Finding Fibonacci in the Helicone - SCOPES Digital Fabrication

Lesson Details

Subjects
Age Ranges
Standards
Fab-Safety.2, Fab-Modeling.2, Fab-Fabrication.1, Fab-Design.1, HSF.IF.A1, HSF.IF.A2, HSF.IF.A3, HSG.CO.A5
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Author

Daniel Smithwick
Daniel Smithwick
Maker
I am the Product Manager (Master Fabricator status) for the SCOPES-DF project. I collaborate to build foundational and scalable digital fabrication knowledge for the next generation. I received a PhD from MIT for my dissertation on Physical Design Cognition and previously… Read More

Summary

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.

 

 

What You'll Need

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)

 

The Instructions

Understandings

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).

 

The Fibonacci Sequence & the Golden Ratio

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)

 

 

Show example of how the new Google logo was designed using

Golden Ratio, https://www.goldennumber.net/google-logo-design-golden-ratio/

 

Closure: Recap concepts of the Fibonacci sequence and Golden ratio. Assess student basic understanding using KWL Chart.

 

K: What do I know about Fibonacci Sequence and Golden Ratio?

W: What do I want to find out?

L:  What have I learned?

Spiral & the Golden Ratio

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,

 

https://www.youtube.com/watch?v=nt2OlMAJj6o

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

 

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

Understanding and Fabricating the Helicone

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.

Warm Up: Play with and disassemble an existing Helicone to uncover its mechanical operations. Discuss why nature produces spirals to solve spatial distribution problems.