9.4 Symmetry in two and three dimensions

Symmetry in Two Dimensions

Symmetry describes how a figure maps onto itself through reflections or rotations. Understanding symmetry connects directly to transformations you've already studied, since every symmetry is a transformation (a reflection or rotation) that leaves the figure unchanged.

Line and Rotational Symmetry

Line symmetry (also called reflectional symmetry) occurs when a figure can be divided into two congruent halves by a line called the line of symmetry. If you fold the figure along that line, the two halves coincide exactly. A figure can have zero, one, or many lines of symmetry.

Rotational symmetry occurs when a figure can be rotated about its center by some angle less than 360° and map onto itself. The smallest such angle is called the angle of rotation. To find it for a regular polygon, divide 360° by the number of sides:

$\text{angle of rotation} = \frac{360°}{n}$

where $n$ is the number of sides.

Symmetry Order in Figures

The order of rotational symmetry is the number of times a figure maps onto itself during a full 360° rotation (including the 360° rotation itself).

Here's a reference for the most common figures:

Notice the pattern: a regular $n$-gon always has exactly $n$ lines of symmetry and rotational symmetry of order $n$. A non-regular figure like a rectangle has fewer of both because its sides aren't all congruent.

Symmetry in Three Dimensions

Plane Symmetry in 3D

In three dimensions, the analog of a line of symmetry is a plane of symmetry. A 3D figure has plane symmetry when a plane divides it into two halves that are mirror images of each other. Reflecting one half across that plane produces the other half exactly.

Think of it this way: a line of symmetry works in 2D because it's a 1D "cut" through a 2D figure. A plane of symmetry works in 3D because it's a 2D "cut" through a 3D figure. The idea is the same, just one dimension higher.

Types of 3D Symmetry

3D figures also have rotational symmetry about axes (not just a center point). An axis of symmetry is a line through the figure such that rotating the figure around that line by some angle less than 360° maps it onto itself.

Below are the symmetry properties of several common solids. For an Honors course, focus on the cube and tetrahedron; the others follow similar logic.

Cube (6 faces, 8 vertices, 12 edges)

• 13 axes of rotational symmetry:
  • 3 axes through centers of opposite faces (order 4, so 90° rotations)
  • 4 axes through opposite vertices, i.e., long diagonals (order 3, so 120° rotations)
  • 6 axes through midpoints of opposite edges (order 2, so 180° rotations)
• 9 planes of symmetry:
  • 3 planes each parallel to a pair of opposite faces (cutting through the midpoints of four edges)
  • 6 planes each passing through two opposite edges diagonally

Regular Tetrahedron (4 faces, 4 vertices, 6 edges)

• 7 axes of rotational symmetry:
  • 4 axes from a vertex to the center of the opposite face (order 3, so 120° rotations)
  • 3 axes through midpoints of opposite edges (order 2, so 180° rotations)
• 6 planes of symmetry, each passing through one edge and the midpoint of the opposite edge

Sphere

• Infinite planes of symmetry (any plane through the center)
• Infinite axes of rotational symmetry of infinite order (any line through the center)

The sphere is the 3D equivalent of the circle: maximum possible symmetry.