The Mathematical System of Perspective in Art

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1 Claire Girardi Girardi 1 Professor Yolande Petersen Math 101, MW 11:40-1:05 April The Mathematical System of Perspective in Art When we look out our window, we can see a kind of picture. Our perspective determines how this picture appears to us. We can see how objects that are closer to us appear larger, while objects further away appear smaller. Our eyes recognize the depth and dimensions of what we see. Now imagine taking that view from the window and trying to recreate it on a flat surface. How can one make a flat surface appear to be a three-dimensional world? Linear perspective is a mathematical system that allows artists to recreate the world as our eyes see it on a flat surface. We will learn about the essential elements of linear perspective as well as the application of this system through the art of Masaccio who is credited as the first painter to fully demonstrate the system of linear perspective. Linear perspective, also known as geometric, mathematical or optical perspective, is a system that recreates the world as seen by the human eye on a two-dimensional flat surface ( Linear Perspective in Painting ). By organizing space and depth from one point of view, this system replicates the optical effects of recession and dimension. Linear perspective is an essential tool for creating realistic art. This system is used by representational artists who strive to recreate the depth and solidity of the natural world. Linear perspective is composed of three essential elements: the horizon line, the vanishing point, and orthogonals. An important step in linear perspective is to set up a perspective based on how the artist or observer will view the scene. This is done by drawing the horizon line, which is a line that signifies the horizon and the eye level of the viewer. Anything placed above the horizon line will be above the viewer s eye level and anything placed below the horizon line will be below the viewer s eye level ( Linear Perspective in Painting ). Now that the eye level of the artist or viewer has been established, a vanishing point is drawn on the horizon line to establish a point where converging lines recede and meet. These converging lines, also known as orthogonals, are drawn to depict how objects recede in size the further away they are from the viewer. One-point and two-

2 Girardi 2 point perspective are the most common types of linear perspective. One-point linear perspective has only one vanishing point and two-point linear perspective has two vanishing points ( Linear Perspective in Painting ). Figure 1.1 illustrates how these elements are drawn in terms of one-point linear perspective. Figure 1.1 Linear perspective is a fairly recent innovation in artistic history. Prior to the early renaissance, artists of the Byzantine and Gothic ages focused on symbolic representation rather than accurate representations of life ( The History of Medieval Art ). Figure 1.2 For example, figure 1.2 is a 13th century painting by Paolo Veneziano before the development of linear perspective. The focus of this painting is religious symbolism rather than realistic depiction. Notice how it is apparent that this scene was painted on a flat surface as there are no dimensions or depth in this painting.

3 Girardi 3 Moving forward in art history, the early renaissance brought about an emphasis on the depiction of realism in art. Architect and painter Leon Battista initially established the basic rules and guidelines of linear perspective, yet Italian painter Tommaso Cassai Masaccio is credited as the first painter to effectively apply these rules of perspective ( Tommaso Cassai Masaccio Style and Technique ). Tommaso Cassai Masaccio is recognized for his skill in portraying three-dimensional objects, pictorial space, vanishing points and realistic depictions of figures in real time and space ( Tommaso Cassai Masaccio Style and Technique ). Tribute of Money and Holy Trinity are Masaccio s two most notable works. These paintings are great examples of the application of one-point linear perspective. Figure 1.3 Tribute of Money Tribute of Money in figure 1.3, was painted in 1425 and tells the story of a tax collector who attempts to collect money from Jesus Christ and his twelve apostles, despite that they have no money to give him ( Masaccio, The Tribute Money in the Brancacci Chapel ).

4 Girardi 4 Figure 1.4 Figure 1.4 reveals how Masaccio applied the elements of perspective in Tribute of Money. By recognizing the position of the horizon line, we can see that the viewer s eye level is focused on the situation between Christ, the apostles and the tax collector. Another important aspect of this painting is how Masaccio directed the orthogonals towards Christ s head. By doing this, he made Christ s head the vanishing point and therefore the main focus of the painting ( Masaccio, The Tribute Money in the Brancacci Chapel ). Masaccio used the rules of perceptive to guide the viewer s eyes and convey the overall religious theme of this painting.

5 Girardi 5 Figure 1.5 Holy Trinity Holy Trinity (figure 1.5) was painted by Masaccio in 1428 and is noted for the detailed use of one-point linear perspective.

6 Girardi 6 Figure 1.6 Notice how in figure 1.6 the orthogonals run along the edges of the ceiling panels within vault. This gives the illusion that the vault has depth and regression. The orthogonals then meet the vanishing point at the ledge where the two donors are kneeling. Masaccio set the vanishing point low, making the viewer s eye level at base of the cross ( Masaccio s Holy Trinity ). This gives the illusion that Christ is towing above the viewer. Another important piece to notice in figure 1.6, is the area below the vanishing point where we see a skeleton laying on a tomb. Masaccio makes the tomb appear as an external projection yet he joins it with the chapel space by placing the vanishing point in between the two spaces ( Masaccio s Holy Trinity ). Tribute of Money and Holy Trinity are prime examples of how linear perspective allows an artist to add depth and dimension to a flat surface. This system of perspective also enables the placement of figures, objects and architecture in a way that agrees with the space around them. By combining the mathematics of geometry

7 Girardi 7 and the art of illusion, Masaccio influenced how artists are able to emphasize the themes of their paintings while creating the illusion of a window into a three-dimensional world.

8 Works Cited Girardi 8 The Crucifixion. Digital image. Artsy. National Gallery of Art, Washington D.C., n.d. Web. 4 Apr Glazier, Garth. Convergence Lines. Digital image. Illustration Fundamentals: Instructor Garth Glazier. N.p., 8 June Web. 4 Apr Holy Trinity. Digital image. Khan Academy. N.p., n.d. Web. 4 Apr "Linear Perspective in Painting." Linear Perspective in Painting. N.p., n.d. Web. 04 Apr "Masaccio, The Tribute Money in the Brancacci Chapel." Khan Academy. N.p., n.d. Web. 4 Apr "Masaccio s Holy Trinity." Italian Renaissance. N.p., n.d. Web. 4 Apr Saucedo, Jarod. "The History of Medieval Art." Bright Hub Education. N.p., 26 Feb Web. 04 Apr "Tommaso Cassai Masaccio Style and Technique." Artble. N.p., n.d. Web. 04 Apr The Tribute of Money. Digital image. Annasgarbi. N.p., n.d. Web. 4 Apr The Tribute of Money. Digital image. Khan Academy. N.p., n.d. Web. 04 Apr. 2016