The Polyhedra

1. The Polyhedra Abbey Lind Alex Rockoff William Moreton

2. Renaissance • Inspiration • Demonstrated their mastery of perspective. • Deep religious and philosophical truths • Symmetries to go off of

3. Prehistoric Times • Egyptians • While forming the Great Pyramids, obviously knew about tetrahedrons. • Also knew about octahedrons and cubes

4. Prehistoric Continued • Dodecahedrons: toys in Scotland

5. Polyhedra can be found in many works completed during the 1900’s, such as these:

7. Present Day Polyhedra • Now examples of polyhedrons can be found in forms such as computer graphics

8. Definition • Comes from the greek word “poly,” meaning “many” and hedron,” meaning “face.” • Solid bounded by polygons

9. Basic Polyhedra: Prism • A polyhedron with two parallel, congruent bases • The remaining faces, lateral faces, are parallelograms • Right: bases are perpendicular to lateral edges, and lateral edges are rectangles, ex. Buildings • Oblique: anything else Right Oblique

10. Basic Polyhedra: Rectangular Parallelepipeds • Bases are all parallelograms • A right parallelepiped is a right prism with parallelograms as bases • A rectangular parallelepiped has bases that are rectangles, ex. A box

11. Basic Polyhedral: Cubes • A rectangular parallelepiped with each side equal • Also called the hexahedron, or “six-sided.” • Considered a regular polyhedron because all its faces are congruent regular polygons

12. Basic Polyhedral: Pyramid • Always named for the shape of the base • Example: triangular pyramid, quadrangular pyramid • Most common: • Regular:base is a regular polygon and the altitude passes through the center. Lateral faces are all congruent isosceles triangles • Term you may not know: • Frustum: the portion of the pyramid between the base and a plane section parallel to its base.

13. The Platonic Solids • Only five regular solids are possible: 1. Tetrahedron 2. Hexahedron 3. Octahedron 4. Dodecahedron 5. Icosahedron

14. Platonic Solids • Kepler • Made up the shape of the universe • The platonic solids were enclosed in a sphere (outer heaven) • Nested together, and the spheres inside of them=orbits of planets

15. Properties of Platonic Solids • Each can be circumscribed by a sphere, and each vertex will touch it • A sphere can also be inscribed in each Platonic solid, and it will touch each face at its center • Interesting fact: how an Icosahedron forms a golden rectangle

16. Semi-Regular Polyhedra • First kind: truncated, or Archimedean solids • Cut off corners of Platonic solids

18. Archimedean Solids

19. Semi-Regular Polyhedra • Second: star polyhedra • Extend faces of Platonic solids

21. StarPolyhedra

22. Class activity!

23. Euler’s Theorem for Polyhedrons • Vertices – edges + faces = 2 • This formula was announced by Euler in 1752 • Later led to become known as the field of topology

24. Euler’s Theorem for Polyhedrons

25. Euler’s Soccer Ball Proof • V-E+F=2 = F=E-V+2 • F = P + H (Pentagons + Hexagons) • E = (5P +6H) / 2 • V = (5P + 6H) / 3 • r = e - v + 2

26. Euler’s Soccer Ball Proof Cont. • r = e - v + 2 (Euler's formula) • P + H = (5P+6H)/2 - (5P+6H)/3 + 2 (Plug in values for r, e, and v) • 6P + 6H = 15P + 18H - 10P - 12H + 12 (Multiply both sides by 6) • 6P + 6H = 5P + 6H + 12 (Group terms) • P = 12 (Subtract 5P+6H from both sides) • The ball must contain exactly 12 pentagons

27. Homework • For a cube, how many faces, bases, and vertices can be found? • What is a defining characteristic of a prism? • Using Geometer’s sketchpad, create, print, and assemble your own polyhedra (can be same design as we did in class, or can be different- see page 298 in textbook for more ideas!)