

4 p q

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;

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tmini = - 10 p ;

tmaxi = 10 p ;

8x@t_ D, y@t_ D< =

8Ha - bL * Cos@qD + b * Cos@Ha - bL ê b * qD, Ha - bL * Sin@qD - b * Sin@Ha - bL ê b * qD<;

hypocycloid =

ParametricPlot@8x@qD, y@qD<, 8q, tmini, tmaxi<,

PlotStyle -> 88Blue, [email protected]<<,

AspectRatio -> Automatic, AxesLabel -> 8"x", "y"<D

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hypocycloid =




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hypocycloid =




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hypocycloid =




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;

b = 1tmini = - 10 p ;

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8x@t_ D, y@t_ D< =


hypocycloid =




1

-1.5 -1.0 -0.5 0.5 1.0 1.5x

-1.5

-1.0

-0.5

0.5

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y



the geometric pattern of perception of the moon

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