Overview

Summary

The Fibonacci Sequence manifests in nature and visual arts. Through videos, lectures and images, students will learn what the Fibonacci Sequence is, where and how it appears in nature, and its role in the visual arts in various cultures. Students will also learn about the Golden Mean/Ratio and Golden Spiral, an important concept in art history. Students will identify works of art in which the Golden Spiral or Golden Ratio appear.

Learning Objectives

Students will:

• Learn about the origin of the Fibonacci Sequence
• Learn about the Golden Mean/Ratio and Golden Triangle, important concepts in art history
• Be able to identify the Fibonacci Sequence in nature and visual arts

Teaching Approach

• Thematic
• Project-Based Learning
• Arts Integration

Teaching Methods

• Discovery Learning
• Discussion
• Experiential Learning
• Reflection

Assessment Type

Observation

Preparation

Lesson Setup

Teacher Background

Teachers should familiarize themselves with the definition and examples of the Fibonacci Sequence as it applies to art and the Golden Ratio.

Prior Student Knowledge

Students should have some background in geometry and visual arts.

Physical Space

Classroom

Staging

Since images to be shown to students are on the computer, make sure your computer is connected to an LCD projector and that there is a screen in the room.

Accessibility Notes

Students who are deaf or hard of hearing will need handouts/reading materials to explain the Fibonacci Sequence and Golden Ratio.

Instruction

Engage

1. Write the following sequence of numbers on the board and ask students if they know what it is. Ask them if they can identify the pattern and ask what the next number in the sequence would be.

0,1,1,2,3,5,8,13,21,34,55,89,144,….

2. Explain that this sequence is called the Fibonacci Sequence. The first two numbers of the sequence are always 0 and 1. Each following number is the sum of the previous two (so, the next number in the sequence after 144 would be 233 because it is the sum of 89 and 144).

You could also explain that in mathematical terms, in order to determine a given number F_n of the sequence, you could use the recurrence relation (F_n= F_n-1 + F_n-2) when F₀= 0 and F₁= 1.

3. Describe the history of the Fibonacci sequence. It was named after a Medieval Italian monk (and mathematician) named Leonardo de Pisa (known as Fibonacci --- a combination of filius Bonaccio, which means “son of Bonaccio” in Italian). The Fibonacci sequence was introduced to Western Europe in his 1202 book, Liber Abaci. Though the Fibonacci sequence is attributed to Leonardo de Pisa, it also appeared in other cultures in India and Northern Africa before the publication of his book.

4. Explain that the Fibonacci sequence appears in nature and, as a result, has highly influenced visual arts. Using your computer and an LCD projector, show students this video.

5. To reinforce the notion of the Fibonacci sequence in nature, you may want to pull some examples from this site to share with students.

• Nature by Numbers [video]


• Fibonacci Numbers and Nature

Build Knowledge

1. Explain that artists have always been influenced by what they see in nature. The Fibonacci Sequence is related to the Golden Ratio, a concept that appears in both nature and visual arts.

2. Have students calculate the Golden Ratio with calculators: 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, 55/34, 89/55 ...They should get 1, 2, 1.5, 1.6666..., 1.6, 1.625, 1.615384615..., 1.619047619..., 1.617647059..., 1.618181818...

Explain that these ratios will eventually reach the Golden Ratio, (√5 + 1)/2, a number approximately equal to 1.6180339887498948482. The Greek letter Phi is used to refer to this ratio. In the Fibonacci Sequence (0, 1, 1, 2, 3, 5, 8, 13 ...), each term is the sum of the two previous terms (for instance, 2+3=5, 3+5=8 ...). As you go farther and farther to the right in this sequence, the ratio of a term to the one before it will get closer and closer to the Golden Ratio.

3. Show students the graph showing the relationship between the Fibonacci sequence and golden ratio.

4. A similar concept is the Golden Spiral, a logarithmic spiral whose growth factor is related to the golden ratio.

5. Show students this video of how to construct a Fibonacci Spiral: http://library.thinkquest.org/27890/theSeries6a.html

6. Explain that the Golden Ratio and Golden Spiral have informed the work of artists throughout history. Talk about the ways in which the Fibonacci Sequence informed the work of M.C. Escher. M. C. Escher went to Alhambra, Spain and was inspired by the tile patterns he saw there to develop his signature style of images repeating and creating themselves (Islamic design). These tile patterns are also examples of geometric repetition and the Fibonacci Sequence in visual art. Tiles at Alhambra. These designs are based on the Golden Spiral.

For repetition in Escher’s work: http://www.mcescher.com/ (go to “picture gallery” and then “symmetry”)

Recommended Resources

• Alhambra Closeup
  
  
• M.C. Escher



• One calculator for each student

Apply

1. Ask students to think about the different representations of the Fibonacci Sequence and Golden Ratio they’ve seen. Ask them to identify pieces of art that use these themes or formulas. Assign them to find a piece online that corresponds to these ideas.

You may want to assign students specific pieces. Examples of pieces of art that use the Fibonacci Sequence and Golden Ratio can be found at:

http://www.geom.uiuc.edu/~demo5337/s97b/art.htm or http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibInArt.html#daVinci

2. Ask students to prepare a short presentation on the piece they have been asked to research. The presentation should include the history of the piece, how the Fibonacci/Golden Ratio is used, and information about the artist.

Recommended Resources

• http://www.geom.uiuc.edu/~demo5337/s97b/art.htm


• http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibInArt.html#daVinci

Reflect

1. Have students present their findings about the Fibonacci sequence in an existing work of art. They should have an image of their piece to show the class as well as the information listed above (history of the piece, relation to Fibonacci Sequence/Golden Ratio, information about the artist).

2. End the lesson with a discussion about now the Fibonacci sequence has influences art across cultures and time periods (use student projects to exemplify this).

Assessment

Assess the students' final projects. Do they reflect an understanding of the Fibonacci Sequence, Golden Mean and Golden Spiral?

Key Vocabulary

• Fibonacci Sequence
• Nature
• Golden Mean/Ratio
• Golden Spiral
• Visual Art
• Pattern

Extending the Learning

Arrange a field trip to a local art museum. Assign students to research different pieces found in the museum and identify those that use the Fibonacci Sequence or Golden Mean/Ratio. Have students serve as “guides” and talk about these pieces while you are at the museum.

Engage students in a lesson about engineering and architecture.  Identify the presence of the Fibonacci Sequence or Golden Mean/Ratio in processes related to these disciplines.

Standards

The National Standards For Arts Education:

Visual Arts

Grade 9-12 Visual Arts Standard 2 : Using knowledge of structures and functions

Grade 9-12 Visual Arts Standard 3 : Choosing and evaluating a range of subject matter, symbols, and ideas

Grade 9-12 Visual Arts Standard 4 : Understanding the visual arts in relation to history and cultures

Grade 9-12 Visual Arts Standard 5 : Reflecting upon and assessing the characteristics and merits of their work and the work of others

Grade 9-12 Visual Arts Standard 6 : Making connections between visual arts and other disciplines

Science

Science Standard 11 :
Understands the nature of scientific knowledge

Mathematics

Math Standard 5 :
Understands and applies basic and advanced properties of the concepts of geometry

Language Arts

Language Arts Standard 1 :
Uses the general skills and strategies of the writing process

Common Core/State Standards

Select state and grade(s) below, then click "Find" to display Common Core and state standards.