The Sidewalk Problem

In the Renaissance, many European artists became very interested in the problem of representing three dimensional scenes "illusionistically", as if seen through a window. A very simple element of a scene they might have wanted to depict would be a sidewalk made of square blocks, extending off from the viewer to the distance. A child might approach this problem, reasonably enough, with something like the following:
They know that the blocks are square, and squares have four right angles and four equal sides, so they draw them with those properties. Renaissance artists realized though that the image we might trace on a window of such a sidewalk does not match this picture.
  • The sides of the sidewalk don't appear parallel, but instead seem to converge to a point in the infinite distance, commonly called a vanishing point.
  • Also, the dimensions of the blocks don't stay constant, but seem to get smaller and smaller, the more distant the block we look at becomes.
An attempt to draw an image that matches these observations unsystematically might look more like this:
The development of the vanishing point was a crucial step in the right direction, but it left a lot to be understood. How exactly should the distances between the horizontal segments shrink as we look further along the sidewalk? Some artists took a rule of thumb that this diminution should be a geometric progression: each gap should be a fixed ratio (say 2/3) of the gap below it. This would look like the following:
Leon Battista Alberti developed a construction which give a precise and correct spacing. By adding an imaginary side view, he is able to make marks at the correct levels on his canvas.