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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.

**Materials**

- cardboard sheets
- 1/8″ wood sheets (optional)
- glue (white or wood)
- 12″ wood dowel, 1/4″ diameter

**Fab Machines**

- laser cutter

**Software**

- 2D vector design (e.g. Inkscape, Illustrator, CorelDRAW)

Core Objectives

*Students will understand that…*

- Many things in the natural, physical and virtual worlds draw on the Fibonacci sequence
- The Fibonacci sequence is part of everyday life
- Digital fabrication is a engaging way to explore and simulate math concepts such as the Fibonacci sequence
- Examples in contemporary art and culture are relevant to mathematics

*Students will know how to…*

- Create the Fibonacci sequence.
- Define the golden ratio as phi, and derive it from Fibonacci pairs.
- Draw the golden spiral.
- Recognize examples of the Fibonacci sequence in nature.
- Follow steps to digital fabricated a Fibonacci and/or Golden Ratio inspired prototype of a John Edmark Helicone.
- Construct a Helicone (kinetic sculpture) using the Fibonacci sequence

*Students will be able to…*

- Explain the general term of a mathematical sequence. Specifically, the famous Fibonacci sequence which is a numerical pattern in which each term is found by adding the previous two terms.
- Describe examples of the Golden ratio, as well as its symbol, other names, and its irrational numerical representation.
- Explore the relationship between Fibonacci numbers and the Golden Ratio
- Gain understanding of what makes a rectangle a Golden rectangle
- Discover architectural structures that have the Golden Ratio in them. (Examples include Greek statues, the Parthenon, the Egyptian Pyramids, and the Mona Lisa).

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.

**Warm-Up:** Explain Fibonacci sequence and how it relates to the puzzle that Fibonacci posed: how many pairs of rabbits will there be in one year?

Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.

**Explain the solution as follows:** At the end of the first month, they mate, but there is still only 1 pair. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs. At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs (which is the number of pairs in month n − 2) plus the number of pairs alive last month (n − 1). This is the nth Fibonacci number.

**In whole class: **Review Arthur Benjamin’s the magic of Fibonacci number,

https://www.youtube.com/watch?v=SjSHVDfXHQ4

Tell students that the Golden Ratio, or Phi, is 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 and architectural designs. In art, this constant is used to quantify aesthetic beauty, such as in da Vinci’s Mona Lisa. Show examples using the following link (www.geom.uiuc.edu/~demo5337/s97b/art.htm)