1. Ideas for Exercises for the K-12 Classroom
SYMMETRY:
Ideas for Exercises for the K-12 Classroom
Part I: Rotation and Reflection Symmetries
in the Alphabet
C. Y. Jones, Columbia University, January 2014,

2. A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Symmetry
The Letters of the Alphabet
The letters of the alphabet can be used to demonstrate symmetry.
All but seven letters (G, J, K, L, P, Q, R) of the Latin alphabet have some type of symmetry.
The possible symmetry elements are axes and planes.
Perform symmetry operations on the letters to test for symmetry elements in the letter.
The possible symmetry operations are rotation about an axis and reflection through a plane.
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
C. Y. Jones, Columbia University, January 2014,

3. A B X Symmetry Rotational Symmetry
Some objects are symmetrical about a rotation axis through the center of the object.
Rotation about the axis by a specified number of degrees leaves the object indistinguishable from itself before the rotation.
Rotations both parallel and perpendicular to the plane of the letter are possible.
180o rotation
about horizontal axis in the plane of the letter
180o rotation
about axis in the plane of the letter
180o rotation
about axis in the plane of the letter
A B X
180o rotation
about horizontal axis in the plane of the letter
180o rotation
about axis perpendicular to the plane of the letter
C. Y. Jones, Columbia University, January 2014,

4. H I N O S X Z Symmetry Rotational Symmetry: Perpendicular to the Plane
Seven letters are symmetrical about a rotation axis perpendicular to the plane of the letter
The rotation axis ( ) passes through the center of the letter.
Rotation about the axis by 180o leaves the letter indistinguishable from itself before the rotation.
180o
H I N O S X Z
C. Y. Jones, Columbia University, January 2014,

5. Symmetry Rotational Symmetry: in the Plane A H I M O T U V W X Y
Within the plane of the letter, both horizontally and vertically oriented axes are possible.
The rotation axis ( ) is in the plane of the letter.
11 letters are symmetrical about a rotation axis oriented vertically.
7 letters are symmetrical about a rotation axis oriented horizontally.
Rotation about the axis leaves a letter indistinguishable from itself before the rotation.
vertical:
horizontal:
A H I M O T U V W X Y
180o
B C D E H I O
180o
C. Y. Jones, Columbia University, January 2014,

6. Symmetry B C D E H I O X Plane of Symmetry: Horizontal
Eight letters are symmetrical across a horizontal plane perpendicular to the plane of the letter.
Reflection through the plane leaves a letter indistinguishable from itself before the reflection.
B C D E H I O X
C. Y. Jones, Columbia University, January 2014,
C. Y. Jones, Columbia University, January 2014,

7. Symmetry A H I M O T U V W X Y Plane of Symmetry: Vertical
11 letters are symmetrical across a vertical plane perpendicular to the plane of the letter.
Reflection through the plane leaves the object indistinguishable from itself before the reflection.
A H I M O T U V W X Y
C. Y. Jones, Columbia University, January 2014,
C. Y. Jones, Columbia University, January 2014,

8. Symmetry B C D E H B C D E H Use of Symmetry to Generate Letters
Symmetry operations can be used to generate a letter from the unique part of itself.
This unique part contains all of the essential information about the shape of the letter.
Crystallographers call this small part the asymmetric unit.
The asymmetric units of five letters with a horizontal mirror plane are shown below.
Reflection of the asymmetric unit across a horizontal mirror plane, or rotation about a horizontal axis in the plane, gives the complete letter
All the essential information is contained within the asymmetric unit.
B C D E H
B C D E H
C. Y. Jones, Columbia University, January 2014,

9. Symmetry Use of Symmetry to Generate Letters A H I M O T A H I M O T
Symmetry can be used to generate a letter from the unique part of itself.
This unique part contains all of the essential information about the shape of the letter
Crystallographers call this small part the asymmetric unit.
The asymmetric units of six letters with a vertical mirror plane are shown below.
Reflection of the asymmetric unit through a vertical plane, or rotation through a vertical axis in the plane of the letter, gives the complete letter:
All the essential information is contained within the asymmetric unit.
A H I M O T
A H I M O T
C. Y. Jones, Columbia University, January 2014,

10. H Symmetry Crystallography: Reflection s sv sh
Crystallographers use a special notation for symmetry elements.
A reflection plane is given the symbol lower-case Greek “sigma.” s
If an object has a mirror plane parallel to s symmetry axis, write sv (v for vertical).
If an object has a mirror plane perpendicular to s symmetry axis, write sh (h for horizontal).
The capital letter H contains vertical ( ) and horizontal ( ) reflection planes.
Both planes are perpendicular to the letter. sv H sh
C. Y. Jones, Columbia University, January 2014,

11. Symmetry Crystallography: Rotation
Crystallographers use a special notation for symmetry elements.
A proper rotation about is given the symbol capital C in italics with a subscript: Cn.
The proper rotation takes place about the principal axis of the object.
If the object has more than one rotation axis, the principal axis is the one with the largest value of n.
The subscript n is an integer called the order of the rotation.
The order n tells how many equal rotations it takes to rotate the object by 360o, where each rotation leaves the object indistinguishable from the last orientation.
C. Y. Jones, Columbia University, January 2014,

12. H Symmetry Crystallography: Rotation C2 C2 C2
For example, for the letter H has a principal rotation axis ( ) with n = 2; rotations of 180o leave the H indistinguishable
360o / n = 360o / 2 = 180o
The letter H has two C2 axes ( ) that are perpendicular to the principal axis
Notice that the letter H has a Cn axis plus n C2 axes perpendicular to the principal axis C2 H C2 C2
C. Y. Jones, Columbia University, January 2014,

13. Symmetry Crystallography: Point Group
The collection of symmetry elements present in the structure of an object makes up the point group of the object. For example:
- The letter H has a proper rotation axis.
- The rotation axes of the letter H are
a principal C2 axis
2 mutually perpendicular axes of type C2, perpendicular to the principal axis
- The reflection planes of the letter H are
a reflection plane perpendicular to the principal axis (sh )
a second reflection plane parallel to the principal axis
Based on this set of symmetry elements, the point group of the letter H is D2h.
C. Y. Jones, Columbia University, January 2014,