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Platonic Solids

Platonic Solids. Regular Polyhedra. Platonic Solids. http://www.youtube.com/watch?v=C36h00d7xGs What are Platonic Solids? A Platonic Solid is a 3D shape where: 1. Each face is the same regular polygon 2. The same number of polygons meet at each vertex. Pythagoras & Plato.

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Platonic Solids

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  1. Platonic Solids Regular Polyhedra

  2. Platonic Solids • http://www.youtube.com/watch?v=C36h00d7xGs • What are Platonic Solids? • A Platonic Solid is a 3D shape where: • 1. Each face is the same regular polygon • 2. The same number of polygons meet at each vertex.

  3. Pythagoras & Plato • The Regular Polyhedron were first studied by a group of Greek mathematicians under the leadership of Pythagoras in the 6th century B.C • Regular Polyhedron: A solid having faces (surfaces) in the shape of a regular polygon. All of its faces, edges, and corners are identical. • The reason they are called the Platonic solids is because they were named after Plato. Plato is the person who gave instructions on how to make the models of Platonic solids.

  4. Plato • Plato claimed that the atoms of the 4 elements of ancient science had the shape of regular polyhedra. • The elements that made up the Earth were believe to be: • Earth (Cube) • Air (Octahedron) • Fire (Tetrahedron) • Water (Icosahedron)

  5. The ideas that Plato had about the elements of the Earth and their shapes may seem naïve now but he was correct that many things in nature are in the shape of Regular Polyhedron. • Examples:

  6. The Platonic Solids • There are only 5 Platonic Solids: • 1. Tetrahedron • 2. Hexahedron • 3. Octahedron • 4. Dodecahedron • 5. Icosahedron

  7. The Tetrahedron • 3 Triangles meet at each vertex • 4 Faces • 4 Vertices • 6 Edges • A 4 sided dice is a great example. • http://www.mathsisfun.com/geometry/tetrahedron.html

  8. The Hexahedron (Cube) • 4 Squares meet at each vertex • 6 Faces • 4 vertices • 12 Edges • http://www.mathsisfun.com/geometry/hexahedron.html

  9. The Octahedron • 4 Triangles meet at each vertex • 8 Faces • 6 Vertices • 12 Edges • http://www.mathsisfun.com/geometry/octahedron.html

  10. The Dodecahedron • 3 pentagons meet at each vertex • 12 Faces • 20 Vertices • 30 Edges • http://www.mathsisfun.com/geometry/dodecahedron.html

  11. The Icosahedron • 5 Triangles meet at each vertex • 20 Faces • 12 Vertices • 30 Edges • http://www.mathsisfun.com/geometry/icosahedron.html

  12. Models of Regular Polyhedrons • In a painting done in 1525, Albrecht Dürer a German artist, showed how to make paper models of the five regular polyhedrons. • He suggested drawing the diagrams on thick paper which is what we will be doing.

  13. Making the Models • http://www.learner.org/interactives/geometry/platonic.html

  14. Question 1 • How many Platonic Solids are there?

  15. Question 2 • Platonic Solids are also called regular _______

  16. Question 3 • Platonic Solid have all the same regular polygon, and the same number of ______ meet at each _______ .

  17. Question 4 • Which Platonic Solid has 12 faces?

  18. Question 5 • Which Platonic Solid did Plato associate with fire?

  19. Question 6 • Which Platonic Solid did Plato associate with air?

  20. Question 7 • Which Platonic Solid did Plato associate with water?

  21. Question 8 • Which Platonic Solid did Plato associate with earth?

  22. Question 9 • Which Platonic Solid did Plato associate with the universe?

  23. Question 10 • What is the measure of each angle of a regular pentagon?

  24. Question 11 • What is the minimum number of faces meeting at a vertex of a polyhedron?

  25. Question 12 • How many Platonic Solids consist of equilateral triangles?

  26. Question 13 • How many equilateral triangles meet at a vertex of an icosahedron?

  27. Question 14 • Why can’t we make a Platonic Solid with 6 equilateral triangles meeting at a vertex?

  28. Question 15 • Which Platonic Solid consists of squares? Why is their only one?

  29. Question 16 • Which Platonic Solid consists of regular pentagons? Why is their only one?

  30. Question 17 • Why can’t we make a Platonic Solid with regular hexagon faces?

  31. Question 18 • Why are there only 5 Platonic Solids?

  32. Question 19 • Why is the relationship between tessellations and Platonic Solids?

  33. Question 20 • What is Euler’s formula?

  34. Question 21 • Why are Platonic Solids often used for dice?

  35. Question 22 • What are some real-life examples of Platonic Solids?

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