View source for Deltahedron

{{short description|Polyhedron made of equilateral triangles}}
{{distinguish|Deltohedron|text="[[Deltohedron]]", a term sometimes used to refer to the set of trapezohedra}}

[[File:Deltahedron.svg|thumb|upright=1.6|The eight convex deltahedra. First row: [[regular tetrahedron]], [[Triangular bipyramid#As a Johnson solid|triangular bipyramid]], [[regular octahedron]], [[Pentagonal bipyramid#As a Johnson solid|pentagonal bipyramid]]. Second row: [[gyroelongated square bipyramid]], [[regular icosahedron]], [[triaugmented triangular prism]], [[snub disphenoid]].]]
A '''deltahedron''' is a [[polyhedron]] whose faces are all [[equilateral triangle]]s. The deltahedron was named by [[Martyn Cundy]], after the Greek capital letter [[Delta (letter)|delta]] resembling a triangular shape Δ.<ref>{{multiref
|{{harvp|Cundy|1952}}
|{{harvp|Cromwell|1997|p=[https://archive.org/details/polyhedra0000crom/page/75/mode/1up 75]}}
|{{harvp|Trigg|1978}}
}}</ref>

Deltahedra can be categorized by the property of [[Convex set|convexity]]. The simplest convex deltahedron is the [[regular tetrahedron]], a pyramid with four equilateral triangles. There are eight convex deltahedra, which can be used in the applications of chemistry as in the [[polyhedral skeletal electron pair theory]] and [[chemical compound]]s. There are infinitely many concave deltahedra.

== Strictly convex deltahedron ==
A polyhedron is said to be ''convex'' if a line between any two of its vertices lies either within its interior or on its boundary, and additionally, if no two faces are [[Coplanarity|coplanar]] (lying in the same plane) and no two edges are [[Collinearity|collinear]] (segments of the same line), it can be considered as being strictly convex.<ref>{{multiref|{{harvp|Litchenberg|1988|p=262}}|{{harvp|Boissonnat|Yvinec|1989}}}}</ref> 

Of the eight [[convex set|convex]] deltahedra, three are [[Platonic solid]]s and five are [[Johnson solid]]s. They are:<ref>{{multiref
|{{harvp|Trigg|1978}}
|{{harvp|Litchenberg|1988|p=263}}
|{{harvp|Freudenthal|van der Waerden|1947}}
}}</ref> 
* [[regular tetrahedron]], a pyramid with four equilateral triangles, one of which can be considered the base.
* [[triangular bipyramid]], [[regular octahedron]], and [[pentagonal bipyramid]]; [[bipyramid]]s with six, eight, and ten equilateral triangles, respectively. They are constructed by identical pyramids base-to-base.
* [[gyroelongated square bipyramid]] and [[regular icosahedron]] are constructed by attaching two pyramids onto a square antiprism or pentagonal antiprism, respectively, such that they have sixteen and twenty triangular faces.
* [[triaugmented triangular prism]], constructed by attaching three square pyramids onto the square face of a triangular prism, such that it has fourteen triangular faces.
* [[snub disphenoid]], with twelve triangular faces, constructed by involving two regular hexagons in the following order: these hexagons may form a bipyramid in [[Degeneracy (mathematics)|degeneracy]], separating them into two parts along a coinciding diagonal, pressing inward on the end of diagonal, rotating one of them in 90°, and rejoining them together.

The number of possible convex deltahedrons was given by {{harvtxt|Rausenberger|1915}}. Rausenberger named these solids as the ''convex pseudoregular polyhedra''.<ref>{{multiref
|{{harvp|Rausenberger|1915}}
|{{harvp|Litchenberg|1988|p=263}}
}}</ref>

Summarizing the examples above, the deltahedra can be conclusively defined as the class of polyhedra whose faces are [[equilateral triangle]]s.<ref>{{multiref
|{{harvp|Cundy|1952}}
|{{harvp|Cromwell|1997|p=[https://archive.org/details/polyhedra0000crom/page/75/mode/1up 75]}}
|{{harvp|Trigg|1978}}
}}</ref> Another definition by {{harvtxt|Bernal|1964}} is similar to the previous one, in which he was interested in the shapes of holes left in irregular close-packed arrangements of spheres. It is stated as a convex polyhedron with equilateral triangular faces that can be formed by the centers of a collection of congruent spheres, whose tangencies represent polyhedron edges, and such that there is no room to pack another sphere inside the cage created by this system of spheres. Because of this restriction, some polyhedrons may not be included as a deltahedron: the triangular bipyramid (as forming two tetrahedral holes rather than a single hole), pentagonal bipyramid (because the spheres for its apexes interpenetrate, so it cannot occur in sphere packings), and regular icosahedron (because it has interior room for another sphere).{{sfnp|Bernal|1964}}

Most convex deltahedra can be found in the study of [[chemistry]]. For example, they are categorized as the ''closo'' polyhedron in the study of [[polyhedral skeletal electron pair theory]].{{sfnp|Kharas|Dahl|1988|p=[https://books.google.com/books?id=ur7Nqe4ueBYC&pg=PA8 8]}} Other applications of deltahedra&mdash;excluding the regular icosahedron&mdash;are the visualization of an [[atom cluster]] surrounding a central atom as a polyhedron in the study of [[chemical compounds]]: regular tetrahedron represents the [[tetrahedral molecular geometry]], triangular bipyramid represents [[trigonal bipyramidal molecular geometry]], regular octahedron represents the [[octahedral molecular geometry]], pentagonal bipyramid represents the [[pentagonal bipyramidal molecular geometry]], gyroelongated square bipyramid represents the [[bicapped square antiprismatic molecular geometry]], triaugmented triangular prism represents the [[tricapped trigonal prismatic molecular geometry]], and snub disphenoid represents the [[dodecahedral molecular geometry]].<ref>{{multiref
|{{harvp|Burdett|Hoffmann|Fay|1978}}
|{{harvp|Gillespie|Hargittai|2013|p=[https://books.google.com/books?id=6rDDAgAAQBAJ&pg=PA152 152]}}
|{{harvp|Kepert|1982|p=7–21}}
|{{harvp|Petrucci|Harwood|Herring|2002|p=413&ndash;414|loc=See table 11.1.}}
|{{harvp|Remhov|Černý|2021|p=[https://books.google.com/books?id=Hpc9EAAAQBAJ&pg=PA270 270]}}
}}</ref> The regular icosahedron along with some other deltahedra appears in the geometry of [[boron hydride clusters]].<ref>{{harvp|Cotton|Wilkinson|Murillo|Bochmann|1999|p=142}}</ref>

== Non-convex deltahedron ==
[[File:Stella octangula.svg|thumb|upright=0.8|[[Stella octangula]] is a non-convex deltahedron]]
A ''non-convex deltahedron'' is a deltahedron that does not possess convexity, thus it has either coplanar faces or collinear edges. There are infinitely many non-convex deltahedra.<ref>{{multiref
|{{harvp|Trigg|1978}}
|{{harvp|Eppstein|2021}}
}}</ref> Some examples are [[stella octangula]], the [[excavated dodecahedron]], and [[Boerdijk–Coxeter helix]].<ref>{{multiref
|{{harvp|Pedersen|Hyde|2018}}
|{{harvp|Weils|1991|p=[https://archive.org/details/ThePenguinDictionaryOfCuriousAndInterestingGeometry/page/n93/mode/1up?view=theater 78]}}
}}</ref>

There are subclasses of non-convex deltahedra. {{harvtxt|Cundy|1952}} shows that they may be discovered by finding the number of varying vertex's ''types''. A set of vertices is considered the same type as long as there are subgroups of the polyhedron's same group [[Transitive group|transitive]] on the set. Cundy shows that the [[great icosahedron]] is the only non-convex deltahedron with a single type of vertex. There are seventeen non-convex deltahedra with two types of vertex, and soon the other eleven deltahedra were later added by {{harvtxt|Olshevsky}},<ref>{{multiref
|{{harvp|Cundy|1952}}
|{{harvp|Olshevsky}}
|{{harvp|Tsuruta|Mitani|Kanamori|Fukui|2015}}
}}</ref> Other subclasses are the [[Isohedral figure|isohedral]] deltahedron that was later discovered by both {{harvtxt|McNeill}} and {{harvtxt|Shephard|2000}},<ref>{{multiref
|{{harvp|McNeill}}
|{{harvp|Shephard|2000}}
|{{harvp|Tsuruta|Mitani|Kanamori|Fukui|2015}}
}}</ref> and the ''spiral deltahedron'' constructed by the strips of equilateral triangles was discovered by {{harvtxt|Trigg|1978}}.<ref>{{multiref
|{{harvp|Trigg|1978}}
|{{harvp|Tsuruta|Mitani|Kanamori|Fukui|2015}}
}}</ref>

== References ==
=== Footnotes ===
{{reflist|30em}}
=== Works cited ===
{{refbegin|30em}}
* {{citation
 | last = Bernal | first = J. D. | author-link = John Desmond Bernal
 | issue = 1382
 | journal = [[Proceedings of the Royal Society of London]] | series = Series A, Mathematical and Physical Sciences
 | jstor = 2415872
 | pages = 299–322
 | title = The Bakerian Lecture, 1962. The Structure of Liquids
 | volume = 280
 | year = 1964| doi = 10.1098/rspa.1964.0147 | bibcode = 1964RSPSA.280..299B | s2cid = 178710030
}}.
* {{citation
 | last1 = Boissonnat | first1 = J. D.
 | last2 = Yvinec | first2 = M.
 | title = Proceedings of the fifth annual symposium on Computational geometry - SCG '89
 | chapter = Probing a scene of non convex polyhedra
 | date = June 1989
 | pages = 237–246
 | doi = 10.1145/73833.73860
 | isbn = 0-89791-318-3
 | doi-access = free
 }}.
* {{citation
 | last1 = Burdett | first1 = Jeremy K. 
 | last2 = Hoffmann | first2 = Roald
 | last3 = Fay | first3 = Robert C.
 | title = Eight-Coordination
 | journal = [[Inorganic Chemistry (journal)|Inorganic Chemistry]]
 | year = 1978
 | volume = 17 | issue = 9 | pages = 2553–2568
 | doi = 10.1021/ic50187a041
}}.
* {{sfn whitelist|CITEREFCottonWilkinsonMurilloBochmann1999}}{{Cotton&Wilkinson6th}}
* {{citation
 | last = Cromwell | first = Peter R.
 | title = Polyhedra
 | year = 1997
 | url = https://archive.org/details/polyhedra0000crom
 | publisher = [[Cambridge University Press]]
}}.
* {{citation
 | last = Cundy | first = H. Martyn | author-link = Martyn Cundy
 | title = Deltahedra
 | journal = [[Mathematical Gazette]]
 | volume = 36 | pages = 263–266
 | year = 1952
 | issue = 318 | doi = 10.2307/3608204
 | jstor = 3608204
}}.
* {{citation
 | last1 = Cundy | first1 = H. Martyn | author1-link = Martyn Cundy | author1-mask = 3
 | last2 = Rollett | first2 = A.
 | chapter = 3.11. Deltahedra
 | title = Mathematical Models
 | title-link = Mathematical Models (Cundy and Rollett)
 | edition = 3rd
 | location = Stradbroke, England
 | publisher = Tarquin Pub.
 | pages = 142–144
 | year = 1989
}}.
* {{citation
 | last = Eppstein | first = D. | author-link = David Eppstein
 | year = 2021
 | title = On Polyhedral Realization with Isosceles Triangles
 | volume = 37 | pages = 1247–1269
 | journal =  Graphs and Combinatorics
 | issue = 4 | publisher = Springer
 | doi = 10.1007/s00373-021-02314-9
| doi-access = free
 | arxiv = 2009.00116
 }}
* {{citation
 | last1 = Foulds | first1 = L. R.
 | last2 = Robinson | first2 = D. F.
 | journal = Discrete Applied Mathematics
 | volume = 1 | issue = 1–2 | year = 1979 | pages = 75–87
 | title = Construction properties of combinatorial deltahedra
 | doi = 10.1016/0166-218X(79)90015-5
| doi-access = free}}.
* {{citation
 | last1 = Freudenthal | first1 = H. | author1-link = Hans Freudenthal
 | last2 = van der Waerden | first2 = B. L. | author2-link = Bartel Leendert van der Waerden
 | journal = [[Simon Stevin (journal)|Simon Stevin]]
 | mr = 21687
 | pages = 115–121
 | title = On an assertion of Euclid
 | volume = 25
 | year = 1947
}}.
* {{citation
 | last = Gardner | first = Martin | author-link = Martin Gardner
 | title = Fractal Music, Hypercards, and More: Mathematical Recreations from Scientific American
 | location = New York
 | publisher = W. H. Freeman
 | pages = 40, 53, and 58–60
 | year = 1992
}}.
* {{citation
 | last1 = Gillespie | first1 = Ronald J.
 | last2 = Hargittai | first2 = István
 | year = 2013
 | title = The VSEPR Model of Molecular Geometry
 | publisher = [[Dover Publications]]
 | isbn = 978-0-486-48615-4
}}.
* {{citation
 | last = Kepert | first = David L.
 | contribution = Polyhedra
 | doi = 10.1007/978-3-642-68046-5_2
 | publisher = Springer
 | title = Inorganic Chemistry Concepts
 | year = 1982| volume = 6
 | pages = 7–21
 | isbn = 978-3-642-68048-9
}}.
* {{citation
 | last1 = Kharas | first1 = K. C. C.
 | last2 = Dahl | first2 = L. F.
 | editor-last1 = Prigogine | editor-first1 = I.
 | editor-last2 = Rice | editor-first2 = S. A.
 | year = 1988
 | title = Evolution of Size Effects in Chemical Dynamics Part 2: Advances in Chemical Physics Volume LXX
 | contribution = Ligand-Stabilized Metal Clusters: Structure, Bonding, Fluxionarity, and the Metallic State
 | url = https://books.google.com/books?id=ur7Nqe4ueBYC
 | page = 8
 | publisher = [[John Wiley & Sons]]
| isbn = 978-0-470-14180-9
 }}.
* {{citation
 | last = Litchenberg | first = D. R.
 | year = 1988
 | title = Pyramids, Prisms, Antiprisms, and Deltahedra
 | journal = The Mathematics Teacher
 | volume = 81
 | issue = 4
 | pages = 261–265
 | doi = 10.5951/MT.81.4.0261
 | jstor = 27965792
}}
* {{citation
 | last = McNeill | first = J.
 | title = Isohedral Deltahedra
}}
* {{citation
 | last = Olshevsky | first = George
 | title = Breaking Cundy's Deltahedra Record
 | url = https://www.interocitors.com/polyhedra/papers/Polytopics%2028%20-%20Breaking%20Cundy's%20Deltahedra%20Record.pdf
}}
* {{citation
 | last1 = Pedersen | first1 = M. C.
 | last2 = Hyde | first2 = S. T.
 | title = Polyhedra and packings from hyperbolic honeycombs
 | year = 2018
 | volume = 115 | issue = 27 | pages = 6905–6910
 | doi = 10.1073/pnas.1720307115
 | journal = [[Proceedings of the National Academy of Sciences of the United States of America|Proceedings of the National Academy of Sciences]]
| doi-access = free
 | pmid = 29925600
 | pmc = 6142264
 | bibcode = 2018PNAS..115.6905P
 }}
* {{citation
 | last1 = Petrucci | first1 = R. H.
 | last2 = Harwood  | first2 = W. S.
 | last3 = Herring  | first3 = F. G.
 | year = 2002
 | title = General Chemistry: Principles and Modern Applications
 | publisher = Prentice-Hall
 | edition = 8th
 | isbn = 978-0-13-014329-7
}}
* {{citation
 | last = Pugh | first = Anthony
 | year = 1976
 | title = Polyhedra: A visual approach
 | publisher = University of California Press Berkeley
 | location = California
 | isbn = 0-520-03056-7
 | pages = 35–36
}}.
* {{citation
 | last = Rausenberger | first = O.
 | title = Konvexe pseudoreguläre Polyeder
 | journal = Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht
 | volume = 46 | pages = 135–142
 | year = 1915
}}.
* {{citation
 | last1 = Remhov | first1 = Arndt
 | last2 = Černý | first2 = Radovan
 | editor-last1 = Schorr | editor-first1 = Susan
 | editor-last2 = Weidenthaler | editor-first2 = Claudia
 | year = 2021
 | title = Crystallography in Materials Science: From Structure-Property Relationships to Engineering
 | contribution = Hydroborate as novel solid-state electrolytes
 | publisher = [[de Gruyter]]
 | isbn = 978-3-11-067485-9
}}.
* {{citation
 | last = Shephard | first = G. C. | author-link = G. C. Shephard
 | year = 2000
 | title = Isohedral Deltahedra
 | journal =  Periodica Mathematica Hungarica
 | volume = 39 | pages = 86&ndash;100
 | doi = 10.1023/A:1004838806529
}}.
* {{citation
 | last = Trigg | first = Charles W.
 | year = 1978
 | title = An Infinite Class of Deltahedra
 | journal = Mathematics Magazine
 | volume = 51 | issue = 1 | pages = 55&ndash;57
 | doi = 10.2307/2689647
 | jstor = 2689647
}}.
* {{citation
 | last1 = Tsuruta | first1 = Naoya
 | last2 = Mitani | first2 = Jun
 | last3 = Kanamori | first3 = Yoshihiro
 | last4 = Fukui | first4 = Yukio
 | year = 2015
 | title = Random Realization of Polyhedral Graphs as Deltahedra
 | journal = Journal for Geometry and Graphics
 | url = https://www.heldermann-verlag.de/jgg/jgg19/j19h2tsur.pdf
 | volume = 19 | issue = 2 | pages = 227&ndash;236
}}.
* {{citation
 | last = Weils | first = David
 | year = 1991
 | title = The Penguin Dictionary of Curious and Interesting Geometry
 | publisher = Penguin Books
 | isbn = 9780140118131
}}.
{{refend}}

== External links ==
* {{citation
 | title = The Cundy Deltahedra or Biform Deltahedra
 | url = https://www.interocitors.com/polyhedra/Deltahedra/Cundy/index.html
}}
* {{MathWorld
 | title = Deltahedron | id = Deltahedron
 | mode = cs2
}}

[[Category:Deltahedra|*]]
[[Category:Polyhedra]]