Section 9.3: Symmetries and Tilings

Welcome to Section 9.3! Today, we'll delve into the fascinating world of symmetries and tilings. We'll explore how symmetry appears in various forms and how we can use shapes to create patterns that cover a plane. Get ready to see math in a new light!

Learning Objectives

  • Identify various types of symmetries in art, architecture, and nature.
  • Understand rotational symmetries in the plane.
  • Recognize reflectional symmetry of planar figures.
  • Learn about regular tilings of the plane.
  • Explore irregular tilings.

Rotational Symmetry

A planar figure has rotational symmetry about a point if it remains exactly the same after a rotation about that point of less than 360 degrees. For example:

  • A square has 90-degree rotational symmetry.
  • A rectangle has 180-degree rotational symmetry.

Consider a pentagram (five-pointed star). To find its rotational symmetries, we need to determine the angles at which it remains unchanged after rotation. Since the star points divide 360 degrees into five equal angles, each angle is:

$$ \frac{360 \text{ degrees}}{5} = 72 \text{ degrees} $$

Therefore, a pentagram has 72-degree rotational symmetry. The other rotational symmetries are multiples of 72 degrees (144, 216, and 288 degrees).

Reflectional Symmetry

A planar figure has reflectional symmetry about a line L if the figure is identical to its reflection through L. Imagine flipping the plane about a line; if the figure looks the same, it has reflectional symmetry.

For example, a shape might have reflectional symmetry about both horizontal and vertical lines. It can also have rotational symmetry of 180 degrees.

Tilings and Tessellations

A tiling or tessellation of the plane is a pattern of repeated figures that cover up the plane without any gaps or overlaps. There are different types of tilings:

  • Regular Tiling: A tiling made of repeated copies of a single regular polygon, meeting edge to edge, with every vertex having the same configuration.

A regular polygon is a polygon (a closed figure made of three or more line segments) in which all sides are of equal length and all angles have equal measure. For a regular tiling, the angles that meet at a vertex must sum to 360 degrees. For example, you can create a regular tiling with equilateral triangles!

Let's explore some more advanced tilings and patterns in the next section. Keep practicing and have fun with geometry!