© 2010 Pearson Education, Inc. All rights reserved Motion Geometry and Tessellations Chapter 14

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide NCTM Standard: Motion Geometry Young children come to school with intuitions about how shapes can be moved. Students can explore motions such as slides, flips, and turns by using mirrors, paper folding, and tracing. Later, their knowledge about transformations should become more formal and systematic. In grades 3–5 students can investigate the effects of transformations and begin to describe them in mathematical terms. Using dynamic geometry software, they can begin to learn the attributes needed to define a transformation.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide NCTM Standard: Motion Geometry In the middle grades, students should learn to understand what it means for a transformation to preserve distance, as translations, rotations, and reflections do.... At all grade levels, appropriate consideration of symmetry provides insights into mathematics and into art and aesthetics (p. 43).

Slide Copyright © 2010 Pearson Addison-Wesley. All rights reserved Tessellations of the Plane  Regular Tessellations  Semiregular Tessellations  Tessellating with Other Shapes

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Study of Regular Division of the Plane with Reptiles by Maurits C. Escher. Tessellations of the Plane A tessellation of a plane is the filling of the plane with repetitions of figures in such a way that no figures overlap and there are no gaps.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Tessellations with regular polygons are both appealing and interesting because of their simplicity. Tessellation with squares Tessellation with equilateral triangles and hexagons. Regular Tessellations

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Which regular polygons can tessellate the plane? To understand this, look at the tessellation using hexagons. 120° Around a vertex of any regular tessellation, we must have an angle sum of 360° (3 × 120 = 360). The interior angles of a hexagon measure: Regular Tessellations

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide If we have three pentagons surrounding the vertex of a tessellation, the sum of the angles around the vertex is 3 × 108° = 324° < 360°. We do not have enough pentagons to surround the vertex. Regular Tessellations

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide On the other hand, if we include a fourth pentagon, the sum of the angles around the vertex exceeds 360°. So, there cannot be a regular tessellation using pentagons. Regular Tessellations

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Regular Tessellations

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Regular Tessellations Since 360 divided by a number greater than 120 is smaller than 3, and the number of sides of a polygon cannot be less than 3, no regular polygon with more than six sides can tessellate the plane.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Semiregular Tessellations When more than one type of regular polygon is used and the arrangement of the polygons at each vertex is the same, the tessellation is semiregular.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Tessellating with Other Shapes Although there are only three regular tessellations (square, equilateral triangle, and regular hexagon), many tessellations are not regular. One factor that must always be true is that the sum of the measures of the interior angles around a vertex must equal 360°.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Tessellating with Other Shapes A regular pentagon does not tessellate the plane. However, some nonregular pentagons do.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Tessellating with Other Shapes The following two tessellations were discovered by Marjorie Rice, and the problem of how many types of pentagons tessellate remains unsolved today.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Tessellating with Other Shapes Consider any polygon known to tessellate a plane, such as rectangle ABCD (a). On the left side of the figure draw any shape in the interior of the rectangle (b). Cut this shape from the rectangle and slide it to the right by the slide that takes A to B (c). The resulting shape will tessellate the plane.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Tessellating with Other Shapes A second method of forming a tessellation involves a series of rotations of parts of a figure. Start with an equilateral triangle ABC (a), choose the midpoint O of one side of the triangle, and cut out a shape (b), being careful not to cut away more than half of angle B, and then rotate the shape clockwise around point O.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide Tessellating with Other Shapes Complete the tessellating shape and tessellate the plane with it.