- Today:
- Review: Egyptian division
- Platonic Solids
Today's Question of the day:
$\frac{17}{25}$
It covers two kinds of symmetry that are very important: rotational and reflective.
So let's take a look at the definitions.
Part of your homework: to do the problems on the four pages of this symmetry handout. (If a problem says something like "on sheet F1-2", don't worry about it.)
- What is the most symmetric rectangle?
- What are regular polygons?
- How many regular polygons are there?
- What kinds of symmetry do they possess?
- Which regular polygons can be used exclusively to "tile" the plane, like a bathroom floor?
- 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."
- As an example of the application of a regular polygon,
consider some regular convex hexagonal
graph paper, which represents a tiling of the plane
by a regular convex polygon (and which we used to make our Leibniz hexacomb).
- 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.
- How many
Platonic solids are there, do you suppose?
- What's wrong with six sides, and beyond?
- 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 
- Let's make some Platonic solids in 3D. I suggest two different methods to construct them:
- We'll start by trying to make some, using some kits ("geomags"); and
- Platonic solids in 2-D (paper template). This set will be your "cheat sheet" for the upcoming exam.
# 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?
