Exploring 3D space with a computer – Part 3: building models from plans
Adrian Oldknow
Folding
So we want to fold the nearest blue rectangle up around the edge AD. We have already seen that we can do this using the rotation transformation. The problem is to determine the maximum rotation needed to get to the correct position. In this case we know it will be 120º, but that is not the way (yet) we specify a rotation in Cabri 3D. In fact we know we need to stop folding when we reach the top point C in the previous figure. Of course, if we can `fold' one of the rectangles up, then we can use a reflection in a plane of symmetry of the object to fold the other up.
So here is the working model for folding up the rectangular sides. The point E should slide on a circular arc starting at E and finishing at C , but Cabri 3D does not support arcs (yet). Instead we use the point F on the segment EC as a `slider' to control the motion. We rotate the rectangle around AD by the angle defined by E and F . We reflect the result in the plane passing through C, O and the vertex of the green triangle.
You can download the file as 
prism-fold-sides.cg3.
You can also manipulate the model directly in this webpage.
The final task is to make the triangular ends fold up. We use exactly the same technique as for the rectangular sides. This time the point H slides on the segment GC and controls the rotation of triangle ABG about segment AB . The other end is the reflection of the folded up triangle in the plane which is the perpendicular bisector of AD .
So now you know all my tricks! It just remains to tidy up the unwanted bits by hiding objects, and making others very small. There are three points which control the position and dimensions of the box - and these are shown in pale blue. There are two points which control the folding of the sides and ends - and these are shown in yellow.
The file prism-fold-all.cg3 gives the final model, which you can also manipulate from this webpage below.
The Maths Channel programme also had a nice animation of wrapping six cheeses in a container in the shape of a hexagonal prism. Here is my attempt to copy this using our Cabri 3D techniques. Drag A to move the `cheeses'. When they fit nicely together on top of the hexagonal base, drag B to fold up the container. Points C and D control the dimensions of the cheeses, and the box. The file can be downloaded as wrapper4.cg3.
Well that concludes this instalment of our Cabri 3D " cordon bleu " monthly cookery course. Next time we look at how to build scale models.
