Truncated triangular pyramid number explained

A truncated triangular pyramid number^[1] 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.^[2]

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.^[3] Dense packing of convex objects is related to problems like the arrangement of molecules in condensed states of matter^[4] and to the best way to transmit encoded messages over a noisy channel.^[5] 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.^[6]

It is hypothesised that a regular tetrahedron might possibly be the convex body having the smallest possible packing density.^[3] In contrast to this, the densest known packing of truncated tetrahedra can have an exceptionally high packing fraction φ = 207/208 = 0.995192...^[7]

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.^[8] Semiconductors are one of the most active areas of cluster research.^{[9] [10]}

Examples

Tetrahedral Number 20 yields Truncated Triangular Pyramid Number 7 by truncating Tetrahedral number (or triangular pyramidal number) 4,4,4 and 1 from its vertices

Tetrahedral Number 35 yields Truncated Triangular Pyramid Number 19 by truncating Tetrahedral number (or triangular pyramidal number) 4 from each of the vertices

Tetrahedral Number 286 yields Truncated Triangular Pyramid Number 273 by truncating Tetrahedral number (or triangular pyramidal number) 4,4,4 and 1 from its vertices

Tetrahedral Number 560 also yields Truncated Triangular Pyramid Number 273 by truncating Tetrahedral number (or triangular pyramidal number) 84,84,84 and 35 from its vertices

Tetrahedral Number 816 yields Truncated Triangular Pyramid Number 689 by truncating Tetrahedral number (or triangular pyramidal number) 56,35,35 and 1 from its vertices

Tetrahedral Number 969 yields Truncated Triangular Pyramid Number 833 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

In other fields

1. Truncated triangular silver nanoplates synthesized in large quantities using a solution phase method^[11]
2. Theoretical study of hydrogen storage in a truncated triangular pyramid molecule^[12]
3. Packing and self-assembly of truncated triangular bipyramids^[13]

Notes and References

1. Truncated triangular pyramid numbers).
2. Web site: Weisstein . Eric W. . Truncated Tetrahedral Number . 2024-10-23 . mathworld.wolfram.com . en.
3. 10.1073/pnas.0601389103 . Packing, tiling, and covering with tetrahedra . 2006 . Conway . J. H. . Torquato . S. . Proceedings of the National Academy of Sciences . 103 . 28 . 10612–10617 . free . 16818891 . 1502280 . 2006PNAS..10310612C .
4. Web site: Torquato, S. (2002) Random Heterogeneous Materials: Microstructure and Macroscopic Properties (Springer, New York) . 10.1115/1.1483342 .
5. Book: Conway, J. H. & Sloane, N. J. A. (1998) Sphere Packings, Lattices and Groups (Springer, New York). . 978-1-4757-2249-9 . Conway . J. H. . Sloane . N. J. A. . 9 March 2013 . Springer .
6. Web site: Hales, T. C. (2005) Ann. Math. 162, 1065–1185. . 20159940 .
7. https://pubs.aip.org/aip/jcp/article/135/15/151101/190281/Communication-A-packing-of-truncated-tetrahedra
8. Magic numbers in polygonal and polyhedral clusters . 1985 . 10.1021/ic00220a025 . Teo . Boon K. . Sloane . N. J. A. . Inorganic Chemistry . 24 . 26 . 4545–4558 .
9. Clusters . 1996 . 10.1126/science.271.5251.889 . Brauman . John I. . Science . 271 . 5251 . 889 . 1996Sci...271..889B .
10. Califano . Marco . 2024-02-13 . Tetrahedral vs Spherical Nanocrystals: Does the Shape Really Matter? . Chemistry of Materials . en . 36 . 3 . 1162–1171 . 10.1021/acs.chemmater.3c01643 . 0897-4756.
11. Synthesis and Characterization of Truncated Triangular Silver Nanoplates . 10.1021/nl025674h . 2002 . Chen . Sihai . Carroll . David L. . Nano Letters . 2 . 9 . 1003–1007 . 2002NanoL...2.1003C .
12. Theoretical study of hydrogen storage in a truncated triangular pyramid molecule consisting of pyridine and benzene rings bridged by vinylene groups . 10.1007/s00339-018-1841-9 . 2018 . Ishikawa . Shigeru . Nemoto . Tetsushi . Yamabe . Tokio . Applied Physics A . 124 . 6 . 418 . 2018ApPhA.124..418I .
13. Packing and self-assembly of truncated triangular bipyramids . 10.1103/PhysRevE.88.012127 . 2013 . Haji-Akbari . Amir . Chen . Elizabeth R. . Engel . Michael . Glotzer . Sharon C. . Physical Review E . 88 . 1 . 012127 . 23944434 . 1304.3147 . 2013PhRvE..88a2127H .