Posters hung on the wall

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Presentation transcript:

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

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

FRIEZE symmetries There are 7 types of infinitely long linear friezes B 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

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

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

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

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

Find the Symmetry Groups ( C. or D Find the Symmetry Groups ( C? or D? ) for these hubcaps: ( look at 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 @ # $ & 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 @ # $ & 0 1 2 3 4 5 6 7 8 9 No symm.

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

Pair-up corresponding Frieze Patterns (1, 2, 3…) with (A, B, C …) : ( study Poster B ) 1 = …. 2 = …. 3 = …. 4 = …. 5 = …. 6 = …. 7 = ….

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