4. Symmetry & Group Theory Symmetry, Art & Science often used as an implied symbol of the essential symmetry of the human body, and by extension, to the universe as a whole. Leonardo Da Vinci’s Vitruvian Man (1490) Canon of Properties 4. Symmetry & Group Theory Symmetry & Nature 1 4. Symmetry & Group Theory Symmetry & Nature 4. Symmetry & Group Theory Symmetry & Nature 2 4. Symmetry & Group Theory Symmetry & Nature 4. Symmetry & Group Theory Symmetry & Nature (Spiral Symmetry) Apex episcopus Nautilus pompilus 3 4. Symmetry & Group Theory Symmetry & Nature (Spiral Symmetry) Symmetry & Architecture (Spiral Symmetry) Great Mosque of Genome (Larry Young) Samarra (Iraq) 4 4. Symmetry & Group Theory Symmetry & Nature (Bilateral / Mirror Sym) 4. Symmetry & Group Theory Symmetry & Architecture Taj Mahal (Agra, India) Giza Pyramid Field (Cairo, Egypt) 5 4. Symmetry & Group Theory Symmetry & Architecture (Mirror Symmetry) Eiffel Tower (Paris, 325 m, 10.1 tons) 4. Symmetry & Group Theory Symmetry & Landscaping (Mirror Symmetry) Toukouji (Japan) Trocadero Palace (Paris, from Eiffel t) 6 4. Symmetry & Group Theory Super-High Symmetry in Super-Low “Life Forms” 4. Symmetry & Group Theory Super-High Symmetry in Super-Low “Life Forms” 7 4. Symmetry & Group Theory Super-High Symmetry in Super-Low “Life Forms” 4. Symmetry & Group Theory 2-fold axes 8 4. Symmetry & Group Theory 3-fold axes 4. Symmetry & Group Theory 5-fold axes 9 4. Symmetry & Group Theory 2-fold axes 4. Symmetry & Group Theory 3-fold axes 10 4. Symmetry & Group Theory 5-fold axes 4. Symmetry & Group Theory 2-fold axes 11 4. Symmetry & Group Theory 3-fold axes 4. Symmetry & Group Theory 5-fold axes 12 4. Symmetry & Group Theory The Icosahedral Symmetry of Viruses some nomen- clature… 4. Symmetry & Group Theory The Icosahedron Type Platonic Face polygon triangle Faces 20 Edges 30 Vertices 12 Faces per vertex 5 Vertices per face 3 Symmetry Icosahedral group (Ih), order 120 Dual polyhedron dodecahedron Connecting the centers of all the pairs of adjacent faces of Properties regular, convex any platonic solid produces another (smaller) platonic Dihedral Angle 138.1896852° solid. The number of faces and vertices is interchanged, while the number of edges of the two is the same. 13 4. Symmetry & Group Theory The Icosahedron Type Platonic Face polygon triangle Faces 20 Edges 30 Vertices 12 Faces per vertex 5 Vertices per face 3 Symmetry Icosahedral group (Ih), order 120 Dual polyhedron dodecahedron total # of Properties regular, convex symmetry Dihedral Angle 138.1896852° operations! 4. Symmetry & Group Theory The Dodecahedron Type Platonic Face polygon pentagon Faces 12 Edges 30 Vertices 20 Faces per vertex 3 Vertices per face 5 icosahedral (I ) Despite appearances, when Symmetry group h a dodecahedron is order 120 inscribed in a sphere, it occupies more of the Dual polyhedron icosahedron sphere's volume (66.49%) than an icosahedron Properties regular, convex inscribed in the same sphere (60.54%). 14 4. Symmetry & Group Theory The Cube (Hexahedron) Type Platonic Face polygon square Faces 6 Edges 12 Vertices 8 Faces per vertex 3 Vertices per face 4 octahedral (O ) Symmetry group h order 48 Dual polyhedron octahedron Properties regular, convex, zonohedron 4. Symmetry & Group Theory The Octahedron Type Platonic Face polygon triangle Faces 8 Edges 12 Vertices 6 Faces per vertex 4 Vertices per face 3 octahedral (O ) Symmetry group h order 48 Dual polyhedron cube Properties regular, convex 15 4. Symmetry & Group Theory The Tetrahedron Type Platonic Face polygon triangle Faces 4 Edges 6 Vertices 4 Faces per vertex 3 Vertices per face 3 tetrahedral (T ) Symmetry group d order 24 Dual polyhedron tetrahedron (self-dual) Dihedral angle 70° 32' = arccos(1/3) Properties regular, convex 4. Symmetry & Group Theory The Platonic Solids Edges Faces meeting Symmetry Name Picture Faces Edges Vertices per face at each vertex group tetrahedron 4 6 4 3 3 Td cube 6 12 8 4 3 O (hexahedron h octahedron 8 12 6 3 4 Oh dodecahedron 12 30 20 5 3 Ih icosahedron 20 30 12 3 5 Ih 16.