Editing Trapezohedron (section)

==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}}
|}