1 / 22

Introduction to Platonic Solids: From Plato to Kepler

Explore the history of Platonic Solids from Plato to Kepler, their mathematical properties, and the proof of their existence through specific polyhedrons like Regular Tetrahedron, Octahedron, Isosahedron, and more.

lcordell
Download Presentation

Introduction to Platonic Solids: From Plato to Kepler

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. Platonic Solids

  2. Plato(427 BC – 347 BC) • Greek philosepher • 387 BC based a school in Athina • Died 80 years old

  3. Archeological discovery from Scotland 2000BC

  4. Platonic Solids in the antic

  5. Johannes Kepler(27.12.1571 Weil der Stadt – 15.11.1630 Regensburg) • German mathematican and astronomer • Several years lived in Prague as a guest of Rudolf II

  6. Karlova ulice 4, Prague, house of Johannes Kepler

  7. Solar System Model

  8. Platonic Solid • Convex polyhedron • All faces are equal regular polygon (K vertecies) • In each vertex of the polyhedron the same number of edges intersects (L edges)

  9. Which Platonic Solids exist? • V … number of vertecies • F … number of faces • E … number of edges • Euler formula: For each convex polyhedron V + F = E +2

  10. Which Platonic Solids exist? • The conditions of Platonic Solids • K.F = 2.E F = 2E/K • L.V = 2.E V = 2E/L • Substitution into Euler formula 2E/L + 2E/K = E +2 1/L + 1/K = 1/E + 1/2

  11. For which numbers holds1/K+1/L = 1/E +1/2 • K and L are integers bigger or equal to 3 • For K = 3 • L=3, E=6: 1/3+1/3 = 1/6+1/2 • L=4, E=12: 1/3+1/4 = 1/12+1/2 • L=5, E=30: 1/3+1/5 = 1/30+1/2 • For L=>6 not possible • For K = 4 • L=3, E=12: 1/4+1/3 = 1/12+1/2 • For L=>4not possible • For K = 5 • L=3, E=30: 1/5+1/3 = 1/30+1/2 • Pro L=>4not possible • For K => 6 not possible even for L=3

  12. Důkaz neexistence více než 5ti pravidelných mnohostěnů • From previous equatation we know that only following polyhedrons can exist as a Platonic Solids • Ti proof their existence we must construct them

  13. Regular Tetrahedron • 4 3-angle faces, 4 verticies, in each 3 edges, total 6 edges

  14. Regular Tetrahedron • 4 3-angle faces, 4 verticies, in each 3 edges, total 6 edges

  15. Regular Octahedron • 8 3-angle faces, 6 verticies, in each 4 edges, total 12 edges

  16. Regular Octahedron • 8 3-angle faces, 6 verticies, in each 4 edges, total 12 edges

  17. Regular Isosahedron • 20 3-angle faces, 12 vertecies, in each 5 edges, total 30 edges

  18. Regular Isosahedron • 20 3-angle faces, 12 vertecies, in each 5 edges, total 30 edges

  19. Regular 6-hedron • 6 4-angle faces, 8 vertecies, in each 3 edges, total 12 edges

  20. Cube • 6 4-angle faces, 8 vertecies, in each 3 edges, total 12 edges

  21. Regular Dodecahedron • 12 5-angle faces, 20 vertecies, in each 3 edges, total 30 edges

  22. Regular Dodecahedron • 12 5-angle faces, 20 vertecies, in each 3 edges, total 30 edges

More Related