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Platonic Solids

Platonic Solids. Plato (427 BC – 347 BC). Greek philosepher 387 BC based a school in Athina Died 80 years old. Archeological discovery from Scotland 2000BC. Platonic Solids in the antic. Johannes Kepler (27.12.1571 Weil der Stadt – 15.11.1630 Regensburg).

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Platonic Solids

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  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

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