3D Geometry

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.

Pieces of 3D

             

Which of these solids has more triangular faces?

Note: the figure on the right is an octahedron, not a square pyramid.

Pieces of 3D

             

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.

Pieces of 3D

             

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?

Pieces of 3D

             

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.

Pieces of 3D

             

What face shape is gained by slicing off one corner of an icosahedron?

Pieces of 3D

             

Below is a dodecahedron transforming into an icosahedron.

Which shape has more edges?

Pieces of 3D

             
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