Discovering Phi: The Golden Ratio
Students discover the mathematical constant phi, the golden ratio, through hands-on activities. They measure dimensions of "natural objects"—a star, a nautilus shell and human hand bones—and calculate ratios of the measured values, which are close to phi. Then students learn a basic definition of a mathematical sequence, specifically the Fibonacci sequence. By taking ratios of successive terms of the sequence, they find numbers close to phi. They solve a squares puzzle that creates an approximate Fibonacci spiral. Finally, the instructor demonstrates the rule of the Fibonacci sequence via a LEGO® MINDSTORMS® EV3 robot equipped with a pen. The robot (already created as part of the companion activity, The Fibonacci Sequence & Robots) draws a Fibonacci spiral that is similar to the nautilus shape.
The Fibonacci spiral and an unfolding fern frond.Copyright (left) 2008 User:Dicklyon, Wikipedia (PD) and (right) 2007 Karora, Wikimedia (PD) http://en.wikipedia.org/wiki/File:Fibonacci_spiral_34.svg http://commons.wikimedia.org/wiki/File:Tree_fern_frond_at_Akatarawa.jpg
Phi is arguably one of the most important mathematical constants. From the great pyramids to the Parthenon, this number appears in the shapes and scales of many engineering designs and architectural feats. In art, this constant is used to quantify aesthetic beauty, such as in da Vinci's Mona Lisa, or even the face of a beautiful person. This activity builds upon the omnipresence of this number to introduce students to discrete mathematics, and extends the idea of a mathematical sequence to basic programming using the EV3 MINDSTORMS software.
After this activity, students should be able to:
- Explain the general term of the Fibonacci sequence.
- Describe examples of phi in nature.
- Identify phi as the limit of the ratio of terms of the Fibonacci sequence.