Editing Trapezohedron (section)

{{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]].