# Mathematical Knots A A SYMMETRY 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 C6 D5 C20 D8 D20 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 GLIDE - AXIS n HOR. MIRROR n*

VERT. MIRROR *nn C2 - ROTATION 22n 2*n *22n CSymmetry Groups of Finite 3D ObjectsC 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 Dn 22n n=5 Cnh n* n=2 Dnd 2*n n=3 Cnv *nn n=5 Dnh *22n n=4 DSymmetry Groups of Finite 3D ObjectsD Spherical Symmetries

3D-models by Henry Segerman Symmetry Elements Mirror lines Rotation points Glide axis Soccer Ball Symmetry David Swart, Waterloo, Canada [email protected] http://archive.bridgesmathart.org/2015/bridges2015-151.pdf CSymmetry Groups of Finite 3D ObjectsC 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 Dn 22n n=5 S2n n n=6 Cnh n* n=2 Dnd 2*n n=3 Cnv *nn n=5 Dnh *22n n=4

DSymmetry Groups of Finite 3D ObjectsD Spherical Symmetries Oriented Tetrahedron: T 12 elem.: 4*C3, 3*C2 (332) Oriented Double-Tetrahedron: Th 24 elem.: 4*C3, 3*C2, 3*M, I (3*2) Straight Tetrahedron: Td 24 elem.: 4*C3, 3*C2, 6*M Oriented Octahedron (Cube): O 24 elem.: 3*C4, 4*C3, 6*C2

(*332) (432) Straight Octahedron (Cube): Oh (*432) 48 elem.: 3*C4, 4*C3, 6*C2, 3*Me, 6*Mf, I Oriented Icosa-(dodeca)-hedron: I (532)