Leonardo of Pisa (c.1170–c.1250), also known as Fibonacci, is often considered the most talented mathematician of the Middle Ages. In this lesson students discover the mystical nature of the famous Fibonacci sequence. Students also explore the mathematical constant phi, the golden ratio, known as “a precious jewel” of geometry. Students gain a basic understanding of these two important concepts through the digital fabrication of John Edmark’s mathematically inspired kinetic sculptures – the Helicone. By twisting the Helicone back and forth you can toggle between two states: one that reveals a double-helix spiral; and the other state that reveals a pinecone form. Through iteration and customization students can learn to apply mathematical concepts in vector design programs creating interlocking Helicone pieces ready for laser cutting.
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TEACHER NOTES: In his book, Liber Abaci, Fibonacci posed and solved a problem involving the growth of an idealized (biologically unrealistic) rabbit population, assuming that: a newly born pair of rabbits, one male, one female, are put in a field; rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. Although most students are familiar with the concept of pi, the Golden ratio is not widely known. The Golden ratio is represented by a Greek letter and symbol, just like pi. The symbol is the Greek letter, and is pronounced Phi, named after the Greek sculpture known as Phidias. Numerically, the Golden ratio is irrational and is noted by the ratio of (1 + √5)/ 2.
TEACHER NOTE: Before class review Gary Meisner, Spiral and the Golden Ratio, https://www.goldennumber.net/spirals; provide essential questions.
Warm-Up: In whole class, explain to students that if you sum the squares of any series of Fibonacci numbers, they will equal the last Fibonacci number used in the series times the next Fibonacci number. This property results in the Fibonacci spiral, based on the following progression and properties of the Fibonacci series:
12 + 12 + 22 + 32 + 52 = 5 x 8
12 + 12 + . . . + F(n)2 = F(n) x F(n+1)
In whole class: provide an overview of spirals and Fibonacci numbers, using part 1of Vi Hart’s video on Khan Academy, Doodling in math: spiral, Fibonacci, and being a plant, https://www.khanacademy.org/math/math-for-fun-and-glory/vi-hart/spirals-fibonacci/v/doodling-in-math-spirals-fibonacci-and-being-a-plant-1-of-3
Extension: As an extension, the teacher can show the students some basic examples of Fibonacci sequence where it is found in nature,
As another extension, show Part 2 of Vi Hart’s video on Khan Academy, Doodling in math: spiral, Fibonacci, and being a plant, https://www.khanacademy.org/math/math-for-fun-and-glory/vi-hart/spirals-fibonacci/v/doodling-in-math-class-spirals-fibonacci-and-being-a-plant-2-of-3
Enrichment Activity: In self-directed learning or small group personalized learning have students review and discuss Khan Academy, Golden Ratio, https://www.khanacademy.org/math/geometry-home/geometry-lines/the-golden-ratio/v/the-golden-ratio
TEACHER NOTES: The Helicone can be understood along two dimensions: how it works and why it works. How it works is best explored by handling it – spinning it back and forth – and also by taking it apart to see the underlying mechanics. Notice that each layer of the Helicone has mechanical stops glued on its top and its bottom. Together, these stops control the range of motion possible when spinning it back and forth. Explain to students that the magic of the Helicone is largely determined by the angle through which the mechanical stops allow for motion between the two states. In one state – the double-helix state – the angle between each layer is 4 degrees. In the other state – the pinecone state – the angle is 137.5 degrees. This is the Golden Angle, derived from the Golden Ratio, and it begins to answer why Helicones work. Pinecones use the Golden Angle to evenly distribute its seed bracts for maximum light exposure. More advanced students may be encouraged to explore further manipulations of the original Helicone part from the SVG file. Some students may also notice that the bottom and top layers of the Helicone are slightly different than the middle layers.