Reflectional and Rotational Symmetry

Regular octagonal pyramid

Reflectional Symmetry:

4 - the plane through the vertex opposite the base and through midpoint of opposite edges on the base

4- the plane through the vertex opposite the base and through opposite vertices on the base

Rotational symmetry:
-axis through vertex opposite base and through center of base
-one axis order 8

Tetrahedron

F: 4 E: 6 V: 4

Reflectional Symmetry:

6 planes- the plane from an edge through the midpoint of the opposite edge

Rotational Symmetry:

vertex center opposite face-4 axes order 3

midpoint opposite edge- 3 axes order 2

Regular Triangular Prism

-Reflectional Symmetry: 3 planes (plane one vertex, corresponding vertex opposite base, midpoint opposite edge base)

-Rotational Symmetry: 1 axes order 2 (center to opposite)

Cube

F: 6 E: 12 V: 8

Reflectional Symmetry:

• 3 planes (through centers of opposite faces)

• 6 planes (through opposite edges/diagonals) 9 total

Rotational Symmetry:

• axes through centers of opposite faces → order 4

• axes through opposite vertices → order 3

• axes through midpoints of opposite edges → order 2

Trapezoidal Prism (only iscoceles)

Reflectional Symmetry:

• 1 plane (through the center of the prism, perpendicular to the bases, splitting the trapezoids evenly)

Rotational Symmetry:

• axis through the center of the prism (along its length)

• one axis order 2 (180° rotation)

Rectangular Prism

Reflectional Symmetry: 3 planes (through center splitting length, width, height)

Rotational Symmetry:

• axes through center (length, width, height directions)

• each axis order 2

Icosahedron

F:20 E:30 V:12

• Reflectional planes: 15

• Rotational axes:

  • 6 axes (order 5)

  • 10 axes (order 3)

  • 15 axes (order 2)

Octahedron

F:8 E: 12 V:6

• Reflectional planes: 9

• Rotational axes:

  • 3 axes (order 4)

  • 4 axes (order 3)

  • 6 axes (order 2)

Dodecahedron

F:12 E:30 V:20

• Reflectional planes: 15

• Rotational axes:

  • 6 axes (order 5)

  • 10 axes (order 3)

  • 15 axes (order 2)