Polyhaus’ patent-pending designs are based upon polyhedral truncation.

Truncation is the removal of portions of polyhedral solids falling outside a set of symmetrically placed faces.

The operation […] displaces points along the edges of a polyhedron’s face by a ratio r<=1/2, where r is the fraction of the edge length at which to truncate.

https://mathworld.wolfram.com/Truncation.html

Semiregular polyhedrons are those whose faces are all regular polygons and their corners are alike. https://mathworld.wolfram.com/SemiregularPolyhedron.html

Archimedean Solids are the 13 convex polyhedra that have a similar arrangement of nonintersecting regular convex polygons of two or more different types arranged in the same way about each vertex with all sides the same length.

https://mathworld.wolfram.com/ArchimedeanSolid.html

Truncated Tetrahedron: a Semiregular 8-face Polyhedron made of 4-triangular faces and 4-hexagonal faces. 4{3} + 4{6}

Truncated Octahedron: a Semiregular 9-face Polyhedron made of 4-hexagonal faces, 4-triangular faces and 1-octagonal face. 4{6} + 4{3} + 1{8}

Truncated Cube: a Semiregular 14-face Polyhedron made of 8-triangular faces and 6-octagonal faces. 8{3} + 6{8}