The first illustration depicts a tetrahedron inside the cube:

The next illustration depicts 2 opposing tetrahedrons inside the cube. The vertices of the 2 tetrahedrons exist at the same point in space as the vertices of the cube.

Looking at this drawing you can see how theverticesandmid-edgesof a tetrahedron can be projected outward to meet a sphere that encloses the cube and touches its 8 vertices. These become the vertices of the dome.The next illustrations show the

mid-edgesprojected to aspherethe same radius as the vertices. Blues lines are added to show the hemisphere level.

This drawing shows the alignment of great circles and chords.

Octahedral geometry (yellow) becomes apparent - it's the same as the 2 frequency Class II Octahedral dome

To thePAPER MODELof the Tetra-DomeThe Tetra-Dome top half is subdivided and made into a 14 ft diameter 2 x 4 dome with a 4 ft. vertical Base Option and door retrofit. It can make good enclosed space, low cost and functional.

Some diamond crystals have similar structure on a molecular level- see

American Museum of Natural History

and

Earth Science Australia

The Platonic and Archimedean Solids (VRML)