Exploring 3D space with a computer - Part 2: a set of solids
Adrian Oldknow
Cube Tool Continued
If we saw through the cube the cut face may be one of many different shapes. In the next model we will take three points P,Q,R on the edges of the cube to define a plane by which to cut the cube. Can you visualise which edges will be cut by the plane, and the shape they will form?
The `Cube' tool creates a solid cube with eight vertices. In the figure we have made the cube style `empty' and then constructed each of the 12 edges as a segment joining two vertices.
Here is the resulting solid and the cut face - did you expect this?
Explore ways of cutting which give you triangles (e.g. scalene, right-angled, isosceles, equilateral, obtuse), or quadrilaterals (e.g. square, rectangle, rhombus, parallelogram, kite), pentagons or hexagons?
You can manipulate the model below, or download the files 
Cube sawing.cg3 and Cube sawn.cg3.
A real-life application of sawing through solids is in the process of faceting gem stones. So you might like to explore designs posted on web-sites such as:
http://www.gemsociety.org/ http://www.gemcutter.com/ or http://www.faceters.com/
and see if you can model them in Cabri 3D.
A mathematical application of sawing through solids is identifying planes of symmetry of solids. For example there are some obvious ways of sawing through a cube which divide it into two parts, one of which is the reflection of the other in the plane defining the cut.
Here are two simple ways of cutting a cube. Can you say how many different ways each cut can be made?
The first method uses the midpoints of four of the parallel edges. The second uses a pair of opposite edges. You can download the files `Cube symmetry one.cg3' and `Cube symmetry two.cg3' [Hyperlinks] . Check that the reflection in the green polygon of the convex polyhedron (i.e. the half-cube) does fill the vacant space in the cube.
A third possibility is shown here. This uses two opposite vertices and the midpoints of a pair of parallel edges to define a quadrilateral.
What shape is the quadrilateral?
Does it divide the cube into two equal halves?
Does the reflection in the quadrilateral of the bottom half fill the vacant top half?
You can download the file `Cube symmetry three.cg3' here [Download] .
To complete this exploration into mathematical solids we will use some techniques to create stellations of polyhedra. The example here is a simple stellation of the regular pentagon using the `Pyramid' tool.
See for example: http://mathworld.wolfram.com/SmallStellatedDodecahedron.html
