Note: The following essay appeared in Visual Proof: The Experience of Mathematics in Art, Hood Museum of Art Catalog, Dartmouth College, 1999. It is a discussion of the following painting,

Richard Anuszkiewicz, Lunar, 1967,

which cannot be included here due to copyright restrictions.

The Calculus of Op Art

Jody Trout, Assistant Professor of Mathematics, Dartmouth College

Richard Anuszkiewicz is one of the leading exponents of Optical Art, or Op Art, which came to prominence in the mid-1960s. Op Art (a play on the term Pop Art) usually refers to a collection of geometrical abstractions that shock and disrupt vision and create a sense of movement or vibration, changes in spatial direction, or infinite regress. Mainly intended to evoke responses of the human eye, it is less concerned with what viewers think or believe they see than it is with what the artist, employing an array of geometrical and optical devices, wants the viewers to perceive. These works are often too quickly identified with the visual illusions that are found in perceptual psychology textbooks.

The effects in Op Art depend primarily on precise geometry and often dazzling color contrasts. Progressive repetition and manipulation of geometrical patterns combined with intense color combinations provide the fuel for the optical energy that seems to emanate from their interaction, which continues to delight even after one recognizes the true nature of the illusory phenomena. Artists who were part of the Op Art movement often played with this interaction of colors to heighten their optical illusions.

This energy is evident in the piece Lunar. The geometry is at first glance simple and clear: a diamond inscribed in a square. To create this illusion of a diamond, Anuszkiewicz uses two methods; the first, and more obvious, is the optical effect of the interaction of the three colors in this work: reddish-orange, blue, and green. All of the squares in this piece are reddish-orange. Those within the diamond are bordered by green, and those outside it are bordered by blue. These borders of different colors change the viewer's perception of the reddish-orange and create the illusion that the color of the diamond is different from that of the area around it.

The second method the artist uses to create the diamond relies upon his manipulation of the grid, consisting of the reddish orange squares oriented to the horizontal and vertical. In effect, he creates a diamond without using any diagonal lines. The key to our perception of the diamond shape are the tiny squares along its perimeter.

The geometry at work in Lunar is quite subtle, and its explanation relies on the mathematics of integral calculus. The majority of the squares in Lunar all have the same dimensions. Let us investigate an inductive mathematical process that can help explain how the diagonal area of the diamond can arise from the horizontally and vertically arranged squares. Starting with such a grid of isometric (i.e., same dimension) squares, color the edges of all of the squares that lie completely inside the diamond green and all the edges of the squares that lie completely outside the diamond blue. This is the first step. Call the sum of the areas of the green-edged squares A(1). By construction, A(1) < A, where A is the exact area of the diamond. (One could also slightly change the tints of these two types of squares to be more true to the piece itself, but we will not do this for simplicity. We will also not treat the squares at the border of the piece.)

Now take all of the reddish-orange squares that remain. Each of them overlaps on the border of the diamond because the sides are not horizontal or vertical. Divide each of these squares into quarters by dividing each side in half. Color the edges of the new, smaller squares that lie completely inside the diamond green and the edges of those that lie completely outside blue. This is the second step. Call the sum of the areas of all green-edged squares we have so far A(2). Note that A(1) < A(2) < A. However, we cannot stop here because the sides of the diamond are diagonal. There are still squares that lie neither completely inside nor outside the diamond.

Continuing this process inductively, we obtain an infinite sequence of steps consisting of subdividing into fourths each of the reddish-orange squares that lie along the sides of the diamond from the previous step and coloring the edges of the ones that then lie inside green and the edges of the ones that lie outside blue. The sum of the areas of the green-edged squares collected at the nth step is denoted A(n) and is called the nth Riemann sum. It is an approximation to the area A of the diamond. Note that by construction, we have the string of inequalities

0 < A(1) < A(2) < A(3) < ... < A(n) < .... <= A.

Thus, the sequence of numbers {A(n) : n = 1, 2, 3, ... } is a bounded increasing sequence. By a theorem of calculus, the area A of the diamond is the limiting value of these Riemann sums A(n) as n tends to infinity,

lim A (n) = A as n -> infinity

Intuitively, this means that we can make the approximations to the area A as close as we want by choosing a step n large enough and using A(n). This limiting process (called integration) is the standard method in the calculus for computing areas and volumes of complicated objects (such as ellipses and doughnuts) from simpler objects with easily determined areas (such as squares or cubes). The ancient Greeks used a similar form of this process to compute the areas of circles and ellipses.

Returning to our work of art, we see that the perception of the diagonal sides of a diamond is based on the illusion created by an infinite integration process. The eye is deceived into believing that the blue- and green-edged squares get smaller ad infinitum as they approach the now obviously nonexistent boundary of the diamond. The color contrast between the blue- and green-edged squares heightens this illusion because one believes there must be an actual linear border between these two adjacent regions of the plane.

The Calculus of Op Art

Jody Trout, Assistant Professor of Mathematics, Dartmouth College

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