Wednesday, March 01, 2006

Verbogeometry -- installment I --- Cartesian Coordinate System

I would like to step through my paper on verbogeometry in small installments if possible. The paper is laid out so that I discuss a mathematical idea and then show how it relates to verbogeometry. The first installment is an overview on how the Cartesian coordinate system works and how it is applied to a remedial physics problem. Next installment will be about the mechanics of verbogeometry.

VerbogeometryThe confluence of words and analytic geometry

1. Cartesian Coordinate System:
Before we can talk about verbogeometry, we must have a look of some pertinent elements in analytic geometry. Concerning this paper on verbogeometry, you should know enough analytic geometry to plot points and a few basic equations on a Cartesian coordinate system. However, I would like to believe that anyone should be able to enjoy some understanding of verbogeometry. If you need a refresher on the Cartesian coordinate system, you may want to look online. Wikipedia has a nice overview of the System.When we look at the two-dimensional axis of a Cartesian coordinate system, we can see that by picking a point somewhere on the plane defined by this axis that there is a relationship of this point back to the origin. This relationship is understood by the nomenclature of the coordinate pair (x,y) where x and y are distances along each axis from the origin. Furthermore, if we draw lines from a newly created point, perpendicular to both the x-axis and the y-axis and taking into consideration the axis system in the background then we will make a rectangle. (See figure. 1)


The area of a rectangle is product of the lengths of its sides and in the upcoming example the product of the values for the x and y coordinates of this point. Of course this one is a special case because we used the axis origin as our starting point. Example: Let us arbitrarily pick a point defined by the x-y coordinates of (11,13) and draw lines perpendicular to the axes to illuminate what I just said. The area of any rectangle is equal to its height multiplied by its base (The product of the lengths of its sides). We have a green rectangle delineated on our axis system. (See figure. 1) The height of our rectangle is 13 units and it base it 11 units. The area of our rectangle is 143 square units … or 13 X 11 = 143 This later example is one of pure mathematics. However if we want to use math as a language then we will have to apply concepts or words to our axis system. Let us look at a typical physics problem of distance, velocity and time displayed on a two dimensional axis system. Let us assign the y-axis to be levels of velocity in units of miles per hour and the x-axis to be amounts of time in units of hours. (See figure. 2)

Figure 2.

The distance of an object traveling in space is equal to the velocity of the object multiplied by the average time it is traveling in space or d=vt. Let us use the same pure mathematical example as before but by our contextualizing the axis and assigning the y-axis to represent velocity and the x-axis to represent time, our original point from the last example (11,13) has a new meaning. To reiterate … the point before was in the realm of pure mathematics but now the point represents a moment in time of a speeding object. The object is traveling 13 miles per hour and has been traveling for 11 hours. So to calculate, (d = vt), the distance the object has traveled we must multiply the velocity by the time or 13 miles per hour times 11 hours which equals 143 miles.In essence, what we have done has been to assign a concept by using words (velocity and time) to our axis system. Moving our attention up or down on the y-axis displays different values of velocity. Moving left and right on, the x-axis displays different amounts of duration or time. Physical experimentation can easily verify the veracity of this equation. In addition, the same experimentation verifies the verbal concepts and their relationship to each other that we have assigned to our axis system. We can see the relationship between the concepts of distance, velocity and time spread out on a two-dimensional plane via our axis system. It is important to note that these concepts occupy orthogonal spaces as well as all equations in the form of A = BC. Furthermore, the equation makes sense again because it matches our perceptions of the event and the axis system is a model of our experience. Although the use of the latter equations was for the purpose of denotation, Verbogeometry is more interested in the use of equations for connotation and witnessing dissimilar concepts forced into orthogonal spaces to create metaphor.

No comments:

Visit the National Gallery of Writing