these tessellations are found in crystal arrangements. The alternated cubic honeycomb is of special importance since its vertices form a cubic close-packing of spheres. The space-filling truss of packed octahedra and tetrahedra was apparently first discovered by Alexander Graham Bell and independently re-discovered by Buckminster Fuller (who called it the octet truss and patented it in the 1940s). . Octet trusses are now among the most common types of truss used in construction. Frieze forms If cells are allowed to be uniform tilings, more uniform honeycombs can be defined: Families: ×: [4,4,2] Cubic slab honeycombs (3 forms) ×: [6,3,2] Tri-hexagonal slab honeycombs (8 forms) ×: [(3,3,3),2] Triangular slab honeycombs (No new forms) ××: [∞,2,2] = Cubic column honeycombs (1 form) ×: [p,2,∞] Polygonal column honeycombs (analogous to duoprisms: these look like a single infinite tower of p-gonal prisms, with the remaining space filled with apeirogonal prisms) ××: [∞,2,∞,2] = [4,4,2] - = (Same as cubic slab honeycomb family) The first two forms shown above are semiregular (uniform with only regular facets), and were listed by Thorold Gosset in 1900 respectively as the 3-ic semi-check and tetroctahedric semi-check. Scaliform honeycomb A scaliform honeycomb is vertex-transitive, like a uniform honeycomb, with regular polygon faces while cells and higher elements are only required to be orbiforms, equilateral, with their vertices lying on hyperspheres. For 3D honeycombs, this allows a subset of Johnson solids along with the uniform polyhedra. Some scaliforms can be generated by an alternation process, leaving, for example, pyramid and cupola gaps. Hyperbolic forms There are 9 Coxeter group families of compact uniform honeycombs in hyperbolic 3-space, generated as Wythoff constructions, and represented by ring permutations of the Coxeter-Dynkin diagrams for each family. From these 9 families, there are a total of 76 unique honeycombs generated: [3,5,3] : - 9 forms [5,3,4] : - 15 forms [5,3,5] : - 9 forms [5,31,1] : - 11 forms (7 overlap with [5,3,4] family, 4 are unique) [(4,3,3,3)] : - 9 forms [(4,3,4,3)] : - 6 forms [(5,3,3,3)] : - 9 forms [(5,3,4,3)] : - 9 forms [(5,3,5,3)] : - 6 forms Several non-Wythoffian forms outside the list of 76 are known; it is not known how many there are. Paracompact hyperbolic forms There are also 23 paracompact Coxeter groups of rank 4. These families can produce uniform honeycombs with unbounded facets or vertex figure, including ideal vertices at infinity: References John H. Conway,