1. Posters hung on the wall
_MSRI _ SYMMETRY_
Posters hung on the wall

2. SYMMETRY A A A DEFINITION:
An object is symmetric, if there exists a transformation ( mirroring, rotation, translation, … ) that maps the object back onto itself.
Sometimes small details have to be overlooked in order to see the overall symmetry.
First we focus on just 2-dimensional images and shapes:
MIRROR-SYMMETRY (M) CYCLIC SYMMETRY (Cn) BOTH: “DIHEDRAL” (Dn) C3 D5 D8 C6
C20
D20

3. FRIEZE symmetries There are 7 types of infinitely long linear friezesB B
FRIEZE symmetries There are 7 types of infinitely long linear friezes
They are characterized by the presence or absence of these symmetry elements:
Conway
Notation:
FUNDAMENTAL
DOMAIN nn n× n*
*nn
22n
2*n
*22n
GLIDE - AXIS
HOR. MIRROR
VERT. MIRROR
C2 - ROTATION

4. Symmetry Groups of Finite 3D Objects “Cylindrical Symmetries”
Each frieze pattern ( see Poster B ) can be wrapped around a cylinder or a sphere, with n repetitions around the equator, resulting in 7 infinite families of symmetries:
Cn nn n=3
S2n n× n=6
Cnh n* n=2
Cnv *nn n=5
Dn 22n n=5
Dnd 2*n n=3
Dnh *22n n=4
All 3D-print models by
Henry Segerman
All soccer-ball images by
David Swart

5. Symmetry Groups of Finite 3D Objects “Spherical Symmetries”
Oriented Tetrahedron: T (332) 12 elem.: 4*C3, 3*C2
Oriented Double-Tetrahedron: Th (3*2) 24 elem.: 4*C3, 3*C2, 3*M, I
Straight Tetrahedron: Td (*332) 24 elem.: 4*C3, 3*C2, 6*M
Oriented Octahedron (Cube): O (432) 24 elem.: 3*C4, 4*C3, 6*C2
Straight Octahedron (Cube): Oh (*432) 48 elem.: 3*C4, 4*C3, 6*C2, 3*Me, 6*Mf, I
Oriented Icosa-(dodeca)-hedron: I (532) 24 elem.: 6*C5, 10*C3, 15*C2
Straight Icosa-(dodeca)-hedron: Ih (*532) 48 elem.: 6*C5, 10*C3, 15*C2, 15*Me, 6*Mf, I
All soccer-ball images by
David Swart

6. Work-sheets handed out to participants
MSRI _ SYMMETRY
Work-sheets handed out to participants

7. Find the symmetries (mirror-lines, rotation points) in these designs:

8. Find the Symmetry Groups ( C. or D
Find the Symmetry Groups ( C? or D? ) for these hubcaps: ( look at Poster A )

9. 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 @ # $ &
Cut apart these characters and sort them into the letter bins based on their symmetries. ( see Poster A) :
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 @ # $ &
No symm.

10. Try to classify the following Frieze Patterns: ( study Poster B )
pqpqpqpqpqpqpqpqpq = ….
pdpdpdpdpdpdpdpdpd = ….
pppppppppppppppppp = ….
pbpbpbpbpbpbpbpbpb = ….
bdbdbdbdbdbdbdbdbdbd pqpqpqpqpqpqpqpqpqpq
bbbbbbbbbbbbbbbbbbbb pppppppppppppppppppp
bdbdbdbdbdbdbdbdbdbd qpqpqpqpqpqpqpqpqpqp
= ….

11. Pair-up corresponding Frieze Patterns (1, 2, 3…) with (A, B, C …) : ( study Poster B )
1 = … = … = … = … = … = … = ….

12. Identify the Symmetry-Groups of these 3D Objects:
= ….
= ….
= ….
= ….
= ….
= ….
= ….
= ….
= ….
= ….
= ….
= ….
= ….
= ….
= ….