Truncated triangular pyramid number

A truncated number is not the same as the volume or area of the truncated shape.{{huh?|date=October 2024}}

Instead numbers relate more to the problem of how densely given solid objects can pack in space.{{cite journal | doi=10.1073/pnas.0601389103 | title=Packing, tiling, and covering with tetrahedra | date=2006 | last1=Conway | first1=J. H. | last2=Torquato | first2=S. | journal=Proceedings of the National Academy of Sciences | volume=103 | issue=28 | pages=10612–10617 | doi-access=free | pmid=16818891 | pmc=1502280 | bibcode=2006PNAS..10310612C }} Dense packing of convex objects is related to problems like the arrangement of molecules in condensed states of matter{{Cite journal |title=Torquato, S. (2002) Random Heterogeneous Materials: Microstructure and Macroscopic Properties (Springer, New York) |journal=Applied Mechanics Reviews |date=July 2002 |volume=55 |issue=4 |pages=B62–B63 |doi=10.1115/1.1483342 |url=https://doi.org/10.1115/1.1483342 |last1=Haslach |first1=HW }} and to the best way to transmit encoded messages over a noisy channel.{{Cite book |title=Conway, J. H. & Sloane, N. J. A. (1998) Sphere Packings, Lattices and Groups (Springer, New York). | isbn=978-1-4757-2249-9 |url=https://books.google.com/books?id=hoTjBwAAQBAJ | last1=Conway | first1=J. H. | last2=Sloane | first2=N. J. A. | date=9 March 2013 | publisher=Springer }} 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.{{Cite web |title=Hales, T. C. (2005) Ann. Math. 162, 1065–1185. |jstor=20159940 |url=https://www.jstor.org/stable/20159940}}

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...{{cite journal | url=https://pubs.aip.org/aip/jcp/article/135/15/151101/190281/Communication-A-packing-of-truncated-tetrahedra | doi=10.1063/1.3653938 | title=Communication: A packing of truncated tetrahedra that nearly fills all of space and its melting properties | date=2011 | last1=Jiao | first1=Yang | last2=Torquato | first2=Salvatore | journal=The Journal of Chemical Physics | volume=135 | issue=15 | pmid=22029288 | bibcode=2011JChPh.135o1101J }}

Truncated numbers are also relevant to cluster science in inorganic chemistry. Central to the chemical and physical study of clusters is an 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.{{Cite journal |title=Magic numbers in polygonal and polyhedral clusters |date=1985 |doi=10.1021/ic00220a025 |url=https://doi.org/10.1021/ic00220a025 |last1=Teo |first1=Boon K. |last2=Sloane |first2=N. J. A. |journal=Inorganic Chemistry |volume=24 |issue=26 |pages=4545–4558 |url-access=subscription }} Semiconductors are one of the most active areas of cluster research.{{Cite journal |title=Clusters |date=1996 |doi=10.1126/science.271.5251.889 |url=https://doi.org/10.1126/science.271.5251.889 |last1=Brauman |first1=John I. |journal=Science |volume=271 |issue=5251 |page=889 |bibcode=1996Sci...271..889B |url-access=subscription }}{{Cite journal |last=Califano |first=Marco |date=2024-02-13 |title=Tetrahedral vs Spherical Nanocrystals: Does the Shape Really Matter? |url=https://pubs.acs.org/doi/10.1021/acs.chemmater.3c01643 |journal=Chemistry of Materials |language=en |volume=36 |issue=3 |pages=1162–1171 |doi=10.1021/acs.chemmater.3c01643 |issn=0897-4756}}

Tetrahedral Number 35 {{OEIS|id=A000292}} yields Truncated Triangular Pyramid Number 19 {{OEIS|id=A051937}} by truncating Tetrahedral number (or triangular pyramidal number) 4 from each of the vertices

Tetrahedral Number 286 {{OEIS|id=A000292}} yields Truncated Triangular Pyramid Number 273 {{OEIS|id=A051937}} by truncating Tetrahedral number (or triangular pyramidal number) 4,4,4 and 1 from its vertices

Tetrahedral Number 560 {{OEIS|id=A000292}} can also yield Truncated Triangular Pyramid Number 273 {{OEIS|id=A051937}} by truncating Tetrahedral number (or triangular pyramidal number) 84,84,84 and 35 from its vertices OR its corresponding closest lesser Truncated Triangular Pyramid Number series number 451 also {{OEIS|id=A051937}} by truncating Tetrahedral number (or triangular pyramidal number) 35,35,35 and 4 from its vertices

Tetrahedral Number 969 {{OEIS|id=A000292}} yields Truncated Triangular Pyramid Number 833 {{OEIS|id=A051937}} by truncating Tetrahedral number (or triangular pyramidal number) 56,35,35 and 10 from its vertices

However, Tetrahedral Number 3276 {{OEIS|id=A000292}} does not yield its corresponding closest lesser Truncated Triangular Pyramid Number series number 3059 {{OEIS|id=A051937}} by truncating any combination of symmetric or asymmetric smaller Tetrahedral number (or triangular pyramidal number) from its vertices - because the difference between 3276 and 3059 = 217 which is part of Pollock tetrahedral numbers conjecture series of numbers which are a sum of more than 4 tetrahedral numbers {{OEIS|id=A000797}}.

Again, Tetrahedral Number 5984 {{OEIS|id=A000292}} does not yield its corresponding closest lesser Truncated Triangular Pyramid Number series number 5713 {{OEIS|id=A051937}} by truncating any combination of symmetric or asymmetric smaller Tetrahedral number (or triangular pyramidal number) from its vertices - because the difference between 5984 and 5713 = 271 which is again part of Pollock tetrahedral numbers conjecture series of numbers which are a sum of more than 4 tetrahedral numbers {{OEIS|id=A000797}}.

But other Tetrahedral Numbers - whether in-between or above/below such known exceptions - again yield corresponding closest lesser Truncated Triangular Pyramid Number series number - like 5456 {{OEIS|id=A000292}} yields Truncated Triangular Pyramid Number 5194 {{OEIS|id=A051937}} by truncating Tetrahedral number (or triangular pyramidal number) 84,84,84 and 10 from its vertices OR 11480 {{OEIS|id=A000292}} yields Truncated Triangular Pyramid Number 11137 {{OEIS|id=A051937}} by truncating Tetrahedral number (or triangular pyramidal number) 220,84,35 and 4 from its vertices

and so on.

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

1. Truncated triangular silver nanoplates synthesized in large quantities using a solution phase method{{cite journal | url=https://pubs.acs.org/doi/10.1021/nl025674h | doi=10.1021/nl025674h | title=Synthesis and Characterization of Truncated Triangular Silver Nanoplates | date=2002 | last1=Chen | first1=Sihai | last2=Carroll | first2=David L. | journal=Nano Letters | volume=2 | issue=9 | pages=1003–1007 | bibcode=2002NanoL...2.1003C | url-access=subscription }}
2. Theoretical study of hydrogen storage in a truncated triangular pyramid molecule{{cite journal | url=https://doi.org/10.1007/s00339-018-1841-9 | doi=10.1007/s00339-018-1841-9 | title=Theoretical study of hydrogen storage in a truncated triangular pyramid molecule consisting of pyridine and benzene rings bridged by vinylene groups | date=2018 | last1=Ishikawa | first1=Shigeru | last2=Nemoto | first2=Tetsushi | last3=Yamabe | first3=Tokio | journal=Applied Physics A | volume=124 | issue=6 | page=418 | bibcode=2018ApPhA.124..418I | url-access=subscription }}
3. Packing and self-assembly of truncated triangular bipyramids{{cite journal | url=https://doi.org/10.1103/PhysRevE.88.012127 | doi=10.1103/PhysRevE.88.012127 | title=Packing and self-assembly of truncated triangular bipyramids | date=2013 | last1=Haji-Akbari | first1=Amir | last2=Chen | first2=Elizabeth R. | last3=Engel | first3=Michael | last4=Glotzer | first4=Sharon C. | journal=Physical Review E | volume=88 | issue=1 | page=012127 | pmid=23944434 | arxiv=1304.3147 | bibcode=2013PhRvE..88a2127H }}

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Category:Figurate numbers

Category:Tetrahedra