Exploring 3D space with a computer – Part 3: building models from plans
Adrian Oldknow
Cube Challenge
Before we go on to the other 2D representation let's see how you managed with square in the cube challenge.
Did you find that A and B must divide their edges in the same ratio? Actually that ratio turns out to be 3:1 - which you may be able to prove e.g. by using coordinates. So here is what I believe to be the solution - using the `midpoint' tool twice. You can download the file `Square in cube.cg3' [Download] .
My calculations give the sides of the square as ¾ Ö 2 - so its area is just 12.5% more than that of a face.
We have just seen that we can produce a variety of 2D images from a 3D model created in Cabri 3D. Now we will explore `going in the other direction' - i.e. working from a `flattened' version of a model, known as a `net', to folding it up to produce a 3D solid.
First we can see that in the last of the icons, Cabri 3D gives us a kit of regular shapes, known to geometers as the `Platonic solids'. There are just five of these and we already explored, in part 2, that you can make a regular octahedron by joining up the midpoints of the faces of a cube - and vice versa - so that this pair of solids are known as `duals'. We will now see why there are only five regular solids, which are duals of which, and which of these solids fill space!
So first we can spread some regular polygons around on our `table' - i.e. Cabri's ground plane. Here we have a pink equilateral triangle, a green square and a blue regular pentagon. A line perpendicular to the `table' has been drawn through one corner of the triangle, which has then been rotated four times about it to make a nearly complete hexagon. Can you now imagine cutting out the shape made by the five triangles and folding them up so that the two free edges of the first and last triangle now join up?
Did you imagine a pentagonal pyramid lying on one of its triangular faces?
Let's see how we can start to model `folding up' in Cabri 3D. Basically we have to rotate part of the model around one edge of one of the triangular faces. Try dragging point D in the model below to see the basic idea. You can download the file as 
triangles-fold.cg3 .
