4.5 Platonic Solids Wednesday, February 25, 2009

Symmetry in 3-D  Sphere – looks the same from any vantage point  Other symmetric solids? CONSIDER REGULAR POLYGONS

Start in The Plane  Two-dimensional symmetry  Circle is most symmetrical  Regular polygons – most symmetrical with straight sides

2D to 3D  Planes to solids  Sphere – same from all directions  Platonic solids Made up of flat sides to be as symmetric as possible Faces are identical regular polygons Number of edges coming out of any vertex should be the same for all vertices

Five Platonic Solids  Cube Most familiar  Tetrahedron  Octahedron  Dodecahedron  Icosahedron

Powerful?  Named after Plato  Euclid wrote about them  Pythagoreans held them in awe

VerticesEdgesFacesFaces at each vertex Sides of each face Tetrahedron Cube Octahedron Dodecahedron Icosahedron

Vertices V Edges E Faces F Faces at each vertex Sides of each face Tetrahedron Cube Octahedron Dodecahedron Icosahedron

Vertices V Edges E Faces F Faces at each vertex Sides of each face Tetrahedron Cube Octahedron Dodecahedron Icosahedron

Some Relationships  Faces of cube = Vertices of Octahedron  Vertices of cube = Faces of Octahedron

Duality  Process of creating one solid from another  Faces Vertices

Euler's polyhedron theorem  V + F - E = 2

Archimedean Solids  Allow more than one kind of regular polygon to be used for the faces  13 Archimedean Solids (semiregular solids)  Seven of the Archimedean solids are derived from the Platonic solids by the process of "truncation", literally cutting off the corners  All are roughly ball-shaped

Truncated Cube

Archimedean Solids

Soccer Ball – 12 pentagons, 20 hexagons

Solid (pretruncating) Truncated Vertices EdgesFaces Tetrahedron Cube Octahedron Dodecahedron Icosahedron

Solid (pretruncating) Truncated Vertices EdgesFaces Tetrahedron12188 Cube Octahedron Dodecahedron Icosahedron329060

Solid (post-truncating) Truncated Vertices EdgesFaces Tetrahedron81812 Cube Octahedron Dodecahedron Icosahedron609032

Some Relationships  New F = Old F + Old V  New E = Old E + Old V x number of faces that meet at a vertex  New V = Old V x number of faces that meet at a vertex

Stellating  Stellation is a process that allows us to derive a new polyhedron from an existing one by extending the faces until they re-intersect

Two Dimensions: The Pentagon

Octagon

How Many Stellations?  Triangle and Square  Pentagon and Hexagon  Heptagon and Octagon  N-gon?

Problem of the Day  How can a woman living in New Jersey legally marry 3 men, without ever getting a divorce, be widowed, or becoming legally separated?