1. Polyhedrons or Polyhedra A polyhedron is a solid formed by flat surfaces. We are going to look at regular convex polyhedrons: “regular” refers to the fact that every face, every edge length, every facial angle, and every dihedral angle (angle between two faces) are equal to all the others that constitute the polyhedron. “convex” refers to the fact that all of the sides of the shapes are flat planes, i.e., they are not “concave”, or dented in.

2. Characteristics of Regular Convex Polyhedra Each face is congruent to all others Each face is regular Each face meets the others in exactly the same way So how many regular polyhedra are there?

3. The History of the Platonic Solids April 11, 2005

4. Video Pull out your video chart quiz and fill it in as the video is played. The answers will not be as obvious as our last videos, so pay attention! You will need this information for your quiz on Friday!

5. A History of Platonic Solids There are five regular polyhedra that were discovered by the ancient Greeks. The Pythagoreans knew of the tetrahedron, the cube, and the dodecahedron; the mathematician Theaetetus added the octahedron and the icosahedron.

6. These shapes are called the Platonic solids, after the ancient Greek philosopher Plato; Plato, who greatly respected Theaetetus' work, speculated that these five solids were the shapes of the fundamental components of the physical universe

7. Tetrahedron The tetrahedron is bounded by four equilateral triangles. It has the smallest volume for its surface and represents the property of dryness. It corresponds to fire.

8. Hexahedron The hexahedron is bounded by six squares. The hexahedron, standing firmly on its base, corresponds to the stable earth.

9. Octahedron The octahedron is bounded by eight equilateral triangles. It rotates freely when held by two opposite vertices and corresponds to air.

10. Dodecahedron The dodecahedron is bounded by twelve equilateral pentagons. It corresponds to the universe because the zodiac has twelve signs corresponding to the twelve faces of the dodecahedron.

11. Icosahedron The icosahedron is bounded by twenty equilateral triangles. It has the largest volume for its surface area and represents the property of wetness. The icosahedron corresponds to water.

12. The Archimedean Solids April 7, 2003

13. The 13 Archimedean Solids All these solids were described by Archimedes, although, his original writings on the topic were lost and only known of second- hand. Various artists gradually rediscovered all but one of these polyhedra during the Renaissance, and Johannes Kepler finally reconstructed the entire set. A key characteristic of the Archimedean solids is that each face is a regular polygon, and around every vertex, the same polygons appear in the same sequence, e.g., hexagon- hexagon-triangle in the truncated tetrahedron. Two or more different polygons appear in each of the Archimedean solids, unlike the Platonic solids which each contain only a single type of polygon. The polyhedron is required to be convex.

14. Truncated Tetrahedron Truncated Octahedron Truncated Cube Cuboctahedron Great Rhombicuboctahedron Small Rhombicuboctahedron Snub Cube Truncated Icosahedron Truncated Dodecahedron Icosidodecahedron Great Rhombicosidodecahedron Small Rhombicosidodecahedron Snub Dodecahedron

15. Truncated Polyhedrons The term truncated refers to the process of cutting off corners. Truncation adds a new face for each previously existing vertex, and replaces n-gons with 2n- gons, e.g., octagons instead of squares. truncated cube cube

16. Snub Polyhedrons The term snub can refer to a process of replacing each edge with a pair of triangles, e.g., as a way of deriving what is usually called the snub cube from the cube. The 6 square faces of the cube remain squares (but rotated slightly), the 12 edges become 24 triangles, and the 8 vertices become an additional 8 triangles.

17. April Project http://www.scienceu.com/geometry/classroom /buildicosa/index.html This is the website that contains directions to your April Project: “Building an Icosahedron”. Your group is going to build one big Platonic solid, the icosahedron. You will have two class periods to work together. This project is due April 30 th.