- You do have a homework
assignment due today
- Reminder: you will have an exam after fall break, Friday the 21st. It will cover everything from our last exam to Pascal's triangle.
- Question of the week: What makes some solids so Platonic?
- Review and extension of Platonic
solids
- From the October 7th, 2011 New York Times
- What can you do with a dodecahedron?
- Icosahedral
- amoeboid protozoa Circogonia icosahedra
- Viruses:
- The Cliff Notes Version...
"An array of viruses. (a) The helical virus of rabies. (b) The segmented helical virus of influenza. (c) A bacteriophage with an icosahedral head and helical tail. (d) An enveloped icosahedral herpes simplex virus. (e) The unenveloped polio virus. (f) The icosahedral human immunodeficiency virus with spikes on its envelope."
- Bacteriophages and Virions
- The Cliff Notes Version...
- Dodecahedral
- Viruses:
- Viruses:
- 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."
- There are exactly five platonic solids:
Tetrahedron Hexahedron
or CubeOctahedron Dodecahedron Icosahedron 
- Today we will create a set of Platonic solids in 3-d, using
a paper template.
One of the advantages of making a nice set of these is that you will be able to use them, with anything you want written on them, for the next exam. They
- must be constructed -- built! no flat versions -- from an 8.5x11 piece of paper (just like the one I give you in class); and
- You can't undo them during the test.
- Let's 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?
- From the October 7th, 2011 New York Times
- Duality
- Each Platonic solid has a "twin" -- called its dual -- which we can discover from the table.
- Duality in nature:
- Let's spin some duals....
- Each Platonic solid has a "twin" -- called its dual -- which we can discover from the table.
- Java applets allow one to play with Platonic solids
- A Treatise on Symmetry
- hexagonal graph paper (make your own!)
- Platonic solids in 2-D (paper template)
- concave versus convex (an animation)