The Cubical Pyramid

The cubical pyramid is constructed by tapering a 3D cube along the W-axis. It consists of a cubical cell, which is its base, and six square pyramidal cells, its sides, which converge on its apex.

Parallel Projection

The following diagram shows a parallel projection of the cubical pyramid:

The cubical base is visible, as well as the 6 square pyramidal cells, which all converge at the apex to the right. The blue lines outline the pyramidal cell farthest away from the viewer.

Perspective Projections

The next diagram shows a perspective projection of the cubic pyramid. This viewpoint happens to look directly at its cubical base, and hence the projection has a cubical envelope.

The center of the cube in this projection is the apex of the cubical pyramid. The blue lines outline the 6 square pyramids attached to each face of the cube, with their apices meeting at the apex of the cubical pyramid. The following diagrams show these square pyramid cells:

The next projection shows the cubical pyramid rotated slightly in the ZW plane. We see that its apex now projects outside its cubical base.

The last projection shows the cubical pyramid rotated 90 degrees in the ZW plane, so that its base lies on the XYW hyperplane.

We see that its cubical base now projects onto a flat square, and it now has a square pyramidal envelope.

Geometric Relationships

Eight cubical pyramids can be joined together with their apices sharing a common vertex, to form a hypercube.

Eight cubical pyramids can also be attached at their bases to the cubical cells of a hypercube: the square pyramidal cells pair up to form octahedra, resulting in a 24-cell.