Finding Fibonacci in the Helicone – SCOPES Digital Fabrication

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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.

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.

Standards

  • (Fab-Safety.2): I can operate equipment in a Fab Lab following safety protocols.
  • (Fab-Modeling.2): I can construct compound shapes and multi-part components ready for physical production using multiple representations.
  • (Fab-Fabrication.1): I can follow instructor guided steps that link a software to a machine to produce a simple physical artifact.
  • (Fab-Design.1): I can be responsible for various activities throughout a design process within a group under instructor guidance.
  • (HSF.IF.A1): Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
  • (HSF.IF.A2): Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
  • (HSF.IF.A3): 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.
  • (HSG.CO.A5): Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

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