Exploring 3D space with a computer - Part 2: a set of solids
Adrian Oldknow
Cube Tool
Now we will use some of the built-in regular solids to investigate their duals. First we will start with a cube. We just need two points A, B in the ground plane. Select the `Cube' tool from the ninth icon and build a cube with its base face centered on A and with B as a vertex. Use the `Midpoint' tool from the fifth icon to construct the midpoints of each face by selecting a pair of diagonally opposite points. Join these midpoints to show the edges of the polygon which has the midpoints of the cube's faces as its vertices. Do you recognise it?
In order to show it as a polyhedron we will `saw' off the eight corners of the cube in turn to reveal its faces. First we use the `Triangle' tool from the fourth icon to define a face of the dual shape by using three of the face midpoints of the cube. Then use the `Cut polyhedron' tool from the eighth icon. First select `this cube'. Then select the triangle to see the message `..Cut by a half space defined by this equilateral triangle'. A new convex polyhedron will be formed with the triangular pyramid cut away from the current
vertex. Repeat this process for each face in turn until you have an object defined by eight equilateral triangles. Now hide (or `mask') each triangle in turn and you will be left with the dual solid of the cube.
In the screen shot the dual solid has been left as with a solid surface style, and a new `transparent' cube created on points A, B to make the duality clear.
Can you imagine, and then create, the dual solid of the octahedron? Can you find a relationship between the edge-lengths of the original cube, its dual, and the dual-of-the-dual?
What are the dual solids of the other regular polyhedra:
tetrahedron, octahedron, dodecahedron, icosahedron?
You can download a set of Cabri 3D files showing the stages in the cutting process:
 Cube.cg3 |
Cube cut.cg3 |
| Cube cut two.cg3 |
Cube cut three.cg3 |
| Cube cut four.cg3 |
Cube cut five.cg3 |
| Cube cut six.cg3 |
Cube cut seven.cg3 |
| Cube cut eight.cg3 |
Cube dual.cg3 |
The working model of the final stage is below.
