I love these octahedrons. They are one of the most simple multi-axis turnings.

They start with a sphere which I have been demonstrating for many years. I skipped quickly through the sphere portion in this video to focus more on the multi-axis turning.

The sphere is oriented on three axes. If you think of a cube, these are the perpendicular through the faces of the cube.

After boring out the holes along the axes, these are transformed into tubes. Finally, at the center, a smaller cube forms. This can be carved away into pleasing curves.

For a truncated octahedron, pick one or more axes before boring. Bore out only enough to form a solid tube instead of a hollow tube. Part off or carve away the solid tube.

BTW – I saw an original non-truncated octahedron demonstrated by Max Brosi at AAW – Raleigh. He deserves credit for confirming my octagon measures using drafting techniques instead of my use of algebra. Then showing an octahedron. A confirming demonstration.

Enjoy.