The golden rectangle is a geometric concept found in many aspects of the natural world as well as in architecture, art, and popular culture. This lesson is designed to be part of a geometry curriculum discussing the mathematical and aesthetic qualities of basic shapes, and of their use in society and nature.

PA Academic Standards:

Mathematics: 2.9.8.J – Analyze geometric patterns and develop descriptions of the patterns
Arts: 9.4.B – Aesthetic Interpretation

NJ Academic Standards:

Mathematics: G.MG.1 – Apply geometric concepts in modeling situation
Visual Arts: 1.4.B – Aesthetic Responses & Critique Methodologies

Grade Level:

Approximately grade 8, depending on curricular structure

Art Images Required:

Most images listed can be found by searching the collection at the Museum's website. Images that are available from ARTstor are also indicated; typing this exact search phrase will direct you to the specific image from the ARTstor database.

• Choir Screen from the Chapel of the Château of Pagny, France, ARTstor search: "PMA_.1930-1-84a–d
• Sugar Cane by Diego Rivera, ARTstor search: "1943-46-2"
• At the Moulin Rouge: The Dance by Henri de Toulouse-Lautrec, ARTstor search: "PMA_.1986-26-32"
• Breaking Home Ties by Thomas Hovenden, ARTstor search: "PMA_.1942-60-1"

Background:

This lesson plan is designed to be part of a geometry curriculum discussing the mathematic and aesthetic qualities of basic shapes. The geometric concept of the golden rectangle can be found in many aspects of the natural world as well as in architecture, art, and popular culture. A golden rectangle consists of proportions that have been long considered visually and psychologically pleasing.

• Review the idea of the golden mean with the class. (NOTE: You can find extensive information on the ratio provided by Middlebury Community College in Vermont at http://community.middlebury.edu/~harris/Humanities/TheGoldenMean.html.)

Lesson Process:

• Review the terms mentioned above. Before examining ways in which the golden rectangle is used in art it is necessary to know how to create one. Instruct students on how to draw a golden rectangle using a piece of paper, a pencil, a ruler, and a compass with the four-step process described below.
  1. Begin by drawing a square.
  2. Extend the top and bottom lines of the square to the right.
  3. Place the needle of the compass at the midpoint of the square's baseline and draw an arc from the upper-right corner of the square down to where it crosses over the extended baseline. Use this point as the new corner of the rectangle
  4. Complete the rectangle by drawing a vertical line connecting the extended baselines. This is a golden rectangle.
  
• View the Choir Screen from the Chapel of the Château of Pagny. Have students locate examples of rectangles in this decorative work. How can they tell whether the shapes are actually golden rectangles?
  • Print and distribute copies of the Choir Screen to the class. Instruct students to find a rectangular shape and trace it with a pencil. Using the shorter side of the rectangle as one side of a square, have students draw a square inside the rectangle. Follow the steps listed above to see whether the rectangle fits the dimensions of a golden rectangle. (OPTION: There is an algebraic formula for a golden rectangle which can also be used to make this determination. See Extension.)
  • You may bring in other examples of architectural objects, many of which will reveal examples of golden rectangles.
• View Diego Rivera's Sugar Cane. Examine the composition of the mural, noting its shape and looking specifically for rectangles. Where can they be found? (NOTE: The front balcony of the building where the man is reclining on the hammock.)
  • Observe where the mural can be divided into sections. (NOTE: Use the two girls on the left or the young man in the foreground on the right as a point of reference.) Follow the same steps as above to see if these sections are also golden rectangles.
  • Look for other rectangles in Rivera's mural. (NOTE: Measuring shapes within the composition of a painting is not as exact as the shapes in architecture, so students should see if the shape is approximately a golden rectangle.)
• Why would an artist use a golden rectangle when designing a choir screen? Why would a painter use a golden rectangle when organizing the general composition of a mural?

Assessment:

• Golden rectangles can be found all around us—in design, advertising, and of course, art. Find several examples and organize these into a mini-portfolio. The portfolio can include magazine clippings, drawings, print-outs of digital photos, etc. In all cases, use a colored marker to draw the rectangles you see.
• Find additional information explaining why the golden rectangle is so aesthetically pleasing. Compile this research into an essay that explains this effect in a way your classmates can understand.

Remediation:

• View At the Moulin Rouge: The Dance by Henri de Toulouse-Lautrec. Follow the same process as above to discover golden rectangles in the painting.
• Look at Breaking Home Ties by Thomas Hovenden, using the same process as above.

Extension:

The golden rectangle can be represented mathematically by describing the ratio of one side to the other, indicated by the following formula: or approximately 1:1.618. Use this formula to create a golden rectangle and also to check to see if other rectangles discovered in art and architecture fit the proper ratio.

For more information, please contact Education: School & Teacher Programs by phone at (215) 684-7580, by fax at (215) 236-4063, or by e-mail at .