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

Vertices Edges Faces Faces at each

vertex

Sides of each face

Tetrahedron

Cube

Octahedron

Dodecahedron

Icosahedron

VerticesV

EdgesE

FacesF

Faces at each

vertex

Sides of each face

Tetrahedron 4 6 4 3 3

Cube 8 12 6 3 4

Octahedron 6 12 8 4 3

Dodecahedron 20 30 12 3 5

Icosahedron 12 30 20 5 3

VerticesV

EdgesE

FacesF

Faces at each

vertex

Sides of each face

Tetrahedron 4 6 4 3 3

Cube 8 12 6 3 4

Octahedron 6 12 8 4 3

Dodecahedron 20 30 12 3 5

Icosahedron 12 30 20 5 3

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

Edges Faces

Tetrahedron

Cube

Octahedron

Dodecahedron

Icosahedron

Solid(pretruncating)

Truncated Vertices

Edges Faces

Tetrahedron 12 18 8

Cube 14 36 24

Octahedron 14 36 24

Dodecahedron 32 90 60

Icosahedron 32 90 60

Solid(post-truncating)

Truncated Vertices

Edges Faces

Tetrahedron 8 18 12

Cube 24 36 14

Octahedron 24 36 14

Dodecahedron 60 90 32

Icosahedron 60 90 32

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?