Illustrates the congruence mappings of translation, reflection, and rotation with physical examples of 2-D and 3-D figures, and outlines methods for creating Escher-type tessellations.Subsections: Translations; Reflections; Rotations; Composition of Mappings; Congruence; Mapping Figures onto Themselves; Escher-Type Tessellations; Problem-Solving ApplicationOne-page Math Activity: Rotating, Reflecting, and Translating Figures on Grids
The following steps illustrate a method of altering the sides of an equilateral triangle to obtain a non polygonal figure that will tessellate. These steps can be carried out with pencil and paper or by computer software programs.
Step 1 Draw a curve from A to B.
Once the lines of the original figure have been erased, the figure that remains will tessellate.
Starting Points for Investigations
If the preceding figure is used to form a tessellation, which of the transformations (rotation, translation, reflection) will map the preceding tessellation onto itself?
Step 3 produces a curve on one side of the triangle that is said to have point symmetry because it can be rotated onto itself by a 180˚ rotation. Suppose Step 3 is used to produce a curve with point symmetry on all three sides of a triangle. Will the resulting figure tessellate?
Suppose Step 3 is used to create a curve with point symmetry on each of the six sides of a regular hexagon. Will the resulting figure tessellate?