Editing Trapezohedron

{{Short description|Polyhedron made of congruent kites arranged radially}}
{{redirect-distinguish|Deltohedron|Deltahedron}}
{{Infobox polyhedron
| name          = Set of dual-uniform {{nowrap|{{mvar|n}}-gonal}} trapezohedra
| image         = Pentagonal trapezohedron.svg
| caption       = Example: dual-uniform [[pentagonal trapezohedron]] ({{math|1=''n'' = 5}})
| type          = dual-[[Uniform polyhedron|uniform]] in the sense of dual-[[Semiregular polyhedron|semiregular]] polyhedron
| euler         = 
| faces         = {{math|2''n''}} [[Congruence (geometry)|congruent]] [[Kite (geometry)|kites]]
| edges         = {{math|4''n''}}
| vertices      = {{math|2''n'' + 2}}
| vertex_config = {{math|V3.3.3.''n''}}
| schläfli      = {{math|{ } ⨁ {''n''}<nowiki/>}}<ref>[[Norman Johnson (mathematician)|N.W. Johnson]]: ''Geometries and Transformations'', (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: ''Finite symmetry groups'', 11.3 Pyramids, Prisms, and Antiprisms, Figure 11.3c</ref>
| wythoff       = 
| coxeter       = {{CDD||node_fh|2x|node_fh|2x|n|node}}<BR>{{CDD||node_fh|2x|node_fh|n|node_fh}}
| conway        = {{math|dA{{sub|''n''}}}}
| symmetry      = {{math|D{{sub|''n''d}}, [2{{sup|+}},2''n''], (2*''n''),}} order {{math|4''n''}}
| rotation_group = {{math|D{{sub|''n''}}, [2,''n'']{{sup|+}}, (22''n''),}} order {{math|2''n''}}
| surface_area  = 
| volume        = 
| dual          = (convex) uniform {{nowrap|{{mvar|n}}-gonal}} [[antiprism]]
| properties    = [[Convex set|convex]], [[face-transitive]], regular vertices<ref>{{Cite web|title=duality|url=http://maths.ac-noumea.nc/polyhedr/dual_.htm|access-date=2020-10-19|website=maths.ac-noumea.nc}}</ref>
| vertex_figure = 
| net           = 
}}

In [[geometry]], an {{nowrap|'''{{mvar|n}}-gonal'''}} '''trapezohedron''', '''{{mvar|n}}-trapezohedron''', '''{{mvar|n}}-antidipyramid''', '''{{mvar|n}}-antibipyramid''', or '''{{mvar|n}}-deltohedron'''<ref name=":1">{{cite web |last1=Weisstein |first1=Eric W. |title=Trapezohedron |url=https://mathworld.wolfram.com/Trapezohedron.html |access-date=2024-04-24 |website=MathWorld}} Remarks: the faces of a delt'''o'''hedron are delt'''o'''ids; a (non-twisted) kite or deltoid can be [[Dissection (geometry)|dissected]] into two [[isosceles triangle]]s or "deltas" (Δ), base-to-base.</ref>{{sup|,}}<ref name=":2">{{cite web |last=Weisstein |first=Eric W. |title=Deltahedron |url=https://mathworld.wolfram.com/Deltahedron.html |access-date=2024-04-28 |website=MathWorld}}</ref> is the [[dual polyhedron]] of an {{nowrap|{{mvar|n}}-gonal}} [[antiprism]]. The {{math|'''2'''''n''}} faces of an {{nowrap|{{mvar|n}}-trapezohedron}} are [[Congruence (geometry)|congruent]] and symmetrically staggered; they are called [[#Symmetry|''twisted kites'']]. With a higher symmetry, its {{math|2''n''}} faces are [[Kite (geometry)|''kites'']] (sometimes also called ''trapezoids'', or ''deltoids'').{{sfn|Spencer|1911|p=575, or p. 597 on Wikisource, CRYSTALLOGRAPHY, 1. CUBIC SYSTEM, TETRAHEDRAL CLASS, footnote: « [Deltoid]: From the Greek letter δ, Δ; in general, a triangular-shaped object; also an alternative name for a trapezoid ». Remark: a twisted kite can look like and even be a trapezoid}}

The "{{nowrap|{{mvar|n}}-gonal}}" part of the name does not refer to faces here, but to two arrangements of each {{mvar|n}} [[Vertex (geometry)|vertices]] around an axis of {{nowrap|{{mvar|n}}-fold}} symmetry. The dual {{nowrap|{{mvar|n}}-gonal}} antiprism has two actual {{nowrap|{{mvar|n}}-gon}} faces.

An {{nowrap|{{mvar|n}}-gonal}} trapezohedron can be [[Dissection (geometry)|dissected]] into two equal {{nowrap|{{mvar|n}}-gonal}} [[Pyramid (geometry)|pyramids]] and an {{nowrap|{{mvar|n}}-gonal}} [[antiprism]].

==Terminology==
These figures, sometimes called delt'''o'''hedra,<ref name=":1" /> are not to be confused with [[Deltahedron|delt'''a'''hedra]],<ref name=":2" /> whose faces are equilateral triangles.

[[#Symmetry|''Twisted'']] ''trigonal'', ''tetragonal'', and ''hexagonal trapezohedra'' (with six, eight, and twelve ''twisted'' [[Congruence (geometry)|congruent]] kite faces) exist as crystals; in [[crystallography]] (describing the [[crystal habit]]s of [[mineral]]s), they are just called ''trigonal'', ''tetragonal'', and ''hexagonal trapezohedra''. They have no plane of symmetry, and no center of inversion symmetry;{{sfn|Spencer|1911|p=581, or p. 603 on Wikisource, CRYSTALLOGRAPHY, 6. HEXAGONAL SYSTEM, ''Rhombohedral Division'', TRAPEZOHEDRAL CLASS, FIG. 74}}<sup>,</sup>{{sfn|Spencer|1911|p=577, or p. 599 on Wikisource, CRYSTALLOGRAPHY, 2. TETRAGONAL SYSTEM, TRAPEZOHEDRAL CLASS}} but they have a [[center of symmetry]]: the intersection point of their symmetry axes. The trigonal trapezohedron has one 3-fold symmetry axis, perpendicular to three 2-fold symmetry axes.{{sfn|Spencer|1911|p=581, or p. 603 on Wikisource, CRYSTALLOGRAPHY, 6. HEXAGONAL SYSTEM, ''Rhombohedral Division'', TRAPEZOHEDRAL CLASS, FIG. 74}} The tetragonal trapezohedron has one 4-fold symmetry axis, perpendicular to four 2-fold symmetry axes of two kinds. The hexagonal trapezohedron has one 6-fold symmetry axis, perpendicular to six 2-fold symmetry axes of two kinds.{{sfn|Spencer|1911|p=582, or p. 604 on Wikisource, CRYSTALLOGRAPHY, 6. HEXAGONAL SYSTEM, ''Hexagonal Division'', TRAPEZOHEDRAL CLASS}}

[[Crystal system|Crystal arrangements]] of atoms can repeat in space with trigonal and hexagonal trapezohedron cells.<ref name=":0">[http://www.metafysica.nl/turing/promorph_crystals_2.html Trigonal-trapezohedric Class, 3 2 and Hexagonal-trapezohedric Class, 6 2 2]</ref>

Also in crystallography, the word ''trapezohedron'' is often used for the polyhedron with 24 [[Congruence (geometry)|congruent]] non-twisted kite faces properly known as a ''[[deltoidal icositetrahedron]]'',{{sfn|Spencer|1911|p=574, or p. 596 on Wikisource, CRYSTALLOGRAPHY, 1. CUBIC SYSTEM, HOLOSYMMETRIC CLASS, FIG. 17}} which has eighteen order-4 vertices and eight order-3 vertices. This is not to be confused with the ''dodecagonal trapezohedron'', which also has 24 congruent kite faces, but two order-12 apices (i.e. poles) and two rings of twelve order-3 vertices each.

Still in crystallography, the ''deltoid dodecahedron''{{sfn|Spencer|1911|p=575, or p. 597 on Wikisource, CRYSTALLOGRAPHY, 1. CUBIC SYSTEM, TETRAHEDRAL CLASS, FIG. 27}} has 12 congruent non-twisted kite faces, six order-4 vertices and eight order-3 vertices (the ''[[rhombic dodecahedron]]'' is a special case). This is not to be confused with the ''[[hexagonal trapezohedron]]'', which also has 12 congruent kite faces,{{sfn|Spencer|1911|p=582, or p. 604 on Wikisource, CRYSTALLOGRAPHY, 6. HEXAGONAL SYSTEM, ''Hexagonal Division'', TRAPEZOHEDRAL CLASS}} but two order-6 apices (i.e. poles) and two rings of six order-3 vertices each.

==Forms==
An '''{{mvar|n}}-trapezohedron''' is defined by a [[Regular skew polygon|regular zig-zag skew]] {{math|'''2'''''n''}}-gon base, two symmetric [[Apex (geometry)|apices]] with no [[Degrees of freedom|degree of freedom]] right above and right below the base, and [[quadrilateral]] faces connecting each pair of [[Adjacent side (polygon)|adjacent]] basal edges to one apex.

An {{mvar|n}}-trapezohedron has two apical vertices on its polar axis, and {{math|2''n''}} basal vertices in two regular {{mvar|n}}-gonal rings. It has {{math|'''2'''''n''}} [[Congruence (geometry)|congruent]] [[Kite (geometry)|kite]] faces, and it is [[Isohedral figure|isohedral]].

{{Trapezohedra}}

=== Special cases ===
* {{math|1=''n'' = 2}}. A degenerate form of trapezohedron: a geometric figure with 6 vertices, 8 edges, and 4 degenerate [[Kite (geometry)|kite]] faces that are visually identical to triangles. As such, the trapezohedron itself is visually identical to the regular [[tetrahedron]]. Its dual is a degenerate form of [[antiprism]] that also resembles the regular tetrahedron.

* {{math|1=''n'' = 3}}. The dual of a ''triangular antiprism'': the kites are rhombi (or squares); hence these trapezohedra are also [[zonohedra]]. They are called '''[[rhombohedra]]'''. They are [[cube]]s scaled in the direction of a body diagonal. They are also the [[parallelepiped]]s with congruent rhombic faces.[[File:Gyroelongated triangular bipyramid.png|thumb|A {{math|60°}} rhombohedron, [[Dissection (geometry)|dissected]] into a central regular octahedron and two regular tetrahedra]]
** A special case of a rhombohedron is one in which the rhombi forming the faces have angles of {{math|60°}} and {{math|120°}}. It can be decomposed into two equal regular tetrahedra and a regular [[octahedron]]. Since parallelepipeds can [[Tessellation|fill space]], so can a [[Tetrahedral-octahedral honeycomb|combination of regular tetrahedra and regular octahedra]].

* {{math|1=''n'' = 5}}. The [[pentagonal trapezohedron]] is the only polyhedron other than the [[Platonic solid]]s commonly used as a [[Dice|die]] in [[roleplaying games]] such as ''[[Dungeons & Dragons]]''. Being [[Convex set|convex]] and [[Isohedral figure|face-transitive]], it makes [[fair dice]]. Having 10 sides, it can be used in repetition to generate any decimal-based [[Uniform distribution (discrete)|uniform probability]] desired. Typically, two dice of different colors are used for the two [[Numerical digit|digits]] to represent numbers from {{math|00}} to {{math|99}}.

==Symmetry==
The [[symmetry group]] of an {{mvar|n}}-gonal trapezohedron is {{math|1=D{{sub|''n''d}} = D{{sub|''n''v}}}}, of order {{math|4''n''}}, except in the case of {{math|1=''n'' = 3}}: a cube has the larger symmetry group {{math|O<sub>d</sub>}} of order {{math|1=48 = 4×(4×3)}}, which has four versions of {{math|D<sub>3d</sub>}} as subgroups.

The [[Point groups in three dimensions#Rotation groups|rotation group]] of an {{mvar|n}}-trapezohedron is {{math|D<sub>''n''</sub>}}, of order {{math|2''n''}}, except in the case of {{math|1=''n'' = 3}}: a cube has the larger rotation group {{math|O}} of order {{math|1=24 = 4×(2×3)}}, which has four versions of {{math|D<sub>3</sub>}} as subgroups.

Note: Every {{mvar|n}}-trapezohedron with a [[Regular skew polygon|regular zig-zag skew]] {{math|'''2'''''n''}}-gon base and {{math|2''n''}} congruent non-twisted kite faces has the same (dihedral) symmetry group as the [[Dual uniform polyhedron|dual-uniform]] {{mvar|n}}-trapezohedron, for {{math|''n'' ≥ 4}}.

One degree of freedom within symmetry from {{math|D<sub>''n''d</sub>}} (order {{math|4''n''}}) to {{math|D<sub>''n''</sub>}} (order {{math|2''n''}}) changes the congruent kites into congruent quadrilaterals with three edge lengths, called ''twisted kites'', and the {{mvar|n}}-trapezohedron is called a ''twisted trapezohedron''. (In the limit, one edge of each quadrilateral goes to zero length, and the {{mvar|n}}-trapezohedron becomes an {{mvar|n}}-[[bipyramid]].)

If the kites surrounding the two peaks are not twisted but are of two different shapes, the {{mvar|n}}-trapezohedron can only have {{math|C<sub>''n''v</sub>}} (cyclic with vertical mirrors) symmetry, order {{math|2''n''}}, and is called an ''unequal'' or ''asymmetric trapezohedron''. Its dual is an ''unequal {{mvar|n}}-[[antiprism]]'', with the top and bottom {{mvar|n}}-gons of different radii.

If the kites are twisted and are of two different shapes, the {{mvar|n}}-trapezohedron can only have {{math|C<sub>''n''</sub>}} (cyclic) symmetry, order {{mvar|n}}, and is called an ''unequal twisted trapezohedron''.

{| class=wikitable
|+ style="text-align:center;"|Example: variations with hexagonal trapezohedra (''n'' = 6)
!Trapezohedron type
!colspan=2|Twisted trapezohedron
!Unequal trapezohedron
!Unequal twisted trapezohedron
|- style="text-align:center;"
![[List of finite spherical symmetry groups|Symmetry group]]
|colspan=2|D<sub>6</sub>, (662), [6,2]<sup>+</sup>
|C<sub>6v</sub>, (*66), [6]
|C<sub>6</sub>, (66), [6]<sup>+</sup>
|- style="text-align:center;"
!Polyhedron image
|[[File:Twisted_hexagonal_trapezohedron.png|160px]]
|[[File:Twisted_hexagonal_trapezohedron2.png|160px]]
|[[File:Unequal_hexagonal_trapezohedron.png|160px]]
|[[File:Unequal_twisted_hexagonal_trapezohedron.png|160px]]
|- style="text-align:center;"
!Net
|[[File:Twisted_hexagonal_trapezohedron net.png|160px]]
|[[File:Twisted_hexagonal_trapezohedron2 net.png|160px]]
|[[File:Unequal_hexagonal_trapezohedron net.png|160px]]
|[[File:Unequal_twisted_hexagonal_trapezohedron net.png|160px]]
|}

==Star trapezohedron==
A '''star {{math|''p''/''q''}}-trapezohedron''' (where {{math|2 ≤ ''q'' < '''1'''''p''}}) is defined by a [[Regular skew polygon|regular zig-zag skew]] [[Star polygon|star {{math|'''2'''''p''/''q''}}-gon]] base, two symmetric [[Apex (geometry)|apices]] with no [[Degrees of freedom|degree of freedom]] right above and right below the base, and [[quadrilateral]] faces connecting each pair of [[Adjacent side (polygon)|adjacent]] basal edges to one apex.

A star {{math|''p''/''q''}}-trapezohedron has two apical vertices on its polar axis, and {{math|2''p''}} basal vertices in two regular {{mvar|p}}-gonal rings. It has {{math|'''2'''''p''}} [[Congruence (geometry)|congruent]] [[Kite (geometry)|kite]] faces, and it is [[Isohedral figure|isohedral]].

Such a star {{math|''p''/''q''}}-trapezohedron is a ''self-intersecting'', ''crossed'', or ''non-convex'' form. It exists for any regular zig-zag skew star {{math|'''2'''''p''/''q''}}-gon base (where {{math|2 ≤ ''q'' < '''1'''''p''}}).

But if {{math|{{sfrac|''p''|''q''}} < {{sfrac|3|2}}}}, then {{math|(''p'' − ''q''){{sfrac|360°|''p''}} < {{sfrac|''q''|2}}{{sfrac|360°|''p''}}}}, so the dual star antiprism (of the star trapezohedron) cannot be uniform (i.e. cannot have equal edge lengths); and if {{math|1={{sfrac|''p''|''q''}} = {{sfrac|3|2}}}}, then {{math|1=(''p'' − ''q''){{sfrac|360°|''p''}} = {{sfrac|''q''|2}}{{sfrac|360°|''p''}}}}, so the dual star antiprism must be flat, thus degenerate, to be uniform.

A [[Dual uniform polyhedron|dual-uniform]] star {{math|''p''/''q''}}-trapezohedron has [[Coxeter-Dynkin diagram]] {{CDD|node_fh|2x|node_fh|p|rat|q|node_fh}}.

{| class=wikitable
|+ style="text-align:center;"|Dual-uniform star ''p''/''q''-trapezohedra up to ''p''&nbsp;=&nbsp;12
|- align=center
!5/2||5/3||7/2||7/3||7/4||8/3||8/5||9/2||9/4||9/5
|- align=center
|[[Image:5-2_deltohedron.png|50px]]
|[[Image:5-3_deltohedron.png|50px]]
|[[Image:7-2_deltohedron.png|60px]]
|[[Image:7-3_deltohedron.png|60px]]
|[[Image:7-4_deltohedron.png|60px]]
|[[Image:8-3_deltohedron.png|60px]]
|[[Image:8-5_deltohedron.png|60px]]
|[[Image:9-2_deltohedron.png|60px]]
|[[Image:9-4_deltohedron.png|60px]]
|[[Image:9-5_deltohedron.png|60px]]
|- align=center valign="top"
|{{CDD|node_fh|2x|node_fh|5|rat|2x|node_fh}}
|{{CDD|node_fh|2x|node_fh|5|rat|3x|node_fh}}
|{{CDD|node_fh|2x|node_fh|7|rat|2x|node_fh}}
|{{CDD|node_fh|2x|node_fh|7|rat|3x|node_fh}}
|{{CDD|node_fh|2x|node_fh|7|rat|4|node_fh}}
|{{CDD|node_fh|2x|node_fh|8|rat|3x|node_fh}}
|{{CDD|node_fh|2x|node_fh|8|rat|5|node_fh}}
|{{CDD|node_fh|2x|node_fh|9|rat|2x|node_fh}}
|{{CDD|node_fh|2x|node_fh|9|rat|4|node_fh}}
|{{CDD|node_fh|2x|node_fh|9|rat|5|node_fh}}
|}

{| class=wikitable
|- align=center
!10/3||11/2||11/3||11/4||11/5||11/6||11/7||12/5||12/7
|- align=center
|[[Image:10-3_deltohedron.png|60px]]
|[[Image:11-2_deltohedron.png|60px]]
|[[Image:11-3_deltohedron.png|60px]]
|[[Image:11-4_deltohedron.png|60px]]
|[[Image:11-5_deltohedron.png|60px]]
|[[Image:11-6_deltohedron.png|60px]]
|[[Image:11-7_deltohedron.png|60px]]
|[[Image:12-5_deltohedron.png|60px]]
|[[Image:12-7_deltohedron.png|60px]]
|- align=center valign="top"
|{{CDD|node_fh|2x|node_fh|10|rat|3x|node_fh}}
|{{CDD|node_fh|2x|node_fh|11|rat|2x|node_fh}}
|{{CDD|node_fh|2x|node_fh|11|rat|3x|node_fh}}
|{{CDD|node_fh|2x|node_fh|11|rat|4|node_fh}}
|{{CDD|node_fh|2x|node_fh|11|rat|5|node_fh}}
|{{CDD|node_fh|2x|node_fh|11|rat|6|node_fh}}
|{{CDD|node_fh|2x|node_fh|11|rat|7|node_fh}}
|{{CDD|node_fh|2x|node_fh|12|rat|5|node_fh}}
|{{CDD|node_fh|2x|node_fh|12|rat|7|node_fh}}
|}

==See also==
{{Commonscat|Trapezohedra}}
*[[Diminished trapezohedron]]
*[[Rhombic dodecahedron]]
*[[Rhombic triacontahedron]]
*[[Bipyramid]]
*[[Truncated trapezohedron]]
*[[Conway polyhedron notation]]
*[[The Haunter of the Dark]], a short story by [[H.P. Lovecraft]] in which a fictional ancient artifact known as The Shining Trapezohedron plays a crucial role.

==References==
{{reflist}}
*{{cite book | author= Anthony Pugh | year= 1976 | title= Polyhedra: A visual approach | publisher= University of California Press Berkeley | location= California | isbn= 0-520-03056-7  }} Chapter 4: Duals of the Archimedean polyhedra, prisma and antiprisms
* {{cite EB1911|wstitle= Crystallography |volume= 07 | pages = 569&ndash;591 |last1= Spencer |first1= Leonard James |author-link= Leonard James Spencer}}

==External links==
*{{mathworld |urlname=Trapezohedron |title=Trapezohedron}}
*{{mathworld | urlname = Isohedron | title = Isohedron}}
*[http://www.korthalsaltes.com/model.php?name_en=square%20trapezohedron Paper model tetragonal (square) trapezohedron]

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