1 / 27

The Polyhedra

The Polyhedra. Abbey Lind Alex Rockoff William Moreton. Renaissance. Inspiration Demonstrated their mastery of perspective. Deep religious and philosophical truths Symmetries to go off of. Prehistoric Times. Egyptians

fergal
Télécharger la présentation

The Polyhedra

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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:

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

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

  8. 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

  9. 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

  10. 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

  11. 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.

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

  13. 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

  14. 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

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

  16. Archimedean Solids

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

  18. StarPolyhedra

  19. Class activity!

  20. 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

  21. Euler’s Theorem for Polyhedrons

  22. 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

  23. 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

  24. 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!)

More Related