3D Geometry

Pieces of 3D

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?

             

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 cuboctohedron.

  • 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?

             
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