- Homework, Section 4.7 due today
-
- For today you were to read section 4.5 (platonic solids); there is an associated homework set:
- Section 4.5, pp. 284-: #1, 4, 5, 11, 12, 15 (due Monday, 3/30)
- Read section 7.6 for Monday.
- Golden Rectangle (definition): a rectangle whose side
lengths are in the ratio
- We started with the Fibonacci spiral:
(see this site)
12 + 12 + 22 + 32 + 52 + 82 + 132 + . . . + F(n)2 = F(n) x F(n+1)
Note: The Fibonacci series spiral on the left is slightly different from the perfect spiral generated by Phi (1.61804...) because of the approximations early in the series leading to Phi. (1, 1, 2, 3, 5, 8 and 13 produce ratios of 1, 2, 1.5, 1.67, 1.6 and 1.625)
- Ratios of successive Fibonaccis give rise to what is known as the "golden mean"
-
The Golden Section in Art and Architecture
(In particular, St. Jerome)
- What sizes do U.S. flags come in? (How about Togolese Flags?)
- One of the themes in Section 4.5 is that of
symmetry. Beautiful symmetries of geometrical objects.
- In the plane of 2-Dimensions,
- What is the most symmetric rectangle?
- What are regular polygons?
- How many are there?
- What kinds of symmetry do they possess?
- Application of the septagon
- Section 4.5: Platonic solids
- Regular polyhedra start with regular polygons -- which ones play a role in the regular polyhedra?
- Activity: We're going to be using "geomags" today to explore three-dimensional objects that are also regular -- regular polyhedra. What objects can you create in three dimensions whose faces are all the same, and which possess a lot of symmetry (faces, edges and vertices are all interchangeable)?
- We can use our geomags to construct a set of Platonic solids: some
of them are "structurally stable", and some of them are not. Maybe a
better way to construct the solids is via a Platonic solid template in 2-D
Activity: as a group, construct a set of platonic solids from the paper templates.
- Why can't a hexagon be used to create a regular polyhedron?
- The sphere is the "missing link" in the Platonic family: it is the
ideal to which the Platonic solids aspire.
The Five Convex Regular Polyhedra (Platonic solids) -- thanks Wikipedia! Tetrahedron Hexahedron
or CubeOctahedron Dodecahedron Icosahedron 
- What kinds of symmetry do these objects possess?
- Application of the dodecahedron
- Activity: We've made them in 3-d. Now use the geomags
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?
- The concept of Duality
- Who's dual with whom?
- Who's dual alone?
- What kinds of symmetry do these objects possess?
- Two dimensional representations of Platonic Solids
- Put your eye to the face of an object and draw it.
- Is it possible to draw each objects without any edges crossing?
- Pictures of Duality
- Regular polyhedra start with regular polygons -- which ones play a role in the regular polyhedra?
