Discovering Euler's formula


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Discovering Euler’s formula on Polyhedrons

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Welcome to re discover Euler’s formula, which had fascinated mathematicians for a long period of time. This formula giving a relationship between the vertices, edges and faces of a solid had inspired the great mathematician Lakatos of the 20th century to write a book on mathematical philosophy as a mathematical discovery of Euler’s formula extending to all kinds of platonic solids. However, you are going to limit yourself to just the regular prisms and pyramids in this section. Leonard Euler was himself a Swiss mathematician of the 18th century.

 

You have to have a basic knowledge of polygons and their properties as a pre requite skill before doing this activity. Also refer to a mathematical dictionary to know about a polyhedron, its vertices, edges and faces. And all you have to do now, is to follow instructions to make a polyhedron(prisms and pyramids), count the number of vertices, edges and faces, tabulate them, analyze  and find a relationship between them.

 

Materials required: Marshmallows, spaghetti sticks/toothpicks

 

Building a triangular prism

Build two congruent triangles using the marshmallows and the spaghettis. Keep them parallel to each other and connect them using 3 spaghettis that are 6 cm long.

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Building a rectangular prism

Build two congruent rectangles using the marshmallows and the spaghettis. Keep them parallel to each other and connect them using 4 spaghettis that are 6 cm long.

 

Building a pentagonal prism

Build two congruent pentagons using the marshmallows and the spaghettis. Keep them parallel to each other and connect them using 5 spaghettis that are 6 cm long.

 

Continue the same procedure to build a hexagonal prism, a heptagonal prism and an octagonal prism.

 

Building a tetrahedron (pyramid with a triangular base)

Build a triangle with the marshmallows and the spaghetti sticks. Poke 3 spaghettis of the same lengths on to the 3 marshmallows of the triangle and connect them at the top by bending them towards the center using another marshmallow.

Building a square pyramid

Build a square with the marshmallows and the spaghetti sticks. Poke 4 spaghettis of the same lengths on to the 4 marshmallows of the square and connect them at the top by bending them towards the center using another marshmallow.

Building a pentagonal pyramid

Build a pentagon with the marshmallows and the spaghetti sticks. Poke 5 spaghettis of the same lengths on to the 5 marshmallows of the square and connect them at the top by bending them towards the center using another marshmallow.

Continue the same procedure to build a hexagonal pyramid, a heptagonal pyramid and an octagonal pyramid.

Now use your constructions to record the number of vertices, edges and faces of the various prisms.

 

Polyhedron

Number of vertices

Number of edges

Number of faces

Relationship

Triangular prism

 

 

 

 

Rectangular prism

 

 

 

 

Pentagonal prism

 

 

 

 

Hexagonal prism

 

 

 

 

Heptagonal prism

 

 

 

 

Octagonal prism

 

 

 

 

Triangular pyramid

 

 

 

 

Rectangular pyramid

 

 

 

 

Pentagonal pyramid

 

 

 

 

Hexagonal pyramid

 

 

 

 

Heptagonal pyramid

 

 

 

 

Octagonal pyramid

 

 

 

 

 

 

Congratulations if you were able to find a relationship between the vertices, edges and faces of a polyhedron. If you have not, or if you want to learn more about Euler’s formula, visit  Euler's formula