Triaugmented triangular prism: Difference between revisions

Revision as of 15:18, 20 December 2022

Revision as of 02:23, 27 January 2023

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==Applications==

In the geometry of [[chemical compound]]s, it is common to visualize an [[atom cluster]] surrounding a central atom as a polyhedron—the [[convex hull]] of the surrounding atoms' locations. The [[tricapped trigonal prismatic molecular geometry]] describes clusters for which this polyhedron is a triaugmented triangular prism, although not necessarily one with equilateral triangle faces.{{r|kepert}} For example, the [[lanthanide]]s from [[lanthanum]] to [[dysprosium]] dissolve in water to form [[cation]]s surrounded by nine water molecules arranged as a triaugmented triangular prism.<ref name=Persson2022>{{cite journal |last1=Persson |first1=Ingmar |last2= |first2= |date=2022 |title=Structures of Hydrated Metal Ions in Solid State and Aqueous Solution |url=https://www.mdpi.com/2673-8015/2/3/14 |journal=Liquids |volume=2 |issue=3 |pages=210–242 |doi=10.3390/liquids2030014 |doi-access-date=28free November 2022}}</ref>

In the [[Thomson problem]], concerning the minimum-energy configuration of <math>n</math> charged particles on a sphere, and for the [[Tammes problem]] of constructing a [[spherical code]] maximizing the smallest distance among the points, the minimum solution known for <math>n=9</math> places the points at the vertices of a triaugmented triangular prism with non-equilateral faces, [[Circumscribed sphere|inscribed in a sphere]]. This configuration is proven optimal for the Tammes problem, but a rigorous solution to this instance of the Thomson problem is not known.{{r|whyte}}

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| title = Regular-faced convex polyhedra

| volume = 291

| year = 1971| issue = 5
}}; see Table IV, line 71, p. 338</ref>

<ref name=bsw13>{{citation

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| title = The discrete fundamental group of the associahedron, and the exchange module

| volume = 23

| year = 2013| s2cid = 14722555 }}</ref>

<ref name=burgiel>{{citation

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| pages = 263–266

| title = Deltahedra

| volume = 36| s2cid = 250435684 }}</ref>

<ref name=ff98>{{citation

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| title = Geometric combinatorics

| volume = 13

| year = 2007| s2cid = 11435731
}}; see Definition 3.3, Figure 3.6, and related discussion</ref>

<ref name=francis>{{citation|first=Darryl|last=Francis|title=Johnson solids & their acronyms|journal=Word Ways|date=August 2013|volume=46|issue=3|page=177|url=https://go.gale.com/ps/i.do?id=GALE%7CA340298118}}</ref>

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| title = On the space-filling enneahedra

| volume = 12

| year = 1982| s2cid = 120914105
}}; see polyhedron 9-IV, p. 301</ref>

<ref name=involve>{{citation

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| doi = 10.2140/involve.2009.2.249

| issue = 3

| journal = Involve, aA Journal of Mathematics

| pages = 249–265

| publisher = Mathematical Sciences Publishers

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| title = Convex polyhedra with regular faces

| volume = 18

| year = 1966| s2cid = 122006114 }}; see Table III, line 51</ref>

<ref name=kepert>{{citation

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| publisher = Springer

| title = Inorganic Chemistry Concepts

| year = 1982}}</ref>| volume = 6

| isbn = 978-3-642-68048-9

}}</ref>

<ref name=knill>{{citation

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| title = Minimal-energy clusters of hard spheres

| volume = 14

| year = 1995| s2cid = 26955765 }}</ref>

<ref name=soifer>{{citation

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| title = Unique arrangements of points on a sphere

| volume = 59

| year = 1952| issue = 9
}}</ref>

}}