Making Platonic Solids

This Project is contributed by Sahil XB

• Aim:To make Platonic solids and verification of Euler's formula , F - E + V = 2
  where , F represents number of faces , E represents the number of edges and V represents number of vertices .

Introduction
A platonic solid is a polyhedron all of whose faces are congruent regular polygons and where the same number of faces meet at every vertex.
Platonic solids are 3 – dimensional solids bounded by regular polygons which are equilateral triangle , square and regular Pentagon . They are only five in number , namely cube , Tetrahedron , Octahedron , Dodecahedron and Icosahedron .

• There are 5 platonic solids
  Why??

The Greeks recognized that there are only five platonic solids.

The key observation is that the interior angles of the polygons meeting at a vertex of a polyhedron add to less than 360 degrees. To see this note that if such polygons met in a plane, the interior angles of all the polygons meeting at a vertex would add to exactly 360 degrees.

• About Leonard Euler...
  Born April 15, 1707Basel, Switzerland
  Died September 18 ,1783St Petersburg, Russia
  Nationality Swiss
  Field Mathematics and physics
• Pre requisite Knowledge...
  
  Knowledge of a regular polygon
  and vertices , faces and edges of 3D solids
• Material Required
  Nets of Platonic solids ,a pair of scissors , transparent sheets (OHP sheets) ,a marker , cello tape , compass and a ruler.
• What is a net of a solid?
  A net is a form of skeleton-outline in 2-D which when folded results in a 3-D shape .
  For example . A tetrahedron net a dodecahedron net
  an icosahedron net

an octahedron net

a cube net


My observations

The interior angle of an equilateral triangle is 60 degrees. Thus on a regular polyhedron, only 3, 4, or 5 triangles can meet a vertex. If there were more than 6 their angles would add up to at least 360 degrees which they can't. Consider the possibilities:3 triangles meet at each vertex. This gives rise to a Tetrahedron .

The tetrahedron also has a beautiful and unique property ... all the four vertices are the same distance from each other! And it is the only Platonic Solid with no parallel faces.

Tetrahedron Facts
•It has 4 Faces
•Each face has 3 edges, and is actually an Equilateral Triangle
•It has 6 Edges
•It has 4 Vertices (corner points) and at each vertex 3 edges meet

cube Facts
•It has 6 Faces
•Each face has 3 edges, and is actually a Square
•It has 12 Edges
•It has 8 Vertices (corner points) and at each vertex 3 edges meet
Since the interior angle of a square is 90 degrees,therefore atmost 3 squares meet at a vertex.

A cube is called a hexahedron because it is a polyhedron that has 6 (hexa- means 6) faces.
Cubes make nice 6-sided dice, because they are regular in shape, and each face is the same size.

Octahedron Facts
•It has 8 Faces
•Each face has 3 edges, and is actually an Equilateral Triangle
•It has 12 Edges
•It has 6 Vertices (corner points) and at each vertex 4 edges meet
4 triangles meeting at a vertex,gives rise to an octahedron.

It is called an octahedron because it is a polyhedron that has 8 (octa-) faces, (like an octopus has 8 tentacles)

Icosahedron Facts
•It has 20 Faces
•Each face has 3 edges, and is actually an Equilateral Triangle
•It has 30 Edges
•It has 12 Vertices (corner points) and at each vertex 5 edges meet
5 triangles meeting at a vertex gives rise to an icosahedron.

It is called an icosahedron because it is a polyhedron that has 20 faces (from Greek icosa- meaning 20)

Dodecahedron Facts
•It has 12 Faces
•Each face has 5 edges, and is actually an Equilateral Triangle
•It has 30 Edges
•It has 20 Vertices (corner points) and at each vertex 3 edges meet
3 pentagons meeting at a vertex, gives rise to a dodecahedron.

It is called a dodecahedron because it is a polyhedron that has 12 faces (from Greek dodeca- meaning 12)

• Experimentation with Platonic Solids.

Now I will verify the Euler’s formula .
* F - E + V = 2
where , F represents number of faces , E represents the number of edges and V represents number of vertices .
Firstly, take a solid e.g. an octahedron. Using a pin and thermocol balls , mark the vertices.

It is observed that there are 6 vertices.

Now, count the number of faces.Using a marker ,number them.
It is observed that there are 8 faces.

Now, count the number of edges.

It is observed that there are 12 edges.

Now ,In an octahedron V= 6

F= 8

E= 12

Therefore, V+F-E= 6+8-12 =2

Thus Euler's formula is verified.
Similarly the formula is verified for other solids also.

This is a creative project done by Sahil.He has made beautiful Platonic Solids and experimented on them.