# 'Create beautiful decorative three-dimensional solids'

How is it possible to fit two-dimensional shapes together to make three-dimensional ones? You might realise that six squares can be fitted together to make a cube, but that’s just the start of a long story!  There are many beautiful three-dimensional shapes that can be created by fitting other polygons together.  The polyhedron in this photograph is made completely from pentagons.  You can make this, and many other of these lovely shapes, in this activity.

Here are a few photos of the construction of the first bauble

Resources

You will need some scissors and a stapler.

A computer with access to a printer will be useful if you are going to construct the polygons for yourself.

Coloured paper can make the finished result even more spectacular!

The ready made  pentagons can be found here.

Printable instructions can be downloaded from here  Baubles’ Instructions

#### Constructing Nets

You can construct the shapes needed for the baubles in Geogebra

Watch this video for some help

Description

Here follows an outline of what the task is.

You will probably want to work in pairs to help speed up the cutting out.

#### Plato’s Baubles

1. Pentagons

The first bauble is made from 12 pentagons.

You could construct the pentagons yourself with Geogebra (see the ‘Constructing Nets’ help video) or you could use these ready made ones.

You could even try inserting a suitable image in your pentagon.

Print out the pentagons on paper (consider printing 6 in one colour and 6 in a different one).

Cut out the circles carefully.

Fold the curved parts over so that you are left with a pentagon with tabs like this

Fit the pentagons together, stapling the tabs on the outside to make a pretty bauble.

2. Triangles

The second bauble is made completely from triangles.

Construct the triangles to make a bauble like the animated diagram on the left[1].

Don't forget to include tabs by inscribing the triangles inside a circle.

Think about how many you need before you print them out!

Archimedes’s Baubles

You can make some interesting shapes from combining different polygons together.

3. Hexagons and Triangles

It is possible to make a bauble this shape using hexagons and triangles.

The tabs for the triangle and the tabs for the hexagon will be different sizes, but that doesn't matter.

Clue 1.  You will need to make 4 hexagons, but I won’t tell you how many triangles!

Clue 2. There is something very important about the lengths of the hexagons and triangles.  Can you work out what it is?

Here is short video that might give you a clue about how you could construct the hexagons and triangles that you need in Geogebra.

4. Triangles and Squares

Can you make the following shape into a bauble using triangles and squares.

This time there are no clues about how many of each shape you will need, but I'm sure you can work it out!

5. Faces, Vertices and Edges

For all of your 3D shapes, there is a relationship between the number of faces, vertices and edges.

Try to find it!

This formula was discovered by a very famous and great mathematician, Leonard Euler.  To see how useful it is, see how it is used to design a football

#### Footnotes

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