- Rigid symmetries:
- Translations
- Rotations
- Reflections
A worthy handout, to try yourself
- symmetry of scale: A pattern in the plane is
scalable (or has symmetry of scale) if the tiles
that make up the pattern can be grouped into
super-tiles that still cover the plane, and, if scaled
down, coincide with the original pattern.
"Super-tiles" -- small tiles placed together to make larger copies of themselves.
- Tiling with pentagons
- Penrose pattern:
- The
Penrose Pinwheel Pattern
- Penrose "Kites and
Darts" ("new P2")
- Penrose Rhomb
pattern ("new P3"): "[the diamonds] occur with the same
frequencies (the ratio of thick to thin is the Golden Number (1 +
sqrt(5))/2, which is also the ratio of their areas)...."
(source: Tilings, Diffraction, and Quasicrystals) - Penrose and the toilet paper caper (described in the section)
- The
Penrose Pinwheel Pattern
- Section 4.5: Platonic solids
The Five Convex Regular Polyhedra (Platonic solids) -- thanks Wikipedia! Tetrahedron Hexahedron
or CubeOctahedron Dodecahedron Icosahedron 
- Regular polyhedra start with regular polygons -- so what are
those?
- Now let's make them in 3-d: We're going to be using "geomags" today to
construct a set of Platonic solids (Problem I.1), and use them
to fill in the table on page 275:
# of Vertices Edges Faces faces at each vertex sides at each face Tetrahedron Cube Octahedron Dodecahedron Icosahedron What conclusions can we draw from this data? Is there a pattern?
- Regular polyhedra start with regular polygons -- so what are
those?
- Links:
- Java applets allow one to play with Platonic Solids
- Or use virtual reality software to view them!
- A Treatise on Symmetry
- Penrose tiling applet (kites and darts)
- Tilings, Diffraction, and Quasicrystals
- Tiling software
- Next time: 4.4: symmetry and tilings
Website maintained by Andy Long. Comments appreciated.