Polypro

1. Polypro Presentation by: Kerry Daugherty, Alison Divens, Ryan Warfel, Jennifer Watters, Ashlie Hill, Matt Winner, Lindsey Camarata, Amanda Hauf

2. Polyhedra • Polyhedra are three-dimensional solids which consist of a collection of polygons, usually joined at their polyhedron edges. • The word “polyhedra” is derived from the Greek word poly (many) plus the Indo-European word hedron (seat). • The plural of polyhedron is "polyhedra" (or sometimes "polyhedrons").

3. Euler’s polyhedral formula • A formula relating the number of polyhedron vertices V, faces F, and polyhedron edges E of a simply connected polyhedra. • Discovered independently by Euler (1752) . • Formula states: F+V-E=2

4. There are 5 types of regular polyhedra: • Tetrahedron • Cube • Octahedron • Dodecahedron • Icosahedron

5. Tetrahedron ` • Four triangular faces, four vertices, and six edges. • 4 - 6 + 4 = 2

6. Cube • Six square faces, eight vertices, and twelve edges. • 8 - 12 + 6 = 2

7. Octahedron • Eight triangular faces, six vertices, and twelve edges. • 6 - 12 + 8 = 2

8. Dodecahedron • Twelve pentagonal faces, twenty vertices, and thirty edges. • 20 - 30 + 12 = 2

9. Icosahedron • Twenty triangular faces, twelve vertices, and thirty edges. • 12 - 30 + 20 = 2

10. Why are there only 5 regular polyhedra? • Each vertex has at least three faces that come together, because if only two came together they would collapse against one another and we would not get a solid. • The sum of the interior angles of the faces meeting at each vertex must be less than 360°, for otherwise they would not all fit together.

11. The interior angles of the regular pentagon are 108°, so we can fit only three together at a vertex, giving us the dodecahedron. The interior angles of a hexagon are 120°, so the angles sum up to precisely 360°, and therefore they lie flat, just like four squares, and do not form a solid. Examples

12. The history of the polyhedra • The Greek philosopher Plato followed earlier philosophers in assigning the polyhedra to atoms of nature. • Plato also assigned polyhedra properties. • Johannes Kepler used polyhedra to explain planetary motion

13. Kepler’s theory on polyhedra • Johannes Kepler believed that polyhedra were related to the planets. • Believed that planets moved in circles around the sun. • Theory later proved wrong by Sir Isaac Newton.

14. Using Polypro • Polypro is a software program that can assist in exploring the 5 Platonic solids, along with pyramids and prisms.

15. Polypro • Polypro can be used to view and construct the 5 Platonic Solids, along with prisms and pyramids

16. Prisms • A prism is a polyhedron, with two parallel faces called bases. The other faces are always parallelograms. The prism is named by the shape of its base.

17. Pyramids • A polyhedron is a pyramid if it has 3 or more triangular faces sharing a common vertex. The base of a pyramid may be any polygon.

18. How to start constructing polyhedra • Create a net

19. Finishing your polyhedra • Connect all of the pieces together to form a solid. • You can now perform your own hands-on investigation of polyhedra properties.

20. Constructing the Polyhedron • Children can investigate polyhedrons by the hands-on activities

21. Teaching Polyhedrons and Polypro Elementary ( 4th and 5th grade) students may have difficulties understanding Polypro. They can, however, benefit from the hands-on activities of constructing their own polyhedron.

22. Teaching Polyhedrons and Polypro • Middle School students are able to use and understand Polypro. • They will be able to investigate the program with guidance. • They can draw and create nets for the objects and build a polyhedron. • The teacher can ask the children to be creative in their construction of their polyhedron. For example: Children can create a polyhedron out of marshmallows and toothpicks.

23. Conclusion • The polyhedron and the prism are important concepts in the field of geometry. Polypro is a tactful method in teaching these theories. Elementary and secondary teachers can benefit form using the visual aid of polypro to teach geometry.