Geometry Software, Dynamic

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Geometry Software, Dynamic

Tucked in with the business news of the day was this headline from the December 9, 1996 issue of The Wall Street Journal newspaper: "Teen Math Whizzes Go Euclid One Better." High-schoolers David Goldenheim and Daniel Litchfield had revisited a 2,000-year old challenge from the Greek mathematician Euclid and solved it in a new way. Given an arbitrary segment, the freshmen found a geometric recipe for dividing its length into any number of equal parts. The mathematics community hailed the students' work as "elegant" and "significant."

Goldenheim and Litchfield devised their segment-splitting technique through old-fashioned conjecturing and reasoning. Yet there was nothing traditional about their geometric tools of choice. The duo conducted their experiments without the aid of a compass or even a ruler. Instead, they turned to technology and a new breed of computer software programs known collectively as "dynamic geometry."

At first glance, the word "dynamic" might sound like an odd way to describe geometry. The dictionary defines "dynamic" as "characterized by vigorous activity and producing or undergoing change," but the images in geometry textbooks are immobile, forever frozen in place.

Consider a picture of a triangle. Any illustration represents a particular triangle with specific side lengths and angle measures. Triangles, however, can be small, large, narrow, or wide. No single image captures this generality.

By contrast, a triangle drawn with dynamic geometry software possesses more freedom. With a click and a drag of the mouse, you can tug a corner of the triangle and watch the object adjust before your eyes. The shape remains triangular, but its sides and angles grow and shrink in a smooth, continuous motion. The effect is similar to an animated movie, only here you are the one controlling the movement.

In science, you devise experiments and then test your theories. Mathematics classes often have fostered a more hands-off approach: textbooks state what to prove. With dynamic geometry software, mathematics regains its rightful place as a laboratory science. Any object constructed on the screen allows you to roll up your sleeves and search for patterns.

Suppose you draw an arbitrary quadrilateral, find the midpoints of its four sides, and then connect these points to their adjacent partners (see the leftmost picture below). By doing so, you form a new quadrilateral (represented by dashed segments) nested inside the original. Measuring its sides and angles with the software reveals a surprise. Opposite sides are equal in length and parallel—the very qualities of a parallelogram. A coincidence?

A mouse tug to any corner of the outer quadrilateral reconfigures the construction into new positions. Within seconds, you can view hundreds of quadrilaterals, each with their midpoints connected. In every one, the dashed quadrilateral remains a parallelogram. Even a twisted pretzel shape cannot disturb the parallel sides (see the rightmost picture). Such visual evidence is quite compelling; it is not a watertight proof, and it does not explain why we should expect to see a parallelogram, but it is a useful start.

A Brief History

As director of Swarthmore College's Visual Geometry Project, Eugene Klotz was among the founders of the dynamic geometry movement. His original goals for geometry software were relatively modest. Bothered by the cumbersome nature of ruler-and-compass constructions, Klotz imagined a software package that would make it easier to draw shapes like lines and circles. He comments:

Basic motor skills were keeping students from being able to draw. I thought we needed to have something that allowed people to make the basic constructions. So to me, our software was a drawing tool. You'd make a geometric drawing that was precise and accurate, and scroll over the page to see what was going on.

This vision of geometry software was a non-interactive one: once drawn, objects on the screen could not be reshaped via mouse dragging. The missing "dynamic" element was to come from a student, Nicholas Jackiw, whom Klotz advised during Jackiw's freshman year at Swarthmore.

Jackiw was perhaps an unlikely choice for a mathematics project. He had steered away from mathematics in high school and college, focusing instead on English and computer science. Still, when Jackiw viewed Klotz's geometry proposal, he sensed something was missing. Jackiw's interest in programming computer games provided an unexpected source of inspiration. If a computer game could immerse players in an interactive world, then why not geometry? Jackiw says:

It's the video game aspect that gives me my sense of interactivity when dealing with geometry … Looking at the input devices of video games is a tremendously educational experience. In the old days, you had games with very interesting controls that were highly specific…The video game Tempest had a marvelous inputdevice…The types of games they would write to suit this bizarre and unique device were always interesting experiments in what does this hand motion transport you to in your imagination. I wanted to have a good feel in all of my games.

The mouse of a Macintosh computer was not an ideal input device for games, but it was suitable for virtual environments where objects could be dragged. The illustration program MacDraw, in particular, contained the rudimentary features of dynamic geometry, as one could draw and move a segment with the mouse and change its length. When Jackiw took the basic premise of MacDraw and applied it (with considerable reworking) to geometry, Klotz found the results striking:

I remember how shocked I was when I first saw it. Jackiw had played with a Macintosh long enough to know that you should be able to drag the vertex or a side of a triangle and protrude the figure. I was flabbergasted. I mean, he made the connection, and I didn't.

One of the earliest programs to showcase the graphical capabilities of the computer was Ivan Sutherland's "Sketchpad." A hand-held light pen allowed the user to draw and manipulate points, line segments, and arcs on a cathode ray tube monitor. In honor of Sutherland's work, Klotz named their new program, "The Geometer's Sketchpad." The product received its commercial release in the spring of 1991.

Interestingly, the Swarthmore group was not alone in its thinking. Working in France, Jean-Marie Laborde and his programming team simultaneously developed the software package Cabri Geometry, which also featured dynamic movement. Initially, neither the Sketchpad nor Cabri people knew of the other's existence. When Laborde and Klotz finally met, they marveled at the similarities in their software. Klotz says:

We had just that Fall got into our dragging bit, and were very proud of what we had. We thought, God, people are going to really love this. But Cabri had scooped us, and we had scooped them. It was one of these, you know, just amazing things where … maybe you can sort out the exact moment, maybe there was a passing meteor, or something.

Student Exploration

Dynamic geometry software programs are great for learning geometry, and they can also be fun. The top picture below shows two circles and a segment AB connecting them. As points A and B spin around their respective circles, what path does point M, the midpoint of segment AB, trace? Dynamic geometry makes this investigation simple to perform. The result, shown in the lower picture below, is an attractive spiral.

Other applications of the software display its versatility. Students can build a working clock, model planetary motion, create a spinning Ferris Wheel, and investigate algebra in a geometric way.

Since words alone cannot convey the experience of using dynamic geometry software, students can try programs themselves. Free demonstration copies of various software programs are available on the Internet.

see also Computer-Aided Design; Computer Simulations.

Daniel Scher