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Marc Frantz holds a BFA in painting from the Herron School of Art and an MS in mathematics from Purdue University. He teaches mathematics at Indiana University, Bloomington where he is a research associate. Annalisa Crannell is professor of mathematics at Franklin & Marshall College. She is the coauthor of "Writing Projects for Mathematics Courses".

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"This practical, hands-on, and significant book makes clear the connections between mathematics and art, and demonstrates why artists need to know mathematics. Viewpoints appeals to students' visual intuition and engages their imaginations in a fresh way."--Barbara E. Reynolds, SDS, coauthor of College Geometry: Using the Geometer's Sketchpad

"This entire book is a thing of beauty: the mathematics, the visual art, the writing, the exercises, and the organization. The authors' passion and excitement for their subject matter is apparent on every page. I am in awe."--Robert Bosch, Oberlin College

"The book's emphasis on a workshop approach is good and the authors offer rich insights and teaching tips. The inclusion of work by contemporary artists--and the discussion of the mathematics related to their work--is excellent. This will be a useful addition to the sparse literature on mathematics and art that is currently available for classroom use."--Doris Schattschneider, author ofM. C. Escher: Visions of Symmetry

"Concentrating on perspective and fractal geometry's relationship to art, this well-organized book is distinct from others on the market. The mathematics is not sold to art students as an academic exercise, but as a practical solution to problems they encounter in their own artistic projects. I have no doubt there will be strong interest in this book."--Richard Taylor, University of Oregon

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VIEWPOINTS

Princeton University Press

Contents

Preface.................................................................viiAcknowledgments.........................................................ix1 Introduction to Perspective and Space Coordinates.....................1Artist Vignette: Sherry Stone...........................................92 Perspective by the Numbers............................................13Artist Vignette: Peter Galante..........................................253 Vanishing Points and Viewpoints.......................................29Artist Vignette: Jim Rose...............................................394 Rectangles in One-Point Perspective...................................43What's My Line?: A Perspective Game.....................................555 Two-Point Perspective.................................................59Artist Vignette: Robert Bosch...........................................776 Three-Point Perspective and Beyond....................................85Artist Vignette: Dick Termes............................................1137 Anamorphic Art........................................................117Viewpoints at the Movies: The Hitchcock Zoom............................135Plates follow page......................................................1388 Introduction to Fractal Geometry......................................139Artist Vignette: Teri Wagner............................................1579 Fractal Dimension.....................................................161Artist Vignette: Kerry Mitchell.........................................193Answers to Selected Exercises...........................................197Appendix: Information for Instructors...................................215Annotated References....................................................223Index...................................................................229

Chapter One

Our first perspective activity involves using masking or drafting tape to make a perspective picture of a building on a window (Figure 1.1). It's tricky! One person (the Art Director) must stand rooted to the spot, with one eye closed. Using the one open eye, the Art Director directs one or more people (the Artists), telling them where to place masking tape in order to outline architectural features as seen from the Director's unique viewpoint. In Figure 1.1, this process resulted in a simple but fairly respectable perspective drawing of the University Library at Indiana University-Purdue University Indianapolis.

If no windows with views of architecture are available, then a portable "window" made of Plexiglas will do just as well. In Figure 1.2, workshop participants at the Indianapolis Museum of Art are making masking tape pictures of interior architectural details in a hallway.

Finally, if a sheet of Plexiglas is not available, the window of a display case will also work. In this case, the Art Director directs the Artists in making a picture of the interior of the case (Figure 1.3).

If the masking tape picture from Figure 1.1 is put in digital form (either by photographing and scanning, or by photographing with a digital camera) it can be drawn on in a computer program, and some interesting patterns emerge. (Figure 1.4).

Observation 1. Lines in the real world that are parallel to each other, and also parallel to the picture plane (the window) have parallel (masking tape) images.

Observation 2. Lines in the real world that are parallel to each other, but not parallel to the picture plane, have images that converge to a common point called a vanishing point.

Two such vanishing points, V₁ and V₂, are indicated in Figure 1.4. The correct use of vanishing points and other geometric devices can greatly enhance not only one's ability to draw realistically, but also one's ability to appreciate and enjoy art. To properly understand such things, we need a geometric interpretation of our perspective experiment (Figure 1.5). As you can see from Figure 1.5, we're going to be using some mathematical objects called points, planes, and lines. To begin describing these objects, let's start with points.

It's assumed that you're familiar with the idea of locating points in a plane using the standard xy-coordinate system. To locate points in 3-dimensional space (3-space), we need to introduce a third coordinate called a z-coordinate. The standard arrangement of the xyz-coordinate axes looks like Figure 1.6; the positive x-axis points toward you.

For a point P(x, y, z) in 3-space, we can think of the x, y, and z-coordinates as "out," "over," and "up," respectively. For instance, in Figure 1.6, the point P(4, 5, 6) can be located by starting at the origin (0, 0, 0) and going out toward you 4 units along the x-axis (you'd go back if the x-coordinate were negative), then over 5 units to the right (you'd go to the left if the y-coordinate were negative), and finally 6 units up (you'd go down if the z-coordinate were negative).

We took a look at the standard xyz-system in Figure 1.6 simply because it is the standard system, and you may see it again in another course. However, it will be convenient for our purposes to use the slightly different xyz-coordinate system in Figure 1.7—it's the one we'll be using from now on. In Figure 1.7 we have included sketches of three special planes called the coordinate planes. In this case, we have to think of the x, y, and z-coordinates as "out," "up," and "over," respectively, as indicated in the figure.

A first look at how this coordinate system will be used to study perspective is presented in Figure 1.8. A light ray from a point P(x, y, z) on an object travels in a straight line to the viewer's eye located at E(0, 0, -d), piercing the picture plane z = 0 at the point P' (x', y', 0) and (in our imagination) leaves behind an appropriately colored dot. The set of all such colored dots forms the perspective image of the object and hopefully fools the eye into seeing the real thing.

In the next chapter we will see how to use this coordinate method to make pictures in perspective, much like special effects artists do in the movies. We close this chapter by taking a look at how even the most basic mathematics can help us make better drawings.

A Brief Look at Human Proportions

Most untrained artists will draw the human figure with the head too large and the hands and feet too small (Figure 1.9). To prevent these common mistakes, artists have made measurements and observations, and come up with some approximate rules, some of which may surprise you:

• The adult human body, including the head, is approximately 7 to 7&fra;12 heads tall.

• Your open hand is as big as your whole face.

• Your foot is as long as your forearm (from elbow to wrist).

That last one really is pretty surprising—we have big feet! To see that these principles result in good proportions, take a look at the two versions of the painting by Diego Velazquez in Figure 1.10. In the digitally altered version on the right, we see that the figure is about 7 heads tall, the left hand (superimposed) is as big as the face, and the man's right foot, when superimposed on his right forearm, just about covers it from elbow to wrist.

Artists who understand human proportions also know how to bend the rules to achieve the effects they want. Comic artists are a good example of this (Figure 1.11).

In their book How to Draw Comics the Marvel Way (Simon & Schuster, New York, 1978) Marvel Comics editor Stan Lee and artist John Buscema reveal that Marvel artists generally draw superheroes eight and three-quarters heads tall, for heroic proportions. Popular comic artist Alex Ross, who has drawn for both Marvel and DC Comics, uses these proportions for the DC Comics superhero Atom-Smasher in Figure 1.11, taken from Ross's book Rough Justice: The DC Comics Sketches of Alex Ross (Pantheon, New York, 2010).

Having rules like this helps comic artists to visually distinguish superheroes from ordinary characters. It also helps the artists to draw the same character again and again in a consistent way. Thus we see that although artists are not bound by any one set of mathematical rules, understanding the rules can be very helpful. That's a theme we will see repeatedly throughout this book.

Exercises for Chapter 1

1. Divide your height in inches by the height of your head in inches (you'll have to measure). According to the artists' rule, the answer should be about 7 to 7.5.

(a) What is your actual answer?

(b) For a child, should the answer be greater or smaller than 7-7.5?

2. In each of Parts (a), (b), and (c), we consider a rectangular box with its faces parallel to the coordinate planes in Figure 1.7. Some of the coordinates of the eight corners (A, B, C, D, E, F, G, H) of the box are given; your job is to fill in the rest.

(a) A = (1, 1, 1), B = (1, 1, 5), C = (4, 1, 1), D = (4, 1, 5), E = (4, 7, 1), F = (4, 7, 5), G = (1, 7, 1), H = ( , , ).

(b) A = (1, 2, 3), B = (2, 3, 4), C = (__, __, __), D = (__, __, __), E = (__, __, __), F = (__, __, __), G = (__, __, __), H = (__, __, __).

(c) A = (1, 1, 1), B = (3, 4, 5), C = (__, __, __), D = (__, __, __), E = (__, __, __), F = (__, __, __), G = (__, __, __), H = (__, __, __).

(d) Which of the boxes in Parts (a), (b), and (c) is a cube? How big is it?

3. This exercise involves drawing sequences of straight line segments without lifting your pencil.

(a) Without lifting your pencil, connect the dots in Figure 1.12 in the following order: QPRSRTU. That is, go from Q to P, from P to R, from R to S, from S back to R, etc. Notice that some vertices (dots) get visited more than once, and some edges (such as RS) get drawn more than once. What letter did you draw?

(b) Referring to Figure 1.12, write down a sequence of vertices that draws the letter H. If your straight line path takes you through a vertex, then list it. For example, don't write PT, write PRT instead.

(c) On the left of Figure 1.13 is a simple drawing of a house, and on the right are the vertices of the drawing. List the vertices in an order that duplicates the drawing. Can you do it so that only one edge is drawn twice?

(d) Refer to the box in Problem 2(a). List the vertices of the box (with occasional repetitions) in an order so that if we connect the dots in the same order, we trace every edge of the box at least once. Your path should stay on the edges and not cut diagonally from one corner to another.

Artist Vignette: Sherry Stone

SHERRY STONE is a lecturer in Foundation Studies at Herron School of Art and Design, IUPUI, with a special interest in teaching first-year art students. Degreed in printmaking, she has become a painter and printmaker who has exhibited in the Midwest and on both coasts. She writes on the topic of the education of artists—and anything else that strikes a whim—and if she hadn't decided to study art, she says she would have become either a writer or a very bad ballet dancer.

If you were to ask my freshman art students what they liked to draw when they were younger, many of them would answer Manga comics. They aren't very different from many other generations of young artists who started off by copying comics. The first comic I tried to copy was "Nancy." The drawings were simple and I was really fascinated with her hair; it looked like a helmet with spikes sticking out of it! When I was older, I liked to copy Wonder Woman, who was a much better role model—if a comic book character can be a role model—and I enjoyed her connection to mythology.

My father was a draftsman: the old-fashioned kind, one of those guys who learned to draw with rulers and mechanical instruments like compasses and protractors, not CADs and computers. My first drawing utensils were his turquoise 2H pencils. That's "h" for "hard," which means they could make the sharp, light, accurate lines that draftsmen needed for architectural drawings. I learned not to like them very well. The marks they made were too light no matter how hard you pressed and they had no erasers on the ends.

When I was growing up, he worked for a company that constructed water towers like the ones you see from the interstate that announce the presence of small towns like "Sellersburg" or "Speed" to everyone passing by. Sometimes his company built water towers shaped as unusual objects like ketchup bottles or Dixie cups. They acted as signposts for companies that were so big that they needed their own water tower. I thought those towers were very cool. That was before Claus Oldenberg began making his monumental Pop Art sculptures of everyday objects like baseball bats. Years later, after my dad left, the company built the giant baseball bat that leans against the front wall of the Louisville Slugger company. It's interesting to consider how an object is regarded as art in one context and not in another.

When I was in sixth grade, my dad started moonlighting as a draftsman for the developer who was building houses in our subdivision. That was the year I almost decided to become an architect rather than an artist. I learned linear perspective and I used it to design dozens of dream homes. My interest in being an architect eventually waned: my heart was set on being an artist, and my interests were too broad to be limited to houses. I have a long history of writing poetry and stories and making drawings and paintings. I love to read. Art is a great profession for someone who has a lot of interests. It's an area where the entire realm of your experiences can come together. That is why artists really need to be well educated. It's hard to make art when you have nothing to say.

My interest in architecture was, however, a valuable detour. I learned linear perspective at a time when many kids decide they can't draw. Upper elementary school children want their drawings to look realistic. They are embarrassed by drawings that look childish because they are growing up and they want their drawings to look as mature as they feel. Linear perspective was one tool I could use to make my drawings look like reality.

Consequently, linear perspective has never been much of a mystery to me. Today, I teach linear perspective to wary students in first-year drawing courses. Some really enjoy it and take to it very quickly, while others treat it like a bad math test. That saddens me because it is so useful in understanding the three-dimensional nature of objects you are drawing, even when you are not specifically using it. Art students are an interesting lot, though. Some are little Da Vincis, very analytical and seem more like scientists and philosophers. Many, though, are intuitive souls and are content to feel their way through problems and don't take well to the structure and rules of perspective.

I find that very puzzling. I once taught a drawing workshop for 8- to 10-year-olds in which the coordinator had written perspective into the course description. I had great reservations about it. I decided to teach it by playing a game of "Follow the Leader": they were to draw what I drew, line for line, and guess what we were drawing. They were very excited and followed me perfectly as we drew a house in two-point perspective with inclined planes, auxiliary vanishing points, and doors and windows centered on the walls. And they happily duplicated it with very little help from me! I think about that every time I am faced with an impossibly confused college art student.

My artwork now has very little to do with linear perspective, but I am always aware of it, even if I am drawing from the human form. Any form that can be simplified into a configuration of geometric shapes can be drawn in linear perspective. By considering the body as a series of boxes and cylinders situated on a plane, it is easier to draw the human form as though it is part of a space.

In my recent work, I have utilized photography and computer programs like Photoshop to do preparatory work for my paintings. I have found the distortions of planes and lines caused by viewing the subject through a lens to be very interesting and sometimes quite a departure from the invented environments one would create with linear perspective—although I have been known to purposely distort the rules of perspective for expressive reasons.

Like perspective environments, photography captures environments that appear very real, yet they both walk a line between illusion of reality and abstraction. They both are two-dimensional, striving to create an illusion of three dimensions, but if artists aren't aware of the inherent limitations of the individual processes, they can create very strange illusions. Some artists find this aspect intriguing and freely manipulate these conventions for their own purposes.

For example, the Photorealist painters of the sixties were very interested in the effect of photography on painting. Richard Estes painted many images of store windows. If you were actually standing in front of one of the stores he painted, you would be able to see the merchandise inside because of our eyes' ability to focus on various planes of space and to ignore some visual information in favor of other information. Estes, however, painted the store window as the camera saw it, with many reflections dancing across the glass and very little of the merchandise visible. Even though painters had been using photographs as resource material since the advent of photography, most artists painted from them as though they were working from life, and often would not admit they had used a photograph. Painting an image as the camera saw it—and not only admitting it but also making the work about it—was new.

(Continues...)

Excerpted from VIEWPOINTSby Marc Frantz Annalisa Crannell Copyright © 2011 by Princeton University Press. Excerpted by permission of Princeton University Press. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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