Rich Theory in Perspective

This work, Rich Theory, is meant to be a refutation of Transfinite Theory. The story of Transfinite Theory begins in Ancient Greece. If I were following standard form, this work would begin in Ancient Greece.

However, I wish to take a diversion.

The about me page mentions that my love for mathematics was driven by an interest in the relation between Visual Perspective and Calculus. I was intensely interested in mathematical history from the Florentine Renaissance through the end of the classical age (this include Newton, Gauss, Euler and, arguably, Einstein).

I would like to start with the mathematics that I find enthralling to show that the work is driven by more than just a contrarian attitude. On reading classical mathematics, I am quite convinced that the classical mathematician was on a better track than the modern mind.

So, before I jump into the main thesis of this work, I would like to present a brief introduction to Visual Perspective.

To make the work compelling, I would like to begin with the outlandish claim claim that we owe Western culture to the day when Filippo di Ser Brunelleschi drew a picture of his local church. There will be a more detailed treatment of visual perspective and a more balanced view of the Renaissance later in the work.

The Renaissance Rediscovery of Visual Perspective

In 14xx the Florentine Architect Philippo di ser Brunelleschi stood on the steps of the Duomo and drew a picture of the Baptistery of San Giovanni. This picture launched a cultural movement called the Renaissance.

Now, I realize that it's absurd to attribute a broad social movement to a single act of a single individual; however, I find it hard to understate the broad impact that this one discovery had for Western Culture.

No other discovery captures the unique confluence of scientific, artistic, and spiritual thought that defined the Renaissance. It is a moment when art led science.

I contend that the rediscovery of visual perspective had a broader and more direct impact on society than any other event of the day including Guttenberg's invention of movable type (circa 1439).

While the invention of the printing press increased the number of books available for scholars. Brunelleschi's rediscovery of visual perspective put a picture in every church.

One most not forget that the vast majority of people in Medieval Europe could not read, and that the paintings in churches, public buildings and salons were a major source of education and entertainment.

Prior to the rediscovery of perspective, paintings simply had icons floating about at random on the canvas. The rediscovery of perspective provided a new framework for organizing artwork and gave paintings a new and deeper dimension.

The rediscovery of perspective impacted all areas of life. For example, interest in perspective kindled a demand for new paintings and new architecture. This demand helped spur the development of modern banking.

The Unification of Pictorial Space

The rediscovery of perspective did more than bring a new media to the public. Central to the theory of perspective is an idea that is best described as "The Unification of Pictorial Space."

Perspective creates a mathematical framework that unites all of the elements on a canvas. The theory of perspective tells us why the saints in foreground are larger than the buildings in the background which are larger than the mountains in the distance.

Moving beyond the canvas, the theory of perspective reinforced the ancient idea that there are unifying forces in nature. Perspective creates a mathematical framework to help us understand how each individual has a slightly different view of one unified reality—a realty that can be described and appears to obey mathematical principles.

This idea that there is a discernible mathematical framework to nature is central to the development of the scientific method. The rediscovery inspired generations of scientists. The groundwork laid by Brunelleschi influenced the works of Copernicus and Galileo.

The Mathematics of Perspective

Rushing along, I would like to quickly introduce the mathematics of perspective.

I find the easiest way to begin a study of perspective is to ask the student to imagine that they are an ant on a highway looking down a long narrow road lined with evenly spaced telephone poles (as seen below). For simplicity, the ant is a unit away from the first pole, and the poles are separated by one unit.

In this model, I make the bold assumption that light travels in a straight line. In the model I draw a blue guideline to indicate the path of light. The apparent height of the second pole in relation to the first is determined by the point where the guideline crosses the first pole. As the first pole is equidistant from the ant and second pole, the apparent height of the second pole is one half the first. The apparent height of the third pole is 1/3. The apparent height of the fourth pole is 1/4. The apparent height of the nth pole is 1/n.

Perspective reduces by a dimension.

This really is not surprising. An artist engaged in plein air drawing is projecting a three dimensional object onto a two dimensional surface.

The function f(x) = 1/x (reduction by a dimension) is an extremely interesting equation. The function is undefined at x = 0. The function returns an extremely large value for small values of x (e.g., 1/0.00001 = 100,000). Also note that the function is extremely steep for low values of x, but quickly levels off.

If you are sitting in front of a computer or reading a book, the computer or book will completely dominate your view because it is close. Step back from the monitor, and it will take up a smaller portion of your view.

Reducing by a dimension creates interesting visual effects. For example, a small item in the foreground can completely block a large item in the background. Holding a small coin near your eye can completely block the view of the moon, despite the fact that the moon is substantially larger than a coin.

Back to perspective. In the math class I had designed, the students would spend several weeks calculating how images would appear on a piece of paper and drawing pictures. If the class had computers, it would also work on writing graphics programs.

The class would do things in nice simple steps. For example, in the next picture, we have the viewer stand up.

When the viewer stands up, we find that the lower part of the field of vision works just like the upper part of the field of vision. The relative height of the second pole is still one half the first.

In another exercise the class would pretend that a gust of wind blew the telephone poles over. In this exercise students would learn that the left and right side of our field of vision works like the upper and lower halves of our field of vision.

The samples above talk about the relative size of objects in our field of view. The class would develop a more sophisticated model where they look at the world through a frame and measure the objects in relation to the frame.

The Vanishing Point

In the previous section I mentioned that the upper and lower half of our field of vision seems is ruled by the same mathematical. So to is the left and right side of our field of vision.

Putting this information together we discover a very interesting visual effect called the vanishing point.

What happens is that the relative size of the telephone poles decreases at the rate of 1/x. The space between the poles decreases at the same rate of 1/y. Since both the x and y values are decreasing at the same rate, we see an illusion where a long line of telephone poles appears to approach a vanishing point on the horizon, we we see in figure three.

Artists have developed a variety of drawing techniques where they start a picture by selecting a vanishing point on the horizon. They then compose the picture by drawing guidelines from the vanishing point.

Artists can enhance their creation by using multiple vanishing points. Restoration experts who investigate the works of the grand masters of the Florentine Renaissance can often find indentations on the canvas and signs that the artist drew guidelines from the vanishing point.

Calculus students may notice that the vanishing point really is just a limit of an infinite series. Later in this work, I will show how one can derive the fundamental theorem of Calculus just by studying perspective.

In my opinion, the vanishing point found in visual perspective is the most elegant place to begin a discussion of the nature of infinity. When we project a three dimensional space onto a two dimensional surface, we find these strange vanishing points. The vanishing point has a defined reality on the two-dimensional picture we created; However, it does not exist in the three dimensional space.

The vanishing point that one finds in the study of visual perspective is the result of the relation between the artist and the artist's canvas. There are vanishing points in photography as well. Again the vanishing point the result of the relation of the frame of the picture and the angle of the camera.

If you were to take an empty picture frame and look at things through the frame, you would find all sorts of vanishing points. These vanishing points are the result of your relation with the frame.

A Voyage to the Source

The Renaissance Rediscovery of Visual Perspective is my favorite subject. I would love to spend more time here. My goal in the first chapter was to introduce the concept of the vanishing point.

Brunelleschi's rediscovery of perspective is like many of the great turning points in the history of mankind.

Historian disagree on the precise date that Brunelleschi drew the first visual perspective pictures. The pictures themselves are lost to time. Above all, it is likely that the Greeks had completely developed the mathematics of visual perspectives century's before Brunelleschi's day. For that matter, it is entirely possible that Brunelleschi had found an old manuscript while rummaging through the ruins of Rome that explained the basics of perspective.

We will spend more time on the Renaissance. The next chapters will bring us to discoveries of the ancients in Egypt and Greece.