[azeevreu] Tetrahedron octahedron building blocks

Create toy building blocks based on the tetrahedral-octahedral honeycomb.  In this honeycomb, the polyhedra attach in a checkerboard pattern: no two tetrahedra share a face, nor do two octahedra.  Therefore, we can have asymmetric connectors between them.

Three possible simple asymmetric connectors are magnets, Lego, and velcro.  Asymmetric connectors seem generally mechanically simpler than symmetric connectors that allow anything to attach to anything.  For many connector types, we will need some flexibility in the pieces, because we may want to simultaneously connect to multiple pieces at once when building a structure, e.g., fit a tetrahedron point first into a tetrahedral hole.  Magnetic connectors avoid this problem.

We could also do beams and connectors.

Pointy tetrahedra left on the floor are of course even more hazardous to step on with bare feet than regular Lego.  One could round off vertices quite a bit, even all the way to spheres (though that makes the volume difference even greater, assuming the octahedra keep their shape, because they naturally lay flat).

Fun things to build: an octahedron twice the size, a tetrahedron twice the size.

One could also apply the idea of asymmetric connectors joining pieces based on an underlying checkerboard honeycomb to the cubical honeycomb: color the cubes in a checkerboard pattern.

We can have larger pieces, e.g., beams and plates spanning multiple cells of the honeycomb.  Such pieces will have both kinds of connectors.

Many others have thought of this, including Hop's Delta Blocks, Patent Number 6,152,797. Date of Patent: Nov. 28, 2000 (so probably expired?), which references Escher's Flatworms.  Note that its tetrahedron face tile and octahedron face tile are subtly different (in beveling) so ought to be colored differently to avoid confusion unlike in the virtual mock-ups.  Also, unless the pieces are flexible, these pieces will have the simultaneous attachment problem mentioned above.