Illuminations: Spatial Reasoning Using Cubes and Isometric Drawings: Part Two -- Solid to Isometric -- One to Many

Spatial Reasoning Using Cubes and Isometric Drawings: Part Two -- Solid to Isometric -- One to Many


Some "Escher" Drawings

In this interactive geometry investigation students will explore polyhedra using different representations and perspectives for three dimensional block figures.

Learning Objectives
 

Students will

  • Analyze characteristics and properties of three dimensional geometric shapes and develop mathematical arguments about geometric relationships
  • Use visualization, spatial reasoning, and geometric modeling to solve problems

Materials
 
  • Computer and Internet connection

Instructional Plan

In Activity 3, 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 activity you will examine some isometric drawings that seem to be impossible, and you will investigate one way Escher used to create these "impossible figures."

TASK 1: Consider the isometric drawings below. Imagine trying to build each figure.

Click on each drawing to view it using the drawing tool. Use the "View" tool to help you understand the drawings.

A.
B.
C.


TASK 2: Now try the following:

  • Open each figure in the drawing tool by clicking on the appropriate figure or button below.
  • Delete a single cube from the picture, and use the View tools to explore the results.
  • Refresh the image and delete a different cube. Try this several times with each drawing.
Questions:

  1. Why do you think some people call these figures impossible?

  2. What about isometric drawings creates these false impressions?

  3. 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. Use coordinate geometry to examine special geometric shapes, such as regular polygons or those with pairs of parallel or perpendicular sides.
  2. Use coordinate geometry to represent and examine the properties of geometric shapes.
  3. Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling.
  
1 period   

NCTM Resources

Principles and Standards for School Mathematics
National Council of Teachers of Mathematics Thinkfinity Verizon Foundation
More and Better Mathematics for All Students
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