Geometry
Symmetry and Tessellations
Big Picture
Symmetry has a lot to do with transformations, since symmetric figures are often “immune” to some types of transformations. This means that some symmetric figures don’t change after certain transformations. An interesting application of transformations is tessellations. Tessellation is when a plane can be covered by one or more shapes without any overlaps or gaps.
Key Terms
Line of Symmetry: A line that divides a figure into two congruent halves.
Rotational Symmetry: When a figure can be rotated (less than 360°) and still looks the same way it did before rotation
Center of Rotation: The point at which the figure is rotated around such that the rotational symmetry holds.
Angle of Rotation: The number of degrees a figure is rotated so that it still looks the same.
Tessellation: A tiling over a plane with one or more figures such that the figures fill the plane with no overlaps and no gaps.
Symmetry
Lines of Symmetry:
- A figure can have one line of symmetry, several lines of symmetry, or no lines of symmetry. A figure that has one or more lines of symmetry has line symmetry.
Rotational Symmetry:
- If a shape can be rotated n times around the center of rotation, the angle of rotation is \(\frac{360^{\circ}}{n}\). To find additional angles of rotation, multiply the angle of rotation by 1, 2, 3, ..., n.
Tessellations
In order to tessellate a shape, the the sum of the angles around each point must be 360°. There should be no gaps or overlaps.
If a shape does not tessellate by itself, another shape can be added so that the two shapes together will tessellate.
- A regular tessellation is formed by congruent regular polygons.
- A semiregular tessellation is formed by two or more different regular polygons.
