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Art: packing three dimensions into two

1. problem
We live in a three-dimensional spatial world.¹ Three dimensions because you can move left-right, up-down, and forward-backward. An artist's canvas is two-dimensional: the paint brush cannot move in the forward-backward direction when confined to the canvas.
How then does an artist create a sense of three-dimensional space on a two-dimensional canvas?

2. intuition
We perceive the buildings and road in the painting shown below as receding away from us, even though they are all simply flat objects on our computer screen. Our intuition probably tells us that the reason for this three-dimensional perception is that the buildings on the right (further from us, in depth) are painted smaller than those on the left (closer to us, in depth). Similarly, the road narrows as it recedes away from us.
In as early as the 1400s, artists were conveying a sense of three-dimensional structure on a flat two-dimensional canvas. As we will see below, the key insight that led to this was the understanding of some simple geometry.

Looking East, by Riva degli Schiavoni, 1730

3. math
Shown below is a diagram of how an artist might map a three-dimensional shape onto a two-dimensional canvas. In the lower part of the diagram, the object (red rectangle) is closer to the artist. In this case, the "image" of the object on the canvas is larger than when the object is further. That is, in order to convey a sense of three-dimensional structure, nearby objects should be drawn larger than distant objects. The precise difference in size is determined by geometry.

Notice that in the above diagram we have a pair of similar triangles (gray). The ratio of the sides of similar triangles are equal, meaning that u/f = X/Z or u = fX/Z . That is, the size of the object on the canvas, u, is inversely proportional to its distance Z. This equation, known as perspective projection, specifies how large an object, that is a given distance from you, should be drawn.
Shown below is an example of how to sketch a box under perspective projection to give the correct sense of three-dimensional structure.

Draw a horizontal line (red). This is the horizon, and the ends of the line are the vanishing points. Draw a single vertical line (black). This is the front edge of the box. Connect the ends of this line to the vanishing points.
Draw a vertical line (black) inside of the triangle formed by the lines extending from the left-most vanishing point. This is another edge of the box. Connect the ends of this vertical line to the right-most vanishing point (gray).
Repeat the above step for the other side of the box.
Erase the horizon and guidelines, leaving behind a sketch of a three-dimensional box.

4. summary
An artist's canvas is two-dimensional. And yet, artists are capable of giving us a sense of the three-dimensional space of the scene being painted. This is accomplished by painting nearby objects larger than more distant objects. The rules of geometry guide an artist in achieving the correct proportions in their paintings.

¹ According to some theoretical physicists, we may actually be living in a nine- or ten-dimensional spatial world, but only able to perceive three dimensions.

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Art: packing three dimensions into two
1. problem	We live in a three-dimensional spatial world.¹ Three dimensions because you can move left-right, up-down, and forward-backward. An artist's canvas is two-dimensional: the paint brush cannot move in the forward-backward direction when confined to the canvas. How then does an artist create a sense of three-dimensional space on a two-dimensional canvas?
2. intuition	We perceive the buildings and road in the painting shown below as receding away from us, even though they are all simply flat objects on our computer screen. Our intuition probably tells us that the reason for this three-dimensional perception is that the buildings on the right (further from us, in depth) are painted smaller than those on the left (closer to us, in depth). Similarly, the road narrows as it recedes away from us. In as early as the 1400s, artists were conveying a sense of three-dimensional structure on a flat two-dimensional canvas. As we will see below, the key insight that led to this was the understanding of some simple geometry. Looking East, by Riva degli Schiavoni, 1730
3. math	Shown below is a diagram of how an artist might map a three-dimensional shape onto a two-dimensional canvas. In the lower part of the diagram, the object (red rectangle) is closer to the artist. In this case, the "image" of the object on the canvas is larger than when the object is further. That is, in order to convey a sense of three-dimensional structure, nearby objects should be drawn larger than distant objects. The precise difference in size is determined by geometry. Notice that in the above diagram we have a pair of similar triangles (gray). The ratio of the sides of similar triangles are equal, meaning that u/f = X/Z or u = fX/Z . That is, the size of the object on the canvas, u, is inversely proportional to its distance Z. This equation, known as perspective projection, specifies how large an object, that is a given distance from you, should be drawn. Shown below is an example of how to sketch a box under perspective projection to give the correct sense of three-dimensional structure. Draw a horizontal line (red). This is the horizon, and the ends of the line are the vanishing points. Draw a single vertical line (black). This is the front edge of the box. Connect the ends of this line to the vanishing points. Draw a vertical line (black) inside of the triangle formed by the lines extending from the left-most vanishing point. This is another edge of the box. Connect the ends of this vertical line to the right-most vanishing point (gray). Repeat the above step for the other side of the box. Erase the horizon and guidelines, leaving behind a sketch of a three-dimensional box.
4. summary	An artist's canvas is two-dimensional. And yet, artists are capable of giving us a sense of the three-dimensional space of the scene being painted. This is accomplished by painting nearby objects larger than more distant objects. The rules of geometry guide an artist in achieving the correct proportions in their paintings.