Exploring 3D space with a computer – Part 3: building models from plans
Adrian Oldknow
Create A Vertex
Here D is any point on the normal to the pink triangle at the vertex C . Each of the four green triangles has been rotated around the segment AB by the angle determined by C and D . In a similar way we can fold up the other triangles. Sliding any of the points D,E,F,G will fold around one of the corresponding edges. In the static image there is a visual illusion where you should be able to make the label A appear to be either inside or outside the vertex. But by spinning the view a little it should be clear that we are looking inside the angle A - like viewing an open umbrella from the handle end!
The working model is now shown below for you to manipulate, or to download as the file 
triangles-fold-all.cg3
So we have made a vertex of a shape made up from five equilateral triangles around the vertex. By gluing more triangles to this basic shape we should be able to build up the regular 20-sided solid known as the `icosahedron'.
Cundy & Rollett's book ` Mathematical Models ' gives a net consisting of a central band of ten triangles with further bands each of five triangles above and below.
The pictures here were made with the ATM's Mathematical Activity Tiles (MATs) which can be bought from:
http://www.atm.org.uk/buyonline/products/mat015.html
In order to build a Cabri 3D model of the pentagonal pyramid we must take care to construct one of the correct height. If we start with the pentagon we can construct two spheres each of radius AB on two adjacent vertices A, B as centre. These spheres have a circle as their `intersection curve'. So the top vertex V will be the intersection of the normal to the pentagon through its centre O with this circle - which is also its intersection with either sphere.
Using Cabri's `pyramid' tool and clicking first on the pentagon, then on the vertex V, we can create the desired pentagonal pyramid. In order to use Cabri's `icosahedron' tool we need to create one of the triangular faces - like the one shown in green. If the solid is created `on the wrong' side of the face, then we just we have to reflect it in that face, and hide the original!
The resulting model is in the Cabri file icosaherdon.cg3. We can find its centre as the midpoint of a pair of opposite vertices, and then construct the sphere with this as centre through any vertex, V , say. All twelve vertices should lie on the surface of this `circumsphere'. Can you find the radius of this sphere in terms of the edge length of the triangular faces? (This is hard!)
The icosahedron is also formed by 20 tetrahedra clustered around the centre point C . But these will only be regular tetrahedra if the radius of the circumsphere CV is the same as the edge length of the icosahedron. We can demonstrate this is not the case by getting Cabri 3D to construct a regular tetrahedron on the green triangular face, and then reflecting the vertex P in the face to the point P' - showing that edge length VP is greater than circumradius VC .
As well as having a circumsphere, the icosahedron has an `insphere'. Construct the midpoint M of any pair of adjacent vertices and construct the sphere centre C through M . It touches each of the 30 edges at their midpoints. Can you find the inradius of the icosahedron in terms of the edge length of the triangular faces? (This is hard, too!)
You can manipulate the image below, or download the file icosa-insphere.cg3 .
