Like other 3D figures, cubes have two-dimensional, one-dimensional, and zero-dimensional parts.
These pieces are related. For example, it's no coincidence that a cube has 6 faces, 12 edges, and 8 vertices, while an octahedron has 8 faces, 12 edges, and 6 vertices.
Which of these solids has more triangular faces?
Note: the figure on the right is an octahedron, not a square pyramid.
There are many different kinds of 3D shapes, and whole families of shapes can be made by modifying a basic solid.
Here are just a few of the ways we can do this:
- cut off the corners, which transforms this cube into a cuboctahedron.
- extrude the solid, which changes this cube into a icositettarahedron (24-hedron),
- or take the dual, which exchanges cubes for octahedrons.
This solid is called a cubeoctahedron.
One way to make it is by cutting all 8 corners off a cube. Another way to make it is by cutting all 6 corners off of an octahedron. Does it have more square faces or more triangular faces?
These two different transformations produce the same shape: a cuboctohedron.
Cutting the corners off a cube until the new faces share vertices:
Or, cutting the corners off an octahedron, until the new faces share vertices:
In both cases, we end up with 6 square faces and 8 triangular faces. In the first transformation, we start with 6 squares and gain the triangles by cutting off the 8 corners of the cube. In the second, we start with 8 triangles, and gain 6 squares by cutting off the 6 corners of the octahedron.
What face shape is gained by slicing off one corner of an icosahedron?
Below is a dodecahedron transforming into an icosahedron.
Which shape has more edges?