In this paper, I show that from difference in circumferences of two circles, one can go to the Pythagorean theorem through proof by construction. Then, I show that the initial radius is a function of solely the angle taken out of the initial circle. I also solve for the innate, angular roots of the Pythagorean theorem from the known expressions described in A Geometric Pattern of Perception (Emmerson, 2009). To me, time is angularity and space is breadth, depth and length. The two mutually interpenetrating ideas, but can be totally separated from each other if necessary to express velocity. However, this leads to chaos and “glitches.” By glitches, I really mean the quality of shimmering. Perhaps all the experiential relationships of space-time are certain qualia.
In my new work, available below, I find eight roots, and to me, this symbolizes how the eightfold path is ever-present to us, and that it is a universal, ethical truth whose fundament is present throughout the universe and all being.
However, I also find eight solid roots, and these are present in my work hosted by Issuu below. I hope you find it interesting and compelling.
My tentative conclusion is that the inverse trigonometric functions are complete, and therefore lead to inconsistencies, following the conclusions of Gödel’s Incompleteness Theorems. After solving for the initial radius purely in terms the angle theta, which corresponds to that arc length removed from the initial circle, I show that there are even more expressions for the initial radius purely in terms of theta. I then make substitutions of known results from lemmas of the system present in A Geometric Pattern of Perception by Parker Emmerson ©2009-2010, available for free download, into these different univocal expressions for length, generating beautiful artwork that is of pure, valid mathematics. I then move onto further questions about perception, tackling
It is important to note that it is an area of the retina that is affected by radiant energy, not just a single point. Gibson’s reason for believing that the geometry of transformations is important to visual perception is that, “transformations are usually represented on a plane, however, whereas the retinal image is a projection on a curved surface,” (The Visual World, 153) (Gibson, James, J.. The Perception of the Visual World. Cambridge, Mass.: The Riverside Press, 1950. Print.), and that, “the actual retinal image on a curved surface is related to the hypothetical image on a picture-plane only by such a non-rigid transformation” (The Visual World, 153). A transformation in geometry is the mapping of one point onto another. Isometries, “are defined as the transformations that preserve distance” (The Four Pillars of Geometry, 145) (Stillwell, John. The Four Pillars of Geometry (Undergraduate Texts in Mathematics). 1 ed. New York: Springer, 2005. Print.). In essence, the distance of the initial radius is preserved through the transformation of a circle into a cone so long as the height is orthogonal to the base of the cone and the initial radius is always the slant of the cone. Next, we see the diagram to which Gibson was referring when considering the notion of a transformation onto a picture plane.
(The Visual World, 79).
In being preserved, the initial radius is considered an invariant. Stillwell comments about the picture plane that, “the line from (-1, 1) to (n, 0) crosses the y-axis at y = n/(n+1)” (The Four Pillars of Geometry, 91). This supposes that the eye is approximated like a point and that it is at the position of (-1, 1) in the Cartesian coordinate system. In the “coordinate system” described by The Geometric Pattern of Perception Theorems (Emmerson, 2009), the y-axis in general is described by the height of a cone. In relation to this diagram, in terms of the y intercept, the height of the cone would be changing with respect to both the initial radius (slant of the cone) and the angle taken out of the initial circle (the angle made between the line from the eye to the x-axis changes is a function of solely the angular amount taken out of the initial circle). Further mathematical analysis of optical infinity with relation to the horizon line and geometric system is needed, but perceived difference in circumferences as an arc length will be a useful formula. Gibson says that, “only because light is structured by the substantial environment can it contain information about it” (Ecological Approach, 86) (Gibson, James J.. “Special Terms Used in the Ecological Approach to Vision.” (Appendix) Glossary 1.1 (1977): 1-4. Print.). The basic equation for an arc length as a difference in circumferences describes an even surface layout. Thus, for even surfaces, the equation that delivers that surface may be used as a linguistic device (in combination with rotation, or specifying the “adumbration” of the viewed surface) for describing the structuring of the light in the environment relevant to the perception of even surface layout. The expression for “phenomenal velocity” tells us “how” motion in general is essentially structured, thus this includes the motion of light. However, this still needs specific interpretation.