Kid's Activity: Golden Rectangles

by Melissa Jaramillo and Julie Snyder

Do you know what a rectangle is? Did you say a shape with four sides? Good for you! A rectangle is a four-sided shape and it has four right angles. If the sides are all the same length, then the shape is a special type of rectangle called a square. Look around the room. Can you find four things shaped like rectangles. In my office, the computer tower, my notebook, the bookcase and the doorway are all rectangular.

In our activity we're going to construct a rectangle with unique dimensions -- it's ratio of width to height (the width divided by the height) is approximately 1.618 (the number has a lot more decimal digits). Mom, dad or an older sibling can help out!

What you'll need:

• Paper
• Ruler
• Protractor
• Pencil
• Drawing compass

How to construct:

Begin with a square. Find the middle of a side, either bisect with your compass and mark or fold in half and make a small crease to mark. Divide your square into 2 equal rectangles.
Extend the square's base to the right, almost to the edge of your paper. Then use a drawing compass to draw the arc of a circle with a radius equal to the diagonal of right rectangle (shown to the right). The arc should meet the extended base of the original square you drew.
The original height of your square will be the height of your golden rectangle. The spot where the arc hit the extended base is the width. Add the rest of the lines. You have successfully constructed a golden rectangle!

People have been interested in this special type of rectangle for thousands of years. One reason is the ratio of length to height, called the Golden Ratio and named phi (φ), turns up in some very unexpected places:

• The ratio of the length of the second joint of your finger to the first joint to the length of the first joint to fingertip is phi (variation in growth cause us to say human measurements approximate the Golden Ratio)
• The ratio of the number of clockwise spirals on a pinecone to counterclockwise spirals is phi
• The ratio of length of elbow to wrist to the length of wrist to fingertips is very close to phi
• One full turn in a DNA molecule spiral is 34 angstroms and its width 21 angstroms in width; the ratio is phi
• Some 12th Century Sanskrit poems have a meter based on the Fibonacci sequence (and hence related to the Golden Ratio)
• The frequency of harmonic notes is based on the Fibonacci sequence

Another reason is that you can remove the square from any golden rectangle. That leaves a rectangle. Do you suppose there's something special about this smaller rectangle? Yes, there is! It is also a golden rectangle. If you remove the square from it, the rectangle remaining is a golden rectangle. You could just keep making smaller and smaller golden rectangles until they were so tiny you couldn't draw them!

Look what happens when you draw a shape comprised of arcs across the squares. Do you know what this shape is called? A spiral!

It is a shape with which we are all familiar. This spiral is commonly seen in seashells and many other plants and animals. The chambered nautilus is probably the clearest example. As it gets larger, it retains its identical form, sealing off the old chamber for buoyancy and living in the new and larger chamber. Since the body of the organism grows in the path of a spiral; its form never changes. This form is commonly called the "Golden Spiral."

This spiral is visible in things as diverse as: hurricanes, spiral seeds, a ram's horn, a pine cone, sea-horse tail, growing fern leaves, DNA molecule, waves breaking on the beach, tornados, galaxies, the tail of a comet as it winds around the sun, whirlpools, seed patterns of sunflowers, daisies, dandelions, and the ears of most mammals.

Phi appears pervasively throughout nature. Can you find more examples?