Transcript polypro P1
Polypro Presentation by: Kerry Daugherty, Alison Divens, Ryan Warfel, Jennifer Watters, Ashlie Hill, Matt Winner, Lindsey Camarata, Amanda Hauf 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 IndoEuropean word hedron (seat). The plural of polyhedron is "polyhedra" (or sometimes "polyhedrons"). 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 There are 5 types of regular polyhedra: • Tetrahedron • Cube • Octahedron • Dodecahedron • Icosahedron Tetrahedron Four triangular ` faces, four vertices, and six edges. 4 - 6 + 4 = 2 Cube Six square faces, eight vertices, and twelve edges. 8 - 12 + 6 = 2 Octahedron Eight triangular faces, six vertices, and twelve edges. 6 - 12 + 8 = 2 Dodecahedron Twelve pentagonal faces, twenty vertices, and thirty edges. 20 - 30 + 12 = 2 Icosahedron Twenty triangular faces, twelve vertices, and thirty edges. 12 - 30 + 20 = 2 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. Examples 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. 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 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. Using Polypro Polypro is a software program that can assist in exploring the 5 Platonic solids, along with pyramids and prisms. Polypro Polypro can be used to view and construct the 5 Platonic Solids, along with prisms and pyramids 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. 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. How to start constructing polyhedra Create a net 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. Constructing the Polyhedron Children can investigate polyhedrons by the hands-on activities 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. 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. 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.