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Three Views of Square Symmetry

James E. Hamblin

James E. Hamblin
Shippensburg University

The dihedral group D8 (sometimes called D4), the group of symmetries of the square, is one of the simplest finite groups. It is one of the first examples students see in an introductory Abstract Algebra course, but it is also an example that tends to confuse many students. In these three mathlets, we will explore different aspects of this symmetry group to get a better understanding of its structure.

  • Quilting Patterns. In this mathlet, you will explore different patterns that can be created starting from a single square design. You will practice flipping and rotating squares to get used to the basic symmetry operations. You will also notice that starting with symmetric squares can lead to similarities among these patterns.
  • Checking Symmetries. In this mathlet, you will create a square design and practice flipping or rotating the square to check which symmetries your design has.
  • Composing Symmetry Operations. In this mathlet, you will learn how to "multiply" symmetries and explore the structure of the dihedral group.

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James E. Hamblin, "Three Views of Square Symmetry," Loci (August 2008)