II. Postulates and Strucuture for Notating a Sample of the Ambient Optic Array

(1)C = 2 pr

This is the circumference of our initial circle with radius r.

(2)C2 = 2 pr1This is the circumference of our second circle,the base of the cone with radius r1.

(3)

r^2 = r1^2 + h^2This is the initial radius squared expressed during transition to be theslant of the cone in terms of the height of the cone, h, and the radius

of the base of the cone, r1.

(4)r = ±

,Hr1^2 + h^2L

The initial radius can be calculated through the Pythagorean Theorem. Theinitial radius is always the slant of the cone when there is a cone.

(5)

s = q rIn this expression,we define an arc length to be equal to a radius of an angle measure,q , and that thus,s ê q = r

(6)The arc length is expressed to be the difference of circumferences =Initial Circumference - Circumference of the base of the cone =C - C2 = 2 pr - 2 pr1 = qr

(7)

The radius of the base of a cone is equal to r1,and we can similarly solve for it.

r1^2 ã r^2 - h^2r1 = ±

,Hr^2 - h^2L

(8)In general, we say that h § r. More specifically,when there is a cone, h § r, and when there is an initial circle, h = r.

(9)t = time = t HsecondsL = q ê 2 p , because this ensures thatthere is a maximum height of the cone equal to the initial radius.

(10)

The elapse of 1 second = 1 revolution of the systemt HsecondsL = q ê 2 pThis equation ensures that when we put a revolution equal to 2 pradians into the equation for the angle theta, we will get one second.We also note that any arc length can be described by an angle of distance, q r.

The model of space - time proposed only requires the acceptance of three fundamental truths. Firstly, that the length thatcomes from a change in circumferences can be equal to an arc length, secondly, that the Pythagorean theorem will hold up formaking calculations within the modular, and thirdly, that ellapsed time is in terms of a cyclic continuum in the sense that we canmeasure it in terms of degrees of a circle.

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The model describes the dynamic between subjective and objective reality in a simple and useful manner of perception(observation). The light is always considered to be directed toward the perceiver, whose locus of perception is either the point ofthe cone of the field of vision, or the reception of a combined wavelength of light when the height is equal to the initial radius.This provides a framework for notating how light travels, carrying information of the energy and velocity of light, which in turnprovides the, "immediate basis for visual perception" (The Visual World, 55) when there is an exposed receptor cell to the light.On receptors chemical compounds of receptors, Gibson has said that, "Like the substances used in photographic emulshions,these are capable of reacting differentially to the energy and wavelength of light" (The Visual World, 48), and that "nothing getsinto the eye except radiant energy" (The Visual World, 54). He has also revealed that

The distance away from the area of the array that the perceiver samples during action of the visual system is equal to theheight of the cone. Within the area of the sample, there is a contained amount of information about layout. The problem ofincluding both the wave-length interpretation and the sampling interpretation within the structuring of the ambient array ariseswhen we consider that the wavelength has not gone through the entire transition with respect to the ecological structure of thearea when there is a sampling, but is realistically compiled by the visual system into an image. The problem can be partiallyresolved by considering that the visual system samples very many specific receptions of light or regions of layout from thesampled area within a general sample. These wavelengths give information to the perceiver in terms of saturation, hue, brightnessand thus, contour.

I will now do some algebra to conclude what the height of the cone is in terms of the initial parameters. It can eventuallybe reduced to a single variable when ideas like differentials of distance and frequency are included in the setting up of theequalities. This will lead to numerous nested equations that describe different parameters of the information being deliveredwithin a visual system. We consider that the radius is equal to a single wavelength, not half a wavelength, because we arediscussing the time it takes to travel a distance of one wavelength of light, and that wavelength travels through the height of thecone. Although, the shape of the diameter can later be shown through three-dimensional graphs to have the shape of a sine wave.

Solve@r1^2 + h^2 ã r^2, hD

(11)::h Ø - r2 - r12 >, :h Ø r2 - r1

2 >>

We say that the amount of qr = s, taken out of the circle is the change in the circle's circumference that is the base of thecone. The change is equal to s = 2pr-2pr 1.

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

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

2 Postulates and Structure for Notating A Sample of the Ambient Optic Array by Parker Matthew Davis Emmerson.nb

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

SolveBh == r2 - r12 , r1F

(12)::r1 Ø - r2 - h2 >, :r1 Ø r2 - h2 >>

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

(13)r q == s ã 2 p HrL - 2 p HHrL^2 - h^2L = 2 p HrL - 2 p r1

q == H2 p rL ê r - H2 p r 1L ê rL = HH2 p rL ê rL - 2 p -h2 + r2 ì r

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

(14)::h Ø -4 p r2 q - r2 q2

2 p>, :h Ø

4 p r2 q - r2 q2

2 p>>

SolveBh ==4 p r2 q - r2 q2

2 p, rF

(15)::r Ø -2 p h

4 p q - q2>, :r Ø

2 p h

4 p q - q2>>

Solve@q * r == 2 p HrL - 2 p r1, r1D

::r1 Ø2 p r - r q

2 p>>

A gradient of flow or of motion can be found by determining the right application of a derivative with respect to time forthe above mentioned expressions for distance in an ecological context.

Postulates and Structure for Notating A Sample of the Ambient Optic Array by Parker Matthew Davis Emmerson.nb 3

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In[18]:= RevolutionPlot3DB2 p r - r q

2 p, 8r, -1, 1<, 8q, -2 p, 2 p<F

Out[18]=

In[15]:= SolveB2 p r - r q

2 p==

2 p h

4 p q - q2, rF

Out[15]= ::r Ø4 p2 h

H2 p - qL H4 p - qL q

>>

4 Postulates and Structure for Notating A Sample of the Ambient Optic Array by Parker Matthew Davis Emmerson.nb

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In[17]:= RevolutionPlot3DB4 p2 h

H2 p - qL H4 p - qL q, 8h, -1, 1<, 8q, -2 p, 2 p<F

Out[17]=

Postulates and Structure for Notating A Sample of the Ambient Optic Array by Parker Matthew Davis Emmerson.nb 5

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