The illusion of three dimensions
"After all, one can't leave his shadow lying about and not miss it sooner or later, don't you agree?" said Peter Pan.
In a memorable scene from the book, Peter Pan returned to the place where he lost his shadow and met Wendy, who sewed his shadow back on for him. The shadow was pulled off when the Darlings' dog Nana caught it and a closing window severed it. "It's quite an ordinary shadow," said Mrs. Darling when she rolled it up and put it in the drawer.
We live in a three-dimensional world. But we take for granted how our visual system actually "sees" three dimensions. The seeing is dependent on the brain evaluating information based on visual cues. Information from our two eyes (with slightly different positions and slightly different imagery) is sent to the brain as two flat images. Part of our depth perception is obtained from the differences between the two images (stereopsis). Our visual system also uses other cues such as: perspective; relative motion (near objects move relatively more than far objects when you move your head); light and shadow, including shading; texture; and occlusion (an object blocking another object). All of these visual cues let us know where something is, and what something is.
When it comes to painting, the process is even more complex. Margaret Livingstone (Vision and Art: The Biology of Seeing) said it best when she wrote, "Artists must look at a three-dimensional scene with their two-dimensional retinas and then generate a two-dimensional painting that appears three-dimensional to viewers who look at it with their two-dimensional retinas."
And no, we cannot just look at the already flat, two dimensional retinal image. We don't have a conscious ability to access the retinal image before it is processed by the brain. And artists can use only some of these cues. Painters, for example, use perspective, light and shadow, shading, texture and occlusion.

Detail of Madonna delle Grazie by Bernardo Daddi, 1347
Artists struggled with the problem of shading and modeling light and shadow when painting darker values, such as dark blue or black.The lighter robes in this painting show markedly more shading than the Madonna's robe, which appears flat. The problem was not solved until the Renaissance.
The expression of three-dimensional structure in art can be traced from the use of occlusion in Egyptian art, to the use of shading in Greek and Roman art, and to the development of mathematical perspective in the Renaissance. As each one of these visual cues was interpreted for use in painting, artists refined and expanded their techniques. The use of light and shadow has probably had the most variation in interpretation throughout art history. It is, after all, a primary component of visual processing, and it also has extended variety in its use and interpretation.
When a three-dimensional object is illuminated by a light source, different parts of the object's surface reflect different amounts of light. Perceived lightness by the human visual system is defined as luminance. Luminance is a photometric measure, but artists generally work with a more generalized assessment of light using a value scale. Shading is a variation in reflected light as a result of three-dimensionality. This is basic dimensional information, both in visual processing and in art. But in order to be able to use shading effectively, one needs to be able to see the changes as independent of the object or objects, and also as independent of color. (Hint: Try isolating information by making a circle with your hand and comparing one area to another.) Some parts of the visual system are more sensitive to color, and some to luminance, but we cannot selectively use only part of our visual system.
The good news is that the more we look, the more we see, and the more we learn to see. So we can learn to see gradations of value and we can learn, to a degree, to see how the amount of light, or lack of light, affects color. With that information, we can reasonably judge value gradations and make valiant attempts to interpret the information.

Benois Madonna by Leonardo da Vinci, c. 1478
Leonardo da Vinci was one of the first artists to use value consistently across colors. This painting exhibits mid-range colors and a consistent range of values. He wrote, "Remember, painter, to dress your figures in lighter colors, darker colors will make lesser relief and be little apparent from a distance.

Holy Family by Michelangelo, c. 1503
Several years after Leonardo achieved "tonal unity", Michelangelo used a different approach. This painting exhibits a wide range of luminances and vivid depth.
Being able to see variations in light and shadow and use shading to describe three dimensions is only part of the problem. The range of luminances in an actual scene cannot be matched by the range of values available in artists' pigments. As Livingstone pointed out, the light from a window might be hundreds of times brighter than a shadow under a desk, and the luminance in an outdoor scene can vary by a factor of a thousand. The whitest white paint only reflects about twenty times as much light as the blackest black paint. A real scene is three dimensional and light is reflected from multiple dimensional objects, but a painting is flat so variations in luminance are dependent entirely on differences in how much light is reflected from the surface.
One of the solutions to the problem of the limited range of luminance in artists' pigments takes advantage of the fact our visual systems are more sensitive to differences than to similarities. By using abrupt changes in contrast, along with subtle changes, a painter is able to create the illusion of an extended range of luminance.
In other words, we cannot paint "real". We can only interpret and possibly approximate what we are seeing. It's the intent and focus of the artist that matters the most. This is the art in art.

Portrait of a Gentleman by Giovanni Battista Moroni, c. 1550-60
While the clothing in this painting shows minimal shading, similar to the painting of the Madonna by Daddi, the figure is grounded and connected to the picture plane by the use of cast shadows and the use of both abrupt and subtle value changes.
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