Illuminations: Using Cubes and Isometric Drawings

Using Cubes and Isometric Drawings

Are They Possible?

Students examine some isometric drawings that seem to be impossible and investigate one way Escher used to create these "impossible" figures.

Learning Objectives
 
Students will:
  • Examine isometric drawings that seem to be impossible
  • Investigate one way Escher used to create these figures

Materials
 

Instructional Plan

In previous lessons, students saw that isometric drawings were not always what they appeared to be. A Dutch artist, M.C. Escher (1898-1972), is famous for his use of unusual perspectives to trick the viewer into seeing "Impossible Figures." In this lesson, students will examine some isometric drawings that seem to be impossible, and they will investigate one way Escher used to create these "impossible figures."

Project the following images for the students to see:

A.
B.
C.

As in previous lessons, you may print out the following PDF to create an overhead transparency:

Are They Possible? Images Are They Possible?

Students should attempt to mentally construct each figure before using the isometric drawing tool.

Now, using the Isometric Drawing Tool , students should build each figure. Next, they should use the View feature to look at the figure from different perspectives. Students can open several tools so they can have access to all three drawings.

Now students should try the following:

  • Choose one of the three figures.
  • Delete a single cube from the picture, and use the View feature to explore the results.
  • Recreate the image and delete a different cube. Try this several times with each drawing.

Students can repeat with the other figures and discuss observations with a partner. Alternatively, students can create "impossible figures" and ask a partner to repeat this activity.

Questions for Students
 
  • Why do you think some people call these figures impossible?
  • What about isometric drawings creates these false impressions?
  • Do you think it is ever possible to have an isometric drawing that does not represent any 3-dimensional object? If so, can you draw one either on paper or using the applet? If not, can you explain why any isometric drawing created by the drawing tool is some 3D shape?

NCTM Standards and Expectations
 
Geometry 6-8
  1. Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.
  
1 period   

NCTM Resources

Principles and Standards for School Mathematics

 Activities
National Council of Teachers of Mathematics Thinkfinity Verizon Foundation
More and Better Mathematics for All Students
© 2000 National Council of Teachers of Mathematics
Use of this Web site constitutes acceptance of the Terms of Use

The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring mathematics learning of the highest quality for all students. The views expressed or implied, unless otherwise noted, should not be interpreted as official positions of the Council.