- Fibonacci Spiral Images -- slight reprise
- Return of the Graded Homework
- I'm going to use 10 as a perfect score, because I screwed up one problem (I wrote a base 8 number with an eight in it!).
- You do have a homework
assignment due Friday...
- Reminder: you will have an exam after fall break, Friday the 21st.
- Question of the day(s): What makes some solids so Platonic?
- We begin our discussion with the convex regular
polygons:
"A regular polygon is a polygon which is equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be convex or star."
- Now, let's move on to solids. A cube is an example of a Platonic
solid. It's the one we're most familiar with. The special thing about a
Platonic solid is that
- Each face, each edge, and each vertex is exactly equivalent to every other face, edge, or vertex (respectively), and
- All faces are congruent (identical) convex regular polygons.
- What kinds of regular polygons are possible for faces?
- How many Platonic solids are there, do you suppose?
- What's wrong with six sides, and beyond?
- In order to construct a Platonic solid, you'd have to be
able to lay one out flat, and have some space open up
between the polygons.
- As an example of the problem, consider a hexagonal graph tiling, which represents a tiling of the plane by a regular convex polygon.
- In order to construct a Platonic solid, you'd have to be
able to lay one out flat, and have some space open up
between the polygons.
- Johannes Kepler thought that the orbits of the planets (of which
five were known at the time) were somehow related to the various
Platonic solids, as shown in this
figure
"The six spheres each corresponded to one of the planets (Mercury, Venus, Earth, Mars, Jupiter, and Saturn). The solids were ordered with the innermost being the octahedron, followed by the icosahedron, dodecahedron, tetrahedron, and finally the cube. In this way the structure of the solar system and the distance relationships between the planets was dictated by the Platonic solids."
The Five Convex Regular Polyhedra (Platonic solids) -- thanks Wikipedia! Tetrahedron Hexahedron
or CubeOctahedron Dodecahedron Icosahedron 
- Now let's make a set of Platonic solids in 3-d: We're going to be
using "geomags" to construct them. We'll construct a set of Platonic
solids, and use them to fill in the following table:
# 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?
