Digital Images in Mathematics and Science Instruction: The Golden Rectangle

School Science and Mathematics, Nov 2003 by Garofalo, Joe, Bell, Randy

Today's educators have easy access to a wealth of digital images. They can download images from the Internet at home or at school, and they can capture their own images with a digital camera. Digital cameras, which not long were ago beyond the financial reach of many educators, are now affordable and have become the standard in many households.

Digital images can be used by teachers to enhance science and mathematics instruction in a variety of ways. For example, images can be projected to illustrate real-world instances of concepts described in class, to provide context for a problem-solving activity, or to analyze data. Digital images can also be used effectively as springboards into exploration of abstract concepts.

Look at the image in Figure 1, recorded by SSM editor, Larry Flick, while on a recent camping trip. Can you explain what is going on in this photograph? (This is a long exposure of circumpolar stars). Approximately what time do you think the image was taken? (Sometime during the night). Toward what cardinal direction was the camera pointed? (North-the star in the center is Polaris). Approximately how long was the shutter open (several minutes, as determined by the length of the arc of the star streaks). What caused the long streak that extends from the upper right corner to the lower left corner of the picture? (A satellite). To paraphrase a common saying, a picture can lead to a thousand questions!

Analyzing images can be very simple or complex. Software can facilitate the analysis, including such programs as Photoshop, the freeware program Scion Image (http://www.scioncorp.com/frames/ fr_download_now.htm), and the Geometer's Sketchpad. There is power in analyzing images qualitatively (aesthetics, color, whether it illustrates a concept, what's going on here?) and quantitatively (measuring lengths, ratios, field of view, etc). The activity we highlight in this month's column uses a combination of quantitative and qualitative analyses.

Analyzing Digital Images in Mathematics

Many mathematics teachers and textbooks describe the golden rectangle, that is, the rectangle whose length and width are in the golden proportion 1: (1 + [radical]5)/2 (or 1: [asymptotically =]1.618), as the most aesthetically pleasing of all rectangles (see Figure 2).

They point out that because the golden rectangle is aesthetically pleasing it appears in art and architecture. Others argue that the golden rectangle appears in music and nature, as well. Hence the golden rectangle is often used to show connections between mathematics and other disciplines.

The Internet, digital cameras, and the Geometer's Sketchpad make exploring instances of golden rectangles in their environment easy for students. They can analyze for themselves the kinds of examples textbook authors have typically used to illustrate the golden rectangle. And they can also look for-and capture-occurrences of golden rectangles on their own.

Two well-known works that have been used traditionally to illustrate the golden rectangle are the ancient GreekParthenon in the field of architecture and Leonardo da Vinci's Vitruvian Man in the field of science.

The Parthenon looks as though it conforms to a golden rectangle. To verify this, a student can readily import this image into the Geometer's Sketchpad and try to fit the Parthenon with a constructed dynamic golden rectangle. (Note: To view instructions for constructing a golden rectangle in the Geometer's Sketchpad see http://www.teacherlink.org/content/ math/activities/skpv4-goldenrec/home.html).

One can see in Figure 3 that a golden rectangle can be fitted to the Parthenon. In fact, golden rectangles of various sizes and orientations can be fitted to different parts of the Parthenon.

This analysis of the image raises several natural questions: Did the designers of the Parthenon deliberately use the proportions of golden rectangle in their plans for the building or did they decide that these dimensions were the most aesthetic? Are there any structural reasons why architects would design a building according to the golden rectangle? More importantly, does the Parthenon truly conform to a golden rectangle, or can we fit a golden rectangle to such a structure somewhat arbitrarily?

Analyzing Digital Images in Science

Examples of the golden rectangle abound in architecture and art, but they are just as easily found in the realm of science. Various flower parts, the ratio of body segment lengths in insects, and even the human body have all been shown to reflect the golden rectangle (Keith, Saunders, & Yanik, 2003). Leonardo da Vinci 's drawing of the Vitruvian Man is a particularly well-known example that combines both art and anatomy. just as with the Parthenon analysis, teachers can use Geometer's Sketchpad to analyze the Vitruvian Man for golden rectangles. Note in Figure 4 that two different golden rectangles were fitted to the body in the picture. Students can be challenged to use Geometer's Sketchpad to find many other examples of close fits of the golden rectangle to the human body.