Viewpoints: Mathematical Perspective and Fractal Geometry in Art - Softcover

Frantz, Marc; Crannell, Annalisa

Softcover

ISBN 10: 0691125929 ISBN 13: 9780691125923

Verlag: Princeton University Press, 2011

Inhaltsangabe

An undergraduate textbook devoted exclusively to relationships between mathematics and art, Viewpoints is ideally suited for math-for-liberal-arts courses and mathematics courses for fine arts majors. The textbook contains a wide variety of classroom-tested activities and problems, a series of essays by contemporary artists written especially for the book, and a plethora of pedagogical and learning opportunities for instructors and students. Viewpoints focuses on two mathematical areas: perspective related to drawing man-made forms and fractal geometry related to drawing natural forms. Investigating facets of the three-dimensional world in order to understand mathematical concepts behind the art, the textbook explores art topics including comic, anamorphic, and classical art, as well as photography, while presenting such mathematical ideas as proportion, ratio, self-similarity, exponents, and logarithms. Straightforward problems and rewarding solutions empower students to make accurate, sophisticated drawings. Personal essays and short biographies by contemporary artists are interspersed between chapters and are accompanied by images of their work. These fine artists--who include mathematicians and scientists--examine how mathematics influences their art. Accessible to students of all levels, Viewpoints encourages experimentation and collaboration, and captures the essence of artistic and mathematical creation and discovery. Classroom-tested activities and problem solving Accessible problems that move beyond regular art school curriculum Multiple solutions of varying difficulty and applicability Appropriate for students of all mathematics and art levels Original and exclusive essays by contemporary artists Forthcoming: Instructor's manual (available only to teachers)

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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

Mathematical Perspective and Fractal Geometry in ArtBy Marc Frantz Annalisa Crannell

Princeton University Press

Copyright © 2011 Princeton University Press
All right reserved.
ISBN: 978-0-691-12592-3

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

Introduction to Perspective and Space Coordinates

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...

Details zur Ausgabe

• Verlag: Princeton University Press
• Erscheinungsdatum: 2011
• Sprache: Englisch
• ISBN 10: 0691125929
• ISBN 13: 9780691125923
• Einband: Taschenbuch
• Anzahl der Seiten: 260
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: Nicht verfügbar