Mathematical concept
A truncated triangular pyramid number is found by removing (truncating) some smaller tetrahedral number (or triangular pyramidal number) from each of the vertices of a bigger tetrahedral number.
The number to be removed (truncated) may be same or different from each of the vertices.
Properties.
A truncated number is not the same as the volume or area of the truncated shape.
Instead numbers relate more to the problem of how densely given solid objects can pack in space. Dense packing of convex objects is related to problems like the arrangement of molecules in condensed states of matter and to the best way to transmit encoded messages over a noisy channel. Kepler's conjecture, which postulated that the densest packings of congruent spheres in 3-dimensional space have packing density (fraction of space covered by the spheres) = pi / sqrt 18 = 74.048% was proved by variants of the face-centered cubic (FCC) lattice packing.
It is hypothesised that a regular tetrahedron might possibly be the convex body having the smallest possible packing density. In contrast to this, the densest known packing of truncated tetrahedra can have an exceptionally high packing fraction φ = 207/208 = 0.995192...
Truncated numbers are also relevant to cluster science in inorganic chemistry. Central to the chemical and physical study of clusters is a understanding of their molecular and electronic structures which is determined by the number of atoms in a cluster of given size and shape and their arrangement or disposition. Semiconductors are one of the most active areas of cluster research.
Examples.
Tetrahedral Number 20 (sequence in the OEIS) yields Truncated Triangular Pyramid Number 7 (sequence in the OEIS) by truncating Tetrahedral number (or triangular pyramidal number) 4,4,4 and 1 from its vertices
Tetrahedral Number 35 (sequence in the OEIS) yields Truncated Triangular Pyramid Number 19 (sequence in the OEIS) by truncating Tetrahedral number (or triangular pyramidal number) 4 from each of the vertices
Tetrahedral Number 286 (sequence in the OEIS) yields Truncated Triangular Pyramid Number 273 (sequence in the OEIS) by truncating Tetrahedral number (or triangular pyramidal number) 4,4,4 and 1 from its vertices
Tetrahedral Number 560 (sequence in the OEIS) also yields Truncated Triangular Pyramid Number 273 (sequence in the OEIS) by truncating Tetrahedral number (or triangular pyramidal number) 84,84,84 and 35 from its vertices
Tetrahedral Number 816 (sequence in the OEIS) yields Truncated Triangular Pyramid Number 689 (sequence in the OEIS) by truncating Tetrahedral number (or triangular pyramidal number) 56,35,35 and 1 from its vertices
Tetrahedral Number 969 (sequence in the OEIS) yields Truncated Triangular Pyramid Number 833 (sequence in the OEIS) by truncating Tetrahedral number (or triangular pyramidal number) 56,35,35 and 10 from its vertices
Related numbers.
Certain truncated triangular pyramid numbers possess other characteristics:
273 (number) is also a sphenic number and an idoneal number
204 (number) is also a square pyramidal number and a nonagonal number
References.
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