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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

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.

+ 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)

+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:

+ The Platonic Solids

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

+ 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

+ The Hexahedron (Cube)

4 Squares meet at each vertex 6 Faces 4 vertices 12 Edges http://www.mathsisfun.com/geometry/hexahedron.html

+ The Octahedron

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

+ The Dodecahedron

3 pentagons meet at each vertex 12 Faces 20 Vertices 30 Edges http://www.mathsisfun.com/geometry/

dodecahedron.html

+ The Icosahedron

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

+ 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.

+ Making the Models

http://www.learner.org/interactives/geometry/platonic.html

+Question 1

How many Platonic Solids are there?

+Question 2

Platonic Solids are also called regular _______

+Question 3

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

+Question 4

Which Platonic Solid has 12 faces?

+Question 5

Which Platonic Solid did Plato associate with fire?

+Question 6

Which Platonic Solid did Plato associate with air?

+Question 7

Which Platonic Solid did Plato associate with water?

+Question 8

Which Platonic Solid did Plato associate with earth?

+Question 9

Which Platonic Solid did Plato associate with the universe?

+Question 10

What is the measure of each angle of a regular pentagon?

+Question 11

What is the minimum number of faces meeting at a vertex of a polyhedron?

+Question 12

How many Platonic Solids consist of equilateral triangles?

+Question 13

How many equilateral triangles meet at a vertex of an icosahedron?

+Question 14

Why can’t we make a Platonic Solid with 6 equilateral triangles meeting at a vertex?

+Question 15

Which Platonic Solid consists of squares? Why is their only one?

+Question 16

Which Platonic Solid consists of regular pentagons? Why is their only one?

+Question 17

Why can’t we make a Platonic Solid with regular hexagon faces?

+Question 18

Why are there only 5 Platonic Solids?

+Question 19

Why is the relationship between tessellations and Platonic Solids?

+Question 20

What is Euler’s formula?

+Question 21

Why are Platonic Solids often used for dice?

+Question 22

What are some real-life examples of Platonic Solids?