Post on 01-Jan-2016
description
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
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
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
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