1. Geometry Polyhedra

2. 2 August 16, 2015 Goals Know terminology about solids. Identify solids by type. Use Euler’s Theorem to solve problems.

3. 3 August 16, 2015 Polyhedron A solid that is bounded by polygons. The polygons are faces. An edge is the intersection of two faces. A vertex is the intersection of three or more faces. Face

4. 4 August 16, 2015 Polyhedron

5. 5 August 16, 2015 Polyhedron Views Solid Wire Frame Hidden Line All three views will be used in these presentations, the text and other materials.

6. 6 August 16, 2015 Which of these are Polyhedrons? YES NO YES

7. 7 August 16, 2015 Concave Polyhedra A diagonal, or part of a diagonal, is outside the figure.

8. 8 August 16, 2015 Regular Polyhedra All of the faces are congruent, regular polygons.

9. 9 August 16, 2015 Cross Section The intersection of a solid and a plane. Cross section is a circle.

10. 10 August 16, 2015 Cross Section What is the intersection now? Cross section is a rectangle.

11. 11 August 16, 2015 What would the cross section be? A Square

12. 12 August 16, 2015 Leonard Euler 1707 – 1783 Probably the greatest mathematician of all time. Worked in, and made enormous contributions to, every branch of mathematics.

13. 13 August 16, 2015 Euler’s Formula Count F, the number of faces. 1 2 3 4 5 6 F = 6

14. 14 August 16, 2015 Euler’s Formula Count V, the number of vertices. 1 2 3 4 5 6 V = 8 7 8

15. 15 August 16, 2015 Euler’s Formula Count E, the number of edges. 1 2 3 4 5 6 E = 12 7 8 9 10 11 12

16. 16 August 16, 2015 Euler’s Formula Faces = 6 Vertices = 8 Edges = 12 V + F = E + 2

17. 17 August 16, 2015 Euler’s Formula Faces = 6 Vertices = 8 Edges = 12 6 + 8 = 12 + 2

18. 18 August 16, 2015 Euler’s Formula Faces = 6 Vertices = 8 Edges = 12 6 + 8 = 12 + 2 14 = 14

19. 19 August 16, 2015 Euler’s Formula

20. 20 August 16, 2015 Try another figure… Faces = Vertices = Edges = F + V = E + 2 5 + 5 = 8 + 2 10 = 10

21. 21 August 16, 2015 Euler’s Formula

22. 22 August 16, 2015 Solve: A polyhedron has 8 faces and 12 vertices. How many edges does it have? 18 V + F = E + 2 12 + 8 = E + 2 20 = E + 2 E = 18

23. 23 August 16, 2015 Solve: A polyhedron has 24 vertices and 36 edges. How many faces does it have? 14 V + F = E + 2 24 + F = 36 + 2 24 + F = 38 F = 14

24. 24 August 16, 2015 Solve: A polyhedron has 32 faces and 60 edges. How many vertices does it have? 30 V + F = E + 2 V + 32 = 60 + 2 V + 32 = 62 V= 30

25. 25 August 16, 2015 The Platonic Solids There are only five of them. They are regular, convex polyhedra. First described ca. 350 BC by Plato in Timaeus. Have been found in many ancient cultures.

26. 26 August 16, 2015 The Five Platonic Solids

27. 27 August 16, 2015 Tetrahedron Has four triangular sides. Associated with fire.

28. 28 August 16, 2015 Hexahedron (cube) Has six square sides. Associated with earth.

29. 29 August 16, 2015 Octahedron Has eight triangular sides. Associated with air.

30. 30 August 16, 2015 Dodecahedron Has 12 pentagonal faces. Associated with the heavens.

31. 31 August 16, 2015 Icosahedron Has 20 triangular faces. Associated with water.

32. 32 August 16, 2015 Johannes Kepler In 1596 Kepler published a tract called The Cosmic Mystery in which he envisioned the universe as consisting of nested Platonic Solids whose inscribed spheres determine the orbits of the planets, all enclosed in a sphere representing the outer heavens.

33. 33 August 16, 2015 Dungeons and Dragons

34. 34 August 16, 2015 Public Toilets in South Korea This is not a Platonic Solid. It is a compound polyhedron. Can you find out its correct name?

35. 35 August 16, 2015 Platonic Solid Links Mathworld GSP Icosahedron

36. 36 August 16, 2015 Summary A polyhedron is a solid object. The sides are faces. Regular polyhedra have congruent faces. There are 5 regular polyhedra (the Platonic Solids). Euler’s Formula: F + V = E + 2

37. 37 August 16, 2015 Homework