Let's Spend A Few Moments Studying Perspective

Perspective brings the parts of bodies closer together, or makes them move apart by projecting their parts seen through a plane interposed between the eye and the object.

- Durer also demonstrated his love or perhaps cult for geometry through his brass engraving entitled “Melancholia”, which he executed in 1514 after returning from his second trip to Italy. It seems that he made this work under the impression of the pain that his mother’s death caused him. This engraving, which has been commented on many times and in various ways, Melancholy took the form of Geometry, is winged and holds a compass. She is worried. What is he thinking about? Could he not establish the law that links beauty to mathematics?

At her feet is a sphere, to her right a huge dodecahedron and above her head, on a wall, a magic square of the fourth order can be seen, which shows that in addition to problems of pure geometry, Durer was also more interested in this not an easy problem in number theory. While playing with this square, whose sum of digits is 34, I wondered if Durer did not choose this number as the constant of his square, because that year he was just 43 years old?

- That may be a mere coincidence, but it is not a mere coincidence that you noticed this magic square. I think it can be a very attractive topic for our next meeting. Don’t you agree?

- Okay, but until then, let’s go back to our perspective, in connection with which I’m going to read you a few lines written by Diderot in the Essays on Painting: “The first step towards understanding chiaroscuro is the study of the rules of perspective. Perspective brings the parts of bodies closer, or makes them separate, only by the mere gradual diminution of size, only by the projection of their parts, seen through a plane interposed between the eye and the object, and connected, either on this very plane, or on some other plane imagined, beyond the object. Painters, devote a few moments to the study of perspective: you will be richly rewarded by the ease and confidence with which you will find yourself in the practice of your art !

Can you cultivate an objective perspective about what you want to demonstrate, related to a reality of the same kind as that of material points and the distances between them?

To understand the universe, or how things are in this world, you have to demonstrate that there is a reality independent of the constructions of the theoretical physicist, or independent of the analytical thinking of the authentic mathematician. It is about a reality of the consciousness or the soul in which the dream is nothing more than an event that was or will be manifested. But until then, let’s look at the analytical thinking of a mathematician.

Think for a moment, and you will recognize that the body of a prophet wrapped in all his huge cloak, and his bushy beard, and the hair that falls on his forehead, and the picturesque veil that gives his head a divine character, are in all respects subject to the same principles as and the polyhedron. In the end, one will embarrass you just as little as the other. The more you multiply your ideal number of plans, the more correct and true you will be, and do not fear that you will be cold because of a condition more or less added to your technique.”

- Diderot wrote these Essays after 1765 and, as I see from the accompanying note, they appeared “eleven years after the philosopher’s death (1795)”, so at the end of the 18th century. I deduce from them that then, the enthusiasm and zeal for the study and especially the application of perspective in painting had significantly decreased. So, before we touch on this issue, I think we should heed Diderot’s advice and “spend a few moments studying perspective”, what do you say?

Is the worldview you project based on the assumption that things change based on what happens to you when your objective perspective is constantly shattered by experiences that don’t align with it?

A mathematician divides his mind into three parts: one analytical, one logical and one intuitive. And then something strange happens to him, because he starts to think everything in proportions and concentrations. His life itself becomes a concentration of meanings attributed to the surrounding events, this representing a type of problematization, which produces a more complex intellectual conflict and has certain difficulties to solve, because everything always turns out “wrong”.

- I think it’s good, let’s begin ! From a geometric point of view, two modes of perspective representation are possible: parallel perspective and central perspective. In parallel perspective, bodies in space are projected onto the drawing plane by a beam of rays parallel to a given direction. It is a conventional method, based on the assumption that the observer’s eye is at infinity. The parallel perspective is used to depict bodies in space, in drawings from school textbooks, but also by painters from East Asia: Japan, China, India. in this case, both vertical and parallel lines in space, remaining parallel in the drawing as well (fig. 33).

In the central perspective, things change. In this case, the projection of the object from space is done by a beam of competing rays, the meeting point O, also called the point of view, is assumed to be the eye of the observer. At the same time, it is assumed that the plane we are drawing, P, is transparent, so that the visual rays taken from O to the object pass through it. But it is better to consider an example. Let P be the plane of the painting and O the point of view, and the object to be projected a square ABCD, its plane being inclined to the plane P, and the lines AD and BC parallel to the horizontal plane (fig. 34), namely AD is above the horizon, and ВС below the horizon, the points A and В being nearer to the plane P than C and D. The visual rays OA, OB, ОС, and OD pierce the plane P respectively in a, b, c, d, so that the trapezoid ABCD represents the perspective of the square ABCD in space, in the P plane.

Vision is the distance you travel between two points: a point where the time perspective for evaluating the evolution of the world is discontinuous, and a point where the perspective on reality depends on how you apply mathematical concepts to explore new models of materializing ideas and innovative concepts.

Let's Spend A Few Moments Studying Perspective so that we can more easily understand the analytical thinking of mathematicians who, playing with number theories, discovered new ways to imagine a new reality in a world that is not always recognized. Or, as someone well said: “When you look, look without interpreting. Thus, you will connect directly to concrete reality, not to an abstract reality, formed by your opinions on reality. And this will help you to transcend any illusion.”

* Note: Dumas, Alexandre - The Hand of Providence, Editura Vestala, 1993, București.